MISAO: Ultra-Short-Term Photovoltaic Power Forecasting with Multi-Strategy Improved Snow Ablation Optimizer

Abstract

The increase in installed PV capacity worldwide and the intermittent nature of solar resources highlight the importance of power prediction for grid integration of this technology. Therefore, there is an urgent need for an effective prediction model, but the choice of model hyperparameters greatly affects the prediction performance. In this paper, a multi-strategy improved snowmelt algorithm (MISAO) is proposed for optimizing intrinsic computing-expressive empirical mode decomposition with adaptive noise (ICEEMDAN) and weighted least squares support vector machine for PV power forecasting. Firstly, a cyclic chaotic mapping initialization strategy is used to generate a uniformly distributed high-quality population, which facilitates the algorithm to enter the appropriate search domain quickly. Secondly, the Gaussian diffusion strategy enhances the local exploration ability of the intelligences and extends their search in the solution space, effectively preventing them from falling into local optima. Finally, a stochastic follower search strategy is employed to reserve better candidate solutions for the next iteration, thus achieving a robust exploration–exploitation balance. With these strategies, the optimization performance of MISAO is comprehensively improved. In order to comprehensively evaluate the optimization performance of MISAO, a series of numerical optimization experiments were conducted using IEEE CEC2017 and test sets, and the effectiveness of each improvement strategy was verified. In terms of solution accuracy, convergence speed, robustness, and scalability, MISAO was compared with the basic SAO, various state-of-the-art optimizers, and some recently developed improved algorithms. The results showed that the overall optimization performance of MISAO is excellent, with Friedman average rankings of 1.80 and 1.82 in the two comparison experiments. In most of the test cases, MISAO delivered more accurate and reliable solutions than its competitors. In addition, the altered algorithm was applied to the selection of hyperparameters for the ICEEMDAN-WLSSVM PV prediction model, and seven neural network models, including WLSSVM, ICEEMDAN-WLSSVM, and MISAO-ICEEMDAN-WLSSVM, were used to predict the PV power under three different weather types. The results showed that the models have high prediction accuracy and stability. The MAPE, MAE and RMSE of the proposed model were reduced by at least 25.3%, 17.8% and 13.3%, respectively. This method is useful for predicting the output power, which is conducive to the economic dispatch of the grid and the stable operation of the power system.

1. Introduction

In recent years, excessive energy consumption has caused problems such as ecological environment deterioration and depletion of non-renewable energy sources. Worldwide, populations have begun to seek a new energy development path to gradually replace traditional fossil fuels with renewable energy [1]. Solar energy is valued because of its safety, efficiency, economy, and environmental friendliness. New energy grid-connected power generation, represented by photovoltaic power generation, is integral to future power systems [2]. In 2022, the global photovoltaic installed capacity reached 230 GW, and the cumulative installed capacity will reach 1156 GW, with a year-on-year growth of 35.3%. The output power of PV systems exhibits a high degree of randomness and volatility due to meteorological factors such as solar radiation [3,4]. When the grid is connected to a high proportion of PV, the complexity of large-scale grid scheduling increases, and the stability and reliability of the power system operation are threatened.

PV power prediction has become one of the vital fundamental technologies used to improve operation quality and ensure the stable operation of PV power generation. Accurate prediction of the power output of PV systems in the ultra-short-term is essential for extending the life of storage devices such as batteries, helping dispatchers to develop rational scheduling plans, improving system reliability, and deploying PV power on a large scale. In addition, it helps to reduce the uncertainty of PV generation on the power system, coordinate with other conventional power sources, improve grid scheduling, and increase the utilization of PV generation. In addition, PV power generation prediction is also utilized to enable online PV array fault diagnosis, which allows for the rapid detection of faults or anomalies due to corrosion, high winds, heavy rains, hailstorms, dirt, UV irradiation, thermal cycling, etc. [5]. Therefore, accurate ultra-short-term PV power prediction is an important research area [6].

Various prediction methods have been proposed for PV power prediction [7]. There are four time periods for PV power prediction: long-term prediction is from one month to one year, medium-term prediction is from one week to one month, short-term prediction is from one week or less, and very short-term or ultra-short-term prediction is from one minute to a few minutes. Long-term forecasting is used for planning and decision-making for PV generation, transmission, and distribution and guarantees reliable power system operation. For medium-term forecasting, a model can be used for medium-term decision support for power system dispatches. Short-term forecasts are essential for day-ahead generation planning and improving the power system reliability. Ultra-short-term forecasts are used to guide real-time grid scheduling and battery storage control to ensure the safe operation of the grid. In addition, this method helps improve the quality of the information and make real-time corrections to short-term PV power forecast data. Forecasting methods can be classified into physical, statistical, and machine learning-based methods [8].

The physical approach is based on the principle of PV power generation. It uses mathematical models of solar radiation, temperature, humidity, cloud cover, barometric pressure, and wind speed obtained from numerical weather prediction (NWP). Physical forecasts are based on detailed site geographic information, accurate weather data, and complete PV information. Historical data are not required. However, since the meteorological environment limits solar resources, the complexity of the atmospheric environment will directly lead to a dramatic increase in model complexity. Therefore, simulation and prediction of PV output using physical models cannot provide accurate and meaningful predictions reflecting reality under extremely severe weather conditions [9]. They are not universal, cannot be adapted to changing meteorological conditions, and can only be used for short-term forecasting.

Statistical forecasting methods include time series methods [10], regression analysis [11], gray theory [12], fuzzy theory [13], multi-source data-driven methods [14], and spatio–temporal correlation [15]. Statistical methods include curve fitting, parameter estimation, and correlation analysis of historical data such as solar radiation and PV power generation. Predicting future PV power generation is achieved by establishing a correlation mapping relationship between input and output data. Compared to physical methods, there is no need to have a clear and complete understanding of the complex photovoltaic conversion relationships of PV systems [16]. Therefore, it is characterized by the simplicity of modeling and generalizability across different regions compared to physical methods. However, the premise of the implementation of statistical methods is that a large amount of correct historical data needs to be processed, and there are difficulties in data collection and calculation during the implementation process. Due to the need for a large number of numerical calculations in the forecasting process, the general computer forecasting time is long. Meeting the requirements of forecasting speed (especially at the minute level) is difficult for ultra-short-term PV power forecasting. In addition, the prediction quality of statistical methods is closely related to the data quality and the degree of historical data retention. Therefore, screening and elimination of spurious data have a significant impact on forecast accuracy. Prediction accuracy relies on high-dimensional computation for effectiveness, which increases the computational effort and reduces the prediction speed.

Machine learning can efficiently extract high-dimensional complex nonlinear features and map them directly to the output. Taking advantage of this, machine learning-based forecasting methods have become one of the most widely used methods for predicting different time series [17]. In recent years, deep learning models represented artificial neural networks (ANN), recurrent neural networks (RNN), gated recurrent unit neural networks (GRU), and so on, which have been widely used in the field of PV power prediction [18,19].

The LSSVM model has proven to be an effective advanced prediction model, which has been increasingly used in solving prediction problems to minimize the sum-of-squares error and marginal error of the training data [20,21]. A previous study [22] proposed a hybrid prediction model combining variable mode decomposition (VMD), particle swarm optimization (PSO), and least squares support vector regression (LSSVM), which improves the accuracy of day-ahead prediction by at least 19% compared to the best RecMO strategy. A previous study [23] developed an evolutionary seasonal decomposition LSSVM model to predict the PV power generation in the current month while using a genetic algorithm to optimize the parameters of the LSSVM, and the results showed that the prediction system has better prediction accuracy. Another study [24] used a hybrid prediction model combining wavelet transform, particle swarm optimization, and support vector machine for day-ahead power generation prediction of real microgrid PV systems with a daily average MAPE of 4.2% and NMAE of 0.4%. Although the LSSVM model performs well in solving the prediction problem, it has some limitations. First, the LSSVM model may face the challenge of high computational complexity for large-scale datasets, leading to longer training time or an inability to handle large-scale data. Second, LSSVM models are more sensitive to selecting data features and adjusting parameters, which need to be carefully designed and adjusted to obtain good performance. In addition, the predictive performance of LSSVM models is affected by data noise and outliers, which may require additional processing steps to enhance the robustness of the model. However, traditional optimization algorithms also have some limitations in optimizing the parameters of the LSSVM model, and these algorithms may converge slowly, tend to fall into local optimal solutions, or need help dealing with high-dimensional complex datasets.

PV power output is highly volatile, especially in unstable weather. In this case, a combined prediction model based on signal decomposition is often used, which mainly decomposes the PV power sequence into several sub-sequences, models the prediction, and superimposes the prediction results. Classic decomposition techniques include wavelet transform (WT) [25], empirical mode decomposition (EMD) [26], and ensemble empirical mode decomposition (EEMD) [27]. However, both EMD and its improved methods exhibit apparent modal overlapping phenomena, which affect the prediction performance of the model to a certain extent. Compared with EEMD and WT, intrinsic computing-expressive empirical mode decomposition with adaptive noise (ICEEMDAN) best decomposes sequences [28]. The frequencies of the decomposed components fluctuate with significant frequency differences, representing different information at different timescales within the time series. However, the direct use of a single model to predict all of the components and superimpose the results will cause some errors.

Changes in weather conditions can make PV power generation stochastic and volatile. Utilizing all historical data as training samples, especially those day-by-day records with significant variations in weather conditions, will prolong the model training time and reduce the forecasting accuracy. Therefore, it is necessary to divide the large amount of historical power data by weather type and train them separately [29]. A similar day selection method based on the Levy–Flight Beetle antenna search algorithm was proposed in the literature [30]. Considering the global horizontal radiation (GHR) trend, GHR fluctuations, and other weather factors, similar days are selected as training samples and better prediction results are achieved. Currently, many short-term PV power prediction studies focus on improving model accuracy, often ignoring the impact of weather factors on PV power generation [31]. In order to improve the prediction accuracy of the short-term PV power prediction model, it is necessary to effectively extract the relevant data from the massive data, avoid data redundancy, and reduce the non-stationarity of the PV power series. In this paper, the advantages of various models are synthesized and an ultra-short-term PV prediction model is constructed. The model considers weather factor feature selection, similar day clustering, signal decomposition, and hybrid deep learning. The main contents of this paper are as follows:

(1)

A similar day clustering model is proposed. The K-means clustering method is applied to the historical data to accurately classify the weather conditions into sunny, cloudy, and rainy days. This concise and targeted classification strategy effectively avoids data clutter and redundancy, dramatically enhances the relevance and reliability of the training data, and lays a solid foundation for the accurate construction of the subsequent ultra-short-term PV prediction model.

(2)

To improve the algorithm’s optimization ability, a multi-strategy improved snow ablation optimizer (MISAO) combining cyclic chaotic mapping initialization strategies, Gaussian diffusion strategies, and random follower search strategies is proposed. The parameters of ICEEMDAN are optimized using MISAO. The model decomposes the fluctuating stochastic raw PV power series into multiple subsequences, each with unique frequencies and features, thus reducing the non-stationarity and complexity of the PV power series.

