1. Introduction
The integration of large-scale wind turbines in recent power systems has solved many problems linked to the conventional generating units. Indeed, wind power (WP) resources are considered environmentally friendly and they can play a key role in combating climate change and maintaining environmental biodiversity. From an economic point of view, wind energy is becoming more economically competitive with conventional energy production methods, due to continuous improvements in turbine efficiency and rising fuel prices. However, it is difficult to provide a precise forecast of WP output because it depends on wind speed and climate changes, which are uncertain and unpredictable. Moreover, for low wind speed, wind energy resources may shut down to prevent damage. Therefore, large-scale penetration of wind energy in power networks increases their uncertainty and confronts system operators with various technical challenges. For example, in the case of generation scheduling problems, power outputs of thermal units must be adjusted in order to meet the total system load and to cover WP fluctuations.
The dynamic economic and emission dispatch problem (DEED) is one of scheduling problems in the field of power systems. It aims to find an optimal online allocation of generation outputs of thermal units in order to cover the total load over a time horizon which is often divided into 24 h [
1]. Mathematically, the DEED problem can be formulated as a multi-objective optimization problem (MOP), where total production cost and total emissions are the objective functions to be minimized. In general, the DEED problem constraints comprise generating unit limits, power balance constraint, ramp rate limits (RRLs), valve point loading effects (VPLEs), and prohibited operating zones (POZs) [
2]. In a few works, the spinning reserve (SR) constraints have been also taken into account in the DEED formulation in order to ensure the power system security and reliability [
3]. Indeed, SR is the production capacity available for the network operator in a finite time interval in order to meet the power demand in the event of a sudden generating unit outage or unexpected increase in the power demand.
Recently, the inclusion of WP resources, along with thermal generating units, in the generation scheduling problems, such as dispatch problems and unit commitment problems, has attracted much attention [
4,
5,
6]. Unfortunately, under-estimation and over-estimation of available WP have posed many difficulties in the resolution of such problems. Thus, an adequate modeling of WP resources, when they are incorporated into generation scheduling problems, is required.
Over the past two decades, the development of effective methods to establish reliable forecasting of wind power generation has been of primordial concern for various research works [
7,
8,
9,
10]. These forecasting methods can be broadly divided into two main categories which are statistical methods [
8] and physical methods [
9,
10]. An overview of WP forecasting models has been presented in [
11], where advantages and disadvantages of each model have been introduced. Recently, some attempts for solving the scheduling problems with WP sources have introduced the over-estimation and under-estimation of costs of available WP involved in such problems [
12,
13,
14,
15]. In these attempted approaches, under-estimation and over-estimation of costs of WP have been combined with total production costs of thermal units. Under-estimation happens when scheduled power is less than available WP, while over-estimation means that scheduled power is greater than available WP. Therefore, a reserve cost can be imposed for the over-estimation case and a penalty cost can be imposed for the WP under-estimation. In [
12], economic and environmental impacts of over-estimation and under-estimation of actual WP on the optimal results of the economic environmental dispatch problem with WP have been investigated. In order to describe the randomness of wind speed, the authors have adopted the mixture Gamma–Weibull distribution function. Jadoun et al. [
13] have added the reserve and penalty costs of available power outputs of renewable sources to the fuel cost of thermal units into the problem formulation of the dynamic economic dispatch and, then, an enhanced version of the fireworks algorithm has been used for the minimization of the total cost function. Alternatively, other works have modelled the uncertainty of WP in power scheduling problems using stochastic programming models [
16,
17,
18,
19]. For instance, reference [
16] have deployed a stochastic MOP for the optimal power flow incorporating intermittent WP and distributed load where variations of wind speed and bus loads have been characterized by probability distribution functions (PDF). The authors of [
16] have also considered the maximization of WP penetration along with fuel cost minimization and they have applied a paired-bacteria optimization algorithm for solving the problem. Reference [
17] has developed a model based on the wait-and-see approach for the economic load dispatch with stochastic renewable sources, where the stochastic variables involved in the problem have been evaluated using the two-point estimate technique. In [
19], a multi-stage stochastic programming has been for the optimal dispatch of generation and reserve where the uncertainties have been described by a scenario tree, while the optimal network decision has been considered as a decision tree. Song et al. [
20] have proposed a strategy for the improvement of the energy capture efficiency of wind turbines by using a stochastic model predictive yaw control (SMPYC). To do this, the authors of this reference have described the intermittent characteristics of predicted wind direction by intelligent scenarios generation (ISG).
