# Deep Learning Algorithms to Predict Output Electrical Power of an Industrial Steam Turbine
## 3 Materials and Methods
### Theoretical Interpretation of Correlation Results  
As seen in the above matrix, the parameter << Steam Mass flow rate \(\mathbf{Q_{ms}}\)>> had the highest dependency relationship with the target parameter (\(\mathbf{P_{elec}}\)).  
We demonstrated theoretically that this strong dependence was justified by a physical link between these two parameters, as seen in Figure 3.  
By applying the first law of thermodynamics [20] to the HP steam turbine following the Rankine cycle model, the internal energy is expressed as follows:  
\[\Delta\mathbf{H_{HP}}+\Delta\mathbf{E_{K}}+\Delta\mathbf{E_{P}}=\mathbf{Q}+ \mathbf{W} \tag{6}\]  
where  
\(\Delta\mathbf{H_{HP}}\) is the variation in enthalpy (kJ).  
Figure 3: Simplified diagram showing the most correlated feature parameter of electrical power.  
Figure 2: Correlation matrix of studied parameters.  
\(\Delta\mathbf{E_{K}}\) is the variation in kinetic energy (kJ).  
\(\Delta\mathbf{E_{P}}\) is the variation in potential energy (kJ).  
\(\mathbf{Q}\) is the calorific energy (kJ).  
\(\mathbf{W}\) is the work carried out on the system (kJ).  
By considering the following hypotheses of the Rankine cycle:  
* the transformation is adiabatic (\(\mathbf{Q}=0\));
* the linear velocity is constant (\(\Delta\mathbf{E_{K}}=0\));
* the altitude is constant (\(\Delta\mathbf{E_{P}}=0\)).  
we obtain  
\[\Delta\mathbf{H_{HP}}=\mathbf{W} \tag{7}\]  
Likewise, by multiplying the work by the mass flow rate of the steam \(\mathbf{Q_{ms}}\), we obtain the useful power of the steam \(\mathbf{P_{stm}}\):  
\[\mathbf{P_{stm}}=\mathbf{W}.\mathbf{Q_{ms}}=\Delta\mathbf{H_{HP}}.\mathbf{Q_{ ms}} \tag{8}\]  
Then,  
\[\mathbf{P_{stm}}=\frac{\Delta\mathbf{H_{HP}}.\mathbf{Q_{ms}}}{\mathbf{m}} \tag{9}\]  
where  
\(\mathbf{P_{stm}}\) is the power of the steam (kW).  
\(\Delta\mathbf{H_{HP}}\) is the variation of enthalpy (kJ).  
\(\mathbf{Q_{ms}}\) is the mass flow rate of steam at collector output (t/h).  
\(\mathbf{Q_{ms}}\) is the mass of the steam (tons).  
Otherwise, we calculate the global efficiency of the steam turbine [21] coupled with the generator:  
\[\eta_{\mathbf{G}}=\eta_{\mathbf{tur}}.\eta_{\mathbf{gen}} \tag{10}\]  
\[\eta_{\mathbf{tur}}=\frac{\mathbf{P_{mec}}}{\mathbf{P_{stm}}+\Sigma\mathbf{P_{ L1}}} \tag{11}\]  
\[\eta_{\mathbf{gen}}=\frac{\mathbf{P_{elec}}}{\mathbf{P_{mec}}+\Sigma\mathbf{P_ {L2}}} \tag{12}\]  
where  
\(\eta_{\mathbf{G}}\) is the global efficiency of the turbine generator.  
\(\eta_{\mathbf{tur}}\) is the efficiency of the turbine.  
\(\eta_{\mathbf{gen}}\) is the efficiency of the generator.  
\(\Sigma\mathbf{P_{L1}}\) is the global losses at the level of the turbine (kW).  
\(\Sigma\mathbf{P_{L2}}\) is the global losses at the level of the generator (kW).  
Considering the fact that losses tend to zero, we obtain  
\[\mathbf{P_{stm}}=\frac{\mathbf{P_{elec}}}{\eta_{\mathbf{tur}}.\eta_{\mathbf{ gen}}} \tag{13}\]  
According to Equations (9) and (13), we deduce  
\[\mathbf{P_{elec}}=\mathbf{K}\cdot\mathbf{Q_{ms}} \tag{14}\]  
where K is a positive coefficient that is equal to  
\[\mathbf{K}\ =\frac{\Delta\mathbf{H_{HP}}\cdot\eta_{\mathbf{tur}}.\eta_{\mathbf{ gen}}}{\mathbf{m}} \tag{15}\]  
From Equation (14), we observe that there is a direct proportional relationship between the target variable (\(\mathbf{P_{elec}}\)) and its most correlated feature (\(\mathbf{Q_{ms}}\)).  
Thus, the correlation of 99% found automatically through Python is physically justified.