(3)

Constructing the multivariate prediction model extends the available data dimensions and enhances the model’s predictability. Meteorological factors have a more significant impact on the PV power system, and adding the influence of meteorological factors to the model and building a multivariate prediction model can provide more practical prediction results and more credible data support for relevant information users.

(4)

A hybrid model for ultra-short-term PV power prediction integrating MISAO-ICEEMDAN-WLSSVM is proposed to improve the prediction accuracy of PV power. MISAO can search for the optimal hyperparameter combinations of the WLSSVM, which improves the model’s prediction accuracy.

(5)

To determine the models’ reliability and robustness, the various models’ prediction results were compared with the MISAO-ICEEMDAN-WLSSVM. Simulation experiments were conducted on three significantly different weather-type datasets.

2. Snow Ablation Optimizer (SAO)

Initially introduced by Deng and Liu [32] in 2023, the SAO algorithm is a bionic optimization technique. Figure 1 illustrates the two processes in physics via which snow is transformed into liquid water and steam. The specific processes that are identified are melting and sublimation. Additionally, it is essential to mention that liquid water resulting from snow melting can undergo an additional transformation into vapor through evaporation. Based on this information, the SAO developed four elements to facilitate the search for the best solution, specifically the initialization, exploration, exploitation, and two-population methods.

2.1. Initialization Phase

The SAO algorithm optimizes the population based on the population size, the bounds of the solution space, and the number of dimensions. The starting position of the complete population can be represented as a matrix with $N$ rows and $D$ columns. The dimension of the solution space is denoted as $D$, the lower and upper borders of the search domain are denoted as $Lb$ and $Ub$ correspondingly, and the population size is denoted as $N$. The expression for the initial position matrix is as follows:

$$\begin{gathered} X=Lb+rand\times (Ub-Lb) \\ \begin{array}{c}={\left[\begin{array}{ccccc}{x}_{1,1}& x_{1,2}& \cdots & x_{1,D-1}& x_{1,D}\\ x_{2,1}& x_{2,2}& \cdots & x_{2,D-1}& x_{1,D}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ x_{N-1,1}& x_{N-1,2}& \cdots & x_{N-1,D-1}& x_{N-1,D}\\ x_{N,1}& x_{N,2}& \cdots & x_{N,D-1}& x_{N,D}\end{array}\right]}_{N\times D}\end{array} \end{gathered}$$

(1)

where rand indicates a random number in [0, 1]. Usually, the position vector of the $i-th$ search agent is expressed as: ${\mathrm{X}}_i=\left(x_{i,1},x_{i,2},\cdots ,x_{i,j},\cdots ,x_{i,D}\right)$, $i=1,2,\cdots ,N$, $j=1,2,\cdots ,D$.

2.2. Exploration Phase

During the detection phase, water vapor is formed through the sublimation or evaporation of snow or liquid water. This water vapor then flows randomly in space, and its detection is accomplished by observing its irregular motion. The SAO method capitalizes on Brownian motion’s stochastic and erratic nature, facilitating the exploration of advantageous and promising locations by individuals within the population. The SAO approach utilizes the probability density function of a zero-mean univariate normal distribution to ascertain the step size of the particle motion in the case of conventional Brownian motion. Below is the mathematical representation of Brownian motion:

$$f_{\mathrm{B}}\left(x;0,1\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}\times \exp \left(-{\displaystyle \frac{x^2}{2}}\right)$$

(2)

The position update formula for the detection process in the SAO algorithm is as follows:

$$\begin{array}{c}{X}_{i_{\mathrm{new}}}=Elite\_pool(k)+R{B}_i(j)\times (r_1\times (B(j)-X_i(j))\\ +(1-r_1)\times (\overline{X}(j)-X_{\mathrm{i}}(j)))\end{array}$$

(3)

where $X_{i_{\mathrm{new}}}$ represents the updated position of the individual; $R{B}_i(j)$ represents the vector of random numbers generated by the Brownian motion; $r_1$ represents a random number between 0 and 1; $B(j)$ represents the best solution found by the current population; $X_i(j)$ represents the position of the current individual; and $\overline{X}(j)$ represents the position of the center of mass of the entire population during the update of that position. A random selection of individual from the set $Elite\_pool$ is indicated by $Elite\_pool(k)$, where $k$ is an integer chosen at random from [1, 4]. The following are the mathematical formulas for $\overline{X}(j)$ and $Elite\_pool(k)$:

$$\overline{X}(t)={\displaystyle \frac{1}{N}}\sum _{i=1}^N{X}_i(t)$$

(4)

$$Elite\_pool(t)\in [B(t),X_{second}(t),X_{third}(t),Z_c(t)]$$

(5)

where $B(t)$ denotes the optimal solution of the current population, $X_{second}(t)$ and $X_{third}(t)$ denote the individuals with the second and third highest fitness values in the current population, and $Z_c(t)$ denotes the average of the individuals with the current population’s top 50% fitness values. $Elite\_pool(k)$ is a set consisting of the average values of the optimal solution, the second-best solution, the third-best solution, and the elite individuals in the population. In each position update during the exploration process, a value is randomly selected from this set to assist the position update.

The SAO algorithm calls the top 50% of the population’s fitness individuals elite individuals to make it easier to calculate $Z_c(t)$:

$$Z_c(t)={\displaystyle \frac{1}{N_1}}\sum _{i=1}^{N_1}{X}_i(t)$$

(6)

where $N_1$ denotes the number of elite individuals, so $N_1$ is numerically equal to half of $N$.

2.3. Exploitation Phase

In the development phase of the SAO algorithm, the main focus is on when snow is converted to water through the act of melting; in the SAO algorithm, the snow melting process is represented by the degree-day method with the following equations:

$$M(t)=DDF(t)\times T(t)=\left(0.35+0.25\times {\displaystyle \frac{e^{\frac{t}{t\max }}-1}{e-1}}\right)\times e^{{\displaystyle \frac{-t}{t\max }}}$$

(7)

where $M(t)$ denotes the snowmelt rate; $T(t)$ denotes the average daily temperature; $t$ and $t_{\max }$ are the current and maximum iterations, respectively; and $DDF(t)$ is the degree-day factor, which ranges from 0.35 to 0.6. Figure 2 illustrates the trend of the $DDF$ throughout the iterations. Then, the position update equations for this phase are shown below:

$$\begin{array}{c}{X}_{i_{\mathrm{new}}}=M\times B(t)+R{B}_{\mathrm{i}}(t)\times (r_2\times (B(t)-X_i(t))\\ +(1-r_2)\times (\overline{X}(t)-X_i(t)))\end{array}$$

(8)

where $M$ is the snowmelt rate in the SAO algorithm and $r_2$ is a random number chosen from [−1, 1]. This form gives the non-optimal individuals in the population a better chance to develop more feasible solutions based on the current optimal position.

2.4. Dual-Population Mechanism

This section presents a two-stock mechanism to attain equilibrium between exploration and use in SAO. As previously stated, liquid water from snow can also be transformed into vapor for exploration. As the number of iterations rises, the search agent is more likely to engage in erratic movements with highly dispersed properties to explore the search space. Therefore, during the first iteration, the population $P$ is separated into equal-sized sub-populations, $P_a$ and $P_b$. Sub-population $P_a$ is designated for exploration, while $P_b$ is designated for mining. The dimensions of $P$, $P_a$, and $P_b$ directly correspond with $N$, $N_a$, and $N_b$, respectively. With each cycle, $N_a$’s quantity gradually declines, while $N_b$’s quantity correspondingly increases. The mathematical expression is as follows:

$$N_a=N_a+1,N_b=N_b-1, \mathrm{if} N_a<N$$

(9)

3. Proposed MISAO

3.1. Circle Chaotic Mapping Initialization Strategy

The initial distribution of the population’s location substantially influences both the algorithm’s global convergence speed and the quality of the optimal solution. An algorithm’s optimization efficiency can be enhanced by achieving a population distribution that is both varied and uniform [33]. In the snow ablation optimizer (SAO), the starting position of each individual is determined using a random generation process. Nevertheless, this approach fails to ensure a diverse population. It tends to position the starting point at a considerable distance from the ideal solution, impacting the precision of convergence and the efficiency of the search. Cyclic chaotic mapping theory, initially presented by Hu and Liu [34], is an experimental approach designed to minimize the number of tests conducted. Cyclic chaotic mapping ensures a homogeneous distribution of positions spanning the solution domain.

Furthermore, the order of deviation in the sequence is only determined by the size of the solution set and is not influenced by the dimension of the solution space. This finding offers theoretical justification for tackling intricate issues with high dimensions. Hence, this paper employs a cyclic chaotic mapping approach to initiate the positions of all search agents, augmenting diversity. The fundamental principle is explained in the following manner:

$$w_{i+1}=mod(w_i+0.2-{\displaystyle \frac{0.5}{2\pi }}\cdot sin(2\pi w_i),1)$$

(10)

where $w_i$ denotes the individual in the population and $mod$ denotes the residual function.

3.2. Gaussian Diffusion Strategy

We propose implementing a diffusion approach to increase the size of the exploration population in order to prevent becoming stuck in local optima. Three additional individuals are created by randomly altering the present individual and choosing the best individual to replace the current one using a Gaussian step size. This strategy’s implementation improves the use of persons in the algorithm and intensifies the population’s pursuit of space exploration. The mathematical model reflecting this mechanism is defined as follows [35]:

$$X_i=G(X_{best},\gamma ,s)+\sigma \times (X_{best}-X_i)$$

(11)

where $G(X_{best},\gamma ,s)$ denotes the generation of a random matrix obeying a Gaussian distribution, $\gamma$ denotes the standard deviation, and $s$ denotes the vector that determines the size of the generating matrix. $\sigma$ is a random variable that follows a normal distribution. The computation of $\gamma$ can be described as follows:

$$\gamma ={\displaystyle \frac{\log (g)}{g}}\times |X_i-X_{best}|$$

(12)

To facilitate the use of individual localization and to approach the optimal solution, the parameter $\log (g)/g$ is used to reduce the Gaussian step size as the number of iterations increases.

3.3. Random Follower Search Strategy

The original SAO method relied heavily on the initial member to determine the relocation of population members, resulting in a subsequent dependency. However, this reliance can ensnare the algorithm to a solution only optimal within a limited scope, impeding its ability to break free. This predicament emerges from the strong inclination of people in the population to solely adhere to the most superior individual at present without engaging in adequate diversity and investigation. In order to tackle this issue, the random follower search approach [36] has been implemented to improve the variety of individuals in population intelligence algorithms and facilitate a more comprehensive exploration of the solution space. The utilization of the random follower search approach facilitates the promotion of communication and collaboration among populations. By reducing the dependence on the current most successful individual to update the positions of the population members, the significance of exchanging knowledge and sharing experiences between individuals becomes vital. This compels individuals within the population to interact with one another more proactively, thereby promoting the exchange of knowledge and experience between them. Consequently, the algorithm’s experience speeds up convergence, enhancing efficiency and problem-solving capability.

