## 2 Multi-Area Interconnected Power System with the HESS
### Load Frequency Control  
Power system is a complex nonlinear dynamic system, because the power system is always affected by the load disturbance which results in the load frequency fluctuation [18]. In general, we use the method of near linearization to deal with the power system model. We usually adopt the linearization method near a certain working point to deal with the power system model [19; 20]. A linear model is established to represent the dynamic model of system. The linearized model structure of system of \(i\)th control area in the interconnected power system is shown in Figure 2, which mainly includes the generator, the reheating turbine and the governor.  
The linearized model of the governor is as follows:  
\[G_{gi}=\frac{1}{T_{gi}s+1} \tag{1}\]  
The linearized model of the reheat steam turbine is shown as:  
\[G_{ti}=\frac{cT_{ri}s+1}{(T_{ri}s+1)(T_{ti}s+1)} \tag{2}\]  
The linearized model of the generator is as bellow:  
\[G_{pi}=\frac{K_{pi}}{T_{pi}s+1} \tag{3}\]  
The physical implications of variables in the model of \(i\)th (\(i\) = 1,2,3,4) control area are as follows: \(T_{gi}\), \(T_{ri}\), \(T_{ti}\), and \(T_{pi}\), respectively, represent the time constant of governor, the reheating time constant,  
Figure 1: Topological structure diagram of four-area interconnected power system.  
Figure 2: Linearization model structure of the system in \(i\)th control area.  
the time constant of steam turbine and the time constant of power system. \(\Delta X_{ei}\) indicates the incremental variation of governor valve position. \(\Delta P_{ri}\) is the increment of output thermal power from reheat unit. \(\Delta P_{tiei}\) denotes the deviation of power exchange between tie-lines. \(\Delta P_{di}\) represents the load disturbance. \(\Delta f_{i}\) is the load frequency deviation of system. \(c\) is the reheating coefficient and \(K_{pi}\) is the gain of power system. \(T_{ij}\) represents the power synchronization coefficient of tie-line between the \(i\)th control area and the \(j\)th control area (\(i\
eq j\)). \(K_{AFi}\) is the proportional negative feedback coefficient of the frequency deviation. \(\varphi_{i}\) denotes the frequency deviation constant that satisfies \(\varphi_{i}=\beta_{i}/R_{i}\), where \(\beta_{i}\) is the set value of frequency deviation and \(R_{i}\) is the speed regulation gain.  
To improve the power system with better dynamic performance, we should not only make sure that the frequency deviation (\(\Delta f_{i}\)) of the system tends to zero, but also ensure that the deviation of power exchange between tie lines (\(\Delta P_{tiei}\)) is close to zero under any load disturbance. Considering the two targets above, the concept of area control error (ACE) is proposed in the LFC. The expression of ACE in the \(i\)th control area is formulated as (4):  
\[ACE_{i}=\Delta P_{tiei}+\varphi_{i}\Delta f_{i} \tag{4}\]  
In this paper, there is an internal PI controller in power system for frequency regulation which can be seen in Figure 2. The controlled variable \(u_{lfci}\) is as follows:  
\[u_{lfci}=-\frac{K_{i}}{s}ACE_{i} \tag{5}\]  
where \(K_{i}\) represents the control coefficient of PI regulation in \(i\)th control area.  
Many works have been done to improve the control effect of the LFC and several novel control methods have been applied to the internal controller design in LFC. For example, the application of parameter self-adjusting algorithm to the PID controller [11; 12] and the design of adaptive model predictive controller. In this paper, a hybrid energy storage system is introduced into traditional power system and an H\({}_{\infty}\) robust state feedback controller is designed. The controller calculates the power supply in real time according to the variation of user load in system, and then the power control signal is transmitted to the HESS. The regulation of power in the system is realized by the effective charge and discharge of the HESS, which can ensure the stability of the load frequency.