Solution of optimal power flow with wind power uncertainty using hybridized self-adaptive differential evolution

Abstract

The current research presents an optimal power flow (OPF) solution in an electrical system network with the integration of wind power using enhanced self-adaptive differential evolution method with a mixed crossover (ESDE-MC). A model for wind power cost is considered, which contemplates the random nature of wind speed based on Weibull probability density function. The wind energy cost model incorporates the estimated reserve and penalty cost for wind power shortfall and surplus, respectively. In the present work, the wound rotor induction generator (WRIG) model is being utilized for wind power production. Further, WRIGs reactive power (Q) – voltage (V) formulation is developed and is enhanced by changes to conventional power flows. In order to verify the performance and reliability of the proposed method, modified IEEE 30-bus, and IEEE 57-bus systems are considered to solve the OPF with the minimization of thermal generation cost inclusion of wind power. The proposed ESDE-MC method achieved 2788.69$/h which is 8.4 $/h less value in comparison to the best optimal value in the existed literature for IEEE 57-bu system in less computational time. Like these, for IEEE 30 bus system also propose method is proved best optimal value in comparison to all the other methods in the literature. Furthermore, Kruskal-Wallis test is performed to show the effectiveness of the proposed ESDE-MC method.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-025-06555-z.

Introduction

An optimal power generation is essential for a steady operation, control, and planning of the power industry. In this regard, the author J. Carpentier has introduced optimal power flow in¹ based on the concept of economic dispatch. The objective is to identify the possible combination of real power outputs produced by various traditional power generators, voltage magnitudes, transformer taps, and shunt reactors to optimize the thermal cost for a specified load demand by satisfying certain constraints^2,3. The amount of load demand at different buses varies continuously on random basis. Thus, the generation schedule should be adopted promptly without much implications for the cost of operation. However, from the few decades, traditional power production plants such as thermal, gas, and oil have been the primary sources of power production and are major sources of atmospheric pollution. The rise in power consumption, deterioration of traditional energy resources, and the necessity to reduce greenhouse gases (COX, NOX, SOX) have prompted scientists to concentrate on renewable energy sources (RES)^4,5. In the framework of national power savings and reduction in emissions, RES has shown tremendous promise in lowering fuel use and pollutant emissions⁶. In this regard, wind energy production has continuously grown in the last two decades because of its low cost, pure nature, and abundant availability, among another RES⁷. However, the major challenge of wind generation is its uncertainty and erratic behavior of wind velocity⁸. Therefore, to tackle the uncertain behavior of wind, extra cost factors must be added to optimal power flow (OPF). These components of the expected penalty and reserve costs for not using existing wind power as well as wind power shortage have been discussed in⁹.

Literature survey

The influence of the unpredictable nature of wind speed on total operating cost has been presented in¹⁰. It has been evident from¹⁰ that inaccurate forecasts significantly reduce the value of wind power output. The researchers in¹¹ developed a wind generator model based on a new method that correlates wind speed integration with the regular power system. Further, they have concluded that the deviations are minimal in comparison to other existing methods for correlating wind speed. In¹², an extended OPF issue with wind power integration has been presented using the primal-dual interior-point method. However, this study considered the stationary speed wind generator (SSWG) model at a load bus. Further, it has also been observed that SSWG always consumes reactive power; thus, capacitors are often installed in series with the generators to make up for the reactive power losses.

The authors in¹³ suggested the shuffle frog leap technique to solve the OPF problem by considering the combined, variable, and stationary speed wind generators. It has been suggested in¹³ that the generator bus may be treated as an active power-voltage (PV) bus in power flow calculations because it can compensate for the reactive power. Researchers in¹⁴ developed a model that determines the wind generator’s cost and introduced small signal stability restrictions to solve the OPF problem using self-adaptive evolutionary programming. However, the above work is limited for only the single time period of the OPF problem. A wind power production model based on a relative histogram has been developed in¹⁵ by incorporating it into an expanded conic quadratic OPF. It is observed that the developed model is helpful in analyzing the forecasting error distribution and wind power cost coefficients. An incomplete gamma function (IGF) has been proposed in¹⁶ by considering wind and thermal generators combined effect by creating a load dispatch model. However, this method suffers from reduced run time and difficulty in selecting the initial parameters of the method. In¹⁷, an OPF model that integrates wind energy variability, up and down spinning reserves, and thermal power production costs has been described to minimize the total system production cost. A bacterium foraging algorithm has been suggested in¹⁸ to solve the OPF by modeling the wind generator as a double-fed induction generator (DFIG). The presented algorithm has provided the optimal results under normal and contingency conditions. In¹⁹, two distinct types of asynchronous wind turbines (WTs), namely the active power-reactive power (PQ) and resistance-reactance (RX) model to solve the OPF problem. Due to this type of modeling, the suggested method requires two iterations to process, thus increasing the computation time in solving the load flow for larger power systems. The authors in²⁰ presented one iterative process that considers the reactive (Q)- voltage (V) model of asynchronous wind turbines in solving the OPF process to overcome the problem of convergence speed. Due to the intermittent jump mechanism, local search accuracy and convergence efficiency of the cuckoo search algorithm are low, therefore, Modified cuckoo search (MCS) algorithm for improving search capability and computation time is provided in²¹ and harmony search (HS) which has less adjustable parameters and quick convergence are implemented by authors to solve OPF in²², respectively, to compute the OPF problem by incorporating the wind power generation using a wound rotor induction generator (WRIG). However, the value of pitch adjustment rate is difficulty to find and inappropriate value of PAR leads to suboptimal solution. Therefore, Quasi-oppositional is developed in²³ to solve the optimization problem. In²⁴, a hybrid particle swarm optimization (HPSO) is used to solve the OPF by incorporating wind power and addressing the security limitations by comparing the baseline and contingency scenarios. The suggested HPSO has achieved better optimal results in lieu of its improved diversity in search space and not getting trapped in local optima than classical particle swarm optimization (PSO) and other algorithms reported in the literature for both normal and contingency cases. In²⁵ presented an OPF model integrating wind energy, focusing on minimizing power losses, voltage deviation, and emissions. It employs the equilibrium optimization algorithm (EOA) to solve the complex, multi-objective OPF problem efficiently. The proposed method is validated on standard IEEE test systems, showing improved performance over traditional techniques. The study highlighted the EOA effectiveness in enhancing power system operation with renewable energy integration. In²⁶ utilized novel self-adaptive wild geese algorithm (SAWGA) for solving OPF problems in power systems. The approach incorporated stochastic models of wind and solar power to address variability in renewable energy sources. The algorithm optimizes economic, environmental, and technical objectives, achieving improved solutions compared to conventional methods. explores the use of machine learning techniques for wind speed prediction. The authors evaluated various models to enhance forecasting accuracy, which is vital for applications like renewable energy generation and weather forecasting. Their study demonstrated the effectiveness of machine learning approaches in capturing the complex patterns of wind speed variations²⁷. In²⁸, proposed a hybrid energy management strategy for multi-energy microgrids to address uncertainties in renewable energy sources. It integrated advanced optimization techniques and machine learning models for efficient energy dispatch and load forecasting. The approach reduced renewable energy variability impacts by 15% and improves system efficiency by 20%. Simulation results demonstrate its effectiveness and cost efficiency compared to existing methods.