The detailed mathematical formulation is given below:

$$V_j=\left\{\begin{array}{c}{\overrightarrow{s}}_i^j+c_1\cdot \left(lb+c_2\cdot \left(ub-lb\right)\right),j\le k\hspace{0.33em}\mathrm{and}\hspace{0.33em}rand>0.5\\ {\overrightarrow{s}}_i^j-c_1\cdot \left(lb+c_2\cdot \left(ub-lb\right)\right),j\le k\hspace{0.33em}\mathrm{and}\hspace{0.33em}rand\le 0.5\\ \left(\mathrm{Temp}+{\overrightarrow{s}}_i^j\right)/2,\hspace{0.33em}\mathrm{otherwise}\end{array}\right.$$

(13)

$${\overline{s}}_i=s_i\left(pick\right)$$

(14)

$$pick=randperm(dim)$$

(15)

where the parameters $ub$ and $lb$ denote the upper and lower bounds of the exploration space of the ant colony search agent, respectively; ${\overrightarrow{s}}_i^j$ denotes the $j-th$ dimension of the $i-th$ population member $s_i$ after random combination; $c_1$ is an adaptive step control parameter, which determines the path of the ant colony search agent’s evolution to a certain extent; $c_2$ is a random number in the range of [−1, 1]; $dim$ denotes the dimension of the exploration space of the feasible domain; the parameter $k$ is a random integer with the value range of [1, dim]; and $\mathrm{Temp}$ denotes the current member of search agent exchange, and its corresponding mathematical model is as follows:

$$\mathrm{Temp}=\left\{\begin{array}{c}\hspace{1em}{S}_{best}^j,i=1\\ {\overline{s}}_{i-1}^j,\mathrm{otherwise}\end{array}\right.$$

(16)

where $S_{best}^j$ denotes the location of the $j-th$ dimension currently evaluated as the best quality and ${\overline{s}}_{i-1}^j$ denotes the $j-th$ dimension of the $i-1st$ population member $s_{i-1}$ after the random combination process. The adaptive step control parameter $c_1$ is defined as:

$$c_1=2\times e^{-{\left({\displaystyle \frac{4\cdot fes}{Ma{x}_{fes}}}\right)}^2}$$

(17)

The flowchart of the suggested MSAO is presented in Figure 3.

3.4. Results Comparisons and Analysis of CEC2017 Benchmark Functions

This subsection compares MISAO with seven original highly cited algorithms, namely Particle Swarm Optimization (PSO) [37], Differential Evolution (DE) [38], Genetic Algorithm (GA) [39], Gray Wolf Optimizer (GWO) [40], Whale Optimization Algorithm (WOA) [41], Rime optimization algorithm (RIME) [42], and Subtraction-Average-Basic Optimizer (SABO) [43]. This comparison aims to highlight the algorithm’s superiority and rationality. This part aims to validate the optimization performance of MISAO and showcase its innovation and superiority by comparing it with high-performance upgraded methods. The algorithms being compared are the latest optimization algorithms developed within the past five years. These include the moth-flame optimization based on quantum behavioral simulated annealing algorithm (QSMFO) [44], ACO with Cauchy mutation and greedy Levy mutation (CLACO) [45], Gaussian quasi-bone mutation-augmented SMA (GBSMA) [46], and Drosophila Optimizer with Multiple Swarm Outposts Mechanism (MOFOA) [47].

The test set utilized for comparison was IEEE CEC 2017, as specified in

Table 1. Details of the IEEE CEC2017.

		The Function Name	${\mathit{f}}_{\mathbf{min}}$
Unimodal Functions	F1	Shifted and Rotated Bent Cigar Function	100
	F2	Shifted and Rotated Sum of Different Power Function	200
	F3	Shifted and Rotated Zakharov Function	300
Multimodal Functions	F4	Shifted and Rotated Rosenbrock’s Function	400
	F5	Shifted and Rotated Rastrigin’s Function	500
	F6	Shifted and Rotated Expanded Scaffer’s F6 Function	600
	F7	Shifted and Rotated Lunacek Bi_Rastrigin Function	700
	F8	Shifted and Rotated Non-Continuous Rastrigin’s Function	800
	F9	Shifted and Rotated Levy Function	900
	F10	Shifted and Rotated Schwefel’s Function	1000
Hybrid Functions	F11	Hybrid Function 1 (N = 3)	1100
	F12	Hybrid Function 2 (N = 3)	1200
	F13	Hybrid Function 3 (N = 3)	1300
	F14	Hybrid Function 4 (N = 4)	1400
	F15	Hybrid Function 5 (N = 4)	1500
	F16	Hybrid Function 6 (N = 4)	1600
	F17	Hybrid Function 6 (N = 5)	1700
	F18	Hybrid Function 6 (N = 5)	1800
	F19	Hybrid Function 6 (N = 5)	1900
	F20	Hybrid Function 6 (N = 6)	2000
Composition Functions	F21	Composition Function 1 (N = 3)	2100
	F22	Composition Function 2 (N = 3)	2200
	F23	Composition Function 3 (N = 4)	2300
	F24	Composition Function 4 (N = 4)	2400
	F25	Composition Function 5 (N = 5)	2500
	F26	Composition Function 6 (N = 5)	2600
	F27	Composition Function 7 (N = 6)	2700
	F28	Composition Function 8 (N = 6)	2800
	F29	Composition Function 9 (N = 3)	2900
	F30	Composition Function 10 (N = 3)	3000

. The collection comprises single-peak, multi-peak, hybrid, and composite functions to comprehensively assess the performance of each technique across various optimization situations. The swarm intelligence algorithms were initialized with a population size of 30. The maximum number of evaluations was set at 300,000. Additionally, 30 distinct comparison experiments were conducted to verify the experimental fairness.

3.4.1. Effectiveness Validation of Different Components

This study incorporates three enhancement strategies, including cyclic chaotic mapping initialization, Gaussian diffusion, and random follower search, to enhance the optimization performance of the classic SAO algorithm. In order to validate the effectiveness of each component, six different MISAO-derived algorithms with one or more fusion strategies are designed, as shown in

Table 2. Different MISAO-derived variants with three improvement strategies.

Strategy	MISAO-1	MISAO-2	MISAO-3	MISAO-4	MISAO-5	MISAO-6	MISAO
Circle Chaotic Mapping initialization	1	0	0	1	1	0	1
Gaussian diffusion	0	1	0	1	0	1	1
Random follower search	0	0	1	0	1	1	1

, where 1 represents that the strategy is embedded; instead, 0 indicates that the strategy is not embedded. The performance of SAO, MISAO, and multiple MISAO-derived variants in ablating was simultaneously evaluated using the CEC 2017 test set.

Table 3. Experimental results of MISAO-derived variants on the CEC2017 test suite.

