Improved hybrid sparrow search algorithm for an extreme learning machine neural network for short‐term photovoltaic power prediction in 5G energy‐routing base stations

IET Renewable Power GenerationVolume 17, Issue 2 p. 336-348 ORIGINAL RESEARCHOpen Access Improved hybrid sparrow search algorithm for an extreme learning machine neural network for short-term photovoltaic power prediction in 5G energy-routing base stations Ming Yan, Ming Yan orcid.org/0000-0002-2862-0696 School of Electrical Engineering, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of China Guangxi Key Laboratory of Multimedia Communications and Network Technology, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of ChinaSearch for more papers by this authorWenhao Guo, Wenhao Guo School of Computer and Electronic Information, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of China Guangxi Key Laboratory of Multimedia Communications and Network Technology, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of ChinaSearch for more papers by this authorYongle Hu, Yongle Hu Runjian Co., Ltd, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of ChinaSearch for more papers by this authorFeng Xu, Feng Xu China Unicom Guangxi Branch, China Unicom, Qingxiu, Nanning, Guangxi, 530007 People's Republic of ChinaSearch for more papers by this authorJunjiang Chen, Junjiang Chen School of Computer and Electronic Information, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of China Guangxi Key Laboratory of Multimedia Communications and Network Technology, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of ChinaSearch for more papers by this authorQi Du, Qi Du School of Electrical Engineering, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of China Guangxi Key Laboratory of Multimedia Communications and Network Technology, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of ChinaSearch for more papers by this authorHanbo Zheng, Hanbo Zheng School of Electrical Engineering, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of ChinaSearch for more papers by this authorTuanfa Qin, Corresponding Author Tuanfa Qin [email protected] orcid.org/0000-0002-4014-0396 School of Computer and Electronic Information, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of China Guangxi Key Laboratory of Multimedia Communications and Network Technology, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of China Correspondence Tuanfa Qin, School of Computer and Electronic Information, Guangxi University, Xixiangtang, Nanning, People's Republic of China. Email: [email protected]Search for more papers by this author Ming Yan, Ming Yan orcid.org/0000-0002-2862-0696 School of Electrical Engineering, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of China Guangxi Key Laboratory of Multimedia Communications and Network Technology, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of ChinaSearch for more papers by this authorWenhao Guo, Wenhao Guo School of Computer and Electronic Information, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of China Guangxi Key Laboratory of Multimedia Communications and Network Technology, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of ChinaSearch for more papers by this authorYongle Hu, Yongle Hu Runjian Co., Ltd, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of ChinaSearch for more papers by this authorFeng Xu, Feng Xu China Unicom Guangxi Branch, China Unicom, Qingxiu, Nanning, Guangxi, 530007 People's Republic of ChinaSearch for more papers by this authorJunjiang Chen, Junjiang Chen School of Computer and Electronic Information, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of China Guangxi Key Laboratory of Multimedia Communications and Network Technology, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of ChinaSearch for more papers by this authorQi Du, Qi Du School of Electrical Engineering, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of China Guangxi Key Laboratory of Multimedia Communications and Network Technology, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of ChinaSearch for more papers by this authorHanbo Zheng, Hanbo Zheng School of Electrical Engineering, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of ChinaSearch for more papers by this authorTuanfa Qin, Corresponding Author Tuanfa Qin [email protected] orcid.org/0000-0002-4014-0396 School of Computer and Electronic Information, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of China Guangxi Key Laboratory of Multimedia Communications and Network Technology, Guangxi University, Xixiangtang, Nanning, Guangxi, 530004 People's Republic of China Correspondence Tuanfa Qin, School of Computer and Electronic Information, Guangxi