In²⁹, security constraint OPF is solved using the fuzzy artificial physics optimization (FAPO) method with the inclusion of wind power in normal and contingency phases. The fuzzy model in²⁹ is used to find the value of gravitational constant, which helps to find a better solution than other methods. In³⁰, OPF is solved with a power system integrating with thermal and wind generators whose wind speed is simulated with Weibull distribution. Further, the solution is assessed by considering the reserve costs and the constraints are handled using the penalty function. Using the proposed approach, it is observed that the solution that emission has been reduced with the inclusion of wind generators. The authors in³¹ have extended the concept by considering the unpredictability of wind energy and load demand using the probability distribution function (PDF). The results obtained using the suggested technique considerably reduce nominal power generating costs while reducing the probability of constraint violation. Weibull PDF is used to address wind speed uncertainty in³², and PSO is used to evaluate OPF. The obtained results proved that wind power generation leads to a reduction in thermal cost and transmission loss. In³³ presented a probabilistic OPF approach for wind-thermal coordination, addressing wind energy intermittency. It incorporates uncertainty analysis to ensure reliable system operation while optimizing cost and performance. The study demonstrates the effectiveness of balancing renewable and conventional energy sources under variable conditions. The concept of dynamically controlling the cognitive and social behavior of the swarm is introduced in classical PSO and developed enhanced PSO^34,35 to improve the solution quality. In³⁶ differential search algorithm optimized hybrid power systems with multiple RESs by improving power balance, reducing costs, and ensuring reliability. The paper³⁷ proposed a trigonometric mutation-based backtracking search algorithm (TM-BSA) for solving the OPF problem with renewable energy integration. The method improved optimization by enhancing BSA with trigonometric mutation, achieving better cost minimization and power loss reduction. In³⁸, the authors have tackled the OPF problem by integrating the wind power and the solar power utilizing a success-history-variable adaptive differential evolution (SHADE). The SHADE introduced an effective constraint handling method, which helped to maintain all the operating constraints within definite limits. A modified Jaya (MJAYA) algorithm³⁹ is used for OPF, incorporating RES on a different objective function that handles cost, voltage profile, power loss, and emission. This methodology provided better optimal results than the original JAYA and other techniques given in the paper.

In⁴⁰, a triangular fuzzy membership function-based model is used to determine the optimum proportion of wind power to the overall cost, power loss and emission profile of the thermal generators. Lognormal and Weibull PDF has been used to forecast the uncertain power outputs from solar and wind generators in⁴¹ to compute OPF. The uncertainty in load demand and stochastic nature of wind power has been considered in⁴² and the solution is obtained using the bird swarm technique. OPF formulation in⁴³ considers WT and photovoltaic (PV) power generation as dependent variables, and voltage magnitudes at WT and PV buses are treated as decision variables. OPF is solved with the inclusion of RES by using heap-based evolutionary technique, and different scenarios are conducted to identify the optimal location of PV and wind buses in⁴⁴. The authors introduced the OPF inclusion of wind power in⁴⁵ using Monte-Carlo Scenarios, the stochastic nature of wind speed, wind frequency distributions, and Weibull fitting were achieved. In⁴⁶, the authors solved OPF with wind power, the current study predicts that a wind farm that includes varying wind turbines that are capable of providing pro-active electricity to the grid would have an interrelationship with the wind farm. OPF including wind, thermal, and solar powers is addressed in⁴⁷ using a persistence - extreme learning machine (P-ELM) method. This method utilizes a mix of persistence and extreme learning machine techniques to predict wind speed and the solar insolation that reduces thermal cost and active power loss. From the above literature, it has been identified that the researchers have tackled the OPF problem by considering various methodologies, i.e., without and with integrated wind power, considered wind power with wind speed uncertainty, different types of wind turbines, consumption of reactive power by wind generator, and PV systems. Further, it is identified that the researchers incorporated uncertainty of wind speed in terms of overestimation and underestimation costs, modeled the wind turbines as doubly fed induction generator (DFIG) or WRIG to generate wind energy, and employed classical and evolutionary optimization methods for finding the solutions. Despite the fact that these techniques are capable of providing OPF solutions to a certain degree, the current work has been carried out in order to make additional improvements to the method utilized as well as to accelerate the execution process.

