# Improved Probability Prediction Method Research for Photovoltaic Power Output
## 2 Single Improved Prediction Model
### Improved Sparse Gaussian Process Regression Model
#### 2.1.2 Improved Grey Wolf Optimization Algorithm  
Because the SPGP model is complicated and the concavity and convexity of the optimization problem formula (Equation (7) cannot be judged intuitively, traditional convex optimization methods may not be applicable to the solution of the hyper-parameter in the SPGP model. Moreover, because this method relies too heavily on initial values, it falls easily into local optimum. Therefore, this paper introduces an improved grey wolf optimization algorithm to optimize the hyper-parameter. Grey wolf optimization (GWO) is a heuristic population intelligent optimization algorithm proposed by [PERSON] in 2014 [33]. It has the advantages of simple principles, less parameters and a strong global search ability, but there is still much room for improvement in the algorithm.  
When dealing with the optimization problem, it is assumed that in the D-dimensional search space, the number of grey wolves is N, and the position of the \(i\)-th wolf is defined as \(X_{i}=(X_{i}^{1},\cdots,X_{i}^{D})\), where \(X_{i}^{k}\) indicates the position of the \(i\)-th wolf in the \(k\)-th dimension. This paper will improve the algorithm to obtain an improved grey wolf optimization (IMGWO) according to the following three aspects:  
(1) Opposition-based learning algorithm: The initial population is obtained by randomly generated methods in the GWO algorithm. However, the initial population will affect the search efficiency and the quality of the solution. Therefore, this paper adopts the method of opposition-based learning to generate a diverse initial population. Assuming that there is a number \(x\) in \([m,n]\), the opposite point of \(x\) is defined as \(x^{\prime}=m+n-x\).  
(2) Nonlinear convergence factor: A common problem facing optimization algorithms is how to balance a global search ability with a local search ability. In GWO, the convergence factor decreases linearly from 2 to 0 over the course of iterations. It cannot reflect the actual search process because the process is not linear. Therefore, in this paper, a new non-linear convergence factor is proposed: \(\overset{\rightarrow}{a}=2-2(k/K))^{n}\), where \(k\) indicates the current iteration, \(K\) is the maximum number of iterations, \(n\) is the attenuation order, and \(n\) is 3.  
(3) Dynamic weight updating: In the stage of position updating, GWO uses an equal weight method that ignores the different characteristics of alpha, beta, and delta. The wolves have different importance for the population, so we need different weights to reflect the different importance. In this paper, fitness values are used to describe the importance of wolves. Equation (6) is adopted to update location information:  
\[\overset{\rightarrow}{X}(t+1)= w_{a}\overset{\rightarrow}{X}_{1}+w_{\beta}\overset{ \rightarrow}{X}_{2}+w_{\sigma}\overset{\rightarrow}{X}_{3} \tag{6}\]  
where \(w_{i}=f(X_{i}(t))/(f(X_{\alpha}(t))+f(X_{\beta}(t))+f(X_{\delta}(t)))(i=\alpha, \beta,\delta)\) is the weight coefficient, \(f(X_{i}(t))\) is the fitness value, and \(\overset{\rightarrow}{X}_{i}