## 2 Artificial Neural Networks Topology
### NARX Neural Network  
NARX is a neural network with recurrent dynamic behavior [38; 39; 40] that has been effectively used for time series problems estimation. The main difference between MLP and NARX is that NARX  
Figure 2: The developed deep feed-forward (DFF) network topology.  
allows a weighted feedback connection between layers of neurons. This allows the NARX models to consider lagged (past period) values of variables, which make them suitable for time series analysis. Alternatively, although in the literature, there are many approaches for time series analysis, such as the autoregressive integrated moving average approach and the autoregressive moving average approach; these approaches cannot cope with nonlinear problems [41; 42]. NARX, however, can efficiently be used for modelling time series with non-linear behavior.  
The NARX neural network can be constructed into one of two architectures [31], the parallel architecture (namely close-loop architecture), and the series-parallel architecture (namely, open-loop architecture), see Figure 3.  
This study uses the open-loop architecture. This is because of the pure feed-forward network architecture and the past accurate values availability of the time series. The NARX network behavior, in compliance with the time series forecast, can be mathematically modeled by the equation shown in Equation (4) [31]:  
\[y(t)=f(y(t-1),y(t-2),\ldots y(t-n_{y}),x(t-1),x(t-2),\ldots,x(t-n_{x})) \tag{4}\]  
This equation defines in what way a NARX network is used to forecast a data series \(y(t)\) value. The equation makes use of the previous values of the \(y\) series and another external series \(x\). The \(f()\) unknown function is a mapping function, and it is the purpose of the training phase of the network to find an approximation of this mapping function. This is done through an optimization process of the neuron bias and the network weights.  
As mentioned above, the NARX is a model with non-linear behavior that has the ability to approximate forthcoming values of time series using its former calculated values and some exterior data. In the presented work, NARX is used to estimate the values of the optimum operating voltage and current. As inputs, the NARX uses four inputs with exterior data at time \(t-1\). In addition, the NARX uses two time-series inputs that represent its previous outputs of the estimated optimum operating voltage and current at time \(t-1\). Figure 4 shows the developed NARX network discussed in this work.  
As illustrated in Figure 4, the first layer has six nodes of inputs. These are the panel temperature, the solar radiation, the panel OCV, the panel SCC, and the two output values previously attained from the time series. In the middle (hidden) layer there are five nodes with hyperbolic tangent sigmoid transfer activation function. The last layer consists of two nodes, which calculate the two outputs of the network. These are the optimum operating current and voltage. A linear activation function is used in the nodes of the last layer.  
Figure 3: The two neural network architectures of nonlinear autoregressive network with exogenous inputs (NARX).