Storn and Price developed a numerical optimization called Differential Evolution (DE) in 1990⁴⁸. Differential evolution and their variants have been successfully used in several fields⁴⁹, such as, PV models⁵⁰, water supply pollution source positions⁵¹, non-linear mathematical equation systems⁵², multi-objective economic emission dispatch problems⁵³. The DE algorithm regardless of the original parameter values can discover a near optimum solution and has improved convergence characteristics⁵⁴. In addition, the OPF issue is also solved by DE and DE variations in^55,56. In⁵⁷, a variant of DE, namely success-history adaptive DE has been utilized to find a solution for OPF including wind and solar energy in the system and traditional thermal generators. In⁵⁸, the OPF problem is resolved with the suggested small-population parallel DE. In⁵⁹, DE was used for OPF with three distinct constraints handling methods. The optimal external archive adaptive DE is designed for optimization based on promising performance in the advanced DE version⁶⁰. But owing to dynamic population creation, the disadvantage of JADE has delayed convergence. The EJADE-SP was therefore developed to improve JADE’s performance by introducing sorting mechanism for crossover rate (CR), randomizing the CR and the mutation factor in order to maintain efficiency and diversity, speeds up convergence by dynamic population reduction strategy⁶¹. DE multi criteria trial vector generation methods and control parameters are individually modified for their recent success values in⁶². The effects of operator applications in terms of exploitation and exploration capacities are evaluated in the multi-criteria adaptation scheme and a multi-objective decision process is used to aggregate the impacts accordingly. However, it has been discovered that in DE, solutions move relatively quickly towards the optimum point initially, but then slow down when fine tuning of the parameters is necessary⁶³. DE has a number of control parameters due to which choosing appropriate parameter values is typically a problem-dependent process that necessitates the user’s prior expertise. Despite its critical relevance, there is no uniform approach for establishing of DE control parameters, which are often chosen arbitrarily within certain predetermined limits⁶⁴. Some DE mutation variants are more robust but less efficient in terms of convergence rate⁶⁵. In spite of these concerns, DE has some more limitations namely trapping in local optima, slow convergence in identifying the global optimum solution and numerous changes in self-adaptive control parameters namely, self-adaptive with strategy adaption and self-adaptive DE with neighborhood strategy etc. are reported in^66–68 to achieve its improved performance.

In order to address the aforementioned issues of conventional DE, a novel Enhanced Self-adaptive Differential Evolution (ESDE-MC) technique is introduced in the current work.

The contributions of this paper are as follows:

1. In the proposed technique, a mixed crossover operator explores a new possible search area in ESDE-MC methods and helps the search process progress towards the optimum global area.

2. To provide a balance among global exploration and local exploitation, a dynamic self-adaptive parameter management system for mutation and crossover rates is employed.

3. A wind power integrated power systems have been considered to find solution for OPF problem.

4. The wound-rotor induction generator is used for generating wind power. Further, the uncertainty in wind power generation has been considered by adding additional cost components in the objective function.

5. The reserve cost for wind power shortages and wind surplus penalty charges are also included in these costs.

6. To evaluate the efficiency and robustness of the proposed ESDE-MC approach, modified IEEE 30-bus and IEEE 57-bus systems are considered in the present paper.

The remaining article is organized as below: Sect. 2 exhibits paradigm for wind generation cost modeling. Section 3 demonstrates OPF problem formulation including wind power generation cost. Section 4 reports on the simulation findings and statistical assessment, and Sect. 5 highlights the paper’s contribution.

Paradigm for wind generation cost modeling

Portrayal of wind energy based on Weibull PDF

The parameters of the wind speed changes at a given place usually follow Weibull PDF function and is referred from^69,70. The Weibull PDF characterized by scale (C), shape (K) factors and wind speed v m/s is given in Eq. (1):

1

In the present study, the relationship between wind speed and wind power conversion system (WPCS) is regarded a linear power output and is described in Eqs. (2) & (3) and taken from²¹.

2

3

For winds below and above cut-in speeds, there is no wind power. As long as the wind speed is within the range of rated and cut-off wind speeds, the power output is equal to the rated value. In other words, this suggests that the wind power random variable is discrete in character. In order to do this conversion, it is necessary to use a tool like: So, it is important to transform wind speed PDF to wind power PDF and is given in Eq. (4)¹⁵ for linear portion.

4

Wind generator cost analysis

Due to intermittent nature of wind speed, the energy comes from a wind generator might not have been equivalent to its planned energy. As a result, several situations occur, as stated below⁹.

Reserve cost calculation for wind energy shortage

The real wind energy might be lower in comparison to the anticipated energy, called overestimation. The independent system operator (ISO) acquires energy through reserve capacity in order to satisfy the load, when it is overestimated and the price for the purchase of this power is called reserve cost and mentioned in Eq. (5)⁹:

5

Penalty cost calculation for wind energy excess

Underestimation occurs when real wind energy exceeds anticipated energy. In this situation, there is more wind energy available. Economically, the ISO must utilize the available wind energy that may be accomplished by reducing the output of the traditional power system via AGC and this procedure is not feasible, ISO must pay a reasonable fee to the wind energy supplier, known as a penalty cost, as described in Eq. (6)⁹:

6

Mathematical modelling of induction generator

Induction Generator (IGs) are utilized instead of synchronous generators in traditional power plants as wind energy generators. IGs fall into four major categories: i) squirrel cage IGs, (ii) WRIGs having adjustable rotor resistance, (iii) doubly fed IGs, and (iv) IGs with full-size power converters²¹. The first two types consume reactive power, and that is a function of the actual power produced and the voltage magnitude, but the latter two types regulate both active and reactive powers independently and operate at a higher speed. However, due to high power conversion efficiency, good power quality and lower mechanical strength WRIG is generally preferred⁶¹ in the current work and a mathematical model for it is developed and the wind generator’s equivalent circuit is shown in Fig. 1²¹.

Fig. 1.

Equivalent circuit diagram of IG¹⁹.

WRIG produces active power () and consumes reactive power () are given as Eq. (7) to Eq. (9).

7

8

9

With the existence of a wind power, traditional load flow calculation is difficult to employ and it must be modified. The produced real power is influenced by wind speed, whereas the absorbed reactive power is regulated by terminal voltage and the real power produced at the bus to which the wind generator is attached. As a result, in Newton-Raphson (NR) load flow computations, this bus is treated as a unique kind of load bus, as described by Eq. (10) and Eq. (11)²¹:

10

11

ESDE-MC method to find solution for optimal power flow problem incorporating wind energy

Initialization

An individual that indicates complete solution for OPF issue, is randomly initialized. It comprises of control variables namely, generator bus active powers except slack bus, voltage magnitudes, tap settings, shunt compensators reactance’s, and scheduled wind power that are generated randomly within prescribed limits and individual may be expressed in Eq. (12).

