1. Introduction
The main goal of optimal power flow (OPF) is to satisfy system restrictions while optimizing an objective function like total cost of generation, power losses, and voltage stability [
1]. The OPF problem is a nonconvex, large-scale, and non-linear constrained optimization problem, which has been widely used in power system operation. Because of these features, solving the OPF problem is a very popular and challenging task in power system optimization. A feasible optimization approach must be chosen in order to address such a challenge. Many classic optimization approaches have already been used to address the OPF problem, including linear programming, nonlinear programming, quadratic programming, Newton method, and interior point method [
2,
3,
4]. Many traditional methods are commonly utilized in the industry; however, prior to using these techniques, appropriate theoretical assumptions must be made. As a result, those methods are restricted to handle specific sorts of optimization issues. A population-based metaheuristic method is now used to handle complex OPF problems. Multiple potential solutions are maintained and improved utilizing population-based techniques, which frequently use population features to guide the search. Researchers from all over the world have explored OPF using only thermal power generators by using metaheuristic methods [
5]. Kumari [
6] applied an upgraded genetic algorithm (GA) with quadratic load flow solution to tackle the traditional OPF problem, based on the Pareto evolutionary algorithm. Khunkitti [
7] presented a hybrid dragonfly and particle swarm optimization (PSO) technique in solving traditional OPF in order to minimize fuel expense, emission, and power loss. Basu [
8] proposed a differential evolution (DE) method to solve the traditional OPF problem in power systems using FACTS devices, taking into consideration generating costs, emissions, and power losses. Singh [
9] presented the PSO method combined with an aging leader and challengers to solve OPF problem in IEEE-30 bus and IEEE-118 bus systems. Attia [
10] applied a modified Sine-Cosine algorithm which contained Levy flights to solve the OPF problem in IEEE-30 bus and IEEE-118 bus systems. Shuijia Li [
11] presented an enhanced adaptive DE with a penalty constraint handling strategy that adapts to the situation to solve OPF in an IEEE-30 bus system
Traditional OPF problems only consider thermal power sources; however, rising fuel prices and environmental concerns have prompted countries to consider renewable energy sources such as wind power. Hence, there is a need to consider wind generation cost in the classical OPF problem and get the optimal operation for the system containing wind energy sources. This problem is called a stochastic optimal power flow [
12,
13]. Liu [
14] proposed an economic dispatch problem incorporating wind power energy and used a genetic algorithm method for coordination of thermal and wind dispatching. Miguel [
15] examined the impact of wind’s stochastic nature on system total operating cost considering the effects of variable loads and errors in the forecasting of wind power on the different components of generation cost. Hetzer [
16] proposed an economic dispatch problem incorporated with wind energy to determine the best output power allocation among the many generators available to fulfil the system load. Dubey [
17] proposed a hybrid flower pollination method with a fuzzy selection mechanism to solve SCOPF which included emission and the generator’s valve-point loading effect for a hybrid system including wind energy. Kusakana [
18] presented SCOPF in a hybrid system which includes solar photovoltaic, wind, diesel generators, and batteries to minimize total operating cost at fixed values for over/under estimation cost coefficients. Karam [
19] applied the multi-operator DE method to solve OPF problem incorporating wind and solar power in IEEE-30 bus and IEEE-118 bus systems while taking into account the variable nature of solar and wind power generation. Partha [
20] proposed a success history-based parameter adaptation approach for DE algorithm to solve the OPF problem combining stochastic wind and solar power. Inam [
21] applied the Gray Wolf Optimizer method to solve the OPF problem which combines thermal power, wind energy, and solar energy in IEEE-30 bus and IEEE-57 bus system. Arsalan [
22] applied the Krill Herd algorithm to solve OPF problems considering FACTS devices and wind energy generation under uncertainty using Weibull PDF in IEEE-30 bus and IEEE-57 bus systems. Mohd [
23] applied the Barnacles Mating Optimizer method to solve the OPF problem with stochastic wind energy in modified IEEE 30-bus and IEEE 57-bus systems. The SCOPF literature contribution can be summarized as indicated in
Table 1.
This research presents a stochastic optimal power flow (SCOPF) model that is combined with wind energy sources. There are two components added to the cost of dispatching wind power, which is wind power underestimation cost and overestimation cost. The underestimation cost refers to the expense of employing greater reserve capacity, while the overestimation cost refers to the fact that the system operator is required to acquire additional electricity from wind farms that they had not anticipated being available. The objective of SCOPF is to obtain optimal scheduled power from wind farms and optimal generating power from the thermal unit which minimizes total operation cost.
