sample average approximation (SAA) method——采樣平均近似算法：處理機會約束

參考文獻

[1]Qiu F, Wang J. Chance-Constrained Transmission Switching With Guaranteed Wind Power Utilization[J]. IEEE Transactions on Power Systems, 2015, 30(3): 1270–1278.
DOI：10.1109/TPWRS.2014.2346987

原始機會約束

下面是一個典型的機會約束：風電消納量大於指定比例以及小於可用風電的概率必須大於 \(1-\in\)。

\[P\left\{ \matrix{ \sum\limits_{w \in W} {\sum\limits_{t \in T} {{p_{wt}}} } \ge \alpha *\sum\limits_{w \in W} {\sum\limits_{t \in T} {{{\tilde C}_{wt}}} } \hfill \cr {p_{wt}} \le {{\tilde C}_{wt}}{\rm{ }}\forall t \in T,\forall w \in W \hfill \cr} \right\} \ge 1 - \in \]

利用SAA轉化為MILP約束

上述機會約束可用如下約束代替：

\[\left\{ \matrix{ \sum\limits_{w \in W} {\sum\limits_{t \in T} {{p_{wt}}} } + {z_m}M \ge \alpha *\sum\limits_{w \in W} {\sum\limits_{t \in T} {C_{wt}^m} } {\rm{ }}\ \ \ \forall m \in {\mathop{\rm A}\nolimits} \hfill \cr {p_{wt}} - {z_m}M \le C_{wt}^m{\rm{ }}\ \ \ \forall w \in W,\forall t \in T,\forall m \in {\mathop{\rm A}\nolimits} \hfill \cr \sum\limits_{m \in {\mathop{\rm A}\nolimits} } {{z_m}} \le k \hfill \cr {z_m} \in \left\{ {0,1} \right\}{\rm{ }}\ \ \ \forall m \in {\mathop{\rm A}\nolimits} \hfill \cr} \right.\]

\(C_{wt}^m\ \ \ \forall m \in {\mathop{\rm A}\nolimits}\)為隨機向量\({{{\tilde C}_{wt}}}\)的采樣樣本，\(z_m\)取0或者1分別表示約束\(m\)成立或者被松弛。

這種轉換就是利用\(m\)個樣本中有\(k\)個樣本成立來等效代替概率（當\(m\)足夠大時，這種等效是合理的，對應的概率即為 \(k/m \ \approx \ \in\)），這和SAA的名字也是相符的。

但是這種轉換顯然會導致得到的MILP問題求解效率較低，實際應用的話需要進一步采用某些算法。

Note that SAA does not guarantee a feasible solution and the feasibility of a solution is probabilistic (i.e., we can only claim that the solution is feasible with a certain probability). Obtaining a feasible solution with a high confidence level requires a sufficiently large number of Monte Carlo samples, which results in a MILP that is difficult to solve. Therefore, efficient solution approaches are necessary to apply SAA in a practical situation.