	F1		F2		F3
	AVG	STD	AVG	STD	AVG	STD
SAO	1.4356 × 1010	4.1652 × 109	1.9597 × 1037	7.4719 × 1037	6.2312 × 104	8.7433 × 103
MISAO-1	3.0675 × 106	1.2035 × 106	3.6374 × 1032	1.5265 × 1033	1.6754 × 105	2.9168 × 104
MISAO-2	2.1009 × 109	1.1516 × 109	9.5503 × 1029	5.2109 × 1030	6.7376 × 104	1.4574 × 104
MISAO-3	1.3708 × 107	5.7049 × 107	1.7082 × 1033	7.0088 × 1033	5.9686 × 104	2.5339 × 104
MISAO-4	1.0374 × 109	2.7584 × 108	6.8887 × 1030	3.7723 × 1031	1.2342 × 105	2.9425 × 104
MISAO-5	1.1051 × 106	6.1015 × 105	5.9048 × 1012	6.8204 × 1012	4.0549 × 104	1.2012 × 104
MISAO-6	4.0262 × 106	1.3219 × 106	7.0249 × 1016	9.5988 × 1016	5.0456 × 104	1.9200 × 104
MISAO	2.1933 × 104	2.2253 × 104	2.4396 × 1021	9.2127 × 1021	6.1219 × 104	3.5196 × 103
	F4		F5		F6
	AVG	STD	AVG	STD	AVG	STD
SAO	2.5458 × 103	9.4491 × 102	8.3048 × 102	3.9304 × 101	6.6958 × 102	1.3766 × 101
MISAO-1	4.7403 × 102	3.1313 × 101	6.8997 × 102	3.2949 × 101	6.4940 × 102	8.1975 × 100
MISAO-2	7.8114 × 102	2.5851 × 102	7.0181 × 102	2.6274 × 101	6.3143 × 102	8.8476 × 100
MISAO-3	5.4624 × 102	4.5613 × 101	6.7147 × 102	4.0756 × 101	6.4119 × 102	8.7626 × 100
MISAO-4	6.1316 × 102	8.4760 × 101	6.3167 × 102	4.4956 × 101	6.1396 × 102	4.5371 × 100
MISAO-5	5.1841 × 102	1.8175 × 101	6.8840 × 102	2.1006 × 101	6.0017 × 102	3.4353 × 10−2
MISAO-6	5.2706 × 102	3.2561 × 101	6.0968 × 102	2.9046 × 101	6.1498 × 102	6.1363 × 100
MISAO	5.0148 × 102	1.5002 × 101	5.8517 × 102	1.2773 × 101	6.0116 × 102	8.1538 × 10−1
	F7		F8		F9
	AVG	STD	AVG	STD	AVG	STD
SAO	1.1424 × 103	6.4035 × 101	1.0909 × 103	4.1032 × 101	7.6505 × 103	2.0115 × 103
MISAO-1	9.1294 × 102	4.6527 × 101	9.3927 × 102	2.6927 × 101	6.8644 × 103	1.5740 × 103
MISAO-2	1.1618 × 103	7.0989 × 101	9.7827 × 102	3.6054 × 101	4.3277 × 103	1.3862 × 103
MISAO-3	1.1352 × 103	1.1395 × 102	9.3225 × 102	2.8250 × 101	4.0505 × 103	1.2546 × 103
MISAO-4	9.6689 × 102	2.0132 × 101	9.0470 × 102	3.4856 × 101	2.3935 × 103	1.0566 × 103
MISAO-5	9.2969 × 102	1.5678 × 101	9.9259 × 102	2.2065 × 101	2.2256 × 103	4.0278 × 102
MISAO-6	8.6983 × 102	3.9362 × 101	9.1552 × 102	2.9839 × 101	3.0513 × 103	1.1130 × 103
MISAO	9.1464 × 102	2.0630 × 101	8.8206 × 102	1.1588 × 101	1.1370 × 103	4.9427 × 102
	F10		F11		F12
	AVG	STD	AVG	STD	AVG	STD
SAO	8.8892 × 103	4.4488 × 102	5.5759 × 103	1.8585 × 103	8.7132 × 108	5.7328 × 108
MISAO-1	7.6757 × 103	6.7327 × 102	1.8035 × 103	4.3925 × 102	3.7237 × 107	1.8648 × 107
MISAO-2	8.4955 × 103	6.8812 × 102	1.3905 × 103	1.2317 × 102	1.7383 × 107	1.6605 × 107
MISAO-3	5.5369 × 103	8.4991 × 102	1.3491 × 103	8.1450 × 101	1.6220 × 107	1.4288 × 107
MISAO-4	4.8102 × 103	8.5677 × 102	1.2897 × 103	2.3141 × 102	1.7595 × 107	1.6609 × 107
MISAO-5	4.7715 × 103	3.4400 × 102	1.8265 × 103	3.0893 × 102	3.1885 × 106	2.1386 × 106
MISAO-6	4.5056 × 103	4.1068 × 102	1.3323 × 103	4.9737 × 101	1.4338 × 106	1.4434 × 106
MISAO	4.2828 × 103	6.4064 × 102	1.2463 × 103	3.8981 × 101	1.2447 × 106	8.6153 × 105
	F13		F14		F15
	AVG	STD	AVG	STD	AVG	STD
SAO	5.8571 × 107	1.0065 × 108	1.1990 × 106	9.8111 × 105	2.1574 × 106	3.6503 × 106
MISAO-1	3.3801 × 106	1.8106 × 106	4.7517 × 105	3.2796 × 105	8.0131 × 105	5.6906 × 105
MISAO-2	8.1516 × 105	4.1978 × 106	2.5486 × 104	3.0098 × 104	1.1738 × 104	6.2870 × 103
MISAO-3	1.4797 × 105	2.4732 × 105	1.3499 × 105	1.0236 × 105	2.0211 × 104	1.3024 × 104
MISAO-4	2.7977 × 105	4.8407 × 105	3.8807 × 105	1.8079 × 105	2.2460 × 104	1.5763 × 104
MISAO-5	3.2568 × 104	1.5097 × 104	3.6784 × 104	3.0641 × 104	1.2420 × 104	1.1801 × 104
MISAO-6	2.9037 × 104	2.6140 × 104	4.8585 × 104	6.2615 × 104	1.3685 × 104	1.0849 × 104
MISAO	2.5515 × 104	2.1349 × 104	6.5650 × 104	7.3663 × 104	3.5325 × 103	2.4757 × 103
	F16		F17		F18
	AVG	STD	AVG	STD	AVG	STD
SAO	4.0919 × 103	2.2891 × 102	2.8521 × 103	2.0374 × 102	6.0175 × 106	7.8567 × 106
MISAO-1	2.8586 × 103	3.2794 × 102	2.3739 × 103	2.6183 × 102	3.3062 × 106	1.3364 × 106
MISAO-2	3.4653 × 103	2.8507 × 102	2.2694 × 103	1.4860 × 102	3.2023 × 105	3.0023 × 105
MISAO-3	2.9941 × 103	4.0675 × 102	2.4128 × 103	3.2043 × 102	1.5622 × 106	1.5784 × 106
MISAO-4	2.5897 × 103	2.9668 × 102	2.1976 × 103	2.2014 × 102	1.6698 × 106	2.2653 × 106
MISAO-5	3.0026 × 103	1.6919 × 102	2.1945 × 103	1.0216 × 102	1.2447 × 106	1.8219 × 106
MISAO-6	2.8338 × 103	2.9254 × 102	2.2222 × 103	2.1668 × 102	3.6235 × 105	1.2828 × 106
MISAO	2.4310 × 103	2.7776 × 102	2.0723 × 103	2.1126 × 102	1.4183 × 106	4.3452 × 105
	F19		F20		F21
	AVG	STD	AVG	STD	AVG	STD
SAO	7.2239 × 106	5.5667 × 106	3.1064 × 103	1.3302 × 102	2.5980 × 103	3.4620 × 101
MISAO-1	3.7651 × 104	5.1006 × 104	2.6207 × 103	1.6306 × 102	2.4898 × 103	3.3569 × 101
MISAO-2	1.4293 × 104	1.2085 × 104	2.6935 × 103	1.4011 × 102	2.4696 × 103	2.6380 × 101
MISAO-3	1.9362 × 104	1.7485 × 104	2.7057 × 103	2.1425 × 102	2.4446 × 103	3.4623 × 101
MISAO-4	8.1096 × 103	9.2642 × 103	2.4366 × 103	1.9564 × 102	2.4170 × 103	3.6339 × 101
MISAO-5	5.6148 × 105	4.3506 × 105	2.5305 × 103	1.5118 × 102	2.4906 × 103	2.1467 × 101
MISAO-6	1.1439 × 104	8.9838 × 103	2.5444 × 103	1.7995 × 102	2.4301 × 103	3.6692 × 101
MISAO	7.6499 × 103	6.5555 × 103	2.4066 × 103	1.4461 × 102	2.3775 × 103	2.4169 × 101
	F22		F23		F24
	AVG	STD	AVG	STD	AVG	STD
SAO	4.2848 × 103	7.7451 × 102	3.1781 × 103	8.6918 × 101	3.2481 × 103	7.5466 × 101
MISAO-1	5.1942 × 103	1.9892 × 103	3.1338 × 103	1.1503 × 102	3.2335 × 103	9.4918 × 101
MISAO-2	2.7830 × 103	3.1781 × 102	2.9072 × 103	4.3431 × 101	3.1070 × 103	6.4303 × 101
MISAO-3	5.2942 × 103	1.8909 × 103	2.9089 × 103	8.2676 × 101	3.0764 × 103	8.0174 × 101
MISAO-4	3.6568 × 103	1.8248 × 103	2.7834 × 103	4.5567 × 101	2.9711 × 103	5.0791 × 101
MISAO-5	6.4097 × 103	1.9746 × 103	2.8367 × 103	1.4110 × 101	3.0393 × 103	1.6466 × 101
MISAO-6	5.4154 × 103	1.3964 × 103	2.7923 × 103	4.9717 × 101	2.9508 × 103	3.3197 × 101
MISAO	3.9161 × 103	6.8382 × 102	2.7165 × 103	1.7152 × 101	2.8943 × 103	2.0961 × 101
	F25		F26		F27
	AVG	STD	AVG	STD	AVG	STD
SAO	3.3304 × 103	1.5281 × 102	8.2500 × 103	7.6077 × 102	3.4742 × 103	8.8325 × 101
MISAO-1	2.9098 × 103	2.4403 × 101	5.6720 × 103	2.2334 × 103	3.3588 × 103	1.6419 × 102
MISAO-2	3.0948 × 103	9.1002 × 101	6.2775 × 103	5.1888 × 102	3.3249 × 103	3.1521 × 101
MISAO-3	2.9688 × 103	5.5997 × 101	6.5209 × 103	8.6147 × 102	3.3214 × 103	5.4919 × 101
MISAO-4	2.9778 × 103	2.6367 × 101	4.7782 × 103	4.9687 × 102	3.2667 × 103	3.2376 × 101
MISAO-5	2.8958 × 103	6.0411 × 100	5.5874 × 103	2.8371 × 102	3.2299 × 103	4.8377 × 100
MISAO-6	2.9241 × 103	2.1835 × 101	5.1608 × 103	5.6080 × 102	3.2419 × 103	1.5749 × 101
MISAO	2.8900 × 103	6.1313 × 100	4.1942 × 103	1.4400 × 102	3.2271 × 103	1.6528 × 101
	F28		F29		F30
	AVG	STD	AVG	STD	AVG	STD
SAO	4.2757 × 103	4.0470 × 102	5.6078 × 103	4.5327 × 102	4.3393 × 107	4.2971 × 107
MISAO-1	3.2460 × 103	2.2766 × 101	4.2828 × 103	2.8036 × 102	1.5137 × 105	8.2389 × 104
MISAO-2	3.5605 × 103	1.5369 × 102	4.1966 × 103	2.4731 × 102	2.2857 × 105	3.4655 × 105
MISAO-3	3.3291 × 103	5.4972 × 101	4.3192 × 103	2.7242 × 102	1.4074 × 104	7.0062 × 103
MISAO-4	3.3743 × 103	3.5529 × 101	4.0170 × 103	2.0021 × 102	9.5909 × 105	9.5392 × 105
MISAO-5	3.3039 × 103	3.1869 × 101	4.1399 × 103	1.3268 × 102	5.6943 × 105	4.0365 × 105
MISAO-6	3.2843 × 103	3.2796 × 101	4.0990 × 103	2.2645 × 102	6.6403 × 105	5.6983 × 105
MISAO	3.2424 × 103	2.8334 × 101	3.6660 × 103	2.0021 × 102	1.0601 × 104	3.9129 × 103

showcases the average values (AVG) and standard deviations (STD) obtained by these algorithms for each of the 30 test functions. The AVG serves as an indicator of the central tendency of the algorithmic performance, providing a summary measure of the typical outcome. Meanwhile, the STD quantifies the dispersion or volatility of the results, offering insights into the reliability and consistency of the algorithms’ performance. The bold values in each column denote the most optimal performance among the compared algorithms for the corresponding test function. The average ranking results are documented in

Table 4. Average ranking of different MISAO-derived variants.

	Mean Rank	Final Ranking
SAO	7.08	8
MISAO-1	5.42	6
MISAO-2	4.83	5
MISAO-3	5.43	7
MISAO-4	4.67	4
MISAO-5	3.10	2
MISAO-6	3.52	3
MISAO	1.80	1

for a more thorough analysis.

Table 3 demonstrates that the mature MISAO significantly outperformed the other comparison approaches by utilizing three strategies. It achieved the highest mean fitness and standard deviation in 22 out of 30 test instances. The MISAO-5 method fared the best in four cases, followed by MISAO-2 in three cases and MISAO-6 in one case. Based on the final Friedman mean ranked values, including MISAO variants in each strategy, a notable enhancement in the overall optimization performance was achieved compared to the original SAO. However, these variants still lacked a certain level of robustness. The order of the contributing effect of each strategy was as follows: random follower search > Gaussian diffusion > cyclic chaotic mapping initialization. For single-peaked functions F1 and F3, except MISAO, the solution of MISAO-5 is the closest to the theoretical value. This suggests that the cyclic chaotic mapping initialization strategy and the random follower search strategy contribute to improving SAO’s local search capability. For the multi-peak functions (F4~F10), combining the cyclic chaotic mapping initialization strategy and the stochastic follower search strategy remains effective in overcoming local optimal stagnation. Furthermore, incorporating the specifically developed Gaussian diffusion method also enhances the algorithm’s search depth to a certain degree, increasing the likelihood of finding the global optimum. MISAO-2 has superior performance on F14, F18, and F22 among the hybrid and combinatorial functions (F11~F30). Similarly, MISAO-5 beats the other benchmark functions, specifically on F17 and F27. Overall, MISAO demonstrates better performance than the other benchmark functions. These findings demonstrate that the enhanced approach outlined in this study can effectively reconcile the benefits and drawbacks. The significance of the stochastic follower search approach in complex optimization tasks cannot be overstated. Maintaining a balance between exploring new possibilities and using existing resources is crucial. Simultaneously, the utilization of the cyclic chaotic mapping initialization approach and the Gaussian diffusion technique both improve the overall exploration of the system and broaden the range of the search. The individual effect of a single technique may not be readily apparent, but the combined application of these three strategies has a substantial impact on enhancing the overall performance of SAO.

3.4.2. Comparison and Analysis of Algorithms

The results of the experimental comparison of MISAO with other classical algorithms are shown in

Table 5. The means and standard deviations obtained during the experiment.

