## 2 System Model of Power Grid  
### Dynamic Model of Power Grid  
The electric power network consists of \(n\) interconnected power subsystems as shown in Figure 1. It assumes that all power subsystems are same. Each power subsystem consists of a distributed energy source and load, including gas turbine generators, wind power generations and battery arrays [34; 35] in the system. These power generating machines supply electric power to meet the demands. Gas turbine, wind power, and battery power output are controllable. Mass loads are considered in the dynamic model of the power grid. Battery electric storage systems and wind power systems, which are connected to the power net by power electronic interface, are controllable. In mathematics, the frequency control method is equivalent to the tie-line bias control (TBC) method as a frequency control in electrical power systems in consideration of tie-line frequency [36; 37; 38; 39]. For each subsystem, the block diagram is shown in Figure 2. The meaning of each parameter in the block diagram is given as follows. * \(K_{i}\) and \(B_{i}\) are TBC gain and frequency bias, respectively. * \(T_{gi}\) and \(T_{di}\) are the governor and gas turbine constant, respectively. * \(M_{i}\) and \(D\) are the inertia and damping constant, respectively. * \(R_{gi}\) and \(T_{ij}\) are the regulation and synchronizing constant, respectively. * \(\Delta x_{gi}\) is a governor input of a gas turbine generator. There are six power notations as follows:  
* \(\Delta P_{gi}\) is an output of the gas turbine generator. * \(\Delta P_{Wi}\) is an output of wind power generation. * \(\Delta P_{Li}\) is the load fluctuation except controllable load. * \(\Delta P_{Bi}\) is an output of the battery electric storage system. * \(\Delta P_{ji}\) and \(\Delta P_{ij}\) is the tie-line power low deviation. * \(\Delta P_{ij}-\Delta P_{ji}\) is the output power of area \(i\), which is delivered to area \(j\). * \(\Delta P_{i}\) in Equation (1) shows the electric power generation of subsystem i and the supply error margin of power consumption. Figure 1: Power grid structure. Frequency deviation \(\Delta f_{i}\) can be calculated by the supply error margin shown in the block diagram. The power \(\Delta P_{i}\) for subsystem \(i\) is:  
\[\Delta P_{i}=\Delta P_{gi}+\Delta P_{Wi}+\Delta P_{Bi}-\Delta P_{Li}+\Delta P_{ ij}-\Delta P_{ji} \tag{1}\]  
In a mathematical form, for subsystem \(i\), if the set of neighbored subsystem is denoted as \(D_{i}\), the dynamics of each subsystem can be described by using continuous time-state equation:  
\[\dot{x}_{i}=A_{i}x_{i}+B_{i}u_{i}+\sum_{j\in D_{i}}A_{ji}x_{j}+E_{i}w_{i} \tag{2}\]  
\[A_{i}=\left( \begin{array}{ccccc}0&-\sum_{j\in D_{i}}T_{ji}&0&0&0\\ 1/M_{i}&-D_{i}/M_{i}&1/M_{i}&0&0\\ 0&0&-1/T_{di}&1/T_{di}&0\\ 0&-1/(T_{gi}R_{gi})&0&1/T_{gi}&K_{i}/T_{gi}\\ 1&-B_{i}&0&0&0\end{array} \right),\]  
\[B_{i}=\left( \begin{array}{ccccc}0&0\\ 1/M_{i}&1/M_{i}\\ 0&0\\ 0&0\\ 0&0\\ 0&0\end{array} \right),\]  
\[A_{ji}=\left( \begin{array}{ccccc}0&0&0&0&0\\ \frac{T_{ij}}{M_{i}\sum_{k\in D_{i}}T_{ik}}&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\end{array} \right),j\in D_{i}.\]  
Figure 2: Subsystem structure. where \(x=(\Delta P_{outi},\Delta f_{i},\Delta P_{gi},\Delta x_{gi}|L_{AR_{i}})^{T}\), and \(\Delta P_{outi}\), \(\Delta f_{i}\), \(\Delta_{gi}\), \(\Delta x_{gi}\), and \(U_{AR_{i}}\) are output power deviation, frequency deviation, gas generator output power deviation, governor input of gas turbine generator, and regional demand for subsystem \(i\), respectively. \(w_{i}\) is disturbance of the dynamic system, which is bounded. The power output of subsystem \(i\) is \(\Delta P_{outi}=\sum_{j\in D_{i}}\Delta P_{ij}\). The tie-line power flow deviation of \(i\) is expressed as \(\Delta\sum_{j\in D_{i}}p_{ji}=\sum_{j\in D_{i}}T_{ij}(\Delta f_{j}B_{i}\Delta f _{i})\), when the adjoining area is \(j\). The regional demand is defined by \(U_{AR_{i}}=\int AR_{i}dt\), where \(AR_{i}=\Delta\sum_{j\in D_{i}}P_{ji}\).