University, Xixiangtang, Nanning, People's Republic of China. Email: [email protected]Search for more papers by this author First published: 26 September 2022 https://doi.org/10.1049/rpg2.12600AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Given the advancements in solar power generation and fifth-generation (5G) technologies, it is crucial to reduce energy consumption based on accurate predictions of the photovoltaic power requirements of 5G base stations (BSs) connected to renewable energy sources. To ensure sufficient prediction performance even under complex weather conditions and promote two-way flow of energy and information in the power supply system of the 5G BS, this work first proposes a reference scenario for a 5G BS with an energy router. Then, the comprehensive grey relational analysis is used to determine the optimal photovoltaic input variables. Finally, the improved logistic distribution, Laplacian distribution with inverse incomplete gamma-function weight factor, and nonlinear mutation perturbation strategy are introduced to improve the ability of the sparrow search algorithm to avoid locally optimal solutions for an extreme learning machine of the short-term photovoltaic power prediction model. The proposed model is applied to actual measured data for validation, and the results show that this model has the lowest mean percentage error as well as highest prediction accuracy under different weather conditions. The coefficient of determination for the proposed prediction model is greater than 99%; therefore, it is expected to improve the photovoltaic prediction performances of 5G energy-routing BSs as well as promote sustainable development and sharing of energy and information. 1 INTRODUCTION Over the last decade, mobile data traffic has grown rapidly owing to the popularisation of fifth-generation (5G) communication technologies and exploration of sixth-generation (6G) communication technologies [1]. Meanwhile, the reality is that cellular network operators incur significant energy consumption, high energy costs, and increased carbon emissions. Compared with the current fourth-generation (4G) networks, 5G networks require a higher number and density of base stations (BSs), whose energy consumption will be correspondingly three to five times greater than those of the 4G BSs [2]. Energy routers are regarded as the primary network units for implementing energy internet [3-5]; therefore, energy routers are expected to become core components of future microgrids and 5G BS power supply systems [6]. Previous research [7] has demonstrated the advantages of energy routers with a focus on traditional power grids, which cannot meet the needs of microgrids involving multiple distributed energy sources. In another work [8], the authors proposed flexible scheduling of the microgrid through the energy router, and an algorithm was used to achieve maximum benefit from the overall microgrid system. However, 5G BS power supply systems with access to a variety of distributed energy sources are different from the current microgrids. The load type in the 5G BS is relatively fixed, the energy consumption has obvious regularity, and the possibility of power failure should be avoided as much as possible. Previous studies have not considered the use of energy routers for energy conservation in BSs. In the process of searching for a solution to the energy consumption problem, solar energy has been gradually favoured by several countries owing to its minimal environmental impact and low cost of power generation [9]. However, because of the uncertain and intermittent characteristics of photovoltaic (PV) output power, accurate predictions of the short-term PV power requirements of 5G BSs remain a technical challenge [10, 11]. On the one hand, to increase the accuracy of short-term PV power predictions, input variables with strong correlations should be selected. In ref. [12], the authors used the grey correlation analysis to determine the importance of each input variable for PV power prediction; however, ignoring the degree of numerical similarity will affect these correlations to a certain extent. On the other hand, as a machine learning algorithm widely used with nonlinear fitting models, an extreme learning machine (ELM) has certain advantages for short-term PV power prediction [13, 14]. Jing et al. [15] considered the uncertainty of historical data and obtained a more accurate prediction ability compared with other neural networks by optimising the ELM network architecture. However, the random selection of ELM parameters often leads to the results falling into local optima, which ultimately affects the prediction performance. In recent years, intelligent