sample average approximation (SAA) method——采样平均近似算法：处理机会约束

参考文献

[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

原始机会约束

下面是一个典型的机会约束：风电消纳量大于指定比例以及小于可用风电的概率必须大于 \(1-\in\)。

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

利用SAA转化为MILP约束

上述机会约束可用如下约束代替：

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

\(C_{wt}^m\ \ \ \forall m \in {\mathop{\rm A}\nolimits}\)为随机向量\({{{\tilde C}_{wt}}}\)的采样样本，\(z_m\)取0或者1分别表示约束\(m\)成立或者被松弛。

这种转换就是利用\(m\)个样本中有\(k\)个样本成立来等效代替概率（当\(m\)足够大时，这种等效是合理的，对应的概率即为 \(k/m \ \approx \ \in\)），这和SAA的名字也是相符的。

但是这种转换显然会导致得到的MILP问题求解效率较低，实际应用的话需要进一步采用某些算法。

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.