# 考虑回水效应的一库多级式水电站群短期优化调度模型A Short-Term Optimal Scheduling Model of Cascaded Hydropower Stations with One Reservoir Considering the Effect of Backwater

DOI: 10.12677/JWRR.2019.83030, PDF, HTML, XML, 下载: 477  浏览: 1,600  国家科技经费支持

Abstract: The hydraulic coupling among cascaded hydropower stations with one reservoir is close, and the backwater from the downstream reservoir is easy to form a jacking effect on the tailwater level of the upstream reservoir, which affects the generation efficiency and planning. Based on the unit of the basic dispatching unit, this paper puts forward a short-term optimal scheduling model of cascaded hydropower stations with one reservoir, which takes the effect of backwater into consideration. Through the interval division of the downstream reservoir water level, the three-dimensional curve of tailwater level-discharge-reservoir water level is described as three typical scenarios, and the linear description is realized by piecewise linear method. The developed model converts into a mixed integer linear programming model, which can be solved by commercial software Lingo. Based on a case study with Tianshengqiao cascaded hydropower stations located in Hongshui River, the study shows that the proposed model considering backwater effect can realize the refined calculation and improve the economic benefits of scheduling.

1. 引言

2. 模型描述

2.1. 目标函数

$E=\mathrm{max}\underset{i=1}{\overset{I}{\sum }}\underset{n=1}{\overset{Ni}{\sum }}\underset{t=1}{\overset{T}{\sum }}{p}_{i,n,t}\Delta t$ (1)

2.2.约束条件

1、线性约束

1) 水量平衡约束

${V}_{i,t}={V}_{i,t-1}+3600\left({I}_{i,t}+I{F}_{i,t}-{Q}_{i,t}\right)×\Delta t$ (2)

${Q}_{i,t}=\underset{n=1}{\overset{{N}_{i}}{\sum }}{q}_{i,n,t}+{S}_{i,t}$ (3)

2) 始、末库容限制

${V}_{i,1}={V}_{i}^{Initial}$ (4)

${V}_{i,T}={V}_{i}^{End}$ (5)

3) 水库边界条件

$\underset{_}{{V}_{i}}\le {V}_{i,t}\le \stackrel{¯}{{V}_{i}}$ (6)

$\underset{_}{{Q}_{i}}\le {Q}_{i,t}\le \stackrel{¯}{{Q}_{i}}$ (7)

$0\le {S}_{i,t}\le \stackrel{¯}{{S}_{i,t}}$ (8)

4) 机组出力限制

$\left\{\begin{array}{l}{w}_{i,n,t}\underset{_}{{p}_{i,n}}\le {p}_{i,n,t}\le {w}_{i,n,t}\stackrel{¯}{{p}_{i,n}}\\ {w}_{i,n,t}\in \left\{0,1\right\}\end{array}$ (9)

5) 机组发电流量限制

${w}_{i,n,t}\underset{_}{{q}_{i,n}}\le {q}_{i,n,t}\le {w}_{i,n,t}\stackrel{¯}{{q}_{i,n}}$ (10)

6) 机组开、停机状态约束

${x}_{i,n,t}-{\stackrel{˜}{x}}_{i,n,t}={w}_{i,n,t}-{w}_{i,n,t-1}$ (11)

${x}_{i,n,t}+{\stackrel{˜}{x}}_{i,n,t}\le 1$ (12)

${x}_{i,n,t},{\stackrel{˜}{x}}_{i,n,t}\in \left\{0,1\right\}$ (13)

7) 机组开、停机持续时间约束及最大开机次数约束

${x}_{i,n,t}+\underset{\lambda =t+1}{\overset{t+{T}_{i,n}^{On}-1}{\sum }}{\stackrel{˜}{x}}_{i,n,\lambda }\le 1$ (14)

${\stackrel{˜}{x}}_{i,n,t}+\underset{\lambda =t+1}{\overset{t+{T}_{i,n}^{Off}-1}{\sum }}{x}_{i,n,\lambda }\le 1$ (15)

$\underset{t=1}{\overset{T}{\sum }}{x}_{i,n,t}\le {X}_{i,n}^{Max}$ (16)

8) 机组发电水头约束

${h}_{i,n,t}=\frac{{Z}_{i,t-1}+{Z}_{i,t}}{2}-{Z}_{i,t}^{d}-{h}_{i,n,t}^{loss}$ (17)

2、非线性约束

1) 水位库容约束

${Z}_{i,t}={f}_{i,zv}\left({V}_{i,t}\right)$ (18)

2) 尾水位泄量约束

${Z}_{i,t}^{d}={f}_{i,zqz}\left({Q}_{i,t},{Z}_{i+1,t}\right)$ (19)

3) 机组动力特性约束

${p}_{i,n,t}={f}_{i,n,pqh}\left({q}_{i,n,t},{h}_{i,n,t}\right)$ (20)

4) 机组振动区限制

$\left({p}_{i,n,t}-\stackrel{¯}{{p}_{i,n}^{k}}\right)\left({p}_{i,n,t}-\underset{_}{{p}_{i,n}^{k}}\right)\ge 0$ (21)

3. 求解算法

3.1. 水位库容约束处理

$\left\{\begin{array}{l}{Z}_{i}^{a}={f}_{i,zv}\left({V}_{i}^{a}\right)\\ \underset{_}{{V}_{i}}={V}_{i}^{0}<{V}_{i}^{1}<\cdots <{V}_{i}^{a}<\cdots <{V}_{i}^{A}=\stackrel{¯}{{V}_{i}}\end{array}$ (22)