algorithms have often been used as a strategy to optimise machine learning algorithms. For instance, the differential evolution algorithm is a classical algorithm to simulate biological evolution. Ni et al. [16] analysed a PV power prediction model based on an improved differential evolution algorithm, and the model was optimised with an ELM network to obtain favourable results; however, this approach required high computational complexity. Particle swarm optimisation (PSO) is a classical algorithm to simulate the predation behaviours of birds and has higher efficiency as well as fewer parameters that require adjusting. In ref. [17], a PSO approach was used to optimise the key parameters of the least-squares support vector machine and overcome the sensitivity to the initial values; the advantages of this method were also proved by comparisons. The slime mould algorithm (SMA) is a simple optimisation method based on the trophic growth processes of slime mould. In ref. [18], the SMA was used to optimise the random parameters of the ELM network, which optimised the recognition rate of the model to a certain extent. The harris hawks optimisation (HHO) is a more complex meta-heuristic algorithm inspired by the preying habits of the Harris eagle on rabbits. Setiawan et al. [19] found that the HHO can optimise the kernel function of support vector regression (SVR); their results showed that the optimisation capability of SVR was enhanced after HHO optimisation but that the time efficiency was reduced. A large number of other newly proposed meta-heuristic algorithms have been studied and widely used in the field of machine learning, such as the weighted mean of vectors algorithm (INFO) [20], monarch butterfly optimisation (MBO) [21], pelican optimisation algorithm (POA) [22], moth search algorithm (MSA) [23], arithmetic optimisation algorithm (AOA) [24], colony predation algorithm (CPA) [25], wild horse optimiser (WHO) [26], and sparrow search algorithm (SSA). Among these, the SSA was proposed by literature Xue and Shen [27] in 2020. Li et al. [28] showed that the SSA could be used for parameter optimisation of long short-term memory (LSTM) and convolutional neural networks (CNNs). Compared with the single neural network model, the convergence speed of the SSA is higher and prediction effects are better. Jia et al. [29] studied the optimisation of gate recurrent unit (GRU) using the SSA and advanced data processing; the proposed hybrid model was shown to have better adaptability and accuracy. Inspired by the above works, the present study offers the following important contributions. (1) Based on the existing energy router technology, an overarching framework is developed for 5G energy routing BSs to realise real-time information sharing, energy distribution, and energy utilisation. (2) The optimal input variables for an ELM PV prediction model are determined using the comprehensive grey relational analysis, which strengthens the correlation between the predicted and actual power outputs. (3) An improved sparrow search algorithm (ISSA) is proposed to avoid locally optimal solutions. The proposed ISSA can improve the overall performance of the prediction model under different weather conditions by optimising the random parameters of the ELM network. The remainder of this paper is organised as follows. Section 2 describes the reference scenario for predicting the PV output power. Section 3 presents the prediction model and optimised algorithm, while Section 4 summarises the simulation results and analyses the PV output predictions. Section 5 then provides the conclusions of this study and presents some future prospects. 2 REFERENCE SCENARIO Currently, 5G BS power supply systems are significant components of the energy internet; however, the energy internet is limited by the low efficiency of distributed energy, and BSs lack the necessary components to transform energy into other forms. Thus, the use of 5G BS power supply systems with large-scale renewable energy access can threaten the stability and reliability of the grid. Based on conventional power supply systems, we propose a power supply network based on the 5G energy-routing BS. The 5G BSs in different areas act as relatively independent non-synchronous operation units, and each unit is individually responsible for energy management within its region. An energy router typically has the functions of state monitoring, access and interconnection, and safety management for different energy systems and loads based on the flow of energy and information. At present, the power supply systems in 5G BSs are mainly powered by the grid, and the energy flow is unidirectional. If a failure leads to a lengthy power