## 3 Multi-Area AC State Estimation Model Using ADMM  
The power system network is a wide-area interconnected system that is typically partitioned into regions based on geographical location. The system states of each region are monitored and supervised by a local control center. The overall PSSE goal is to estimate the states in each control region in an optimal way. If each control center estimates their local states without communicating with neighboring regions, then the estimated solution is sub-optimal. At the same time, if there are measurements pertaining to lines connecting boundary buses between areas or injection measurements associated with those boundary buses, then there is a need for communication between those control centers in order to utilize those measurements. For illustration purposes of interconnected power network, the IEEE 14 bus system is partitioned into four areas as shown in Figure 1. As seen, area 1 has a connection with area 2 through the tie-lines connecting buses 5 and 2 (in area 1) with buses 3 and 4 (in area 2), respectively. At the same time, area 1 has a connection with area 3 through the transmission line connecting bus 5 with bus 6. Therefore, towards optimal estimation, area 1 would require to communicate with areas 2 and 3. This could be done through information sharing of states of buses 3, 4, and 6 if there is a power flow measurements in those lines and/or power injections at buses 2 and 5. Similar scenario is applied for other areas. Hence, for a multi-area state estimation, each area would augment shared states with neighboring areas to their local states when performing local SE. The advantage of ADMM algorithm in the state estimation process is pertained to the decomposition of the process of state estimation while constraining the estimates of bordering buses if they are shared with neighboring areas. In that way, communication is limited only between neighboring regions, which leads to estimating the system states in fully distributed manner and without a central process. In addition, upon convergence, global estimation is optimal and carried out with minimal computations (and requirements of information sharing). The framework for the multi-area state estimation is initialized with the partition of the grid into \(K\) control centers. Each control center is aiming to solve the following model:  
\[z_{k}=h_{k}(x_{k})+e_{k} \tag{21}\]  
where \(z_{k}\in\mathbb{R}^{M_{k}}\) is the measurement vector of region \(k\), \(x_{k}\in\mathbb{R}^{N_{k}}\) is the state variables vector of region \(k\) associated with the measurement in \(z_{k}\), \(h_{k}(x_{k}):\mathbb{R}^{M_{k}}\rightarrow\mathbb{R}^{N_{k}}\) is a non-linear  
Figure 1: \(IEEE\)-14 bus system divided into four areas. Dotted lassos shows the the boundary buses between areas. differentiable function that relates the states to the measurements, \(e_{k}\) is the measurement error vector assumed with zero mean, standard deviation \(\sigma\) and having Gaussian probability distribution, \(M_{k}\) is the number of measurements in region \(k\), and \(N_{k}\) is the number of unknown state variables in region \(k\). The model in (1) is non-linear. Hence, a linearization is required in order to solve the non-convex optimization problem. Using Gauss-Newton approximation, as described in Section 2.1, one can write the following model:  
\[\bigtriangleup z_{k}=H_{k}\bigtriangleup x_{k}+e_{k} \tag{22}\]  
Solving for \(\bigtriangleup x_{k}\), one can obtain the following:  
\[\bigtriangleup\hat{x}_{k}=(H_{k}^{T}R_{k}^{-1}H_{k})^{-1}H_{k}^{T}R_{k}^{-1} \bigtriangleup z_{k} \tag{23}\]  
Then, the estimated state can be updated simply as:  
\[x_{k}^{t+1}=x_{k}^{t}+\Delta x_{k}^{t+1} \tag{24}\]  
The process is solved iteratively until a convergence criterion is achieved. In [15], a framework for applying ADMM for distributed state estimation is established considering a multi-area linear measurement model. In some applications, as fault diagnosis of power systems, a more accurate model is required.