12

The complete initial population for NP population size represented in Eq. (13).

13

OPF problem formulation with wind generation cost

The objective function consists of thermal generator costs, overestimation and underestimating costs owing to wind power uncertainty. The basic quadratic cost curve to depict the fuel costs of thermal generators is expressed in Eq. (14)⁴⁶.

14

The cost of wind generation, considering the impact of over and under estimations, expressed in Eq. (5) and Eq. (6) were added to Eq. (14) to reflect the total cost of thermal and wind production that must be optimized for efficient scheduling, which is expressed as the objective function in Eq. (15)²¹.

15

subject to the following constraints²¹

16

17

18

19

20

21

22

In the present work, the penalty factor method is²¹ used to handle the inequality constraints. These inequality restrictions are included with the objective model as quadratic penalty factors. As a result, the fitness function with penalty factors is expressed in Eq. (23):

23

where f indicates objective function value. P_p, P_v, P_Q, P_s are the penalty coefficients of the respective constraints. Here, these values are chosen high to eliminate the infeasible individuals during the iterative process. P_G1, indicates slack bus active power output.

Mutation

The mutation is to generate a donor vector from the parent vector to locate the associated child vector in a better position. In literature, DE/rand/1 mechanism is extensively used however, this mechanism suffers from slow convergence rate. To address this situation, numerous researchers have shown that DE/best/2 is one of these approaches and has a high potential similar to DE/rand/1, and is rapidly converging because of its inclusion in the evolutionary search for the best solution^71,72. The current research thus used a DE/best/2 mutation strategy, and is expressed in Eq. (24).

24

Proposed mixed crossover

The mixed crossover (MC) is suggested in this work that is depending on the choice of eigenvector or binomial crossovers and is explained as below:

Binomial crossover (BC)

Binomial crossover is being used to increase chromosomal diversity by generating a trial vector via transfer of knowledge among target and donor vectors. The amount of individual’s moves from target and donors’ vector is decided by crossover probability and is given Eq. (25)⁶⁷:

25

Covariance matrix-based Eigen crossover

In general, DE crossovers exhibit rotationally variable and invariant strategic behavior. The chromosomes formed via rotationally invariant crossover (RIC) usually to imitate the operational landscape that accelerates convergence. The RIC, on the other hand, falls short of exploring the search space, yielding a local optimum. In addition, the rotationally variant crossover may span the search area and discover an almost optimal solution, which results in a little delayed convergence. As a consequence, this research employs covariance matrix-based EC to improve the efficiency of RIC through moving the reference system to follow the operational landscape. The covariance matrix, which is made up of variance and covariance, in particular, depicts the variety of the population as well as the interactions between the variables. As a result, using the covariance matrix systematically should be highly effective for easing DE’s dependency on the coordinate system and removing the interactions between variables. In the present work, the covariance matrix learning includes two core techniques: Eigen decomposition of the covariance matrix and the coordinate transformation. Here, first covariance matrix of the population is generated and then eigenvectors of such covariance matrix are produced. Thereafter, the donor vector and target vector are translated into Eigen reference system (ERS). This method guarantees that the chromosomes are represented in ERS instead of initial reference system (IRS). Next, binomial crossover is performed in ERS to generate a trial vector later it is moved to original coordinate system. The trial vector produced in ERS is very effective in identifying optimal solutions instead of initial reference system. The methodological flow for EC is explained as follows²⁴:

• i)
  Evaluate the covariance matrix of the whole individuals. Convert to canonical structure to determine eigenvectors.

  26

  where is an orthogonal matrix, and the column of this matrix denotes the eigenvector . denotes diagonal matrix composed of Eigen values.

• ii)
  Move the donor and target vectors into ERS.

  27
• iii)Implement BC in ERS to generate trial vector .
• iv)Finally, move the from ERS to IRS.
  28

Mixed crossover (MC)

Because of the difficulties in exploring the search space, the rotationally invariant nature of EC will not successfully lead the search strategy heading a best solution. To mitigate the possibility of ineffective behavior of the RIC a mixed crossover is introduced to combine the original crossover operator with its EC. The procedure for MC is as follows:

29

where P is an eigenvector ratio created here between zero and one that will be used to establish the relative strength of the EC or BC. This technique yields effective minimizations for generic functions that may be solved by arbitrarily interleaving both the EC operator (P = 1) and the BC operator (P = 0).

Self-adaptive factor adjusting

Like many evolutionary techniques, DE is also sensitive for tuning factors namely F and rates⁶⁸. The changing values of these factors may influence algorithm performance and cause the solution to get trapped in sub optimal. So, a self-adaptive factor method is included in current study. Here, every starting target vector F and values are produced at random between lower and upper values. Then, to create a trail vector, related attributes are employed to each target vector . If successfully passes to next iteration, similar attributes are passed on to the following iteration. Instead, attributes of are produced at random among lower and higher values.

Selection

It compares the fitness values of respective trail and target vectors and the fitness value is the greatest pass to the following generation⁶⁷.

30

Procedural steps for proposed ESDE-MC approach for solving OPF problems

• Step 1. Set ESDE-MC factors such as , maximum and minimum bounds of mutation rate (), and crossover rate () and maximum iterations (G_max).

• Step 2. An individual that provides a full solution to the OPF issue is arbitrary generated inside the search area that incorporates CVs such as active power outputs excluding reference bus, scheduled wind power, generators voltages, transformers tap, and shunt reactors. As a result, a single individual is expressed in Eq. (10).

• Step 3. Create for every starting target vector among lower and higher bounds.

  31

• Step 4. Run the load flows using in Eq. (12) and Eq. (13) to get voltages and angles at each bus.