outage, the BS equipment are inoperable, leading to significant losses. With the introduction of energy routers in 5G BS power supply systems, the large-scale utilisation of renewable energy and efficient transmission of power and information can be achieved more effectively. Figure 1 shows a power supply network model consisting of 5G energy-routing BSs and an energy management and trading system. The energy-routing BSs are classified into central and micro energy-routing BSs. Each central energy-routing BS is equipped with an active antenna processing unit (AAU), a baseband unit (BBU), an energy storage device, an energy router, a variety of distributed energy generation devices, and other loads; each micro energy-routing BS only contains an AAU, lithium batteries, and other energy storage devices. The central energy-routing BS provides DC power to the micro energy-routing BSs. Conventionally, because the BS equipment, particularly the BBU, cannot operate effectively at high temperatures, the BS generally requires air conditioning for cooling, thereby adding to the energy consumption. The proposed power supply network can reduce the demand for air conditioning and hence the power consumption of the BS [30]. Finally, the energy-routing BSs and their energy management and trading systems transfer information via 5G communication, thus realising real-time information sharing as well as energy distribution and utilisation between the BSs [31]. FIGURE 1Open in figure viewerPowerPoint Power supply network model consisting of 5G energy-routing base stations (BSs) As a crucial component of the energy-routing BS, an energy router consists of an energy management module, a communication interface, a power electronics conversion module, and several distributed power supply plug-and-play interfaces. The distributed generation equipment, energy storage devices, and grid interfaces are attached to the DC bus (intermediate link for energy transmission) using a variety of power electronics converters. The use of renewable energies such as solar energy can meet the energy demands of BSs as much as possible, and energy-storage devices such as batteries and charging piles can ensure stability of the power supply. The 5G communication interface is used for information sharing with the other energy-routing BSs and energy-routing centres. Current 5G technologies does not suffer from significant delays in transmitting information, thus making them suitable for a BS power supply system [32]. In addition, when the distributed energy supply from a 5G energy-routing BS can completely meets the load needs, the energy router can transfer any excess energy to other BSs. 3 PREDICTION MODEL Accurate energy production predictions are vital for stable operation of 5G BS power supply systems. This section describes a PV power prediction model based on the modified SSA to improve the PV power prediction accuracies and energy allocation efficiencies for BSs within the power supply system. Parameter selection The power generated at the 5G energy routing BS is greatly affected by the characteristics of the components, panel installation angle, altitude, and weather, and may therefore be inconsistent and/or intermittent. Owing to the complexity of the BS-based power supply system, it is difficult to consider all of the performance parameters in practical engineering applications. However, given that power generation is a highly autocorrelated time series, it may be more accurate and convenient to combine statistical variables (i.e. historical power generation data from PV power stations) with physical variables (i.e. external meteorological factors) to predict PV power generation. In this study, the comprehensive grey relational analysis is adopted to compare the correlations between different PV power prediction input variables by combining the geometric similarities and distance proximities of the feature vectors. First, the difference correlation matrix for the comparison and reference sequences is defined as follows: Δ i ( k ) = s 0 ( k ) − s i ( k ) ( i = 1 , 2 , … , m ; k = 1 , 2 , … n ) , $$\begin{align} {\Delta _i}(k) = {\left| {{s_0}(k) - {s_i}(k)} \right|}(i = 1,2,\ldots,m;k = 1,2,\ldots n), \end{align}$$ (1)where s i $s_i$ (k) represents the kth eigenvalue of the ith comparison sequence, and s0(k) is the eigenvalue of the reference sequence. The degree of geometric similarity is calculated using the grey correlation degree between the reference and comparison sequences: α 1 s 0 ( k ) , s i ( k ) = 1 e Δ i ( k ) . $$\begin{align} {\alpha _1}{\left({{s_0}(k),{s_i}(k)} \right)} = \frac{1}{{{e^{{\Delta _i}(k)}}}}. \end{align}$$ (2) The quotient correlation matrix and its distance