	F1		F2		F3
	AVG	STD	AVG	STD	AVG	STD
MISAO	2.1933 × 104	2.2253 × 104	2.4396 × 1021	9.2127 × 1021	6.1219 × 104	3.5196 × 103
PSO	1.0374 × 109	1.2399 × 109	6.8887 × 1030	3.7723 × 1031	1.1502 × 105	3.8991 × 104
DE	5.1095 × 109	1.8305 × 109	1.7854 × 1037	9.6817 × 1037	2.6127 × 105	7.1828 × 104
GA	4.9827 × 1010	1.6541 × 1010	2.5210 × 1031	5.4317 × 1031	2.8484 × 105	5.7385 × 104
GWO	1.4356 × 1010	4.1652 × 109	1.9597 × 1037	7.4719 × 1037	6.2312 × 104	8.7433 × 103
WOA	5.2771 × 1010	2.7584 × 108	3.5738 × 1039	6.4996 × 1039	9.0248 × 104	1.1754 × 104
RIME	1.1051 × 106	6.1015 × 105	3.6374 × 1032	1.5265 × 1033	1.6754 × 105	2.9168 × 104
SABO	3.3206 × 109	2.8103 × 109	1.0191 × 1040	4.8863 × 1040	1.2342 × 105	2.9425 × 104
QSMFO	2.1009 × 109	1.1516 × 109	9.5503 × 1029	5.2109 × 1030	6.7376 × 104	1.2012 × 104
CLACO	3.0675 × 106	1.2035 × 106	5.9048 × 1012	6.8204 × 1012	4.0549 × 104	1.4574 × 104
GBSMA	4.0262 × 106	1.3219 × 106	7.0249 × 1016	9.5988 × 1016	5.0456 × 104	1.9200 × 104
MOFOA	1.3708 × 107	5.7049 × 107	1.7082 × 1033	7.0088 × 1033	5.9686 × 104	2.5339 × 104
	F4		F5		F6
	AVG	STD	AVG	STD	AVG	STD
MISAO	5.0148 × 102	1.5002 × 101	5.8517 × 102	1.2773 × 101	6.0116 × 102	8.1538 × 10−1
PSO	6.1316 × 102	8.4760 × 101	7.2836 × 102	5.3817 × 101	6.5555 × 102	1.2690 × 101
DE	1.2774 × 103	4.3559 × 102	8.4452 × 102	5.7197 × 101	6.7916 × 102	9.6979 × 100
GA	1.1035 × 104	5.0858 × 103	1.0167 × 103	7.1475 × 101	7.1615 × 102	1.2046 × 101
GWO	2.5458 × 103	9.4491 × 102	8.3048 × 102	3.9304 × 101	6.6958 × 102	1.3766 × 101
WOA	5.5569 × 103	1.2921 × 102	8.6365 × 102	2.1006 × 101	6.7678 × 102	5.6477 × 100
RIME	5.1841 × 102	1.8175 × 101	6.8840 × 102	4.4956 × 101	6.0017 × 102	3.4353 × 10−2
SABO	6.1437 × 102	9.1927 × 101	6.3167 × 102	3.6188 × 101	6.1396 × 102	4.5371 × 100
QSMFO	7.8114 × 102	2.5851 × 102	7.0181 × 102	2.9046 × 101	6.4940 × 102	8.8476 × 100
CLACO	4.7403 × 102	3.1313 × 101	6.8997 × 102	3.2949 × 101	6.3143 × 102	8.1975 × 100
GBSMA	5.2706 × 102	3.2561 × 101	6.0968 × 102	2.6274 × 101	6.1498 × 102	6.1363 × 100
MOFOA	5.4624 × 102	4.5613 × 101	6.7147 × 102	4.0756 × 101	6.4119 × 102	8.7626 × 100
	F7		F8		F9
	AVG	STD	AVG	STD	AVG	STD
MISAO	9.2969 × 102	2.0630 × 101	8.8206 × 102	1.1588 × 101	1.1370 × 103	4.9427 × 102
PSO	1.2128 × 103	1.0587 × 102	9.7770 × 102	2.3574 × 101	8.6126 × 103	1.5509 × 103
DE	1.3124 × 103	8.5029 × 101	1.0730 × 103	6.2022 × 101	1.1048 × 104	3.4590 × 103
GA	2.1723 × 103	3.2267 × 102	1.2478 × 103	5.0006 × 101	1.1425 × 104	3.1382 × 103
GWO	1.1424 × 103	6.4035 × 101	1.0909 × 103	4.1032 × 101	7.6505 × 103	2.0115 × 103
WOA	1.3275 × 103	2.0132 × 101	1.1062 × 103	2.0226 × 101	1.0315 × 104	1.0795 × 103
RIME	9.6689 × 102	1.5678 × 101	9.9259 × 102	3.4856 × 101	2.2256 × 103	4.0278 × 102
SABO	9.1464 × 102	5.6379 × 101	9.0470 × 102	2.2065 × 101	2.3935 × 103	1.0566 × 103
QSMFO	1.1618 × 103	7.0989 × 101	9.7827 × 102	3.6054 × 101	6.8644 × 103	1.3862 × 103
CLACO	9.1294 × 102	4.6527 × 101	9.3927 × 102	2.6927 × 101	4.3277 × 103	1.5740 × 103
GBSMA	8.6983 × 102	3.9362 × 101	9.1552 × 102	2.9839 × 101	3.0513 × 103	1.1130 × 103
MOFOA	1.1352 × 103	1.1395 × 102	9.3225 × 102	2.8250 × 101	4.0505 × 103	1.2546 × 103
	F10		F11		F12
	AVG	STD	AVG	STD	AVG	STD
MISAO	4.2828 × 103	6.4064 × 102	1.2463 × 103	3.8981 × 101	1.2447 × 106	8.6153 × 105
PSO	6.0783 × 103	1.0073 × 103	1.8035 × 103	4.3925 × 102	1.7595 × 107	1.6609 × 107
DE	7.5964 × 103	3.4400 × 102	1.1218 × 104	4.7194 × 103	4.3472 × 108	2.9425 × 108
GA	4.5056 × 103	8.1843 × 102	2.4795 × 104	9.7912 × 102	1.3814 × 108	2.5819 × 109
GWO	8.8892 × 103	4.4488 × 102	5.5759 × 103	1.8585 × 103	8.7132 × 108	5.7328 × 108
WOA	9.1445 × 103	4.1068 × 102	2.5344 × 104	6.0007 × 103	9.8850 × 109	9.4561 × 107
RIME	7.6757 × 103	8.0881 × 102	1.8265 × 103	3.0893 × 102	3.7237 × 107	1.8648 × 107
SABO	8.5300 × 103	1.0504 × 103	2.6138 × 103	1.2015 × 104	5.5632 × 109	1.1342 × 108
QSMFO	8.4955 × 103	8.5677 × 102	1.3905 × 103	1.2317 × 102	1.7383 × 107	1.6605 × 107
CLACO	4.7715 × 103	6.7327 × 102	1.2897 × 103	8.1450 × 101	3.1885 × 106	2.1386 × 106
GBSMA	4.8102 × 103	6.8812 × 102	1.3323 × 103	4.9737 × 101	1.6220 × 107	1.4288 × 107
MOFOA	5.5369 × 103	8.4991 × 102	1.3491 × 103	2.3141 × 102	1.4338 × 106	1.4434 × 106
	F13		F14		F15
	AVG	STD	AVG	STD	AVG	STD
MISAO	2.5515 × 104	2.1349 × 104	6.5650 × 104	7.3663 × 104	3.5325 × 103	2.4757 × 103
PSO	2.7977 × 105	4.8407 × 105	3.8807 × 105	5.8230 × 105	2.2460 × 104	1.5763 × 104
DE	1.0297 × 107	8.1260 × 106	2.5162 × 106	2.5599 × 106	5.8449 × 106	7.0967 × 106
GA	4.1739 × 109	3.6276 × 109	8.1429 × 106	1.1216 × 107	1.9632 × 108	2.7712 × 108
GWO	5.8571 × 107	1.0065 × 108	1.1990 × 106	9.8111 × 105	2.1574 × 106	3.6503 × 106
WOA	2.5142 × 109	1.5301 × 108	2.8264 × 106	1.8079 × 105	6.4704 × 108	2.7147 × 107
RIME	3.3801 × 106	1.8106 × 106	4.7517 × 105	3.2796 × 105	8.0131 × 105	5.6906 × 105
SABO	5.9559 × 106	2.8572 × 107	4.1875 × 105	4.7102 × 105	1.1935 × 106	2.5632 × 106
QSMFO	8.1516 × 105	4.1978 × 106	1.3499 × 105	1.0236 × 105	2.0211 × 104	1.1801 × 104
CLACO	3.2568 × 104	1.5097 × 104	3.6784 × 104	3.0098 × 104	1.2420 × 104	6.2870 × 103
GBSMA	1.4797 × 105	2.4732 × 105	2.5486 × 104	3.0641 × 104	1.1738 × 104	1.3024 × 104
MOFOA	2.9037 × 104	2.6140 × 104	4.8585 × 104	6.2615 × 104	1.3685 × 104	1.0849 × 104
	F16		F17		F18
	AVG	STD	AVG	STD	AVG	STD
MISAO	2.4310 × 103	2.7776 × 102	2.0723 × 103	2.1126 × 102	1.4183 × 106	4.3452 × 105
PSO	3.0098 × 103	3.7055 × 102	2.3624 × 103	2.4549 × 102	2.9849 × 106	4.2654 × 106
DE	4.2364 × 103	7.4903 × 102	2.1945 × 103	1.0216 × 102	2.3048 × 107	2.8803 × 107
GA	4.6939 × 103	6.1868 × 102	2.1976 × 103	2.2014 × 102	2.6407 × 107	3.3508 × 107
GWO	4.0919 × 103	2.2891 × 102	2.8521 × 103	2.0374 × 102	6.0175 × 106	7.8567 × 106
WOA	8.9062 × 103	1.9477 × 103	2.4279 × 104	1.6776 × 104	1.0283 × 107	1.7853 × 106
RIME	3.0026 × 103	1.6919 × 102	2.7520 × 103	3.4184 × 102	3.3062 × 106	2.2653 × 106
SABO	2.5897 × 103	2.9668 × 102	3.3441 × 103	4.1853 × 102	1.6698 × 106	1.8219 × 106
QSMFO	3.4653 × 103	3.2794 × 102	2.3739 × 103	2.6183 × 102	1.5622 × 106	3.0023 × 105
CLACO	2.8586 × 103	2.8507 × 102	2.2694 × 103	1.4860 × 102	1.2447 × 106	1.3364 × 106
GBSMA	2.8338 × 103	2.9254 × 102	2.2222 × 103	2.1668 × 102	3.2023 × 105	1.2828 × 106
MOFOA	2.9941 × 103	4.0675 × 102	2.4128 × 103	3.2043 × 102	3.6235 × 105	1.5784 × 106
	F19		F20		F21
	AVG	STD	AVG	STD	AVG	STD
MISAO	7.6499 × 103	6.5555 × 103	2.4066 × 103	1.4461 × 102	2.3775 × 103	2.4169 × 101
PSO	3.7651 × 104	5.1006 × 104	2.6759 × 103	2.3411 × 102	2.4868 × 103	5.2382 × 101
DE	2.4940 × 107	2.4503 × 107	2.8637 × 103	2.4033 × 102	2.6631 × 103	4.9489 × 101
GA	3.9131 × 108	4.0324 × 108	3.3074 × 103	2.6114 × 102	2.8315 × 103	5.5491 × 101
GWO	7.2239 × 106	5.5667 × 106	3.1064 × 103	1.3302 × 102	2.5980 × 103	3.4620 × 101
WOA	1.4356 × 108	7.3293 × 106	3.0260 × 103	1.8563 × 102	2.6458 × 103	2.1467 × 101
RIME	5.6148 × 105	4.3506 × 105	2.5305 × 103	1.9564 × 102	2.4906 × 103	3.6339 × 101
SABO	3.8317 × 106	8.4477 × 106	2.4366 × 103	1.5118 × 102	2.4170 × 103	3.5004 × 101
QSMFO	1.4293 × 104	1.2085 × 104	2.6935 × 103	1.7995 × 102	2.4898 × 103	3.6692 × 101
CLACO	8.1096 × 103	9.2642 × 103	2.6207 × 103	1.6306 × 102	2.4696 × 103	3.3569 × 101
GBSMA	1.9362 × 104	1.7485 × 104	2.5444 × 103	1.4011 × 102	2.4301 × 103	2.6380 × 101
MOFOA	1.1439 × 104	8.9838 × 103	2.7057 × 103	2.1425 × 102	2.4446 × 103	3.4623 × 101
	F22		F23		F24
	AVG	STD	AVG	STD	AVG	STD
MISAO	3.9161 × 103	7.7451 × 102	2.7165 × 103	1.7152 × 101	2.8943 × 103	2.0961 × 101
PSO	3.6568 × 103	1.9963 × 103	2.8797 × 103	8.2425 × 101	2.9963 × 103	5.2792 × 101
DE	8.0583 × 103	1.6379 × 103	3.1521 × 103	9.0451 × 101	3.2903 × 103	1.0428 × 102
GA	1.0153 × 104	9.4112 × 102	3.5887 × 103	1.7800 × 102	4.0021 × 103	3.0527 × 102
GWO	4.2848 × 103	1.8248 × 103	3.1781 × 103	8.6918 × 101	3.2481 × 103	7.5466 × 101
WOA	1.0441 × 104	6.8382 × 102	3.7575 × 103	5.5987 × 101	3.8584 × 103	5.0791 × 101
RIME	6.4097 × 103	1.9746 × 103	2.8367 × 103	1.4110 × 101	3.0393 × 103	1.6466 × 101
SABO	5.0935 × 103	1.9985 × 103	2.7834 × 103	4.5567 × 101	2.9711 × 103	5.9447 × 101
QSMFO	5.4154 × 103	1.9892 × 103	3.1338 × 103	1.1503 × 102	3.1070 × 103	9.4918 × 101
CLACO	5.1942 × 103	3.1781 × 102	2.9072 × 103	4.3431 × 101	3.2335 × 103	6.4303 × 101
GBSMA	2.7830 × 103	1.3964 × 103	2.7923 × 103	4.9717 × 101	2.9508 × 103	3.3197 × 101
MOFOA	5.2942 × 103	1.8909 × 103	2.9089 × 103	8.2676 × 101	3.0764 × 103	8.0174 × 101
	F25		F26		F27
	AVG	STD	AVG	STD	AVG	STD
MISAO	2.8900 × 103	6.1313 × 100	4.1942 × 103	1.4400 × 102	3.2271 × 103	1.6528 × 101
PSO	2.9778 × 103	3.9919 × 101	6.3264 × 103	1.4005 × 103	3.2914 × 103	5.1901 × 101
DE	2.8958 × 103	6.0411 × 100	8.5254 × 103	1.0374 × 103	3.4413 × 103	9.2001 × 101
GA	6.1888 × 103	1.2584 × 103	1.0211 × 104	9.2768 × 102	4.2253 × 103	2.2631 × 102
GWO	3.3304 × 103	1.5281 × 102	8.2500 × 103	7.6077 × 102	3.4742 × 103	8.8325 × 101
WOA	3.6130 × 103	2.6367 × 101	8.9393 × 103	2.8371 × 102	3.6815 × 103	3.9853 × 101
RIME	3.2546 × 103	9.7431 × 101	5.5874 × 103	4.9687 × 102	3.2299 × 103	4.8377 × 100
SABO	3.0194 × 103	5.8063 × 101	4.7782 × 103	3.3793 × 102	3.2667 × 103	3.2376 × 101
QSMFO	3.0948 × 103	9.1002 × 101	6.2775 × 103	2.2334 × 103	3.3249 × 103	1.6419 × 102
CLACO	2.9098 × 103	2.4403 × 101	5.6720 × 103	5.1888 × 102	3.3588 × 103	3.1521 × 101
GBSMA	2.9241 × 103	2.1835 × 101	5.1608 × 103	5.6080 × 102	3.2419 × 103	1.5749 × 101
MOFOA	2.9688 × 103	5.5997 × 101	6.5209 × 103	8.6147 × 102	3.3214 × 103	5.4919 × 101
	F28		F29		F30
	AVG	STD	AVG	STD	AVG	STD
MISAO	3.2424 × 103	2.8334 × 101	3.6660 × 103	2.0021 × 102	1.0601 × 104	3.9129 × 103
PSO	3.3743 × 103	6.1510 × 101	4.1743 × 103	3.2414 × 102	9.5909 × 105	9.5392 × 105
DE	3.8753 × 103	2.4378 × 102	5.5841 × 103	5.5787 × 102	7.2547 × 107	6.5087 × 107
GA	7.6307 × 103	1.5798 × 103	6.5037 × 103	1.1108 × 103	2.6012 × 108	2.4383 × 108
GWO	4.2757 × 103	4.0470 × 102	5.6078 × 103	4.5327 × 102	4.3393 × 107	4.2971 × 107
WOA	5.0587 × 103	3.1869 × 101	6.6990 × 103	3.5520 × 102	3.0792 × 109	8.7527 × 108
RIME	3.3039 × 103	3.5529 × 101	4.1399 × 103	1.3268 × 102	5.6943 × 105	4.0365 × 105
SABO	3.4998 × 103	1.2305 × 102	4.0170 × 103	2.0736 × 102	1.4405 × 107	1.1861 × 107
QSMFO	3.5605 × 103	1.5369 × 102	4.1966 × 103	2.8036 × 102	6.6403 × 105	5.6983 × 105
CLACO	3.2460 × 103	2.2766 × 101	4.2828 × 103	2.4731 × 102	1.5137 × 105	8.2389 × 104
GBSMA	3.2843 × 103	3.2796 × 101	4.0990 × 103	2.2645 × 102	2.2857 × 105	3.4655 × 105
MOFOA	3.3291 × 103	5.4972 × 101	4.3192 × 103	2.7242 × 102	1.4074 × 104	7.0062 × 103