• Step 5. Compute the fitness function value of every target vector using Eq. (15).

• Step 6. Using construct trial vector for each using Eq. (19) and Eq. (24).

• Step 7. Set any variable to its appropriate boundary value if it is exceeded.

• Step 8. Compute objective function values of every using Eq. (15).

• Step 9. Compare objective function values of every with the respective and better chromosome is added to succeeded iteration . Evaluate for all new target vectors using Eq. (32).

  32

• Step 10. If the present generation exceeds the G_max, stop the search process and display the better value from previous generation as best solution, else move to Step 6.

Simulation results

To assess the performance and viability of ESDE, ESDE -EC and ESDE-MC methods in resolving OPF including wind power by considering modified IEEE 30-bus and IEEE 57-bus systems.

Parameter settings

The wind farm is made up of 20 WRIGs of the same kind, each with a capacity of 2 MW. The wind turbine and IG specifications are taken from²¹. The c & k are both set at 2 and 10, respectively. The contribution of thermal and wind generators to load mostly determined by the factors K_H & K_L. There are many combinations of K_H & K_L that exists. However, in this study, two distinct cases of K_H & K_L are examined and classified as M1 (K_H = 2 $/MWh and K_L = 5 $/MWh) and M2 (K_H = 2 $/MWh and K_L = 10 $/MWh). The lowest and highest ranges of the considered are 0.1 and 1 respectively. At first, the optimal solution to an objective function was found in M1 of a modified IEEE 30-bus network by repeating the proposed methods for 50 to 200 runs with a difference of 50 runs. According to collected results, the objective function value does not improve further after 200 runs. So, in order to ensure consistency across all the methods in the current work, 200 runs are examined. The NP and P are set to 60 and 0.5 respectively. The simulation programs were built using 2.2 GHz core i3 CPU and 4 GB RAM in the MATLAB 7.10 environment.

Characteristics of wind power cost

The variation of different costs namely, wind power, overestimation and underestimation are investigated as shown in Fig. 2. The cost of underestimation is maximum when zero wind power is actually generated and it increases with increase in wind energy. But from the other side, overestimation cost is zero when wind power generation is zero and it rises as wind energy increases. The total cost (TC) of producing wind energy is the sum of the costs of overestimation and underestimating. The interesting observation is that the slope of the wind energy cost curve shown in bold black in Fig. 2 is initially negative and gradually increases to zero and then gradually increasing becomes positive.

Fig. 2.

Wind power cost generating curve with c = 10, k = 2.

Modified IEEE 30-bus system

It has 6 generators, the step size handling of four transformer taps of 0.025 p.u. and two shunt reactors of 0.01 p.u. The complete system data, lowest and highest ranges of all control variables and fuel cost parameters are obtained from²¹. The wind farm is included at bus number 22. The modified IEEE 30 bus system along with wind farm depicted in Fig. 3. The proposed methods were robustly evaluated by 20 independent trials and compared with MCS²¹, modified PSO (MPSO)¹⁹ and harmonic search (HS)²² and respective results, the minimum, the average, the worse total costs (TC) and standard deviation (SD) as well as the mean runtime (MRT) are presented in Table 1. These findings showed that proposed ESDE-MC technique is providing superior statistical results than the proposed ESDE, ESDE-EC, krill herd (KH) stud krill herd (SKH) and the other techniques namely MPSO²¹, MCS²¹and HS²² reported in the literature in acceptable running time. The standard deviation in Table 1 showed that the current technique is more resilient than the other algorithms for identifying optimal solutions. Table 2 provides the thermal, wind and total costs and optimum control variables in the three methods. These results prove that the proposed technique has the lowest thermal power cost and overall cost in comparison to other techniques. The convergence curves of M1 with developed methods are depicted at Fig. 4, and observed that ESDE-EC and ESDE methods have parsimonious characteristics with TC values of 732.7654 ($/h) and 733.2610 ($/h) in the first 85 and 123 runs respectively and thereafter there is no further improvement in TC. On the other hand, the TC acquired by ESDE-MC decreases rapidly in the first 98 runs to 732.5586 ($/h). From this description, it can be observed that the ESDE-MC method at the beginning of the run process offers high convergence and improved performance, in comparison with the other methods, to discover near optimal solutions in an appropriate execution period. The inset illustrated in Fig. 4 depicts the evolution of M1 across a narrow range, starting with the 50th generation and ending with 200th generation, and the y-axis picked again from Y-axis from 730 $/h to 736 $/h and it could assist readers to view and distinguish between the convergence properties of considered hybridized algorithms.

Fig. 3.

Modified IEEE 30-bus system with wind power plant.

Table 1.

Comparison of simulation results with the other algorithms (modified IEEE 30 bus system).

Different cases	Method	Minimum ($/h)	Average ($/h)	Worst ($/h)	SD	AET(s)
M1	MPSO21	735.3632	–	–	–	343.20
	MCS21	733.2596	–	–	–	345.58
	ESDE	733.7883	734.7944	735.7307	0.6344	15.92
	KH	733.2825	733.9824	734.8730	9.4516	25.84
	ESDE-EC	733.1742	733.7639	734.6831	0.4792	26.91
	SKH	732.7054	733.0921	733.6551	0.2726	20.14
	ESDE-MC	732.5586	732.7654	733.2610	0.2080	18.75
M2	MPSO21	787.8148	–	–	–	365.76
	MCS21	787.5013	–	–	–	341.34
	HS22	788.1882	–	–	–	–
	ESDE	788.2707	789.3082	790.0526	0.6179	16.35
	KH	787.7886	788.3366	788.8212	0.4153	25.01
	ESDE-EC	787.6704	788.2641	788.9793	0.4318	21.24
	SKH	787.2152	787.5199	787.8350	0.2128	20.89
	ESDE-MC	786.9649	787.1859	787.5104	0.1430	19.12

Significant values are in bold.

Table 2.

Optimal combination of control variables for all the cases (modified IEEE 30-bus system).