correlation degree are then calculated, which is defined as the set of quotients of all the eigenvalues for the reference and comparison sequences: Δ i 0 ( k ) = s i ( k ) s 0 ( k ) $$\begin{align} \Delta _i^0(k) = \frac{{{s_i}(k)}}{{{s_0}(k)}} \end{align}$$ (3) The grey correlation degree for distance proximity between the reference and comparison sequences is as given in Equation (4): α 2 ( s 0 ( k ) , s i ( k ) ) = e − 1 − Δ i 0 ( k ) $$\begin{align} {\alpha _2}({s_0}(k),{s_i}(k)) = {e^{ - {\left| {1 - \Delta _i^0(k)} \right|}}} \end{align}$$ (4) Finally, the comprehensive grey correlation degree is calculated by combining Equations (2) and (4): α i = 1 m ∑ k = 1 m α 1 ( s 0 ( k ) , s i ( k ) ) · α 2 ( s 0 ( k ) , s i ( k ) ) $$\begin{align} {\alpha _i} = \frac{1}{m}\sum _{k = 1}^m {\sqrt {{\alpha _1}({s_0}(k),{s_i}(k)) \cdot {\alpha _2}({s_0}(k),{s_i}(k))} } \end{align}$$ (5) Two sets of historical PV power data for 5G BSs are selected and compiled in this study: one set from sunny days and the other from cloudy days. The comparison variables include illumination intensity, wind speed, temperature, and relative humidity, while the reference variable is the PV output power. The results of the comprehensive grey correlation analysis are presented in Table 1. TABLE 1. Comparison of the comprehensive grey correlation degree for input variables in PV power prediction Power generation data Meteorological factors Comprehensive grey correlation Sunny days Illumination intensity 0.9552 Temperature 0.5771 Relative humidity 0.3642 Wind speed 0.6157 Cloudy days Illumination intensity 0.9524 Temperature 0.6214 Relative humidity 0.4068 Wind speed 0.6914 Table 1 shows that the correlations of illumination intensity with PV output power for the two groups are above 0.95 and are the strongest correlations among the variables. In contrast, the weakest correlation is with respect to relative humidity on sunny days (0.37). The comprehensive grey correlations for temperature and wind speed are lower than those for radiation intensity but sufficiently large to be suitable for the model. Therefore, the use of illumination intensity, wind speed, and temperature as input variables can produce more accurate results. ELM neural network ELM neural network is a widely used type of network characterised by the random generation of hidden node parameters, leading to fast-training models. The ELM network used in the present study is structured in the form of a single hidden layer feedforward neural network (SLFN), as shown in Figure 2. FIGURE 2Open in figure viewerPowerPoint ELM neural network structure This network includes an input layer, a hidden layer, a context layer, and an output layer. The input weight is the weight between the input and hidden layers. The bias is the threshold for the hidden layer of neurons, whose hidden function can be either linear or nonlinear. The output weight is the weight between the hidden and output layers. The context layer is used to store information within or between the layers to ensure more obvious time-variant characteristics. The non-linear state-space equation for the ELM neural network is as follows: x ( k ) = f θ 1 x c ( k ) + θ 2 μ ( k ) y ( k ) = g [ θ 3 x ( k ) ] f ( k ) = 2 1 − e − 2 k − 1 x c ( k ) = x ( k − 1 ) , , $$\begin{equation} {\left\lbrace { \def\eqcellsep{&}\begin{array}{*{20}{l}}x(k) = f{\left[ {{\theta ^1}{x_c}(k) + {\theta ^2}\mu (k)} \right]}\\[6pt] {y(k) = g[{\theta ^3}x(k)]}\\[8pt] {f(k) = \displaystyle\frac{2}{{1 - {e^{ - 2k}}}} - 1}\\[10pt] {{x_c}(k) = x(k - 1)}, \end{array} } \right.}, \end{equation}$$ (6)where x and x c $x_c$ are the hidden layer node element and feedback state vectors of the n-dimensional nodes, respectively; u is the r-dimensional input vector, k is the current state moment, and y is the node vector for the output layer with dimension n; θ1, θ2, and θ3 are the weight matrices for the connections between the context and hidden layers, input and hidden layers, as well as hidden and output layers, respectively; f(k) is the hidden layer transfer function, and g() is the output layer transfer function. The ELM neural network is trained to solve for the hidden layer, that is, the intermediate layers, nodes, weights, and other information. The ELM neural network is trained using an optimised gradient descent algorithm. In addition, the weight of the input layer and bias of the hidden layer determine the output layer matrix of the ELM network. The parameter selection for the ELM algorithm is relatively simple and does not require adjustment during process. The algorithm overcomes the disadvantages of the traditional gradient descent methods, such as falling into a local minimum, slow model training, and