Chance-constrained programming (CCP) is an interesting subclass of stochastic programming which can be used to describe random variables involved in the problem objective functions or in the constraints. It ensures that the optimal solution must satisfy a certain constraint with a probability more than a pre-specified tolerance set between 0 and 1. The CCP has been applied in various engineering domains [
21,
22]. More recently, CCP-based approaches have been widely adopted for handling uncertainties of renewable energy sources [
23,
24,
25,
26,
27,
28,
29,
30]. In reference [
23], a detailed strategy for modeling the economic load dispatch problem with WP has been introduced where the power balance constraint has been described by a chance constraint. This strategy has drawn the attention of many researchers in the field of power scheduling. For example, Liu [
24] has developed a stochastic optimization model for the economic dispatch, including large scale WP, where the probability of the random WP has been integrated into the problem constraints. However, the author has not focused on the optimization algorithm. In reference [
25], a CCP-based model for the dynamic economic dispatch (DED) problem incorporating WP has been illustrated where a chance constraint representing the probability that the available WP can be sufficient is more than a pre-specified confidence level. Moreover, SR constraints have been integrated into the proposed model to compensate for WP fluctuations. Accordingly, Cheng et al. [
26] have also used CCP to model the stochastic economic dispatch problem in such a way that the probability of the scheduled WP to be greater than a certain percentage of the total load is at least equal to a pre-specified level. Furthermore, energy storage has been added to mitigate the impacts of WP variability. In [
29], a chance constraint describing the probability that the energy balance constraint cannot be met has been added to the constraints set of the economic and emission dispatch problem with WP where that probability has been considered to be less than a certain tolerance pre-specified between 0 and 1. However, the effects of the WP fluctuations have not been investigated. The same strategy has been used by Alshammari et al. [
30] for solving such a problem, but the tolerance is set in a way that the system security, which has been calculated using different fuzzy membership functions, is ensured.
One of the main greatest strengths of the majority of the above strategies is the use of meta-heuristic techniques in the optimization process. Unfortunately, the confidence levels and the tolerances involved in the aforementioned CCP-based models are set in advance, which may lead to an incorrect operation of the power network. In effect, when these parameters are fixed at high values, the risk of insufficient WP at high system load will increase. Moreover, fixing these parameters at small values may decrease the opportunity to properly exploit the WP at low power demand.
In this paper, a new model for the DEED problem incorporating intermittent WP is presented for an efficient operation of wind farms. The suggested model is based on CCP where the wind turbine output is characterised by a chance constraint describing the probability that the energy balance cannot be fulfilled. It is an expansion of the models presented in [
23,
30]. Unlike those references, the tolerance with regard to insufficient supply is not fixed beforehand but is added to the problem decision variables and it is determined according to the system spinning reserve. This may give the system operator an opportunity to predict more wind farm production, as long as the spinning reserve is greater than the scheduled WP.
Recently, several variants and modified versions of the artificial bee colony (ABC) have been introduced for handling various complex optimization problems [
30,
31,
32]. Over the last few years, a particular focus has been given to the use of chaos theory in the improvement of optimization algorithms, including the ABC algorithm [
30,
33]. In fact, various research works have demonstrated that the hybridization of chaos with meta-heuristic techniques can significantly achieve better performance. For instance, in reference [
30], two chaotic maps have been combined with the original ABC algorithm in order to enhance its performance. The first one has been the Ikeda map which has been used for the generation of the initial population. Moreover, all random numbers involved in the various phases of the algorithm have been substituted by chaotic sequences generated by Ikeda map. The second one has been the chaotic Zaslavsky map which has been used in the search process for the optimal solutions in order to ameliorate the global convergence of the ABC algorithm.
In this study, four various chaotic maps are embedded in all phases of the ABC algorithm to further improve and enhance its convergence characteristics. This new algorithm, called hybrid chaotic maps-based artificial bee colony (HCABC), can be considered as an expansion of the IABC presented in [
30].
In summary, the main contributions of this paper are as follows.
