## 3 Mathematical Model of Power System with the HESS in the \(i\)th Control Area  
State space description can effectively reveal and utilize the inner state information of the system. By this way, the complexity of mathematical representation will not be increased when the numbers of variables increase. In this paper, the mathematical model of \(i\)th control area within the four-area interconnected power system is established in the form of state space equation. To facilitate modeling, the linear expression of the reheat steam turbine (Equation (2)) is decomposed as shown in Figure 5. In Figure 5, \(\Delta P_{gi}\) denotes the output power increment of power generation, \(\Delta P_{ri}\) is the output thermal power increment of reheat unit and \(\Delta X_{ei}\) indicates the incremental variation of governor valve position. Thinking about the \(i\)th control area in a multi-area interconnected power system, define the output \(y_{i}=ACE_{i}\), the input \(u_{i}=P_{ci}\), inner state-space vector \(x_{i}\in\mathbf{R}^{6\times 1}\) and the disturbance vector \(w_{i}\in\mathbf{R}^{2\times 1}\). The detailed expressions are as follows and all variables in state-vector \(x_{i}\) are explained in Section 2.1. \[ \begin{array}{c}x_{i}=\left[\begin{array}{cccc}x_{i1}&x_{i2}&x_{i3}&x_{i4} &x_{i5}&x_{i6}\end{array}\right]^{T}=\left[\begin{array}{cccc}\Delta f_{i}& \Delta P_{gi}&\Delta P_{ri}&\Delta X_{ei}&\Delta P_{tiei}&\int ACE_{i}d_{t} \end{array}\right]^{T}\\ \\ w_{i}=\left[\begin{array}{cccc}\Delta P_{ci}&2\pi\sum\limits_{i=1,j\
eq j}^{ n}T_{ij}\end{array}\right]^{T}\end{array} \]  
The state space model of power system in \(i\)th control area can be established as follows:  
\[\left\{ \begin{array}{c}\dot{x}_{i}=A_{i}x_{i}+B_{i}u_{i}+F_{i}w_{i}\\ y_{i}=C_{i}x_{i}\end{array} \right. \tag{6}\]  
where \(A_{i}\in\mathbf{R}^{6\times 6}\), \(B_{i}\in\mathbf{R}^{6\times 1}\), \(F_{i}\in\mathbf{R}^{6\times 2}\) and \(C_{i}\in\mathbf{R}^{1\times 6}\) are parameter matrices of system. Their expressions are as below:  
\[ \begin{array}{c}B_{i}=\left[\begin{array}{cccccc}-\frac{K_{pi}}{T_{pi}}&0& 0&0&0&0\end{array}\right]^{T}\\ \\ C_{i}=\left[\begin{array}{cccccc}-\frac{\dot{B}_{i}}{R_{i}}&0&0&0&0&0\end{array} \right]^{T}\\ \\ F_{i}=\left[\begin{array}{cccccc}-\frac{K_{pi}}{T_{pi}}&0&0&0&0&0\\ 0&0&0&0&-2\pi&0\end{array}\right]^{T}\end{array} \]  
Figure 5: Transfer function of reheat steam turbine in the \(i\)th control area. \[A_{i}=\left[ \begin{array}{cccccc}-\frac{1}{T_{pi}}&\frac{K_{pi}}{T_{pi}}&0&0&- \frac{K_{pi}}{T_{pi}}&0\\ 0&-\frac{1}{T_{ii}}&\frac{1}{T_{ii}}&0&0&0\\ -\frac{c}{T_{gi}}(\frac{1}{K_{i}}+K_{AFi})&0&-\frac{1}{T_{ni}}&\frac{1}{T_{ri}} -\frac{c}{T_{gi}}&0&-\frac{c}{T_{gi}}\\ -\frac{1}{T_{gi}}(\frac{1}{K_{i}}+K_{AFi})&0&0&-\frac{1}{T_{gi}}&0&-\frac{K_{i} }{T_{gi}}\\ 2\pi\sum\limits_{i=1,i+j}^{n}T_{ij}&0&0&0&0&0\\ \frac{\beta_{i}}{R_{i}}&0&0&0&0&0\end{array} \right]\]  
H\({}_{\infty}\) State Feedback Controller Design for Power System with the HESS in the \(i\)th Control Area  
Take the four-area interconnected power system as study object. The control block diagram designed for the \(i\)th control area is shown in Figure 6. \(P_{ci}^{*}\) is a given power instruction calculated by the H\({}_{\infty}\) controller to the HESS. \(P_{ci}\) denotes the power flow provided to power system by the HESS. \(y_{i}\) is the output of power system in the \(i\)th control area which denotes the area control error (ACE). \(x_{i1}\),..., \(x_{i6}\) (\(i=1,2,3,4\)) present six inner state quantities of the \(i\)th control area. The specific description and meaning of each state which have been given in Sections 2 and 3. Combining the mathematical model  
The controller coefficient matrix \(K_{i}\) is designed satisfying the stability condition of \((A_{i},B_{i},C_{2i})\). The matrix \(D_{12i}\) in Equation (7) is required to be column full rank to make sure that the controller \(K_{i}\) is a real rational function matrix.