sensitive selection of the learning rate. Therefore, it is more suitable for short-term PV power predictions. However, the weight of the input layer and bias of the hidden layer in the standard ELM network are given randomly, which can easily cause local minimum values of the results and eventually induce errors in the output weights. Therefore, to address this drawback, an ISSA is proposed herein to optimise the processing. Modified sparrow search algorithm In the SSA, each sparrow has one location attribute, which is used to mark the location of its current search. The sparrows in the population are given one of three roles: predator, entrant, or scout. The sparrow population approaches the optimal solution through iteration, which is the position closest to the food. In general, this algorithm has a stronger ability for local search and higher convergence speed than other algorithms in dealing with unimodal test functions [27]. However, because the sparrows in the SSA directly move to the currently optimal solution, the algorithm has weak global search ability and can easily be trapped into local optima, unlike other classical algorithms such as the PSO. Therefore, several modifications are adopted in the present study to avoid the results of the SSA from falling into local optima 3.3.1 Optimising the population distribution via logistic mapping The sparrow population initialisation has an important influence on the optimal solution. Random generation is the default initialisation method, for which the initial population can be poorly distributed. However, when the initial sparrow population is initialised using chaos mapping, a more effective spatial distribution can be achieved. Chaos mapping has been developed from nonlinear systems and has become an effective optimisation tool having the characteristics of randomness, ergodicity, and sensitivity to the initial values. The logistic chaos mapping approach can be summarised as follows: z m = u × z m × [ 1 − z m ] $$\begin{align} {z_m} = u \times {z_m} \times [1 - {z_m}] \end{align}$$ (7)where z m $ z_{m}$ is the mth chaotic variable, and the system shifts from nonlinear to chaotic state when 3.56 ≤ μ ≤ 4 $ 3.56 \le \mu \le 4$ . When μ = 4 $ \mu = 4$ and z 0 ≠ { 0 , 0.25 , 0.5 , 0.75 , 1 } $ z_{0} \ne \lbrace 0,0.25,0.5,0.75,1\rbrace $ , the logistic chaos mapping can generate chaotic variables with values in the range of [0, 1] but primarily concentrated in [0, 0.1) and [0.9, 1]. However, the uniformity of the chaotic variables generated using the standard logistic chaos mapping is poor, so the ergodicity needs strengthening. By transferring the chaotic variables in the intervals [0, 0.1) and [0.9, 1] to other regions, we propose a logistic chaos mapping with improved ergodicity: z m + 1 n = 0.1 + 0.8 × g , { z m , z m ≤ 0.1 ∪ z m ≥ 0.9 } z m , e l s e $$\begin{equation} z_{{{m}} + 1}^n = {\left\lbrace { \def\eqcellsep{&}\begin{array}{*{20}{l}}0.1 + 0.8 \times g,\lbrace {z_m},{z_m} \le 0.1 \cup {z_m} \ge 0.9\rbrace \\[3pt] {{z_m}, {else}} \end{array} } \right.} \end{equation}$$ (8)where the parameter g is a random number subject to a normal distribution in the range of [0, 1]. The standard logistic chaos mapping and proposed improved approach are compared using 200 chaotic variables each, and the resulting ergodicity is presented in Figure 3. FIGURE 3Open in figure viewerPowerPoint Comparison of the chaotic variable distributions for the standard and modified logistic chaos mappings Figure 3 shows that the modified logistic chaos mapping approach evenly transfers the chaotic variables that are otherwise concentrated in [0, 0.1) and [0.9, 1] to other regions, leading to better ergodicity. 3.3.2 Fusion of a Laplacian distribution and an inverse incomplete γ function to adjust the weight factors We propose a weighted factor adjustment combining a Laplacian distribution with an inverse incomplete gamma function and introduce it to the predator position update operation of the original SSA. The improved predator position update equation for the ISSA is given by ω t = ω min + ω max − ω min m ∫ 0 m e − t t − t / T d t + δ L ( λ 1 , λ 2 ) x i , d t + 1 = x i , d t + ω t × ( f d , g t − x i , d t ) , R 2 < W V x i , d t + G , R 2 ≥ W V $$\begin{eqnarray} \omega _t &=& {\omega _{\min }} + \frac{{{\omega _{\max }} - {\omega _{\min }}}}{m}\int _0^m {{e^{ - t}}{t^{ - t/T}}\text{d}t} + \delta L({\lambda _1},{\lambda _2})\nonumber\\ x_{i,d}^{t + 1} &=& {\left\lbrace { \def\eqcellsep{&}\begin{array}{*{20}{l}}x_{i,d}^t + {\omega _t} \times (f_{d,g}^t - x_{i,d}^t),{R_2} < WV\\[4pt] {x_{i,d}^t + G,{R_2} \ge WV} \end{array} } \right.} \end{eqnarray}$$ (9)where ω t $ \omega _{t}$ is a fusion of the λ distributi