An improved whale optimization algorithm for forecasting water resources demand

Highlights

• We establish logarithmic and hybrid water resource predicting models.

• We modify whale optimization algorithm using social learning and wavelet mutation.

• We use improved algorithm to optimize the parameters of water predicting model.

• We use the CEC2017 test set to verify the effectiveness of the proposed algorithm.

• We test the model with 2013–2016 of Shaanxi, predict water demand for 2017–2021.

Abstract

Water demand forecasting can promote the rational use of water resources and alleviate the pressure on water demand. By analyzing the use of water resources, this paper establishes three models of water demand forecasting, logarithmic model, linear and exponential combination model and linear, exponential and logarithmic hybrid models. In order to accurately estimate the demand for water resources, an improved whale optimization algorithm based on social learning and wavelet mutation strategy is proposed. The new algorithm designs a new linear incremental probability, which increases the possibility of global search of the algorithm. Based on the social learning principle, the social ranking and social influence are used to construct the social network for the individual, and the adaptive neighborhood learning strategy based on the network relationship is established to achieve the exchange and sharing of information between groups. The Morlet wavelet mutation mechanism is integrated to realize the dynamic adjustment of the mutation space, which enhances the ability of the algorithm to escape from local optimization. The latest CEC2017 benchmark functions confirms the superiority of the proposed algorithm. The water consumption from 2004 to 2016 in Shaanxi Province of China is used for the experiment. The results show that the performance of the proposed algorithm for solving the three water resources forecasting model is better in comparison to other algorithms. The prediction accuracy is as high as 99.68%, which verified the validity of the model and the practicality of the proposed algorithm.

Introduction

Water is an indispensable material resource in the world, which plays an important role in industrial, agricultural development and human survival. With the annual growth of the population and the rapid development of society, the consumption of water increases sharply [1], and the shortage of water resources keeps emerging [2]. The effective use of water resources is the key to the sustainable development of water resources. And Water demand forecasting is the primary task of optimal allocation of water resources. Therefore, it is vital to study models and methods for accurately predicting water demand.

Accurately prediction of water used affected by a range of factors, including population, economic growth, and living standards, is a challenging task. At present, there are mainly three types of predictive models [3], namely: nonlinear models, statistical analysis models and grey prediction models [4]. Artificial neural network [5], support vector machine [6] is the representative of nonlinear model; time series analysis model [7], Markov chain model [8] is a typical statistical analysis model. These models have their own advantages and limitations, which affect the accuracy of predictions.

In order to more accurately predict the demand for water resources, many scholars have proposed a new water resource prediction model. Zhang et al. proposed a modified water demand estimation method for drought identification over arid and semiarid regions [9]. Ilic et al. estimated water circulation by using neural fuzzy algorithm [10]. For the water demand forecasting model, the value of parameters has an important impact on the forecast results. So, reasonable calculation of model parameters is the key to water demand forecasting.

The value of water resources prediction model parameters is a typical optimization problem. The traditional analysis-based solution method requires the objective function to have convexity and differentiability. And the specific problem is selected for the appropriate initial value, the effect is better. For the complex optimization problems that the initial value is difficult to select and the objective function is hard to meet the requirements, the stochastic optimization algorithm based on biological evolution mechanism emerged in recent years has become the mainstream algorithm for solving these optimization problems. The algorithm is swarm intelligence model, through the information exchange and sharing to implement the search space of exploration and development. It has the advantages of simple design, less control parameters and fast convergence speed. Meta-heuristic swarm intelligence algorithm for simulating animal foraging behavior and population survival rule is an important stochastic optimization algorithm. The particle swarm optimization algorithm (PSO) [11] which takes the social cognition and historical cognition of particles into behavior is a representative Meta-heuristic swarm intelligence algorithm. It opens a new way to solve optimization problems based on the evolutionary survival principle of animals. Since then, many scholars have proposed swarm intelligence algorithms with different mechanisms through the study of different animal populations, such as grey wolf optimization algorithm (GWO) [12], moth-flame optimization algorithm (MFO) [13], salp swarm algorithm (SSA) [14], ant lion optimizer (ALO) [15], firefly algorithm (FA) [16], teaching-learning-based optimization algorithm (TLB) [17], harris hawks optimization (HHO) [18].

Due to the excellent performance of intelligent algorithms in solving optimization problems, swarm intelligent optimization algorithm have been used to predict water demand in recent years. The Harmony Search Algorithm (HS) [19] is used to optimize the parameters of the ARIMA model based on historical water demand data to predict short-term water demand. Bai proposed a multi-scale urban water demand estimation method [1]. The method uses the adaptive chaotic particle swarm optimization (CPSO) algorithm to search the optimal weighting factor of the correlation vector regression model. Wang proposed a hybrid model based on linear and exponential models [20], using the firefly algorithm (FA) to solve the weight operator in the hybrid model.

