# Load Frequency Control for Multi-Area Power Plants with Integrated Wind Resources
## 4 State Observer Based on Sliding Mode Control Strategies
### Decentralized State Estimator Feedback Integral Sliding Mode Control (DSEFISMC) Design  
Previously, we designed the integral surface and proved the power system asymptotically stability in the sliding motion. Next, a continuous second-order DSEFISMC law is developed to reduce the chattering inherent in the first-order design. We start by defining the second-order sliding manifold \(S_{i}[z_{i}(t)]\) such that the estimated system state trajectories are forcefully driven to zero asymptotically, which is given as  
\[S_{i}[z_{i}(t)]=\dot{\sigma}_{i}[z_{i}(t)]+\epsilon_{i}\sigma_{i}[z_{i}(t)] \tag{28}\]  
and  
\[\tilde{S}_{i}[z_{i}(t)]=\ddot{\sigma}_{i}[z_{i}(t)]+\epsilon_{i}\ddot{\sigma} _{i}[z_{i}(t)] \tag{29}\]  
where \(\epsilon_{i}>0\) is a positive constant. Using Equation (16) yields  
\[ \begin{array}{l}\dot{S}_{i}[z_{i}(t)]=L_{i}[A_{i}\dot{z}_{i}(t)+B_{i}\dot{u} _{i}(t)+\sum\limits_{\begin{array}{c}j=1\\ j\
eq i\end{array}}^{N}H_{ij}\ddot{z}_{j}(t)+T_{i}(\dot{y}_{i}(t)-\dot{u}_{i}(t ))]\\ \par-L_{i}(A_{i}-B_{i}\dot{I}_{i})\dot{z}_{i}(t)+\epsilon_{i}\ddot{\sigma}_{i}[ z_{i}(t)]\end{array}  \tag{30}\]  
Based on the definition of the sliding surface and sliding manifold, the continuous DSEFISMC law for the MAIPS is known as follows  
\[ \begin{array}{l}\dot{u}_{i}(t)=-(L_{i}B_{i})^{-1}\{L_{i}B_{i}I_{i}\dot{z}_{i }(t)+\epsilon_{i}\dot{\sigma}_{i}[z_{i}(t)]+\sum\limits_{\begin{array}{c}j=1 \\ j\
eq i\end{array}}^{N}L_{j}H_{ji}\dot{z}_{i}(t)\\ \par+L_{i}T_{i}(\dot{y}_{i}(t)-\dot{u}_{i}(t))-\delta_{i}sat(S_{i}[z_{i}(t)])\} \end{array}  \tag{31}\]  
We propound a theorem to demonstrate the reachability of the estimated system state trajectories to the manifold in the following. **Theorem 2**. : _Consider Equation (6) with the continuous DSEFISMC law (30). Then, system state trajectory is directed towards the sliding manifold \(S_{i}[z_{i}(t)]\) and once the trajectory hits the sliding manifold \(S_{i}[z_{i}(t)]\) it remains on the sliding manifold thereafter._  
The incremental frequency is displayed in Figure 4 and the tie-line power deviation is shown in Figure 5, whereas the control signal is given in Figure 6.  
Figure 4: Frequency deviation of three-area power system. Figure 5: Tie-line power of three-area power system. Figure 3: Diagram of a three-area power system integrated with a wind plant. **Remark 1**. : _Under the assumed initial conditions of the MAIPS, the proposed SOboSMC acted quickly to converge the frequency error to zero at 3 s, with a maximum overshoot of 0.005 Hz as compared to the 6-s settling time and maximum overshoot of 0.08 Hz seen in [23]. This validates that the MAIPS performance is better when using the proposed SOboSMC._  
Case 2. This simulation was done with the MAIPS with and without wind turbines. The load disturbances of the areas 1, 2 and 3 were assumed to be the same as those in case 1 and [23]. The parametric uncertainty is considered in the form of matched uncertainty \(\Delta A_{i}\), which is expressed as  
\[\Delta A_{1}=\left[ \begin{array}{cccc}0&0&0&0&0\\ 0&0&0&0&0\\ -2.26\cos(t)&2\cos(t)&-2.604\cos(t)&3\cos(t)&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ \end{array} \right]\text{ where }\Delta A_{1}=\Delta A_{2}=\Delta A_{3}.\]  
The wind variation is shown in Figure 7. The frequency error with and without the wind plant can be seen in Figures 8 and 9, whereas the tie-line power error with and without the wind farm are shown in Figures 10 and 11. For the MAIPS without wind farms, the response of the power network is better, pertaining to overshoot and settling time, and is also chattering-free in comparison with the design presented in [23]. Once more, the frequency transient is kept within the operational safety range, which is \(\pm 0.2\) Hz for the safety of the power system frequency [36]. Therefore, the proposed approach shows good control performance for the LFC of an MAIPS with load disturbances, parameter uncertainties and wind variations, without the loss of control accuracy. Figure 6: Control signal of three-area power system.