A new model for the DEED problem with uncertain WP is presented. Unlike other CCP-based strategies that fix the probability of scheduled WP in advance, this study considers that probability as a decision variable. This probability is used to describe the tolerance that energy balance constraint cannot be fulfilled. The value of this tolerance is determined over horizon time according to the variations of the system load and actual system spinning reserve. In the proposed model, all system constraints, such as generation capacity, VPLEs, RRLs, and spinning reserve constraints, are considered. Therefore, to the best of the authors’ knowledge, this study is the first attempt to hourly adjust the WP probability over a horizon time for DEED problem;
A new ABC-based technique is adopted for solving the studied problem. In this technique, referred as HCABC, four chaotic maps—which are Ikeda chaotic map, Zaslavsky chaotic map, Lorenz attractor, and chaotic sine map—are employed to generate the random variables involved in the phases of the ABC method, which are initialization of the colony, employed bee, onlooker bee, and scout bee phases; and
The effectiveness and robustness of the proposed CCP-based strategy, as well as the adopted algorithm, are evaluated on various test cases.
The outline of the rest of this paper is organized as follows. In
Section 2, the classical DEED is formulated.
Section 3 illustrates the stochastic model of the DEED problem with uncertain WP. In
Section 4, the various chaotic maps adopted in this study are presented.
Section 5 illustrates the proposed HCABC algorithm.
Section 6 presents a discussion of the simulation results. Finally, conclusions are presented
Section 7.
3. Stochastic DEED Modeling
To address the stochastic characteristic of wind speed, several models have been proposed [
23,
24,
25,
26,
34]. For example, Song et al. [
34] have proposed a systematic approach for an accurate estimation of the annual energy production of a wind turbine. Four power-curve models, including the linear, general, cubic, and quadratic models, have been investigated in order to show the effectiveness of the proposed approach. However, the linear model remains the most suitable for probability calculation of WP output, which is required in this study. For that purpose, wind power output can be assumed as a piecewise linear function of wind speed at each time
t, as given in (14).
The Weibull distribution function is frequently applied for describing randomness of WP in power scheduling problems [
23,
29,
30]. For example, this function has been used in [
34] to determine the mean of the wind turbine output in one hour. In this study, the Weibull distribution function is adopted to calculate the minimum and maximum values of the tolerance that the power balance constraint cannot be met. Then, this tolerance will be updated at each hour, based on the system spinning reserve. Then, the corresponding WP output can be determined.
The two-parameter Weibull distribution as (15) is used in this study. The cumulative distribution function of the Weibull distribution can be expressed as (16).
According to Equations (14) and (16), and using probability theory for random variables, the probabilities of events
,
and
can be expressed as (17), (18), and (19), respectively.
where
.
Traditionally, the integration of wind power into the dispatch problem has been characterised by its average. Although the implementation of this approach appears easy, it has been criticized for its probabilistic infeasibility [
24]. To mitigate this problem, wind power availability can be described by a chance constraint, as follows.
Equation (20) describes the probability that the power balance constraint cannot be satisfied. It represents the risk degree of unsatisfying generation. According to (19), Equation (20) can be rewritten as the following equation.
It is noteworthy that the acceptable minimum (
) and maximum (
) values of
are
and
, respectively. Thus, values of probability
should satisfy the following constraint.
In this study, the following typical values of wind farm parameters, which are used in several works [
23,
30], are adopted:
;
;
;
; and
. Thus, the bounds of the confidence level
can be as follows.
As wind speed is stochastic, wind power cannot be estimated with high accuracy. Therefore, to ensure the operation reliability of the power system when wind farms are integrated an additional spinning reserve constraint is added to the dispatch problem.
From Equation (25), it can be seen that wind power must satisfy inequality (26).
As a result, wind power availability and spinning reserve constraints are taken into account in the stochastic DEED problem by adding constraints (21), (22), and (25).
In summary, the DEED problem incorporating wind power can be summarized mathematically, as follows.
Note that the vector of decision variables is . Moreover, all these constraints are simultaneously taken into account in the optimization process. If, at least, one of these constraints is not satisfied, the value of the objective function will be at a maximum value, fixed at . Where, and are maximum values of the total production cost and the total emissions, respectively.