In this work, a novel metaheuristic optimization technique called Aquila Optimizer (AO) was proposed to solve the SCOPF problem. First, the traditional OPF problem was solved using the AO algorithm using IEEE-30, IEEE-57, and IEEE-118 buses systems then results were compared to other metaheuristic techniques to prove superiority and accuracy of the proposed method. Second, the SCOPF problem was solved to get optimal scheduled wind power at six scenarios for overestimation and underestimation cost coefficients to minimize total operation costs. The variation of overestimation and underestimation cost coefficients affects the share of the thermal and wind generating power; this work demonstrates the reason for changes in power share in six scenarios. These scenarios studied the effect of changing overestimation and underestimation cost coefficients in the values of optimal fuel and wind generation costs. Moreover, this work presented a study to demonstrate the effect of changing scheduled wind power on the wind generation cost at different values of overestimation and underestimation cost coefficients. A modified IEEE-30 bus system which includes two wind farms rated at 80 MW and 50 MW were used to solve the SCOPF problem using the AO method. Wind speed probability distribution function (PDF) was simulated using the Weibull distribution curve and the AO method was used to determine the optimal parameters of Weibull distribution which minimized root mean square error (RMSE). The wind power PDF of the wind farm was designed based on the Weibull wind speed distribution and it was a mixed discrete and continuous distribution. Modeling of wind speed was based on an hourly wind speed dataset collected from a site in Texas for 5 years [
24].
4. Aquila Optimizer
AO is a population-based metaheuristic optimization technique. This method is based on getting and improving multiple potential solutions utilizing population-based techniques, which frequently use population features to steer the search. Because of their capacity to explore search space and exploit local resources, population-based metaheuristic algorithms are ideal for global searches.
The natural habits of the
Aquila when catching its prey inspired this method. The optimization processes of this algorithm are separated into four categories [
26]:
Select a search area from a vertical stoop
[
26].
The AO method has two stages, called exploration and exploitation, to update the current populations. The exploration stage starts when
. It has two approaches, the first of which is based on Equation (19).
where
is the best global solution and T is the number of total iterations. During the exploration phase, the factor
is used to manage the search, R is random values between 0 and 1, and
is the average of the measurements for each individual.
Use a contour fly and a short attack to explore a diverging search space
[
26].
The AO relies on the Levy flight distribution to update the current person in the second exploration technique, as described in the following equation:
where
is an individual chosen at random,
is a term used to describe the Levy flight distribution which can be formulated as:
where s = 0.01 and b = 1.5 are constants, while u and v are random numbers created between 0 and 1. To imitate the spiral shape, y and x are employed as follows:
where
is a random number between 0 and 20,
= 0.005, and U = 0.00565.
Investigate in a convergent search space with a slow attack and low flight
[
26].
During the search process, two strategies are utilized to replicate individuals’ exploitation abilities. The first technique is based on the best answer (
) and the average of each individual’s position (
), and is written as:
where R is a random number between 0 and 1,
and
indicate the parameters for adjusting the exploitation phase which were fixed at a small value = 0.1 in this work, UL and UB are the lower and upper limits of the optimization problem
Swoop down and grab prey with a stroll
[
26].
The second way of exploitation is based on
,
, and the
quality function.
where the primary goal of QF is to balance search techniques, and it is characterized as:
G1 denotes the many motions used to find the optimal solution, and it is defined as
G2 is a random number descries from 2 to 0, and it is formulated as
6. Conclusions
In this work, the AO method was used to solve the OPF problem to optimize the generation cost of thermal generators. The performance of the proposed method was tested on standard IEEE-30, IEEE-57, and IEEE-118 bus systems and results were compared with other existing optimization techniques from the literature. The results showed that AO method produced the lowest fuel cost among the other algorithms under comparison, which proves the validity and accuracy of the AO method. Then, the OPF problem with wind generation sources was investigated and exploited. Wind speed was simulated using Weibull PDF and the wind power was expressed as a random variable by applying a transformation to wind speed PDF. The optimal parameters of Weibull PDF were estimated using the AO method to get the best fitting of wind speed data, which minimizes RMSE.
SCOPF was applied using the AO method on a modified IEEE-30 bus system which was combined with two wind energy sources. The cost function for SCOPF was made up using two penalty costs: overestimation cost and underestimation cost. In order to demonstrate the effects of penalty costs, six scenarios for penalty and reverse cost were considered in the SCOPF study. From the six scenarios, it was noticed that the best optimal operation occurred when the reverse cost was minimal, which means that the AO method schedules wind farms at their maximum power. The AO method proved accuracy and validation in solving OPF and SCOPF incorporating wind energy sources.
The importance of the study is applying a novel optimization method on a hybrid thermal-wind system using real wind speed data. Six scenarios of this work were applied to study the effect of changing overestimation and underestimation cost coefficients in the optimal scheduled wind power from wind farms, the thermal unit generated power, the resultant fuel, and wind generation costs. Moreover, this work presented a study to demonstrate the effect of changing scheduled wind power on the wind generation cost at different values of overestimation and underestimation cost coefficients. Finally, we conclude that the AO method proves superiority in solving complicated OPF problems in a large system with penetration of wind energy. In future work, multi-objective optimal power with wind energy penetration will be solved to optimize more objective functions along with generation cost.