2.1.1. Variational Mode Decomposition (VMD)
VMD (Variational Mode Decomposition) is an adaptive signal processing technique designed to address the challenges of endpoint ambiguity and mode size in signal decomposition. In comparison to traditional EMD methods, VMD demonstrates superior performance. This algorithm effectively decomposes complex signals into distinct modes, encompassing both low and high-frequency components, thereby mitigating the high complexity, nonlinearity, and non-smoothness inherent in time series data [
38]. VMD employs an iterative optimization approach to adaptively determine the IMF of a signal based on its characteristics, providing enhanced flexibility and robust decomposition capabilities. The specific computational process is elucidated by the mathematical Equations (1)–(4), incorporating key equations for variational optimization modeling and estimating mode bandwidth. This renders VMD a potent tool for handling non-stationary signals and complex temporal data [
39].
In this Equation (1),
represents a set of mode functions and
represents their corresponding frequencies. The objective is to minimize the squared gradients of each mode function, fitting their linear combination to the input signal
. The constraint
ensures that the total sum of the linear combination of mode functions equals the input signal.
Equation (2) introduces the Lagrangian operator to update Equation (1), where
is the Lagrange multiplier.
is a weight parameter, which adjusts the modal function to fit the input signal. By adjusting the Lagrange multiplier, a better fit to the input signal is achieved, enhancing the accuracy of mode decomposition [
40]. This updated equation plays a crucial role in the iterative process of the VMD algorithm, aiding the optimization algorithm in progressively approaching the optimal fit to the input signal.
Equations (3) and (4) represent the update steps of the VMD model. The numerator part represents the input signal minus half of the modal functions other than the current modal function and the Lagrange multiplier. The denominator contains a bandwidth adjustment term, where is a weight parameter, is the frequency, and is the frequency of the current mode function, and where denotes the update of the k-th mode in the frequency domain and represents the corresponding frequency update. This iterative process utilizes the gradient of the optimization problem to progressively update the modes and frequencies.
2.1.2. Improved Snake Optimization Algorithm (ISOA)
The SOA is a new intelligent optimization algorithm proposed by Hashim et al. that is a heuristic algorithm inspired by the collective behavior of snakes in biology. It simulates the behavior of snake groups in activities, such as food searching, migration, and collaborative behavior. The Snake Algorithm is primarily employed for solving optimization problems and signal-processing tasks [
37].
The SOA is characterized by its simple structure and high flexibility; however, it faces challenges, such as susceptibility to local optima and slow convergence speed [
41,
42]. To overcome these drawbacks, the utilization of chaotic initialization enhances the diversity of the initial population, while the adaptive inertia weight factor helps balance exploration and exploitation during the search process. Additionally, the introduction of the Levy flight strategy strengthens global search capabilities and avoids local optima. The combination of these improvement measures aims to enhance the performance of the SOA algorithm, accelerate convergence speed, and increase the probability of finding the global optimum.
SOA has two phases: the exploration and exploitation phases, which are controlled by food quantity (
Q) for conversion. When
Q < 0.25, the SOA is in the exploration phase. On the contrary, when
Q > 0.25, the algorithm is in the exploitation phase. Food quantity is calculated as follows:
where
represents the current iteration,
is the total number of iterations, and
= 0.5. The initial population of the SOA is divided into two parts, denoted as
and
where
represents the male population of snakes and
represents the female population of snakes.
Exploration phase:
Equations (6)–(9) describe the process of position updating in the exploration phase of the SOA, where the snake swarm updates its positions based on fitness levels and random numbers. Firstly,
and
are calculated through exponential functions, where
and
represent fitness levels associated with a random number
and
and
are the fitness levels of individuals in the population. Next, the position update of the snake swarm is controlled by parameters related to the random number
, and
and
are a randomly generated value that represents the current time a random number generated at time t.
is a constant used to control the adjustment amplitude of the algorithm.
and
are the upper and lower bounds, respectively, used to limit the range of
and
and
= 0.05.
Exploitation phase:
In the Exploitation phase of the SOA, the snake swarm employs two different predation strategies, and the specific choice depends on
, and
is a random number that ranges from 0 to 1 (0 <
< 1, 0 <
< 1); if
> 0.6, the snake swarm adopts the battle mode; otherwise, it uses the mating mode. When
> 0.25 and the temperature is greater than 0.6, the SOA is in the hot mode. In this case, the update of the snake swarm’s positions is determined by the following equation:
where
is the position of the individual (male or female),
is the position of the best individuals, and
is constant and equals 2.
