# WAMS-Based Online Disturbance Estimation in Interconnected Power Systems Using Disturbance Observer
## 3 Proposed Disturbance Observer-Based Disturbance Estimation
### Disturbance Observer Design  
To stabilize the power system frequency through UFLS, the information of the unknown power system disturbance should be obtained at first, which mainly comprises of the load variations. Using our proposed scheme we design a disturbance observer to estimate the power system disturbance, \(d_{lx}\). With the estimated results, the proposed UFLS can be implemented more accurately. To design a disturbance estimator we will consider a power system described in (17). In real power systems, the system disturbance problem concerns not only external disturbances, but also influence caused by model uncertainties. To include this fact, the model (17) is further modified and rewritten as  
\[ \begin{cases}\dot{x}=A_{n}x+B_{n}u+F_{d}d_{lx}\\ y=C_{n}x+D_{n}u+D_{d}d_{ly}\end{cases}  \tag{26}\]  
where the nominal system matrices are \(A_{n}\), \(B_{n}\), \(C_{n}\) and \(D_{n}\), \(d_{lx}\) and \(d_{ly}\) are the lumped disturbances on states and outputs, which can be given as in (27) and (28). Equation (26) is similar to the one in (13), the only difference is that we also added the desired outputs and its associated matrices. \[d_{lx}=d_{x}+\left(A-A_{n}\right)x+\left(B-B_{n}\right)u \tag{27}\]  
\[d_{ly}=d_{y}+\left(C-C_{n}\right)x+\left(D-D_{n}\right)u \tag{28}\]  
The lumped disturbances consist of external disturbances and internal disturbances caused by model uncertainties. In order to design the disturbance observer to estimate the lumped disturbances we make use of the input, output and estimated state information. The estimate of \(d_{ly}\) is then obtained from the second equation in (26), and is described as \[\hat{d}_{ly}=y-C_{n}x-D_{n}u \tag{29}\]  
The estimation error \(e_{d_{ly}}\) is then be obtained as  
\[e_{d_{ly}}=\hat{d}_{ly}-d_{ly}=0 \tag{30}\]  
To obtain the disturbance \(\hat{d}_{lx}\), which is of critical concern, we consider the following assumption. **Assumption 1**. : _The lumped disturbances \(\hat{d}_{lx}\) is slowly varying relative to the observer dynamics, that is \(\hat{d}_{lx}\approx 0\). This is true because for the UFLS problem, it can assume the system load variations are slow changing except for the instance of the disturbance._  
Assumption 1 will be consistent in our simulations, and results show that the assumption holds true. However, the method is also feasible for fast time-varying disturbances. When uncertainties are taken into account, the lumped disturbances becomes a function of the states, and can still be accurately estimated if the disturbance observer dynamics are faster than that of the closed-loop system. Consequently, for the system described by (26), the disturbance observer in (31) is designed to estimate the disturbances \(d_{lx}\) of any \(n\)-area power system. \[ \begin{cases}\dot{z}=-LB_{d}\left(z+L\dot{x}\right)-L\left(A_{n}\dot{x}+B_{n}u \right)\\ \hat{d}_{lx}=z+L\dot{x}\end{cases}  \tag{31}\]  
where \(\hat{d}_{lx}=\Delta P\) is the power system disturbance we are interested in, \(z\) is an auxiliary vector and \(L\) is the disturbance observer gain matrix to be designed. Considering the system described by (26) under the lumped disturbances, it meets the requirements of \(Assumption1\). Using disturbance observer (31), its output estimates can asymptotically track the lumped disturbances if the observer gain matrix \(L\) is chosen such that the matrix \(-LB_{d}\) is stable. The eigenvalue theory states that all eigenvalues of the system matrix must have a negative real part for the system to be stable. Likewise, \(-LB_{d}\) must have negative real part. To prove this, the disturbance estimation error of the disturbance observer (31) is defined as  
\[e_{d_{lx}}=\hat{d}_{lx}-d_{lx} \tag{32}\]  
Considering Assumption 1, and combining (26), (31) and (32) gives  
\[