5. Basic Principles of HCABC
The ABC algorithm is a powerful population-based algorithm widely applied for the solution of nonlinear optimization problems. The ABC algorithm was introduced by Karaboga in 2005 [
45]. It mimics the foraging behavior of honeybee colonies. In its basic version, the algorithm performs a kind of neighborhood search combined with a global search, and can be used for both combinatorial and continuous optimization. In this algorithm, a candidate solution of the optimization problem is represented by a food source. Each food source has a quantity of nectar which characterizes its fitness value. The colony’s population is divided into three groups of bees which cover the search space for looking for a food source. The effectiveness and specific properties of the ABC algorithm have been proven in numerous studies [
46]. A review on the ABC technique and its applications in various engineering domains has been presented in [
47]. From literature review, it is found that the ABC technique has been applied for medical image [
48], power system problems [
49], feature selection methods [
50], optimal design of controllers [
51], and so on. For example, an improved version of ABC based on variable coefficients has been proposed in [
48] for image segmentation. In [
49], an ABC-based method has been applied for robust decentralized controller design for power system stabilization. The ABC algorithm has also been applied in [
52] for reactive power dispatch problem where the decision variables have been terminal voltages of generating units, tap transformers positions, and reactive power outputs of compensators.
Mainly, the ABC algorithm comprises four phases, which are initialization step, employed bee phase, onlooker bee phase, and scout bee phase. A detailed description of the ABC algorithm can be found in various research works, such as [
46]. A brief description of ABC phases is presented below.
In the initialization step,
individuals are generated randomly, based on Equation (32). Note that an individual can be an employed bee or an onlooker bee. Each food source represents a candidate solution. For a
D-dimensional optimization problem, a food source or solution
is represented by the vector
.
where
is a random number in
.
and
are lower and upper bounds of the
j-th decision variable
.
In the employed bee phase, each food source
is updated by an employed bee so that a new food source
at the neighborhood of
is generated. To do so, a local search around
is performed as given in Equation (33). Then, a greedy selection between solutions
and
is performed to choose the better one.
where
j and
k are selected randomly from
and
.
is a random number.
In the onlooker bee phase, onlooker bees measure the nectar amount of food sources discovered by employed bees and then choose food sources based on probabilistic selection. Generally, the roulette wheel-based selection mechanism is used. The probability
to select a food source
is related to its nectar amount
and it is calculated as given in (34). Similarly to what it is done in the employed bee phase, selected candidate solutions
are changed to discover new food sources
by using Equation (32). Then, a greedy selection between old and new food sources is applied.
where
is a random number within
.
In the scout bee phase, if any food source was not changed after a pre-specified number of cycles (
LIMIT), it will be removed from the ABC colony and another food source will be created randomly, as given in Equation (36).
where
is a random number within
.
Although the ABC algorithm has showed good results in many optimization problems, it has suffered from some drawbacks, such as low colony diversity, premature convergence, and low convergence rate when solving complex optimization problems [
53]. Similar to other meta-heuristic techniques, these negative points are mainly due to the use of random sequences in the various phases of the algorithm. In fact, it can be seen that four random numbers within
are involved in the various phases of the original ABC. Generally, the randomness property used in the classical ABC is obtained by employing probability distribution, such as uniform distribution and the Gaussian method. However, randomness in the ABC method may get trapped in local optima, especially for multi-modal optimization problems. Because of randomicity and ergodicity of chaotic maps, chaotic sequences generated by a chaotic system changes randomly in the search space and can, eventually, go through the whole space if the number of iterations is relatively high. These characteristics of chaos systems can help the optimization algorithms to escape local optima convergence and ensure the diversity of the search process. Therefore, the search performance of such stochastic optimization method can be improved.
In this study, the idea of employing chaotic sequences in lieu of random numbers is suggested to overcome the aforementioned ABC drawbacks. To do so, sequences
of the Ikeda chaotic map, the Zaslavsky chaotic map, the Lorenz attractor, and the sine map are employed for manipulating the initialization step, employed bee phase, onlooker bee phase, and scout bee phase, respectively. It is noteworthy to point out that sequences
of the Ikeda map and the Lorenz attractor are not within
. Therefore, these sequences are normalized, as follows.
The steps of the suggested hybrid chaotic maps-based ABC (HCABC) technique are summarized in
Figure 5.