$\underset{a=1}{\overset{A}{\sum }}{r}_{i,t}^{a}=1$ (23)

${r}_{i,t}^{a}\in \left\{0,1\right\}$ (24)

${r}_{i,t}^{a}{V}_{i}^{a-1}\le {V}_{i,t}^{a}\le {r}_{i,t}^{a}{V}_{i}^{a}$ (25)

$\underset{a=1}{\overset{A}{\sum }}{V}_{i,t}^{a}={V}_{i}^{a}$ (26)

${Z}_{i,t}=\underset{a=1}{\overset{A}{\sum }}\left[{r}_{i,t}^{a}{Z}_{i}^{a-1}+\frac{{Z}_{i}^{a}-{Z}_{i}^{a-1}}{{V}_{i}^{a}-{V}_{i}^{a-1}}\cdot \left({V}_{i,t}^{a}-{r}_{i,t}^{a}{V}_{i}^{a-1}\right)\right]$ (27)

3.2. 尾水位泄量约束处理

Figure 1. Three-dimensional curves of tail water level - discharge - reservoir water level

3.2.1. 下游水库库水位离散

$\left\{\begin{array}{l}\underset{k=1}{\overset{3}{\sum }}{\beta }_{i+1,t}^{k}=1\\ {\beta }_{i+1,t}^{k}\in \left\{0,1\right\}\\ \underset{k=1}{\overset{3}{\sum }}{\beta }_{i+1,t}^{k}{Z}_{i+1}^{k-1}\le {Z}_{i+1,t}^{k}\le \underset{k=1}{\overset{3}{\sum }}{\beta }_{i+1,t}^{k}{Z}_{i+1}^{k}\end{array}$ (28)

Figure 2. Simplified curves of tail water level-discharge-reservoir water level

3.2.2. 下游水库坝前水位关联尾水位泄量曲线

$-\stackrel{¯}{{Z}_{i}^{d}}\left(1-{\beta }_{i+1,t}^{1}\right)\le {Z}_{i,t}^{d}-{f}_{i,zq|z}^{1}\left({Q}_{i,t}\right)\le \stackrel{¯}{{Z}_{i}^{d}}\left(1-{\beta }_{i+1,t}^{1}\right)$ (29)

$-\stackrel{¯}{{Z}_{i}^{d}}\left(1-{\beta }_{i+1,t}^{2}\right)\le {Z}_{i,t}^{d}-{f}_{i,zq|z}^{2}\left({Q}_{i,t}\right)\le \stackrel{¯}{{Z}_{i}^{d}}\left(1-{\beta }_{i+1,t}^{2}\right)$ (30)

$-\stackrel{¯}{{Z}_{i}^{d}}\left(1-{\beta }_{i+1,t}^{3}\right)\le {Z}_{i,t}^{d}-{f}_{i,zq|z}^{3}\left({Q}_{i,t}\right)\le \stackrel{¯}{{Z}_{i}^{d}}\left(1-{\beta }_{i+1,t}^{3}\right)$ (31)

3.2.3. 尾水位泄量曲线线性化转换

3.2.2节中被选定的尾水位泄量二维曲线仍然为非线性曲线，不可直接运用到MILP模型中参与计算。参考3.1节中水位库容约束的处理方法，离散下泄流量，线性插值得到相应的尾水位，实现尾水位泄量曲线的线性处理。详细步骤此处不再赘述。

3.3. 机组动力特性约束处理

3.4. 机组振动区限制处理

$\left\{\begin{array}{l}\underset{k=1}{\overset{K+1}{\sum }}{\alpha }_{i,n,t}^{k}={u}_{i,n,t}\\ {\alpha }_{i,n,t}^{k}\in \left\{0,1\right\}\\ \underset{k=1}{\overset{K+1}{\sum }}{\alpha }_{i,n,t}^{k}\underset{_}{{p}_{i,n,f}^{k}}\le {p}_{i,n,t}\le \underset{k=1}{\overset{K+1}{\sum }}{\alpha }_{i,n,t}^{k}\stackrel{¯}{{p}_{i,n,f}^{k}}\end{array}$ (32)

4. 应用实例

4.1. 工程背景

Table 1. Characteristic parameters of cascaded hydropower plants

4.2. 计算结果及分析

4.2.1 . 汛期计算结果分析

(a) 天生桥一级 (b) 天生桥二级

Figure 3. Power output and water level of hydropower plants in flood season

(a) 天生桥一级1#机组 (b) 天生桥二级1#机组

Figure 4. Power output of units in flood season

4.2.2. 枯期计算结果分析

(a) 天生桥一级 (b) 天生桥二级

Figure 5. Power output and water level of hydropower plants in dry season

4.2.3. 考虑回水与未考虑回水模型尾水位对比分析

(a) 汛期 (b) 枯期

Figure 6. Comparison of tail water level with and without considering backwater

5. 结语

1) 本文构建了“一库多级”水电站群短期调度MILP模型，重点处理了梯级水头联系：尾水位–泄量–库水位三维曲线。

2) 针对模型中的水位库容关系、尾水位–泄量–库水位关系、机组动力特性曲线以及机组振动区等非线性约束，采用分段线性化方法拟合，保留了原约束的特性，较好地贴近实际工况。

3) 以机组为基本调度单元，能有效满足振动区、机组开停机持续时间及最大开机次数约束等约束，保证了机组的稳定运行。

4) 采用Lingo软件求解红水河梯级实例，尽管模型中的约束和整数变量数目庞大，仍可在可接受的时间范围内求得全局最优解，为涉及回水问题的“一库多级”水电站群短期调度提供一种可靠的解决方法。