# A Deep Learning Method for Short-Term Dynamic Positioning Load Forecasting in Maritime Microgrids
## 2 Review of Artificial Neural Networks
### Recurrent Dynamic Neural Network
Furthermore, \(Td_{x}\) and \(Td_{y}\) are integer numbers of the inputs and outputs of the NN memorial order, where they are made known the tapped delay lines (TDLs) in the RNN, respectively. Hence, the equation value of the output \(y(n,t)\) in (15) is preceded by the previous values of the time delay tapped outputs and inputs. To estimate the output of the feedforward NN- NARX model, a nonlinear activation function f(x) is applied. As a result, for the training process of DP load, the preceding values of the NARX output, and input, a feed-forward series-parallel architecture is used as shown in Figure 5. Consequently, the desired output signal \(\overset{\wedge}{y}(n,t)\) is defined in (16) based on previous values of the output signal and external input signal such as sea state as environment disturbances. Furthermore, the mathematical model of output of NARX is computed as:  
\[\overset{\wedge}{y}(n,t)=f_{out}^{\wedge}(\overset{\wedge}{W_{out}}(f_{ fluid}(W_{hid}(x(n,t-1),\ldots,x(n,t-Td_{x}),y(n,t-1),\ldots,y(n,t-Td_{n}))+b_{hid}))+b_{ out}) \tag{16}\]  
where \(\overset{\wedge}{y}(n,t)\), \(b_{out}\), \(b_{hid}\), \(\overset{\wedge}{W_{out}}\) and \(f_{out}^{\wedge}\) are the predicted value of the outputs, bias vectors, hidden bias, output matrix of weights, and activation functions of the output layers, respectively. Additionally, \(W_{hid}\) and \(f_{hid}\) are the matrix of weights and activation functions of the hidden layers, correspondingly. Typically, the NN feedforward learning algorithm based on LM-BP is formulated as [68]:  
\[y_{n}(w_{ij}+dw_{ij})=y_{n}(w_{ij})+\
abla y_{n}(w_{ij})^{T}dw_{ij}+\frac{1}{2 }*dw_{ij}^{T}\
abla^{2}y_{n}(w_{ij})dw_{ij} \tag{17}\]  
In (17), \(\
abla y_{n}(w_{ij})^{T}\) and \(\
abla^{2}y_{n}(w_{ij})\) are the vector of gradient and the error function of the hessian matrix, hence the \(dw_{ij}\) is defined as:  
\[dw_{ij}=-[\
abla^{2}y_{n}(w_{ij})]^{-1}\
abla y_{n}(w_{ij}) \tag{18}\]where \(\
abla^{2}y_{n}(w_{ij})=J^{T}J+S\). The \(J\) is specified as the Jacobian matrix, which includes the initial network derived signal error, which is related to adjust the weights of the neuron. By considering the directional propagation algorithm, (17) is computed in (17), where \(S\) expresses the second-order derivative function \(\
abla^{2}y_{n}(w_{ij})\). By ignoring the \(S\) in (17), the equation is solved by the Gauss-Newton method. We can also obtain results for the hessian matrix by the following approximation in (19). \[\
abla^{2}y_{n}(w_{ij})=J^{T}J+\mu I \tag{19}\]  
To define the size of the \(\mu\) coefficient of \(I\) in (19) as the unit specify matrix, a scalar quantity is applied. Hence, The LM-BP algorithm is used to solve the BP scaling problem, which frequently tends to minimize the converge of the local error in preference to the global error. Moreover, BP has appropriate performance on the simple training problem; on the other hand, the BP performance will be decreased, by increasing the data and data-sizing complexity, therefore BP cannot perform discontinuously. For DP load predicting, we have faced in the environmental disturbance similar problem.