## 4 Auxiliary Problem Principle Method with Self-Adaptive Penalty Parameter
### The Extension of Self-Adaptive Penalty Parameter for MAED Problem  
Problem (7) is the two-area economic dispatch model. Apparently, Problem (7) can be easily extended to \(N\)-area system. By denoting \({\Theta}\left({x,x_{j}}\right)={\lambda,x_{j}}+B_{x_{j}}x_{j}=0\) for the consistency constraint between area \(i\) and area \(j\), the corresponding lagrangian multiplier and penalty parameter for \({\Theta}\left({x_{i},x_{j}}\right)\) are defined as \({}^{\lambda_{i}}\) and \(c_{ij}\). In consideration of the most complex case, each area exchanges active power with all the other areas, a general formulation for the \(N\)-area model of MAED can be expressed as follows:  
\[\begin{array}{c}{\min\;\sum_{i=1}^{n}{f_{i}\left({x_{i}}\right)+\sum_{i=1}^{n }{\sum_{j=1}^{n}{\left[{\left({\lambda_{i}^{2}}\left({A_{j}x_{i}+B_{j}x_{j}} \right)+\frac{C_{ij}}{2}\left\|{A_{j}x_{i}+B_{j}x_{j}}\right\|}\right^{2}} \right)}}}\\ {\mbox{\emph{subject to:}}}\quad x_{i}\in\Omega_{\[\sum_{i=1}^{n}\sum_{j=i+1}^{n}\left|V_{q}^{k}-v_{q}^{k}\right|_{N_{q}}^{2}-\sum_{i =1}^{n}\sum_{j=i+1}^{n}\left|V_{q}^{k+1}-v_{q}^{k}\right|_{N_{q}}^{2}\geq\sum_{i =1}^{n}\sum_{j=i+1}^{n}\left|V_{q}^{k}-v_{q}^{k+1}\right|_{N_{q}}^{2} \tag{38}\]  
where  
\[v_{q}=\begin{pmatrix}A_{q}x_{q}\\ B_{q}x_{q}\\ \lambda_{q}\end{pmatrix},\ N_{q}=\begin{pmatrix}\left(\beta_{q}-c_{q}\right)I&- c_{q}I&0\\ -c_{q}I&\left(\beta_{q}-c_{q}\right)I&0\\ 0&0&\dfrac{1}{c_{q}}I\end{pmatrix},\ \ i,j=1,2,\cdots,n\text{ and }j>i \tag{39}\]  
For the sake of balance, we only need to adjust penalty parameter such that  
\[\sum_{i=1}^{n}\sum_{j=i+1}^{n}\left[\left(A_{q}x_{q}^{k+1}-A_{q}x_{q}^{k} \right)-\left(B_{q}x_{q}^{k+1}-B_{q}x_{q}^{k}\right)\right]_{N_{q}}^{2}\approx \sum_{i=1}^{n}\sum_{j=i+1}^{n}\left|\lambda_{q}^{k+1}-\lambda_{q}^{k}\right| _{N_{q}}^{2} \tag{40}\]  
In order to achieve balance in (40), we have to create a coordinator server which communicates with all sub-systems for exchanging required information. However, it is impossible for practical applications. In fact, achieving the balance in (40) provided that  
\[\left|\left(A_{q}x_{q}^{k+1}-A_{q}x_{q}^{k}\right)-\left(B_{q}x_{q}^{k+1}-B_{ q}x_{q}^{k}\right)\right|_{N_{q}}^{2}\approx\left|\lambda_{q}^{k+1}-\lambda_{q}^{k} \right|_{N_{q}}^{2},\ \ \ i,j=1,2,\cdots,n\text{ and }j>i \tag{41}\]  
The corresponding self-adaptive strategy for adjusting penalty parameter can be expressed as follows.  
\[c_{q}=\begin{cases}0.5\dfrac{c_{q}}{t_{q}^{k+1}}&\text{if }\ r_{q}^{k+1}>10\\ 2c_{q}&\text{if }\ r_{q}^{k+1}<0.1\\ c_{q}&\text{otherwise}\end{cases} \tag{42}\]  
where  
\[r_{q}^{k+1}=\dfrac{\left|\left(A_{q}x_{q}^{k+1}-A_{q}x_{q}^{k}\right)-\left(B _{q}x_{q}^{k+1}-B_{q}x_{q}^{k}\right)\right|_{N_{q}}}{\left|\lambda_{q}^{k+1}- \lambda_{q}^{k}\right|_{N_{q}}^{2}},\ i,j=1,2,\cdots,n\text{ and }j>i \tag{43}\]  
It is clear that the proposed SAPP only uses existing information to adjust penalty parameter, no extra information is needed. In addition, in comparison with the optimization problem (36), the calculation cost of (42) and (43) is slight and can be neglected. Therefore, the proposed SAPP scheme is implementable for practical applications.