# Neural Network-Based Cost-Effective Estimation of Useful Variables to Improve Wind Turbine Control
## 3 Neural Network Based Estimators
### NN-Based Estimators  
An NN can be described as a model with a layered structure that resembles the human brain, including layers of connected nodes [23]. NNs are inspired by how the human brain works and can include many processing layers with simple elements running in parallel. Typically, an NN contains an input layer, one or more hidden layers and an output layer (Figure 5). The layers are combined through neurons or nodes, and the output of each layer becomes the input to the next layers. Recently, NNs have been applied to various tasks. In this study, we employ an NN as a pattern recognition method to identify a highly nonlinear model. The double-layer perceptron (DLP) feedforward NN shown in Figure 5 is adopted in this study. Although many different types of NN are available, the DLP feedforward structure is one of the most popular structures. The NN structure in the figure describes the following generic nonlinear model [24]:  
\[y(k)=h(u(k-\tau),\ldots,u(k-n_{b}),y(k-1),\ldots,y(k-n_{a})), \tag{10}\]  
where \(\tau\) is the time delay, \(n_{a}\) and \(n_{b}\) correspond to the order of the model, and the nonlinear function \(y:\mathbb{R}^{n_{a}+n_{b}-\tau+1}\rightarrow\mathbb{R}\) represents the nonlinear model. This NN includes \(n_{a}+n_{b}-\tau+1\) inputs, a single output \(y(k)\), \(K\) nonlinear hidden nodes represented by \(\gamma:\mathbb{R}\rightarrow\mathbb{R}\)  
Figure 5: Structure of DLP feedforward NN. and a single linear output element (corresponding to the sum block shown in Figure 5). In addition, the following bipolar sigmoid transfer function is used in this network:  
\[\gamma(x)=\tanh(\alpha x), \tag{11}\]  
where parameter \(\alpha\) is typically set to \(1\). The weights of the first layer are denoted by \(w_{i,j}^{1}\), where \(i=i,\ldots,K,j=0,\ldots,n_{a}+n_{b}-\tau+1\), and the weights of the second layer are denoted by \(w_{i}^{2}\), where \(i=0,\ldots,K\). The output signal is expressed as follows [25]:  
\[y(k)=w_{0}^{2}+\sum_{i=1}^{K}w_{i}^{2}\gamma(z_{i}(k)), \tag{12}\]  
where the sum of input signals of the \(i^{th}\) hidden node is  
\[z_{i}(k)= w_{i,0}^{1}+w_{1,1}^{1}u(k-\tau)+\ldots+w_{i,n_{b}-\tau+1}^{1}u(k-n_ {b})\] \[+w_{i,n_{b}-\tau+2}^{1}y(k-1)+\ldots+w_{i,n_{a}+n_{b}-\tau+1}^{1}y( k-n_{a}). \tag{13}\]  
Subsequently, the NN is trained by minimising the following index:  
\[\sigma=\sum_{k=1}^{n_{p}}(\tilde{y}(k)-y(k))^{2}, \tag{14}\]  
where \(\tilde{y}(k)\) is the output of the NN, and \(y(t)\) denotes the output measurements used to train the NN. Here, the Levenberg-Marquardt [26] algorithm, also referred to as the damped least-squares method, is employed as the nonlinear gradient optimisation algorithm. It uses the Jacobian for calculations, which assumes that the performance is a mean or sum of squared errors. For training the NN, data sets are broken down into three: 70% for training, 20% for testing and 10% for validation. Readers are referred to the literature [1,2,24] for additional details. The procedure used to design an NN-based estimator is summarised as follows:  
1. Turbine 1 (out of five in this paper) is equipped with a sensor, and each remaining turbine (i.e., Turbines 2 to 5) is equipped with a nonlinear NN-based estimator, as shown in Figure 4.
2. The inputs and outputs are measured and collected from Turbine 1 using the single sensor at regular intervals, that is, every 1000 s. Note that the output could be different variables but is limited to wind speed or aerodynamic torque in this paper. To estimate the aerodynamic torque, the inputs are a combination of generator torque, generator speed, fore-aft acceleration (FAA) and tower bending moment (TBM). The output is aerodynamic torque; therefore, a sensor is required to measure aerodynamic torque. To estimate the wind speed, the inputs are a combination of generator torque, TBM and FAA, and the output is wind speed. Therefore, a sensor (i.e., LiDAR) is required to measure the wind speed. Note that different combinations of generator torque, generator speed, TBM and FAA are tested. 3.