## 2 Materials and Methods
### The Proposed Power System
#### 2.1.3 Two Area Interconnected Power System  
A two area interconnected power system was combined with a connection link. Its block diagram, to be modelled by MATLAB-Simulink, is given in Figure 5, without a controller. In the figure, \(U_{1}\) and \(U_{2}\) are the control signals, \(\Delta P_{tic1}\) and \(\Delta P_{tic2}\) are the line power changes, and \(\Delta f_{1}\) and \(\Delta f_{2}\) are the input frequency deviations. A MATLAB-Simulink model of the two area interconnected renewable energy power system is shown in Figure 6, where, \(\Delta f_{i}\) is the frequency of the system, \(R_{i}\) is the regulation constant, \(\Delta P_{Li}\) is the load variations, \(\Delta P_{PVi}\) is the power variation depending on the sunlight of the PV cell, and \(U_{i}\) is the field controller output. The whole system with multi variable parameters is described in terms of Equations (9)-(12). Parameters of two-area power system with PV-SPP are shown in Appendix A.  
\[\dot{x}=Ax(t)+Bu(t)+Ld(t) \tag{9}\]  
\[x(t)=\big{[}\Delta f_{1}\big{(}\Delta P_{gt1}+\Delta P_{gt1}\big{)}\Delta P_{ 1\dot{u}1-2}\Delta f_{1}\big{(}\Delta P_{gt2}+\Delta P_{gt2}\big{)}\Delta P_{ v1}\big{]} \tag{10}\]  
\[u(t)=[u_{1}u_{2}] \tag{11}\]  
\[d(t)=[(P_{L1}+P_{PV1})(P_{L2}+P_{PV2})] \tag{12}\]  
where \(A\) is the system matrix; \(B\) and \(L\) are the input and disturbance matrices; and \(x(t)\), \(u(t)\), and \(d(t)\) are the state, control, and load change disturbance vectors, respectively. The system output depends on the area control error (\(ACE\)), shown in Figure 6. The \(ACE\) is symbolized by Equation (14). \[y(t)=\left[ \begin{array}{c}y_{1}(t)\\ y_{1}(t)\end{array} \right]=\left[ \begin{array}{c}ACE_{1}\\ ACE_{1}\end{array} \right]=Cx(t) \tag{13}\]  
Figure 4: A LFC applied a single area interconnected power system. Figure 5: Block diagram of an uncontrolled two area interconnected power system. \[ACE_{i}=\Delta P_{i{k}j}+b_{i}\Delta f_{i} \tag{14}\]  
where \(b_{i}\) is the frequency bias constant, \(\Delta f_{i}\) is the frequency deviation, \(\Delta P_{i{i}{e}1-2}\) is the change in the tie-line power, and \(C\) is the output matrix. \(b_{i}\) is the expression of the frequency change in the power unit. The transfer functions used for the modelling of a turbine and a generator are given as follows [10]. \[G_{G(s)} =\frac{1}{T_{G(s)}+1} \tag{15}\] \[G_{T(s)} =\frac{1}{T_{T(s)}+1} \tag{16}\]  
In the system, a RTU type turbine was employed for increasing the efficiency of these turbines. They are made up of several turbine blades, in which the steam is reenergized between the two turbines by intermediate heaters reheating the steam. They are usually operated in the large plants [24]. The RTU transfer function Equation (17) is explained as follows [8]:  
\[G_{R(s)}=\frac{\left(K_{R}\ \times\ T_{R}\right)_{(S)}+1}{T_{R(s)}+1} \tag{17}\]  
The frequency stability is defined as the ability to maintain a stable frequency after an imbalance between the generated and demanded power. The power balance fails when any difference occurs on the load side or on the generation side [26]. Here, \(P_{L1}\) represents the load variation of the first  
Figure 6: Two area multi source interconnected with PV power system grid in MATLAB-Simulink block diagram. area. When \(P_{\Delta 1}\) is non-zero, the system frequency changes. The frequency change is defined by the equations given below. \[P_{1}=P_{Gt_{1}}+P_{V1} \tag{18}\]  
\[P_{\Delta 1}=P_{1}+P_{L1} \tag{19}\]  
\[\Delta\omega_{1}=\frac{P_{\Delta 1}}{K_{S1}} \tag{20}\]  
where \(K_{s}\) is defined as the characteristic constant of the system frequency in the first area. Also, the transfer function expressing the load changes affecting the system frequency is clarified as follows [8]:  
Figure 5:0  
To identify the \(ACE\) value, Equation (22) is used. This gives the frequency changes when a load increase of \(P_{L1}\) in the first area occurs for two area interconnected power systems. Equation (23) expresses the power change in the tie line [27].