# The Optimal Dispatch of a Power System Containing Virtual Power Plants under Fog and Haze Weather
## 2 F-H Impacts on Photovoltaic Output and Load Forecasts
### Prediction Model of the Wavelet Neural Network  
There are various prediction models and algorithms suitable for photovoltaic output and load forecasts, such the artificial neural network (ANN) and its modified versions, the support vector machine, the extreme learning machine, _etc._ In this section, the WNN model that combines the ANN and the wavelets theory is selected to be the predictor because it is familiar to our team and it has the following advantages [18; 19; 20; 21; 22]:  
(1) The ANN has characteristics such as self-learning, self-adaption and fault tolerance, and it is a universal function approximator. (2) The motif and the whole structure of the WNN are set according to the theory of wavelet analysis, which could effectively extract the local information of the signal by multiresolution analysis and avoid the blindness of the structure design of the back propagation (BP) neural network. (3) The WNN model has better learning ability and higher accuracy. In the same prediction tasks, the WNN with a simpler network structure possesses faster convergence speed. The concepts of the ANN, the BP neural network and the wavelets theory are explained as follows to form the basic foundation framework of the WNN. (1) Artificial neural network:  
The ANN is a mathematical model that has the ability of parallel distributed information processing, based on the representation of neural activity in the human brain [18]. The most popular design for the ANN is the multi-layer feed forward network, which has an input layer, an output layer and one or more hidden layers. Each layer has a number of nodes or neurons, which consist of multiple inputs and a single output. A weight is associated with each input, while the input signal is multiplied by these weights. The neuron is responsible for the combination of these weighted inputs, and in accordance with an activation function, the output is determined. (2) Back propagation neural network:  
The BP neural network is a feed forward neural network, the signals of which are transmitted forward while the errors in reverse. Its basic learning rule is the steepest descent method, which minimizes the sum of squared errors according to the back-propagation to continuously adjust the weights and thresholds of the network [19]. (3) Wavelets theory:  
The popularity of wavelets with compact supports is due mainly to their relation to the dyadic multiresolution analysis that dominates wavelet research [20]. Historically, the continuous wavelet transform came first, which is defined as below:  
\[f_{r}(a,b)=\frac{1}{\sqrt{a}}\int_{-\infty}^{+\infty}x(t)\phi\left(\frac{t-b}{ a}\right)dt,a>0 \tag{6}\]  
where \(\phi(a,b)\) is a mother wavelet from which a family of wavelet daughters is obtained by scaling and translating it, thus by changing \(b\) (controlling the location) and \(a\) (determining the width of the wavelet and the frequency resolution). (4) Wavelet neural network:  
The WNN, based on the topology of the BP neural network, selects a certain wavelet basis function as the transfer function of nodes in the hidden layer, where the signal propagates forward while the error returns [21; 22]. The basic structure of the WNN is shown in Figure 6.  
Based on the strict theoretical background of wavelet analysis and the BP neural network, the WNN is able to adjust the weight and threshold values of the network and the scalability factors and  
Figure 6: The basic structure of the wavelet neural network (WNN) prediction model. shift factors of the wavelet function, which gives the WNN more sensitive approximation capability and better fault-tolerant capability than the BP neural network. As shown in Figure 6, \(X_{1}\), \(X_{2}\),..., \(X_{n}\) are inputs of the WNN; \(H_{1}\), \(H_{2}\),..., \(H_{l}\) are outputs of the nodes of the hidden layer; \(Y_{1}\), \(Y_{2}\),..., \(Y_{m}\), are outputs of the forecast; and \(n\), \(l\) and \(m\) are the node numbers of the input layer, hidden layer and output layer.