# Load Frequency Optimal Active Disturbance Rejection Control of Hybrid Power System
## 3 The Proposed Control Strategy
### The Principle and Control Algorithm of ADRC  
[PERSON] et al. [14] summarized the fundamental concepts of the ADRC strategy. The ADRC approach employs three key components: the TD, an ESO, and an NLFCL.  
The principle of ADRC is based on its ability to estimate and compensate for disturbances. By incrementally augmenting the input signal throughout the transition process, the disparity in the output is minimized, enabling the system to attain a stable state without exceeding the desired value.  
The second-order block diagram of the ADRC is depicted in Figure 3. In general, the determination of the order of the ADRC can be based on the characteristics of the controlled object. Higher orders of ADRC typically lead to improved control performance, albeit at the expense of increased complexity in implementation. After careful consideration of multiple factors, a second-order system is adopted. The system construction and integration are relatively simple, yet it successfully achieves the desired control effect. The mathematical representation of the second-order LADRC model is given by the following equation.  
\[A\dot{C}\dot{E}=bu_{1}+f(ACE,u_{1},d) \tag{11}\]  
where \(u_{1}\) is the output value, \(ACE\) is the input value, \(b\) is the amplified factor, and \(f\) is the total disturbance of the system, as shown in Figure 3. Assume that,  
\[\left\{\begin{array}{l}z_{1}=ACE\\ z_{2}=A\dot{C}E\\ z_{3}=f\end{array}\right. \tag{12}\]  
The expression can be represented in the state space format, as depicted in Equation (13).  
\[\left\{\begin{array}{l}\dot{z}=Az+Bu_{1}+Ef\\ ACE=Cz\end{array}\right. \tag{13}\]  
where \(A\), \(B\), \(E\), \(C\), and \(z\) are defined as depicted in Equation (14).  
\[A=\begin{bmatrix}0&1&0\\ 0&0&1\\ 0&0&0\end{bmatrix},\;B=\begin{bmatrix}0\\ b_{0}\\ 0\end{bmatrix},\;C=\begin{bmatrix}1&0&0\end{bmatrix}\]  
\[E=\begin{bmatrix}0\\ 0\\ 1\end{bmatrix},\;z=\begin{bmatrix}z_{1}z_{2}z_{3}\end{bmatrix}^{T} \tag{14}\]  
Figure 3: Diagram of the proposed control strategy.  
where \(b_{0}\) is the known parameter of the system.  
The state space observer, known as the linear extended state observer (LESO) of Equation (11), is formulated according to Equation (15).  
\[\left\{\begin{array}{l}\ddot{Z}=Az+Bu_{1}+L\big{(}ACE-A\widehat{\mathcal{C}}E \big{)}\\ A\widehat{\mathcal{C}}E=Cz\end{array}\right. \tag{15}\]  
where \(L\) is the observer gain vector, which can be determined using established methods like the pole placement technique. In order to streamline the tuning process, the observer gains are parameterized as follows:  
\[L=\begin{bmatrix}\beta_{1}&\beta_{2}&\beta_{3}\end{bmatrix}=\begin{bmatrix}3 \omega_{0}&3\omega_{0}^{2}&\omega_{0}^{3}\end{bmatrix} \tag{16}\]  
where \(\omega_{0}\) is the observer bandwidth.  
For the error feedback module, we also use the pole placement method, optimizing all controller poles at the same \(\omega_{c}\) bandwidth.  
In summary, the optimization of three parameters, namely \(\omega_{c}\), \(\omega_{0}\), and \(b_{0}\), is necessary for the implementation of linear active disturbance rejection control.