, where AVG and STD denote the mean and standard deviation of the algorithms after 30 independent runs, respectively, and the optimal values are shown in bold in each column. By comparing and observing the average values (AVG), it is initially clear that for most of the benchmark test functions, MISAO has the smallest average value, which indicates that MISAO obtains relatively high-quality solutions when using other algorithms to optimize the benchmark functions. It is particularly effective in two types of tests: hybrid and composite functions, which suggests that the MISAO algorithm will have a more vital ability to find an optimal solution when faced with complex problems. Also, the standard deviation (STD) of the optimal solution is slight, indicating that MISAO has a high degree of stability when optimizing the benchmark function. To verify the validity of the experimental results of MISAO and to improve the understanding of the experimental results, we further analyzed the experimental data using the Wilcoxon signed-rank test, shown in

Table 6. Wilcoxon signed-rank test results of MISAO and classic algorithms.

	+/−/=	Mean	Rank
MISAO	~	1.82	1
PSO	29/0/1	7.02	8
DE	27/0/3	9.22	11
GA	30/0/0	10.73	12
GWO	26/0/4	8.47	9
WOA	26/0/4	8.65	10
RIME	22/1/7	5.53	4
SABO	29/0/1	6.50	6
QSMFO	29/0/1	6.88	7
CLACO	21/1/8	3.83	3
GBSMA	22/3/5	3.37	2
MOFOA	27/1/2	5.67	5

, where ‘+’ indicates that the performance of MISAO is better than that of other algorithms, ‘−’ indicates that the performance of MISAO is poorer than that of other algorithms, and ‘=’ indicates that MISAO performs as well as other algorithms. Mean denotes the average ranking after 30 iterations of parallelization, and Rank denotes the overall final ranking. According to the results in Table 6, MISAO outperforms its competitors in at least 21 out of 30 functional evaluation trials. The average ratings indicate that MISAO is far superior to its competitors’ algorithms. In addition, the MISAO algorithm has a huge advantage over the algorithms that have been improved in recent years, such as CLACO, GBSMA, and MOFOA.

Figure 4 illustrates some of the results of the different algorithms iterated under the benchmark functions. In order to analyze the iterative process of the algorithms as thoroughly as possible, the nine functions in the figure were selected from a set of 30 functions in the following order: single-peak, multi-peak, hybrid, and combined functions. Figure 4 illustrates how MISAO outperforms other algorithms in pre-searching functions F5, F7, and F21. For functions F16, F23, and F28, MISAO can be updated to a superior solution even later in the optimization search, which shows its strong development potential. In addition, MISAO outperforms competing algorithms in the exploration and development phases for functions F1, F12, and F26. The comparative tests show that MISAO is a very effective metaheuristic optimization algorithm with high robustness and adaptability in solving optimization problems.

To enhance the reliability of the experiments and the validity of the Wilcoxon signed-rank test, Figure 5 presents the findings of the Friedman test, which was also employed for comparative analysis alongside the Wilcoxon test. Figure 5 demonstrates that both methods are effective in validating the experimental results; despite slight differences in the average rankings between the Friedman and Wilcoxon tests, the ranks for each algorithm are nearly identical.

The results from these two testing strategies clearly indicate that MISAO is a highly effective metaheuristic optimization algorithm, outperforming the new methods employed by its competitors.