The whale optimization algorithm (WOA) [21] is a new swarm intelligent optimization algorithm, which simulates the predation behavior of humpback whales. The local exploitation phase is realized by shrinking or spiraling mechanism, and the global exploration is achieved by random search strategy. The algorithm converges quickly, but it is easy to fall into local optimum. Many scholars improve the exploration and development capabilities of the algorithm. An improved whale optimization algorithm (IWOA) [22] improves the update mode of individuals through differential evolution strategy, thus enhances the global optimization ability of the algorithm. Chaotic whale optimization algorithm (CWOA) [23] uses chaotic evolution to calculate the probability of individual update, and improves the computational efficiency of the algorithm. The integration of Lévy flight trajectory-based whale optimization algorithm (LWOA) [24] into Levy flight trajectory enhances the diversity of population and prevents premature convergence. A modified whale optimization algorithm (MWOA) [25] advances the exploitation by quadratic interpolation strategy and improves the quality of solutions. Adaptive whale optimization algorithm (AWOA) [26] improves local optimization ability by introducing adaptive weight to enhance convergence accuracy of the algorithm. Whale optimization algorithm based on chaotic search strategy (WOAC) [27] makes use of the local exploitation and global exploration ability of the nonlinear mixed disturbance equilibrium algorithm with convergence factors and weights. Whale optimization algorithm based on cosine control factor and polynomial mutation (CPWOA) [28] incorporates the cosine control factor, which makes the algorithm slow down the convergence speed in the early iteration for full global exploration, and the polynomial mutation is used to enhance the ability of the algorithm to jump out of the local optimal solution. A hyper-heuristic algorithm is proposed to alleviate the WOA’s drawbacks by automatically choosing a chaotic map and a portion of the population using the differential evolution (DE) algorithm (DEWOA) [29]. Improved whale optimization algorithm introduces Gaussian mutation operator, differential evolution operator and crowding degree factor (WOAGDC) [30] to improve precision and computing speed. A–C parametric WOA (A–C WOA) [31] targets the A and C parameters of the standard WOA specifically through variation of “a” parameter non-linearly and randomly, as well as updating parameter “C” by applying inertia weight strategy. A hybrid metaheuristic algorithm (LAHCAWOA) [32] based on whale optimization algorithm and local search late acceptance hill climbing algorithm for feature weighting to improve the classification performance. In addition, WOA has achieved excellent results in some new fields, such as single mobile robot scheduling [33] and permutation flow shop scheduling problem [34], parameter optimization of the stochastic resonance system [35], closed-loop supply chain network design [36], gold price fluctuations forecast [37], multi-fault diagnosis of rolling element bearings [38], image segmentation [39], [40], quadratic assignment problem [41], data clustering [42] and classification [43].

Inspired by Wang [20], this paper proposes a hybrid water demand forecasting model based on linear, exponential and logarithmic models to predict the water consumption in Shaanxi Province of China. In order to estimate the optimal model parameters, an improved whale optimization algorithm based on adaptive social learning and wavelet mutation (named SMWOA) is proposed. Firstly, by analyzing the exploration probability of the WOA algorithm, this paper gives the reason for the weak exploration ability of the WOA algorithm, and a new linear incremental probability with the number of iterations is designed to improve the exploration ability of the algorithm. Secondly, the adaptive social learning strategy is used to construct the adaptive social network of whale individuals, and the transformation of whale individuals to optimal solution and adaptive social network neighborhood solution is realized according to the probability, which enhances the diversity of the population. Finally, the Morlet wavelet mutation strategy is adopted to realize the dynamic adjustment of the mutation space by utilizing the energy concentration and expansion attributes of the wavelet function, so as to enhance the ability of the algorithm to jump out of local optimization and improve the calculation accuracy of the algorithm. SMWOA algorithm better balances the exploitation and exploration of WOA algorithm, and the effectiveness of the SMWOA algorithm is verified by the challenging standard test set CEC2017 [44]. By optimizing the parameters in the hybrid water demand forecasting model of linear, exponential and logarithmic models and comparing with the results of other models and other improved whale optimization algorithms, the prediction accuracy of the SMWOA for solving the new model is up to 99.68%, which verifies the effectiveness of the proposed model and the excellence of the SMWOA.

The rest of this paper is organized as follows: Section 2 establishes three models of water demand forecasting in Shaanxi Province, China. Section 3 briefly describes the WOA. Section 4 introduces in detail the SMWOA for forecasting water resources demand and the time complexity analysis of the SMWOA. Section 5 by using two kinds of experiments verify the validity of the SMWOA and the new model, and finally summarizes the main findings of this paper and suggests directions for future research.