Control variables	M1					M2
	ESDE-MC	SKH	ESDE-EC	KH	ESDE	ESDE-MC	SKH	ESDE-EC	KH	ESDE
P G1	1.552	156.12	1.54	1.55	1.51	1.627	163.09	1.651	1.500	1.690
P G2	0.436	42.64	0.43	43.55	0.46	0.454	46.26	0.462	41.72	0.416
P G5	0.19	19.74	0.211	19.74	0.19	0.206	20.34	0.192	19.59	0.196
P G8	0.100	10.00	0.100	10	0.10	0.137	12.99	0.120	15.281	0.129
P G11	0.100	10.02	0.100	10	0.11	0.100	10.25	0.100	10.00	0.100
P G13	0.120	12.14	0.120	12.25	0.12	0.120	12.11	0.120	12	0.1200
P sch22	0.400	39.99	0.399	40	0.40	0.263	26.03	0.265	25.49	0.2630
V G1	1.10	1.09	1.10	1.1	1.09	1.10	1.10	1.10	1.1	1.09
V G2	1.08	1.08	1.08	1.092	1.08	1.08	1.08	1.08	1.082	1.07
V G5	1.05	1.05	1.06	1.049	1.03	1.05	1.05	1.049	1.05	1.04
V G8	1.07	1.06	1.07	1.072	1.05	1.07	1.07	1.05	1.063	1.05
V G11	1.10	1.10	1.10	1.1	1.09	1.09	1.09	1.04	1.1	1.10
V G13	1.10	1.08	1.10	1.064	1.08	1.10	1.07	1.07	1.1	1.094
t 6–9	1.03	1.02	1.05	1.05	0.98	1.04	1.02	1.05	1.03	1.050
t 6–10	0.95	0.90	0.93	0.93	1.05	0.94	0.95	1.00	0.93	0.93
t 4–12	1.01	0.96	0.99	0.94	1.05	0.98	1.01	1.02	0.98	0.940
t 28–27	0.96	0.98	0.94	0.95	0.97	0.96	0.975	0.95	0.93	0.94
Q C10	0.120	18.00	0.06	0.05	0.24	0.20	3.00	0.21	0.25	0.00
Q C24	0.150	8.009	0.03	0.04	0.13	0.10	10.00	0.07	0.24	0.11
Total cost ($/h)	732.55	732.754	733.12	733.28	733.78	786.96	787.2152	787.6704	787.78	788.27
Wind cost ($/h)	72.47	72.456	72.39	72.47	72.47	80.84	79.64206	81.4316	77.81	80.57
Thermal cost ($/h)	660.08	660.248	660.78	660.81	661.31	706.12	707.5731	706.2388	709.97	707.69
APL	0.0721	7.273	0.072	0.0742	0.073	0.075	7.6877	0.079701	0.078	0.081
Q wG22	− 0.205	− 20.611	− 0.204	− 0.020	− 0.204	− 0.165	− 15.7741	0.156603	− 0.156	− 0.1596

*All Powers, voltages, transformer taps, reactive powers are in p.u.

Significant values are in bold.

Fig. 4.

Characteristics curves obtained with the proposed techniques of M1 for Test system-I.

The OPF is done for a particular load scenario. The continuous variation of load curve is generally simplified by dividing the overall daily time span into twenty-four number one-hour intervals and assuming the hourly average load remains constant. With the proposed techniques, in the two aforementioned instances, the load bus voltage values that fall within the low and high end of the range are shown in Fig. 5 and it is discovered that these methods are successful at managing the whole range of voltage constraints. The variation of control parameters mutation rate (F) and crossover rate (CR) for both the cases of modified IEEE 30 bus with ESDE-MC algorithm is depicted in Fig. 6 and it is shown that the self-adaptive control methodology tries to change the values of parameters throughout the run of the suggested method by considering the actual search progress, which helped to proposed algorithm for achieving better optimal solution in comparison to other algorithms. The result of investigating the impact of simultaneous variation of cost factors K_L and K_H on the total wind generator cost is illustrated in Fig. 7. The authors have considered the same input parameters of the PDF as mentioned in²¹ to verify and validate the optimal results obtained. Here the general observation is that for a fixed value of coefficient K_H the total wind power cost increases at a larger rate initially but at a smaller one with further increase in parameter K_L. Also, larger the value of K_H larger is the wind power cost.

Fig. 5.

Lower and higher load bus voltages of M1 and M2 with various algorithms for Test system I.

Fig. 6.

Variation of mutation rate (F) and Crossover Rate (CR) in M1 and M2 of Test system I.

Fig. 7.

Impact of cost coefficients (K_L & K_H) on wind power cost.

IEEE 57-bus system

The preceding study is expanded to include a new wind farm on bus number 45 for solving IEEE 57-bus system. It consists of 7 generators, and total load of 1250.80 MW, the step size handling of seventeen transformer taps of 0.025 p.u. and three shunt reactors of 0.01 p.u. The complete system data, lowest, highest ranges of all CVs and fuel cost parameters are taken from²¹. All the proposed methods were used to solve modified IEEE 57-bus system. For 200 runs over 20 trails conducted using the proposed algorithms for M1 and M2, lowest, average and worse TCs, SD and MRT are compared to MCS²¹, MPSO²¹, and details are furnished in Table 3.

Table 3.

Comparison of simulation results with the other techniques (modified IEEE 57 bus system).

Different cases	Method	Minimum ($/h)	Average ($/h)	Worst ($/h)	SD	AET(s)
M1	MPSO21	2937.3	–	–	–	2236.1
	MCS21	2797.1	–	–	–	2459
	ESDE	2796.5964	2801.0326	2804.4350	2.2273	43.27
	KH	2792.7963	2794.5301	2796.6377	1.3579	49.56
	ESDE-EC	2792.3510	2794.4080	2796.5371	1.2837	50.58
	SKH	2790.545	2791.5653	2792.8566	0.6870	48.54
	ESDE-MC	2788.6975	2789.4709	2791.0121	0.6458	46.95
M2	MPSO21	2950	–	–	–	1200
	MCS21	2863.5	–	–	–	541.85
	ESDE	2865.1958	2871.6978	2874.4740	2.4085	44.32
	KH	2862.3142	2865.235	2867.9510	1.7330	48.56
	ESDE-EC	2861.1798	2863.7712	2866.2753	1.5405	51.94
	SKH	2860.0878	2862.1863	2864.5417	1.4879	49.24
	ESDE-MC	2858.1958	2859.3024	2861.7127	1.0198	47.98

Significant values are in bold.