4. MISAO-ICEEMDAN-WLSSVM Model

4.1. Intrinsic Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (ICEEMDAN)

EMD is suitable for decomposing non-stationary and highly random data. It can decompose the original signal into a series of intrinsic mode functions (IMF) with each variable frequency and reflect the local characteristics of nonstationary signals. EEMD and CEEMDAN are further improvements to the EMD algorithm that can solve the mixing problem during EMD. ICEEMDAN is a signal decomposition method proposed by Colominas et al. [20] that extracts IMF components by adding white noise with a mean and variance of 0 and 1, respectively. It can effectively reduce noise interference, mode aliasing, and reconstructed signal distortion in the intrinsic mode functions of EEMD and CEEMDAN algorithms. The specific decomposition process of ICEEMDAN is as follows:

(1)

Add white noise $\left.E_1\left[\begin{array}{c}{\omega }^i\end{array}\right.\right]$ to the original signal $X$:

$$\left.X^{(i)}=X+{\beta }_0 E_1\left[\begin{array}{c}{\omega }^i\end{array}\right.\right]$$

(18)

where ${\omega }^i$ is the $i-th$ white noise that was added;

(2)

Decompose the original signal using the ICEEMAN method to obtain the first IMF component:

$$\left\{\begin{array}{c}{R}_1={\displaystyle \frac{1}{I}}{\displaystyle \sum _{i=1}^I}M[X^i]\\ {\tilde{C}}_1=X-R_1\end{array}\right.$$

(19)

where $R_1$ is the first order residual and ${\tilde{C}}_1$ is the first IMF component.

(3)

Solve the second IMF component:

$$\left\{\begin{array}{c}{R}_2={\displaystyle \frac{1}{I}}{\displaystyle \sum _{i=1}^I}M\left\{R_1\right.+{\beta }_1\cdot E_2\left[{\omega }^i\right]\\ {\tilde{C}}_2=R_1-R_2\end{array}\right.$$

(20)

where $R_2$ is the second order residual and ${\tilde{C}}_2$ is the second IMF component.

(4)

And so on, the $h-th$ IMF component can be obtained as follows:

$$\left\{\begin{array}{l}{R}_h={\displaystyle \frac{1}{I}}{\displaystyle \sum _{i=1}^I}M\left\{R_{h-1}\right.+{\beta }_{h-1}\cdot E_h\left[{\omega }^i\right]\hfill \\ {\tilde{C}}_h=R_{h-1}-R_h\hfill \end{array}\right.$$

(21)

where $R_h$ is the $h-th$ order residual; $h=1,2,3,\cdots ,N$; and ${\tilde{C}}_h$ is the $h-th$ IMF component.

4.2. Weighted Least Squares Support Vector Machine

The LSSVM is excellent for small-sample nonlinear regression prediction problems; therefore, it is commonly used in photovoltaic power prediction [22]. WLSSVM is a new algorithm that was proposed to improve the robustness of the LSSVM algorithm. By assigning different weights to different samples, the WLSSVM effectively reduces the influence of noise in the training samples to enhance the robustness of the model.

WLSSVM is based on the LSSVM optimization problem, where the error $e_i^2$ of each item is weighted by the coefficient $v_i$. The optimization problem can be described as follows:

$$\left\{\begin{array}{c}\min J(\omega ,\epsilon )={\displaystyle \frac{1}{2}}{\omega }^{\mathrm{T}}\omega +{\displaystyle \frac{1}{2}}\gamma {\displaystyle \sum _{i=1}^m}{v}_i{e}_i^2\\ y_i={\omega }^{\mathrm{T}}\phi (x_i)+b+e_i,i=1,2,\cdots ,n\end{array}\right.$$

(22)

where $\omega$ is a vector of weight coefficients, $\gamma$ is a regularization parameter, and $v_i$ is the weight, calculated from the sample training error.

Introducing the Lagrange multiplier, we obtain the following:

$$L={\displaystyle \frac{1}{2}}{\omega }^{\mathrm{T}}\omega +\gamma \sum _{i=1}^{i=n}{v}_i{e}_i^2+\sum _{i=1}^{i=n}{a}_i\left({\omega }^{\mathrm{T}}\phi (x_i)\right.+b+e_i\left.-y_i\right)$$

(23)

where: $a_i$ is the Lagrange multiplier ($i=1,2,\cdots ,n,a_i\ge 0$).

The regression function of WLSSEM can be derived by applying the KKT condition and Mercer criteria:

$$\begin{array}{c}f(x)={\displaystyle \sum _{i=1}^{i=n}}{a}_{i}K(x_i,x_j)+b\end{array}$$

(24)

where: $K(x_i,x_j)=\exp (-|x_i-x_j{|}^2/2{\sigma }^2)$ is the radial basis function.

4.3. Hybrid MISAO-ICEEMDAN-WLSSVM Deep Learning Model

To accurately predict the PV power with strong volatility and stochastic characteristics, the K-means algorithm is used to cluster the PV power data, combining the advantages of ICEEMDAN and WLSSVM, and the hyperparameters of the model are optimized by the multi-strategy improved MISAO, to construct a hybrid MISAO-ICEEMDAN-WLSSVM deep learning model. The flowchart is shown in Figure 6, and the steps are as follows:

(1)

Raw data processing: missing data and anomalies due to various factors are processed by interpolating missing values, and all data are classified into three categories (sunny, cloudy, and rainy days) by the K-means algorithm. The power data of the three weather types are decomposed using ICEEMDAN to obtain a set of IMF components $\left\{{\mathrm{IMF}}_1,{\mathrm{IMF}}_2,\cdots ,{\mathrm{IMF}}_{\mathrm{n}}\right\}$, and the MISAO algorithm is used to optimize the white noise amplitude weights $Nstd$ and the number of noise additions $NE$.

(2)

Modal component prediction: the modal components are normalized and divided into training and test sets, the data are fed into the WLSSVM model for prediction, and the regularization parameter $\gamma$ and kernel parameter $\sigma$ are optimized using the MISAO algorithm.

(3)

Prediction output: after the training, the test set data are used for prediction, after which the components are reconstructed to obtain the final power prediction output.

5. Experiment and Result Analysis

This study selected data from a 20 MW photovoltaic power plant in Hebei, China, as the research object to verify the accuracy of the model. Meteorological and power data from 1 July 2018 to 31 December 2018, including global irradiance, diffuse irradiance, temperature, wind speed, wind direction, and atmospheric pressure, were selected. A time interval of 15 min was selected, from 07:00 to 19:00. The predicted step length was one. The time step of the input data was determined to be 10 by utilizing a trial-and-error approach.

5.1. Data Sets and Data Preprocessing

The training, testing, and validation sets were divided into 60%, 20%, and 20% of the total dataset. Normalization was conducted in this study to address the different magnitudes among the data. We adopted the min–max normalization method (Equation (25)):

$$\begin{array}{c}{Y}_i={\displaystyle \frac{Y-Y_{\min }}{Y_{\max }-Y_{\min }}}\end{array}$$

(25)

Min–max normalization can map the data clearly and directly to the [0, 1] interval, which can prominently exhibit the data’s relative magnitudes and distribution characteristics. It allows the differences and relationships of the data to be presented more intuitively and clearly, facilitating our understanding and analysis of the data. In contrast, mean normalization is mainly adjusted based on the mean of the data and may not fully reflect the extreme values and distribution range of the data in some cases. Although z-score normalization can standardize the data, it is less intuitive and direct than min–max normalization in emphasizing the data’s relative magnitudes and original distribution characteristics. Considering the data’s characteristics and the analysis requirements in this study, min–max normalization can serve our research purposes and data processing requirements more effectively.

5.2. Model Evaluation Indicators

In this study, the mean absolute error (MAE), root mean square error (RMSE), and mean absolute percentage of error (MAPE) were selected to evaluate the accuracy of the prediction of each model, which was calculated as:

$$\begin{array}{c}{e}_{\mathrm{MAE}}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}|y_i-{\tilde{y}}_i|\end{array}$$

(26)

$$e_{\mathrm{RMSE}}=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(y_i-{\tilde{y}}_i\right)}^2}$$

(27)

$$e_{\mathrm{MAPE}}={\displaystyle \frac{100\%}{n}}\sum _{i=1}^n\left|{\displaystyle \frac{{\tilde{y}}_i-y_i}{y_i}}\right|$$

(28)

where $n$ is the number of predicted outcomes, $y_i$ is the actual value, and ${\tilde{y}}_i$ is the model-predicted value.

5.3. Parameter Optimization Results

The MISAO parameters were set as follows: the number of populations was 30, the maximum number of iterations was 30, the optimization range of parameter $\gamma$ was [0.1, 1000], and the optimization range of parameter $\sigma$ was [0.01, 100]. The $Nstd$ search range was [0.15, 0.6], the $NE$ search range was [50, 600], and the minimum average envelope entropy was used as the fitness function. The optimized values of $Nstd$ and $NE$ were [0.461, 452], [0.180, 236], and [0.241, 197] for sunny, cloudy, and rainy weather, respectively.

5.4. Weather Clustering and Data Decomposition

Clustering the data using the K-means algorithm yielded 94, 49, and 41 days of sunny, cloudy, and rainy weather, respectively. Taking sunny weather as an example, Figure 7 shows the power components obtained by the MISAO-ICEEMDAN algorithm for a sunny day. From the figure, it is clear that 12 IMF components and one RES component are obtained from the ICEEMDAN decomposition, and each sequence is arranged according to the frequency from high to low. Among them, IMF1, as the dominant component, has a smooth and relatively stable curve, which can characterize the trend of the original PV power signal well. In contrast, the remaining components accurately show the local characteristics of the power data. Through in-depth analysis, it is found that the fluctuations of different frequency series have a certain regularity. The high-frequency IMF components usually reflect short-term rapid fluctuations, which are likely to be related to transient changes in local meteorology (e.g., short-term cloud cover) or transient responses of the equipment. In contrast, the low-frequency IMF and RES components are more indicative of long-term trends and overall energy levels, which are closely related to the cyclical changes in solar radiation as well as the inherent characteristics of the system. Each mode is independent of the other, effectively avoiding the problem of mode mixing. At the same time, the residuals of the RES component are gradually stabilized, which means that after the ICEEMDAN decomposition, the remaining unexplained part of the IMF component is close to the stable random noise, which strongly verifies the effectiveness of the decomposition. In the prediction model, each power component is set as an output series, and its corresponding meteorological data are used as an input series, which is finally superimposed to obtain the prediction results.

5.5. Result Analysis

To further verify the prediction effect of the proposed model, a total of six comparison models were set up in this study: Temporal Convolution Networks (TCN), Gate Recurrent Unit (GRU), SVM, WLSSVM, SAO-WLSSVM, and MISAO-WLSSVM. The prediction model was visualized and analyzed for three days selected for three weather types, namely sunny, cloudy, and rainy, while keeping the parameters constant, as shown in Figure 8, Figure 9 and Figure 10.

Figure 8 shows the predicted results under sunny conditions. The power variation during sunny weather was moderate, and the overall prediction effect was the best. The prediction results of the improved prediction model were closer to the actual values, and Figure 9 shows the predicted results under cloudy conditions. Owing to continuous changes in cloud cover, solar irradiance also changes with both gentle and abrupt changes. In the abrupt changes, the model in this study performed well, proving its strong anti-interference ability. Figure 10 shows the prediction results under rainy day conditions. The power amplitude is lower during rainy weather, and the power change is more evident owing to the long fluctuation period and significant volatility. The model prediction effect is reduced; however, this model still has some advantages in terms of prediction accuracy and stability.

Using three evaluation metrics (MAPE, RMSE, and MAE) on the prediction effect of the TCN, GRU, SVM, WLSSVM, SAO-WLSSVM, MISAO-WLSSVM, and MISAO-ICEEMDANWLSSVM prediction models for the three weather types, the calculation of the error yields the evaluation results as shown in

Table 7. Comparison of evaluation indexes of prediction results.

