﻿ 基于两阶段–机会约束随机规划的含风电机组组合问题

# 基于两阶段–机会约束随机规划的含风电机组组合问题A Two-Stage Chance-Constrained Stochastic Program for Unit Commitment with Wind Power Output

Abstract: We present a stochastic unit commitment problem with uncertain wind power output. In this paper, the problem is formulated as a jointed two-stage and chance-constrained model in which the random vector is used to describe wind power output, based on the theory of stochastic program-ming. Our model minimizes the unit commitment costs and coal consumption of power system taking into account a certain level of confidence to meet the spinning reserve constraints. We de-signed a new combination sample average approximation algorithm based on the random sampling average approximation method. Finally, the 10 units system simulation experiments are given to verify the validity and reasonableness of the model.

1. 引言

2. 含风电的两阶段–机会约束随机机组组合模型

2.1. 预备知识

$\begin{array}{l}\underset{x\in X}{\mathrm{min}}f\left(x\right)\\ \text{s}\text{.t}\text{.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Pr\left\{G\left(x,\xi \right)\le 0\right\}\ge 1-\epsilon \end{array}$ (1)

$f\left(x\right)$ 为目标函数， $x$ 表示决策向量， $X\in {R}^{n}$ 表示确定性的可行域， $\xi$ 为服从某概率分布的随机向量。 $\epsilon \in \left(0,1\right)$ 称作机会约束规划的风险水平或者置信水平。

$Pr\left\{G\left(x,\xi \right)\le 0\right\}\ge 1-\epsilon$ (2)

$\begin{array}{l}\underset{x\in X}{\mathrm{min}}{c}^{\text{T}}x+E\left[Q\left(x,\xi \right)\right]\\ \text{s}\text{.t}\text{.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Ax=b,x\ge 0\\ Q\left(x,\xi \right)=\underset{y}{\text{min}}f\left(x,y\right)\\ \text{s}\text{.t}\text{.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Tx+Wy\le h,y\ge 0.\end{array}$ (3)

2.2. 目标函数

$F=\underset{u}{\mathrm{min}}\left(\underset{t=1}{\overset{T}{\sum }}\underset{i=1}{\overset{N}{\sum }}{C}_{i,t}+{E}_{\xi }\left[Q\left({u}_{i,t},\xi \right)\right]\right)$ (4)

$Q\left({u}_{i,t},\xi \right)=\underset{P}{\mathrm{min}}\underset{t=1}{\overset{T}{\sum }}\underset{i=1}{\overset{N}{\sum }}{f}_{i}\left({u}_{i,t},{P}_{i,t}\right)$ (5)

2.3. 约束条件

1) 含风电系统的功率平衡约束。

$\underset{i=1}{\overset{N}{\sum }}{P}_{i,t}+{P}_{W,t}\left(\xi \right)={P}_{D,t}$ (6)

2) 含风电系统的旋转备用约束

$\mathrm{Pr}\left(\underset{i=1}{\overset{N}{\sum }}{u}_{i,t}{\stackrel{¯}{P}}_{i}+{P}_{W,t}\left(\xi \right)-{P}_{D,t}-{R}_{t}\ge 0\right)\le \epsilon$ (7)

3. 两阶段–机会约束随机规划模型的采样平均近似方法

3.1. 随机机组组合模型的SAA问题

$\begin{array}{l}\mathrm{min}\text{\hspace{0.17em}}{c}^{\text{T}}u+{E}_{\xi }\left[Q\left(u,\xi \right)\right]\\ \text{s}\text{.t}\text{.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Au\le b,u\in \left\{0,1\right\}\end{array}$ (8)

$\begin{array}{l}Q\left(u,\xi \right)=\mathrm{min}f\left(u,p\right)\\ \text{s}\text{.t}\text{.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Wp+Tu\ge h\left(\xi \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}q\left(u,\xi \right)\le \epsilon \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}p\ge 0\end{array}$ (9)

${\stackrel{^}{q}}_{S}\left(u\right):={S}^{-1}\underset{s=1}{\overset{S}{\sum }}{1}_{\left(0,\infty \right)}\left(G\left(u,{\xi }^{s}\right)\right)$ (10)

$G\left(u,{\xi }^{s}\right)={u}_{i,t}{\stackrel{¯}{P}}_{i}+{P}_{W,t}\left({\xi }^{s}\right)-{P}_{D,t}-{R}_{t}$ (11)

$\mathrm{min}{c}^{\text{T}}u+{S}^{-1}\underset{s=1}{\overset{S}{\sum }}Q\left(u,{\xi }^{s}\right)$

$\begin{array}{l}\text{s}\text{.t}\text{.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Au\le b\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Hp+Tu\ge h\left({\xi }^{s}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{^}{q}}_{S}\left(u\right)\le \gamma \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Q\left(u,{\xi }^{s}\right)\ge f\left(u,p\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\in \left\{0,1\right\},p\ge 0,s=1,\cdots ,S\end{array}$ (12)

$1-P\left({\stackrel{^}{X}}_{S}^{\delta }\subset {X}^{\epsilon }\right)\le |X|{\text{e}}^{-S\gamma \left(\delta ,\epsilon \right)}$ (13)

$S\ge \frac{3{\sigma }^{2}}{{\left({\epsilon }^{\prime }-\delta \right)}^{2}}\mathrm{log}\left(\frac{X}{\alpha }\right)$ (14)

3.2. 两阶段–机会约束随机规划问题的求解

${\stackrel{^}{v}}_{S}$${v}^{*}$ 分别表示SAA问题的最优值和原问题的最优值，对SAA问题做 $M$ 次数值实验，可用如下方法构造问题的统计上下界。

${\stackrel{¯}{v}}_{S}^{M}=\frac{1}{M}\underset{m=1}{\overset{M}{\sum }}{\stackrel{^}{v}}_{S}^{m}$ (15)

${\stackrel{^}{g}}_{{S}^{\prime }}\left(\stackrel{¯}{u}\right)={c}^{T}\stackrel{¯}{u}+\frac{1}{{S}^{\prime }}\underset{s=1}{\overset{{S}^{\prime }}{\sum }}Q\left(\stackrel{¯}{u},{\xi }^{s}\right)$ (16)

${\stackrel{^}{g}}_{{S}^{\prime }}\left(\stackrel{¯}{u}\right)$${c}^{\text{T}}\stackrel{¯}{u}+{E}_{\xi }Q\left(\stackrel{¯}{u},\xi \right)$ 的一个无偏估计，可以看出 ${\stackrel{^}{g}}_{{S}^{\prime }}\left(\stackrel{¯}{u}\right)$ 给出了原问题一个最优值的上界。

3.3. 基于求解SAA问题的算法

$m=1,\cdots ,M$ ，重复下面的步骤

a) 求解对应的SAA问题。

${\stackrel{^}{x}}_{S}^{m}$ 为SAA问题的解向量， ${\stackrel{^}{v}}_{S}^{m}$ 为SAA问题的最优值。

b) 由式(15)计算 ${\stackrel{¯}{v}}_{S}^{M}$

4. 数值实验及分析

4.1. 10场景不同置信水平数值实验

4.2. 10场景置信水平为90%的启停机计划

4.3. 不同场景规模的数值实验

Table 1. Minimal cost and cpu time under different confidence levels

Table 2. Unit commitment plan under confidence level 90% with 10 scenarios

Table 3. Numerical results of different scenario scale

5. 结论

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