These findings show that, compared with other methods, the ESDE-MC provides excellent statistical results in fair run time. The continuous change of load curve is usually streamlined by splitting the day total time into 24 periods of one hour, with an hourly average load simplified by dividing the daily time span into twenty-four number one-hour intervals and assuming the hourly average load remain constant. In this respect the yearly savings in case M1 of IEEE57 x bus would, in case the hourly improvement in costs 8.40 $/h, be 8.4 × 24 × 365 = 73,584 $. The best combination of CVs and total and individual heating and wind generation costs may be found in Table 4. The convergence graph obtained M1 using proposed methods are given in Fig. 8 and the convergence capacity of the ESDE-MC technique is clearly visible. The variation between the F and CR for the both cases of IEEE 57 bus systems with ESDE-MC method is illustrated in Fig. 9. In the aforementioned two instances, the lowest and highest load bus voltages of the three methods are depicted in Fig. 10 and it is understood from this figure, the suggested technique not only providing better optimal solutions with in less execution time but also, it is able to keep the all the equality and inequality constraints within their prescribed limits.

Table 4.

Optimal combination of control variables for all the cases (modified IEEE 57-bus system).

Control variables	M1					M2
	ESDE-MC	SKH	ESDE-EC	KH	ESDE	ESDE-MC	SKH	ESDE-EC	KH	ESDE
P G1	2.19	2.336	2.11	2.10	2.24	2.28	241.44	2.179	2.23	2.33
P G2	1.49	1.50	1.50	1.5	1.45	1.50	150.00	1.50	1.5	1.49
P G3	1.50	1.50	1.50	1.5	1.50	1.49	150.00	1.495	1.5	1.50
P G6	1.20	1.20	1.19	1.2	1.20	1.20	120.00	1.20	1.2	1.20
P G8	2.30	2.26	2.42	2.47	2.23	2.23	223.40	2.43	2.27	2.28
P G9	1.19	1.20	1.20	1.2	1.20	1.20	120.00	1.19	1.2	1.20
P G12	2.44	2.34	2.40	2.34	2.49	2.42	235.58	2.42	2.47	2.38
P sch45	0.39	0.40	0.40	0.4	0.40	0.38	33.78	0.29	0.33	0.35
V G1	1.06	1.03	1.06	1.09	1.06	1.07	1.07	1.04	1.04	1.07
V G2	1.04	1.02	1.05	1.07	1.05	1.05	1.06	1.04	1.03	1.05
V G3	1.02	1.01	1.04	1.04	1.05	1.05	1.04	1.04	1.04	1.02
V G6	1.00	1.01	1.02	1.03	1.04	1.06	1.05	1.03	1.04	0.99
V G8	1.00	1.01	0.99	1.03	1.06	1.06	1.07	1.03	1.06	0.99
V G9	1.01	1.01	1.02	1.03	1.03	1.04	1.04	1.03	1.05	0.99
V G12	0.99	0.99	1.01	1.02	1.02	1.02	1.01	1.00	1	1.00
t 4–18	0.98	1.00	1.04	0.96	1.05	0.95	0.95	0.96	0.97	0.97
t 4–18	1.05	1.05	0.98	0.97	1.04	1.05	1.04	0.99	0.95	1.05
t 21–20	0.96	0.99	0.96	0.97	0.96	1.02	1.01	0.99	0.95	1.03
t 24–25	1.02	1.03	1.05	0.98	1.04	0.96	0.95	1.03	0.98	1.05
t 24–25	0.98	0.95	0.95	0.99	1.05	0.95	0.95	1.03	0.97	1.02
t 24–26	1.01	0.95	0.99	1.03	1.05	0.95	0.95	1.03	1	0.98
t 7–9	0.95	0.95	1.05	0.98	1.00	1.02	1.04	0.98	0.95	0.98
t 34–32	0.96	0.97	0.95	0.99	0.95	1.04	1.00	1.05	0.96	0.99
t 11–41	0.95	1.05	0.99	1.04	1.01	0.98	0.97	0.98	1.04	0.99
t 15–45	1.01	0.95	0.98	0.99	0.97	1.01	0.99	0.98	1.02	1.02
t 14–46	0.99	0.99	1.02	1.05	1.00	1.05	1.05	1.04	0.95	0.98
t 10–51	1.04	1.02	1.03	0.95	0.99	0.96	0.96	1.01	1.01	1.00
t 13–49	0.95	0.95	1.01	1	1.05	0.96	0.95	1.01	0.99	1.05
t 11–43	0.96	0.95	0.96	0.95	1.04	1.01	1.03	0.97	0.97	0.95
t 40–46	1.05	0.95	0.99	1.04	1.03	1.02	1.04	0.99	1.05	0.95
t 39–57	0.97	0.96	0.97	0.98	1.05	0.96	0.95	0.98	0.96	1.02
t 9–55	0.96	0.99	1.04	0.95	0.98	1.02	1.00	0.97	0.97	1.05
Q C18	0.28	9.43	0.26	0.24	0.06	0.11	4.92	0.13	0.15	0.30
Q C25	0.20	15.52	0.29	0.26	0.30	0.23	22.68	0.29	0.17	0.29
Q C53	0.25	5.64	0.16	0.16	0.24	0.090	11.73	0.26	0.23	0.16
Total cost ($/h)	2788.69	2790.5	2792.32	2792.35	2796.59	2858.19	2860.08	2861.17	2862.3142	2865.66
Wind cost ($/h)	72.37	72.473	72.47	72.47	72.47	138.69	111.70	98.86	107.6	119.78
Thermal cost ($/h)	2716.32	2718.0	2719.87	2719.81	2724.13	2719.55	2748.80	2770.30	2754.7	2745.87
APL (p.u.)	0.2258	23.729	0.226	0.229	0.225	0.225	23.37	0.224	0.235	0.2457
QwG45 (p.u.)	− 0.200	− 20.23	− 0.202	− 0.23	− 0.206	− 0.19.70	− 17.72	− 0.185	− 0.187	− 0.1803

*All Powers, voltages, transformer taps, reactive powers are in p.u.