Weather Type	Model	Metrics
		MAPE/%	MAE/MW	RMSE/MW
Sunny	TCN	11.3537	1.9486	0.7029
	GRU	10.7660	1.6261	0.5929
	SVM	9.8237	1.6269	0.6050
	WLSSVM	7.4394	1.3569	0.5363
	SAO-WLSSVM	5.8246	1.0328	0.4139
	MISAO-WLSSVM	4.4907	0.8707	0.3643
	MISAO-ICEEMDAN-WLSSVM	3.3567	0.5309	0.2287
Cloudy	TCN	28.5691	3.0543	1.9109
	GRU	29.1185	2.9264	1.9895
	SVM	24.7850	2.4711	1.6014
	WLSSVM	26.7472	2.3479	1.4691
	SAO-WLSSVM	17.8016	1.9377	1.2764
	MISAO-WLSSVM	11.3291	1.4569	0.9562
	MISAO-ICEEMDAN-WLSSVM	8.0349	1.4569	0.8033
Rainy	TCN	40.1888	5.0142	2.9073
	GRU	34.2816	3.9762	2.3077
	SVM	30.7951	3.5970	2.0810
	WLSSVM	26.0090	3.1009	1.8872
	SAO-WLSSVM	18.0956	2.3070	1.4174
	MISAO-WLSSVM	14.3216	1.9603	1.2211
	MISAO-ICEEMDAN-WLSSVM	9.6233	1.5711	1.0589

.

Table 7 shows that, in comparison to the individual TCN, GRU, SVM, and WLSSVM models, all of the combined prediction models outperform the single model in a variety of weather scenarios. This suggests that the WLSSVM network’s hyperparameters specifically need to be optimized. By comparing the prediction results of the SAO-WLSSVM and MISAO-WLSSVM models, it was found that all of the indicators of the MISAO-WLSSVM model were lower than those of the SAO-WLSSVM model, and the average absolute error percentage MAPE under different weather conditions decreased by 22.9%, 36.4%, and 20.9%, respectively. The average absolute error MAE percentage under different weather conditions decreased by 15.0% and 20.9%, respectively. MAE decreased by 15.7%, 24.8%, and 15.0%, respectively, and the root mean square error MASE decreased by 12.0%, 25.1%, and 13.9%, respectively, under different weather conditions, indicating that MISAO has a better ability to find the optimum and stability than SAO.

In addition, the MISAO-ICEEMDAN-WLSSVM hybrid model obtained the best value among all of the evaluation indicators, in which the indicator MAPE under different weather conditions decreased by 25.3%, 29.1%, and 32.8% compared to that of MISAO-WLSSVM, respectively; the indicator MAE under different weather conditions decreased by 39.0%, 17.8%, and 19.9%, respectively; under different weather conditions, the indicator RMSE decreased by 37.2%, 16.0%, and 13.3%, respectively. The prediction results show that the use of ICEEMDAN decomposition can effectively eliminate redundant noise, extract the main features of the historical PV data, and significantly improve the prediction accuracy of the model, which verifies the superiority and validity of the MISAO-ICEEMDAN modal decomposition.

In order to make a strong argument for the necessity of classifying PV forecasts according to different weather conditions, we selected the same models as in the previous experiment, i.e., TCN, GRU, SVM, WLSSVM, SAO-WLSSVM, MISAO-WLSSVM, and MISAO-ICEEMDAN-WLSSVM, and we started this experiment based on the data from 1 July to 15 July 2018. The experimental metrics are recorded in

Table 8. Performance Indicators Comparison of Prediction Models without Weather Classification.

Model	Metrics
	MAPE/%	MAE/MW	RMSE/MW
TCN	11.3537	1.9486	0.7029
GRU	10.7660	1.6261	0.5929
SVM	9.8237	1.6269	0.6050
WLSSVM	7.4394	1.3569	0.5363
SAO-WLSSVM	5.8246	1.0328	0.4139
MISAO-WLSSVM	4.4907	0.8707	0.3643
MISAO-ICEEMDAN-WLSSVM	3.3567	0.5309	0.2287

, and the prediction results are displayed in Figure 11.

The data in Table 8 show that the key metrics such as MAE, MAPE, and RMSE are significantly higher when not categorized based on weather conditions compared to the previous experiments categorized based on weather. Figure 11 visually presents the prediction result curves of each model in this unclassified experiment. It can be observed that the prediction curves in the unclassified case are more volatile and deviate from the actual values to a more significant extent than the relatively smooth and accurate prediction curves in the previous classification case. It is especially noteworthy that even under such unfavorable experimental conditions, the MISAO-ICEEMDAN-WLSSVM model proposed in this paper still shows relatively optimal performance. This result strongly supports the critical role of weather condition-based classification in improving the accuracy of PV forecasts. It demonstrates the robustness of the proposed model under different experimental settings.

6. Discussion

In order to fully assess the performance of the MISAO-ICEEMDAN-WLSSVM model, it was compared with three existing models, TVF-EMD-ELM (M1), WPD-LFABS-SDS-GRU (M2) and SSA-VMD-Informer (M3).

Table 9. Comparison of the prediction results between the existing models and proposed model.

Model	Description
TVF-EMD-ELM (M1) [48]	The TVF-EMD method decomposes the PV generation data into more stable and constant subsequences. A specially specified set of features (Intrinsic Mode Function (IMF)) is used to train and improve the forecasting Extreme Learning Machine (ELM) model. The modified ELM model is used to evaluate the forecasting effectiveness.
WPD-LFABS-SDS-GRU (M2) [30]	Wavelet packet decomposition extracts the low and high-frequency components hidden in the PV power. The SDS method is combined with the LFBAS optimization algorithm to screen for days similar to the forecast day. A series of GRU networks optimized by LFBAS are used to predict the sub-signals after WPD decomposition and synthesize the final PV power forecast.
SSA-VMD-Informer (M3) [49]	Firstly, the temporal coding of the Informer model is improved; secondly, the original series is decomposed into multiple modal components using VMD; then, the results of VMD are optimized in conjunction with the optimization strategy of SSA to improve the characteristics of the time series data. Finally, the refined data are input into the Informer framework for modeling and prediction.

describes the development process of the existing hybrid models.

Table 10. Comparison with three existing models.

Models	MAPE	MAE	RMSE
TVF-EMD-ELM (M1)	1.9720	13.2230	1.4455
WPD-LFABS-SDS-GRU (M2)	1.4954	9.2411	0.9537
SSA-VMD-Informer (M3)	1.7036	11.1997	1.2406
Proposed model	1.0999	7.0049	0.6970

compares the mean values of the error results of the proposed model with the existing models for the three weather types under the dataset of this paper. The bold values represent the optimal forecasts among all the model’s predictions. As shown in Table 10, the MAE of the proposed model is reduced by 36.25%, 14.82%, and 25.23% compared with M1, M2, and M3, respectively. The MAE of M1 is 1.7254, and its TVF-EMD method is more complicated in parameter selection and has the problem of manual selection, which may be needed to be able to achieve the best decomposition effect for different PV data. The ELM model, when dealing with complex data, finds it difficult to fully explore the deep features, resulting in limited prediction accuracy. The MAE of M3 is 1.4710, and there may be modal aliasing problems during the decomposition process, affecting the feature extraction accuracy. The MAE of M2 is 1.2912, and the wavelet packet transform is time-consuming in selecting the appropriate wavelet basis function and ineffective in dealing with the boundary data, which negatively affects the prediction results. In addition, the combination of multiple algorithms and strategies increases the model complexity and may affect the computational efficiency.

On the other hand, the proposed model adopts the ICEEMDAN method of adaptive decomposition, which avoids complex parameter selection. The similar day model is constructed by K-means clustering, which improves the data quality. The multi-strategy improved MISAO optimizes the model parameters, combining the multivariate prediction considering meteorological factors and the optimal hyperparameter search of WLSSVM, significantly improving the prediction accuracy and adaptability.

The comparative analysis proves that the MISAO-ICEEMDAN-WLSSVM model proposed in this paper performs better in ultra-short-term PV power prediction.

7. Conclusions

Photovoltaic (PV) power generation is greatly affected by external factors, and the output power is random and unstable. In order to improve the prediction accuracy of PV power generation and reduce the impact of PV grid connection on the power system, a new MISAO-ICEEMDAN-WLSSVM prediction model was established. The research results are as follows:

(1)

A MISAO algorithm is proposed by improving the SAO optimization algorithm in three aspects. A series of numerical experiments were conducted using 30 CEC 2017 benchmark functions, and the results show that the MISAO algorithm has a better ability to find the optimal solution and jump out of the local optimal solution.

(2)

The MISAO algorithm was used to optimize the parameters of the ICEEMDAN-WLSSVM model, and the MISAO-ICEEMDAN-WLSSVM prediction model was established.

(3)

The MISAO-ICEEMDAN-WLSSVM model was validated using data from a photovoltaic (PV) power plant in Hebei Province, China, and the prediction results were compared with seven neural network models, including WLSSVM, ICEEMDAN-WLSSVM, and MISAO-ICEEMDAN-WLSSVM, for three different weather types. The results show that the prediction curve of the MISAO-ICEEMDAN-WLSSVM model is closer to the actual value curve with the smallest error.

(4)

Three evaluation indexes, MAPE, MAE and RMSE, were used to evaluate the seven models. The MAPE values of the proposed model were reduced by at least 25.3%, 29.1% and 32.8%; the MAE values were reduced by at least 39.0%, 17.8% and 32.8%; and the RMSE values were reduced by at least 37.2%, 16.0% and 13.3%, respectively, under the three types of weather compared with the other six models. The results showed that the MISAO-ICEEMDAN-WLSSVM prediction model has high prediction accuracy and stability. The proposed prediction model is conducive to accurately predicting the PV output power, making full use of solar energy resources, reducing the impact of the grid-connected PV on the modern power grid power system, promoting the smooth operation of grid scheduling, and maintaining the security and stability of the power system.

Furthermore, there are certain limitations in the prediction aspect of this study. Currently, the one-step forecasting method is employed. One-step forecasting restricts the comprehensive understanding of the long-term variation trend of photovoltaic power and may not fully meet the precise prediction requirements for photovoltaic power changes over a more extended period in some complex scenarios. Future research efforts will be dedicated to exploring multi-step forecasting methods. Intensive study will advance prediction techniques and model architectures, integrate more valuable feature information, and optimize the model training strategies to enhance the accuracy and stability of multi-step forecasting, providing more reliable technical support for optimized management and efficient operation in the field of new energy.

The limitations of this study also include the varying prediction effects of the model under different weather types. Specifically, when it is cloudy and rainy, the forecasting effect is less than when it is sunny. A thorough analysis of how to enhance the impact of weather type on forecasting has yet to be conducted. Moreover, compared to periods of high-power generation, the prediction accuracy is lower in the morning or evening. It is necessary to investigate the enhancement of the forecasting efficacy throughout the day. Future research should examine the effects of different weather conditions on the generation of PV power in more detail.