In the Cold mode of the SOA, when Q > 0.25 and the temperature is less than 0.6, the algorithm adopts a cold mode. In this mode, if
> 0.6, the snake swarm uses the battle mode, and its position update is determined by the following equation:
where
and
represent the positions of the
-th male and female individuals, respectively, and
and
are the best positions in the male and female groups.
and
are coefficients based on fitness levels.
The fitness levels of the best agents in the male and female groups are denoted as and , respectively. Additionally, represents the fitness level of the -th agent in the population.
If
< 0.6, the SOA is in mating mode:
Equations (15) and (16) describe the mating properties of the snake swarm in mating mode, where
represents the position update of the
-th male individual, and
represents the position update of the
-th female individual at time
.
and
refer to the mating ability of males and females, respectively, and they can be calculated as follows:
During the hatching of the egg, the least-fit male,
and the least-fit female,
are substituted as follows:
In the initialization phase of the Snake Algorithm, random number initialization leads to a simple random selection of the search space for the snake group, which fails to fully cover the search space. This results in low solution accuracy and insufficient mid-term global search capabilities. The main idea of chaotic optimization is to utilize the traversal and randomness characteristics of chaos. It maps variables to the value range within the chaotic variable space and linearly transforms the obtained solution back to the optimization variable space, thereby improving algorithm performance. Therefore, this paper adopts the sine map from chaotic mapping for the population initialization in the Snake Algorithm.
Figure 2 shows the population distribution comparison between random initialization and sinusoidal chaotic mapping initialization in the initial stage of the algorithm. The randomly initialized images show that the distribution of initial solution points in the search space is relatively scattered and concentrated, which may lead to insufficient exploration of the search space. On the contrary, the images of sinusoidal chaotic map initialization reveal a more uniform and widely dispersed population layout covering a larger range of the search space, which indicates that sinusoidal chaotic map initialization shows better global exploration potential and more efficient convergence ability.
The principle of the sine chaotic mapping is as follows:
Here, is the sine mapping in the range [0, 1], and and represent the upper and lower bounds of the ith dimension, respectively. Equation (22) is the mathematical model of the selection space for the Bald Eagle Search Algorithm after initialization through the sine chaotic mapping.
The exploration phase of the Snake Algorithm is a crucial process determining the algorithm’s convergence speed. Due to a lack of effective control over the step size, the SOA is prone to deviate from the search direction, missing the optimal value range and leading to the algorithm becoming trapped in local optima. Therefore, in this study, the SOA algorithm is enhanced by incorporating an adaptive inertia weight factor. This adaptive mechanism dynamically adjusts the balance between exploration and exploitation, enhancing the convergence speed of the algorithm, preventing it from becoming trapped in local optima, reducing the need for manual parameter tuning, and strengthening the algorithm’s stability and adaptability. The adaptive mechanism automatically adjusts the weight based on the algorithm’s performance and the iteration process, making the optimization process more efficient and robust. The improved equation and comparison in
Figure 3 are as follows:
where,
represents the non-inertial weight factor,
denotes the maximum number of iterations,
represents the current iteration count, and
represents the updated individual position. Equation (24) suggests that the new position of individual i is determined by a combination of the non-inertial weight factor
, a random factor
, the best position found so far
, and additional displacement terms
.
Levy flight is a type of random walk model characterized by the movement of a point in any dimensional space with a random length and direction, repeating this process. What makes Levy flight distinctive is the heavy-tailed distribution of step lengths, implying a certain probability of generating larger step lengths, potentially resulting in long-distance displacements. Additionally, each trajectory is different, adding randomness and diversity to the Levy flight. Introducing the Levy flight strategy during the development phase of the Snake Algorithm means incorporating the randomness of Levy flight into the algorithm to enhance its exploratory nature and flexibility. This introduction helps the algorithm explore solutions more extensively in the search space, improving its global search capability. The mathematical model of Levy flight is as follows:
where
presents the random step length, and
is the exponent parameter. As the variance of Levy flight grows rapidly without bounds, introducing Levy flight during the development phase of the Snake Algorithm accelerates the search speed and effectively avoids becoming trapped in local optima.