## 3 Multi-Area AC State Estimation Model Using ADMM
Hence, the ADMM framework considering a multi-area non-linear state estimation model is developed in this work. The main characteristic of the power system grid is that some of the states in the state vector \(x_{k}\) are common between region \(k\) and their neighbors due to measurements collected and related to those states. Hence, for neighboring regions \(k\) and \(l\), define \(x_{k}[l]\) (\(x_{l}[k]\)) to be a sub-vector of the states in region \(k\) (\(l\)) that are shared between the two regions. One can define auxiliary variables for each shared state between two regions. For instance, the auxiliary variable \(x_{kl}\) can be introduced to represent \(x_{k}[l]\). Similarly for region \(l\), the auxiliary variable \(x_{lk}\) represents the variable \(x_{l}[k]\). Define for a region \(k\) the set \(N_{k}\) which represents the number of regions that share states with region \(k\). In addition, for each entry in the state vector in region \(k\), i.e., \(x_{k}(i)\), define the set \(N_{k}^{i}\) which consists of the set of regions sharing the state \(x_{k}(i)\) with region \(k\). Since the state is a simple update that relies on the increment (as described in Section 2.1), the augmented Lagrangian (as described in Section 2.2) can be written for the increment of the states as follows:  
\[ \begin{split} L_{p}(\Delta x_{k},\Delta x_{kl},v_{kl}):=& \sum_{k=1}^{K}f_{k}(\Delta x_{k})+\sum_{l\in N_{k}}\left(v_{k,l} \big{(}\Delta x_{k}[l]-\Delta x_{kl}\big{)}+\right.\\ &\left.\left(\frac{\rho}{2}\right)\big{\|}\Delta x_{k}[l]-\Delta x _{kl}\big{\|}_{2}^{2}\right).\end{split}  \tag{25}\]  
ADMM consists of the following iterations:  
\[\Delta x_{k}^{t+1}:=\mathop{\arg\min}\limits_{\Delta x_{k}}L_{p}\Big{(}\Delta x _{k}^{t+1},\Delta x_{kl}^{t},v_{kl}^{t}\Big{)} \tag{26}\]  
\[\Delta x_{kl}^{t+1}:=\mathop{\arg\min}\limits_{\Delta x_{kl}}L_{p}\Big{(} \Delta x_{k}^{t+1},\Delta x_{kl}^{t+1},v_{kl}^{t}\Big{)} \tag{27}\]  
\[v_{k,l}^{t+1}:=v_{k,l}^{t}+\rho\Big{(}\Delta x_{k}^{t+1}[l]-\Delta x_{kl}^{t+ 1}\Big{)} \tag{28}\]  
By applying the minimization of the increment of states as in (23), ADMM iterations can be simplified [15] and outlined in the following steps:  
Step 1:  
\[P=H_{k}\Big{(}H_{k}^{T}W_{k}H_{k}+\rho D_{k}\Big{)}H_{k}W_{k} \tag{29}\]  
\[(II)^{i}=\sqrt{\frac{1-P^{ii}}{P^{ii}}} \tag{30}\]\[(E_{k}^{II})^{ii}=\left(1+\frac{1}{\left(II^{i}\right)^{2}}\right)W_{k}^{ii} \tag{31}\]  
\[\Delta x_{k}^{t+1}:=\left(H_{k}^{T}E_{k}^{II}H_{k}+\rho D_{k}\right)^{-1}\left(H _{k}^{T}E_{k}^{II}\Delta z_{k}+\rho D_{k}p_{k}^{t}\right) \tag{32}\]  
\[x_{k}^{t+1}=x_{k}^{t}+\Delta x_{k}^{t+1} \tag{33}\]  
Step 2:  
\[v_{k,l}^{t+1}:=v_{k,l}^{t}+\frac{\rho}{2}\left(\left(x_{k}^{t+1}[l]-x_{k}^{t}[ l]\right)-\left(x_{l}^{t+1}[k]-x_{l}^{t}[k]\right)\right) \tag{34}\]  
Step 3:  
\[ \begin{split} p_{k}^{t+1}(i)=&\frac{1}{2}\left( \left(x_{k}^{t+1}(i)-x_{k}^{t}(i)\right)+\left(\frac{1}{\left|N_{k}^{i}\right| }\sum_{l\in N_{k}^{i}}x_{l}^{t+1}[i]-x_{l}^{t}[i]\right)\right)\\ &-\frac{1}{\rho\left|N_{k}^{i}\right|}\sum_{l\in N_{k}^{i}}v_{k,l }^{t+1}[i]\end{split}  \tag{35}\]  
where \(D_{k}\) is a diagonal matrix with entry \((i,i)\) is \(\left|N_{k}^{i}\right|\), i.e., the number of regions sharing the state \(x_{k}(i)\) with region \(k\), \(v_{k,l}\) is the Lagrange multiplier associated the constraint between region \(k\) and region \(l\), \(x_{l}[i]\) is the entry in the vector \(x_{l}\) corresponding to the state \(x_{k}(i)\).