Significant values are in bold.

Fig. 8.

Characteristics curves obtained with proposed methods of M1 of Test system I.

Fig. 9.

Variation of F and CR in M1 and M2 of Test system II.

Fig. 10.

Lower and higher load bus voltages of M1 and M2 with various algorithms for IEEE 57-bus system.

Statistical assessment

To observe the robustness of the developed methods to solve OPF with wind power, All of the test systems are put through the renowned Kruskal-Wallis test⁷³. To test the null hypothesis that all populations have the same median, statisticians often resort to a nonparametric technique. Assuming the populations being compared are not regularly distributed, this test may be used to compare three or more samples. Twenty separate simulation runs were performed using the suggested methodologies, yielding findings shown in the box plots of Figs. 11, 12, 13 and 14 respectively for Test System I and II. and it is showed that ESDE-MC technique has a very low chance of rejecting the null hypothesis and a very tiny variation. As a result, all of the above assessment demonstrated that the ESDE-MC technique is productive and useful for addressing OPF problems, including wind power.

Fig. 11.

Box type plots of Kruskal–Wallis test of M1 of Test system I.

Fig. 12.

Box type plots of Kruskal–Wallis test of M2 of Test system I.

Fig. 13.

Box type plots of Kruskal–Wallis test of M1 of Test system II.

Fig. 14.

Box type plots of Kruskal-Wallis test of M2 of Test system II.

Conclusions

Optimal power flow problem, including wind farm has been considered in the present research with the inclusion of intermittent nature of wind power generation, and the discontinuous pattern of wind speed is explained with the Weibull PDF. Wind generation costs, including wind energy surpluses and shortage, are added to the cost of modeling, which are the penalty costs for the intermittent nature of wind speeds. Wind energy is generated using a WRIG and the Q-V methodology is included in load flow program to manage the wind generators. Both, modified IEEE 30-bus and IEEE 57-bus systems with two distinct configurations of reserve and surplus cost factors, have been considered to exhibit the reliability and efficiency of the novel computational methods proposed in the current work. The achieved results with the proposed methods are compared with MCS, MPS KH and SKH methods. These results proved that the proposed ESDE-MC method exhibits better optimal values by satisfying all constraints in less time in comparison to the ESDE and ESDE-EC techniques investigated in this research and other methods. The simulation results clearly depicted that the generation of the clean form of energy from RESs has been enhanced which results in reduction in emission pollution in the hybrid power system by imposing a carbon tax. Apart from this, the statistical assessment showed the proposed method is efficient in comparison to other techniques published in the past work. In view of the aforementioned findings, it is proved that ESDE-MC technique is efficient and superior in terms of faster convergence characteristics and achieving best minimum objective function to solve OPF with inclusion of wind-thermal generators.

Electronic supplementary material

Below is the link to the electronic supplementary material.

List of symbols

target vector in generation

f

Objective function

Sum of stator & rotor reactance

Net real and reactive powers injected atbus accordingly

Cost coefficients ofthermal generator

Different constraints weight factors

values oftarget vector initeration

target and donor vectors in ERS

Number of shunt reactors, transformer taps, transmission lines, wind generators respectively

s

Slip of IG

Voltage and its lower and higher limits atload bus accordingly

Donor vector initeration

Best chromosome ingeneration

Orthogonal matrix, andcolumn of this matrix denotes the eigenvector

trial vector in IRS

Active power of reference bus

Fitness value

target vector initeration

Population size

Rotor resistance

Symbolizes phase angle among and buses;

Weibull Probability distribution function (PDF)

A chosen random number of the population that are mutually distinct

Crossover rate oftarget vector ingeneration

Diagonal matrix composed of eigen values

Susceptance and conductance amongandbuses accordingly

Magnetizing, stator and rotor reactance accordingly

Mutation rate oftarget vector in generation

Voltages at and buses accordingly

Lower or higher limits of reference bus, actual power, load bus voltages, reactive powers of the generators and line MVA accordingly

Reactive power and its lower and higher limits atgenerator bus accordingly

Shunt reactor and its lower and upper limits of shunt reactor respectively

Voltage and its lower and higher limits at generator bus accordingly

Number of buses

w

Designates wind power accessible

Population matrix ingeneration

Specifies planned power

Coefficients for wind energy excess and reserve costs respectively

MVA flow and its utmost MVA flow inTL accordingly

Number of thermal generators

Real power and its lower and higher limits atgenerator accordingly

Tap setting and its lower and higher limits oftransformer tap accordingly

Wind energy penalty cost

Wind energy reserve cost

CV

Number of control variables

n_rand

Random value, that exists among [1,CV], confirming that minimum one individual of gets moved to

Author contributions

Harish Pulluri1, Conceptualization, methodologyTellapati Anuradha Devi2, formal analysisNarsimha Reddy Kuppireddy3,formal analysis Preeti Dahiya4, software, validationCh. Naga sai kalyan5 formal analysis, project administrationB. Srikanth Goud6*, investigation, visualization, supervisionMohammad Shorfuzzaman7, investigation, funding acquisitionAli ELrashidi8,9,* resources, data curation, funding acquisitionWaleed Nureldeen9,resources, data curation, funding acquisitionCh Rami Reddy10,11,writing—original draft preparation, writing—review and editing,

Funding

This research was funded by Taif University, Taif, Saudi Arabia, Project No. (TU-DSPP-2024-50).

Data availability

Data Availability Statement: All the data for producing the results are kept within the manuscript.

Declarations

Competing interests

The authors declare no competing interests.