## 1 Introduction
Processing and calculation are complex [29; 30]; 2) traditional statistical models can address forecasting with linear trends well, but times-series wind speed is always random, chaotic, and nonlinear, and the performance of these models does not satisfy researchers and managers; 3) although artificial-intelligence forecasting methods have some advantages, due to their nonlinear mapping capability compared with traditional statistical models, their main disadvantages and shortcomings are that they are easy to reach local optimal values, and they suffer from overfitting and slow convergence [31]; 4) spatial-correlation methods should consider many influential factors, such as the relationship of the time-series wind-speed data between different sites, original data from several stations, and if the time-series wind speed of the forecasting point and its neighbors is used for forecasting [28]. To overcome the problem of forecasting accuracy and stability and take advantage of single models, a combined model based on multi-objective optimization and combined theory is proposed by us to overcome their defects, including low accuracy and weak stability. Because wind-speed forecasting is a complex study, single-objective optimization is not enough to solve the wind-speed forecasting problem. Solving the single-objective optimization problem (SOP) is a direct task, but multiple-objective problems (like the wind-speed forecasting problem) are often complex, and objective functions are competing (or conflicting) with each other. Due to this reason, there is no single optimal solution, commonly known as a series of Pareto-optimal solutions, which simultaneously optimize all objective functions. A set of optimal solutions, called a Pareto-optimal Set (PS), of which mapping in the target space is named the Pareto front (PF), and not a single optimal solution, is the goal of solving the multiple-objective optimization problem (MOP). In MOPs, the favored solution from the PS is chosen by researchers, but not the optimal solution. Obviously, it is difficult to select one objective function from a number of objective functions and ensure that the selected objective function can achieve a lower mean absolute percentage error (MAPE) and stronger stability. Due to these reasons, multiple-objective optimization can solve the low accuracy and weak stability of the wind-speed forecasting problem through simultaneously searching the goal of multiple functions. In this paper, a new combined model [32] that integrates nonpositive constraint theory [33], four branch models, including two nonlinear hybrid neural networks (Cuckoo Search-Back-PropagationNeural Network (CS-BPNN) and DE-Online Extreme Learning Machine (DE-OSELM)), two linear models (ARIMA and HW), and a nondominated sorting genetic algorithm (NSGA-III) [34], which optimizes the weights of the branch models, is proposed in this paper. Ten-minute time-series wind-speed data from three wind-farm sites were applied to examine our proposed model. We also chose three combined models with two objective functions to test its performance. The major contributions and innovations of this paper are as follows:  
1. According to the observed time-series wind-speed data, the trajectory matrix was constructed, decomposed, and reconstructed to extract signals representing different components of the original time series, such as long-term trend signals, periodic signals, and noise signals, so that the structure of the time series could be analyzed and used for further forecasting. 2. Our proposed combined model is based on MOP theory, which can obtain both accuracy and stability. The wind-speed forecasting problem is a MOP, so the theory overcomes the difficulty of selecting one objective function from multiple functions to obtain higher precision and stronger stability. 3. Due to the wind-speed data having both linear and nonlinear characteristics, linear models (ARIMA and HW) and nonlinear models (BPNN) and OSELM) were chosen to be the branch models of our proposed combined model.