## 2 Gas Turbine
### Gas Turbine Model
#### 2.2.1 Physical Model
The input and output signals are generated per unit (p.u), where the operation signal is divided by the rotor speed nominal signal, N, for standardization. However, the temperature signal unit remains. The model consists of three main control components. The first component is the speed governor. It governs the speed of a system and maneuvers the frequency, exhaust temperature and compressor output as necessary as demanded from the load. The second component is fuel temperature control. It regulates the output temperature, \(T_{M}\), to be lower than the constant maximum or increased for more energy when the demand increases. The third component is the Inlet Guide Vane (IGV) temperature control, which plays a major role in balancing the temperature by opening or closing the air intake. These three control functions are the inputs for the low-value selection, determining the least fuel control actions for the gas turbine operation. The inputs of the model are the load change and ambient temperature. The outputs are represented by the three function blocks, \(f_{1},f_{2}\) and \(f_{3}\). Function block \(f_{1}\) as in Equation (3) represents the exhaust temperature of the turbine by incorporating the fuel flow, \(W_{f}\), rated exhaust temperature, \(T_{R}\), \(IGV\) and rotor speed, \(N\). The parameter \(D\) and \(E\) in the equation is the unknown values, which can be obtained from the operating curves. \[f_{1}=T_{R}+DW_{f}+E(1-N)+3.5(MaxIGV-IGV) \tag{3}\]  
Figure 6: Brayton cycle temperature-entropy diagram. The turbine torque output of the gas turbine with the signals from the fuel flow and the rotor speed is obtained from \(f_{2}\) as in Equation (4). \[f_{2}=\frac{1.15(W_{f}-0.133)}{N} \tag{4}\]  
The additional block of \(f_{3}\) as expressed in Equation (5) represents the exhaust gas flow, which is commonly necessary for the heat recovery stages in the combined cycle. \[f_{3}=N\times{L_{igv}}^{0.257}\Big{[}\frac{288}{T_{a}+273}\Big{]} \tag{5}\]  
where \(T_{a}\) refers to the ambient temperature, \(L_{igv}\) represents the \(IGV\) output and \(N\) is the rotor speed signal in the model. The [PERSON]'s model application is further modified by the IEEE task force by splitting the model into the controls of the gas turbine (the airflow control loop, the temperature control loop and the fuel flow control loop) and the thermodynamics equation properties. The main comparison of IEEE to [PERSON]'s model is the torque and speed calculation as in Figure 8 and the control scheme remained the same. A fixed compression ratio in gas turbine operation is assumed in the derivation. In the structure, \(A\) control block is added as a nonlinear function of the thermodynamic properties, which schedules the airflow. The main equation for the model as depicted in the diagrams is expressed in Equation (6). The parameter needs to be solved by the Newton-Raphson method due to the non-linear nature of Equations (7) and (8). \[T_{R}=T_{f}\Big{[}1-\Big{(}1-\frac{1}{x}\Big{)}\eta_{T}\Big{]} \tag{6}\]  
where \(T_{R}\) is the reference exhaust temperature, \(x\) is the cycle pressure ratio and \(\eta_{T}\) represents the turbine's efficiency. \[x=[PR_{0}W]^{\frac{\gamma-1}{\gamma}} \tag{7}\]  
where \(PR_{0}\) refers to the design cycle pressure ratio, \(\gamma\) is the ratio of specific heat capacities, \(w\) represents the air flow and \(\eta_{C}\) is the compressor's efficiency. \[W=\frac{P_{G}K_{0}}{T_{f}(1-\frac{1}{x})\eta_{T}-T_{i}\frac{(x-1)}{\eta_{C}}} \tag{8}\]  
Figure 7: [PERSON]’s gas turbine dynamic model. \(P_{G}\) refers to the rated power output, \(T_{i}\) is the inlet temperature of the compressor, and \(K_{0}\) is the ratio of net power output and inlet heat capacity. The parameter of \(T_{f}\) equals the firing temperature and is expressed in Equation (9). Figure 7.0  
where \(T_{D}\) is for compressor discharge temperature, \(K_{2}\) is the design combustor temperature rise, and \(W_{f}\) is the fuel flow.