# Long-Term Scheduling of Large-Scale Cascade Hydropower Stations Using Improved Differential Evolution Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Optimization Model

#### 2.1. Objective Function

#### 2.2. Constraints

- Water balance constraint.$$\begin{array}{l}{V}_{i,t+1}={V}_{i,t}+\left({I}_{i,t}-{Q}_{i,t}-{S}_{i,t}\right)\Delta t,\\ {I}_{i,t}={q}_{i,t}+{Q}_{i-1,t}+{S}_{i-1,t}\end{array}$$
- Hydraulic connection.$${Z}_{i,t}^{down}=\{\begin{array}{l}F\left({Q}_{i,t}+{S}_{i,t}\right)\text{}\mathrm{without}\text{}\mathrm{backwater}\text{}\mathrm{effect},\\ F\left({Q}_{i,t}+{S}_{i,t},\text{}{Z}_{i+1,t}\right)\text{}\mathrm{with}\text{}\mathrm{backwater}\text{}\mathrm{effect}.\end{array}$$
- Water level constraint.$${Z}_{i,t}^{min}\le {Z}_{i,t}\le {Z}_{i,t}^{max}$$$$\left|{Z}_{i,t}-{Z}_{i,t+1}\right|\le \Delta {Z}_{i}$$
- Power generating constraint.$${N}_{i,t}^{min}\le {N}_{i,t}\le {N}_{i,t}^{max}\left({H}_{i,t}\right)$$
- Outflow constraint.$${Q}_{i,t}^{min}\le {Q}_{i,t}+{S}_{i,t}\le {Q}_{i,t}^{max}$$
- Water head equation.$${H}_{i,t}=\left({Z}_{i,t}+{Z}_{i,t+1}\right)/2-{Z}_{i,t}^{\mathrm{down}}-{H}_{i,t}^{loss}\left({Q}_{i,t}\right)$$
- Boundary condition.$${\mathrm{Z}}_{i,0}={Z}_{i}^{begin},\text{}{Z}_{i,T}={Z}_{i}^{end}$$

## 3. Overview of iLSHADE

#### 3.1. DE

- “rand/2”:$${\overrightarrow{v}}_{i,G}={\overrightarrow{x}}_{r1,\mathrm{G}}+F\cdot ({\overrightarrow{x}}_{r2,\mathrm{G}}-{\overrightarrow{x}}_{r3,\mathrm{G}})+F\cdot ({\overrightarrow{x}}_{r4,\mathrm{G}}-{\overrightarrow{x}}_{r5,\mathrm{G}})$$
- “best/1”:$${\overrightarrow{v}}_{i,G}={\overrightarrow{x}}_{best,\mathrm{G}}+F\cdot ({\overrightarrow{x}}_{r1,\mathrm{G}}-{\overrightarrow{x}}_{r2,\mathrm{G}})$$
- “best/2”:$${\overrightarrow{v}}_{i,G}={\overrightarrow{x}}_{i,\mathrm{G}}+F\cdot ({\overrightarrow{x}}_{best,\mathrm{G}}-{\overrightarrow{x}}_{i,\mathrm{G}})+F\cdot ({\overrightarrow{x}}_{r1,\mathrm{G}}-{\overrightarrow{x}}_{r2,\mathrm{G}})$$
- “current to best/1”:$${\overrightarrow{v}}_{i,G}={\overrightarrow{x}}_{i,\mathrm{G}}+F\cdot ({\overrightarrow{x}}_{best,\mathrm{G}}-{\overrightarrow{x}}_{i,\mathrm{G}})+F\cdot ({\overrightarrow{x}}_{r1,\mathrm{G}}-{\overrightarrow{x}}_{r2,\mathrm{G}})$$

#### 3.2. iLSHADE

#### 3.2.1. Mutation Strategy “Current to pbest/2-rand”

#### 3.2.2. The PIRFE Strategy

#### 3.2.3. Control Parameters Assignments

## 4. Numerical Experiment

- Using “current to pbest/2-rand” mutation strategy,
- The p value for mutation strategy is computed as ${p}_{G}=\mathrm{rand}[{p}_{\mathrm{min}},{p}_{\mathrm{max}}]$, where ${p}_{\mathrm{min}}=2/NP$ is set such that when ${\overrightarrow{x}}_{pbest,\mathrm{G}}$ is selected, at least 2 individuals are needed, and ${p}_{\mathrm{max}}=0.25$.
- Initial population size ${N}_{init}=15log\left(D\right)\sqrt{D}$, the control parameter of external archive size ${r}^{arc}=2$.
- Historical memory size H = 6; set a final pair of parameters ${M}_{F}[\mathrm{H}]=0.2$ and ${M}_{CR}[\mathrm{H}]=0.8$, other ${M}_{F}$ values are initialized to 0.5 and other ${M}_{CR}$ are initialized to 0.8.
- PIRFE parameter $LEG$ = 50.

## 5. Implementation of iLSHADE for LSLCHS

#### 5.1. Solution Structure and Initial Population

#### 5.2. Constraint Handling

## 6. Case Study

#### 6.1. Description of Case Study

^{2}basin area flowing through the provinces of Qinghai, Sichuan, and Yunnan in western China (See in Figure 5). Along the river, there are four large hydropower stations with large installed capacity, huge regulating storage and high water head. The total installed capacity of the four large hydropower stations is twice more than the Three Gorges Project (the largest hydropower station in the world). The main parameters of these hydropower stations are listed in Table 4.

#### 6.2. Results and Analysis

^{8}KWh) in wet year, 3.03, 7.37, 3.76 (10

^{8}KWh) in normal year, 3.48, 5.68, 1.96 (10

^{8}KWh) in dry year. Obviously, the proposed iLSHADE is superior when solving LSLCHS problem by obtaining the maximal benefit of power production efficiently. In particular, the standard deviation of 51 independent simulations in iLSHADE is 0.02 in wet year, 0.01 in normal year, 0.01 in dry year, which shows that the convergence stability of iLSHADE is better than other algorithms. Meanwhile, it can be seen easily from Figure 6a that iLSHADE can avoid premature convergence effectively, at the same evaluation times keep a fast convergence speed compared to LSHADE, CoDE and JADE. Figure 6b depicts that the ${\phi}_{CRO}\left(\overrightarrow{x}\right)$ of iLSHADE and LSHADE frequent changes and always lower than ${\epsilon}_{CRO}\left(t\right)$, until evaluation times is greater than ${T}_{c}$, ${\phi}_{CRO}\left(\overrightarrow{x}\right)$ is limited to 0.

## 7. Conclusions

^{8}KWh) in a normal year. In particular, the standard deviation of 51 independent simulations in iLSHADE is far lower than other algorithms. Moreover, according to its successful simulation performance with the historical runoff data from 1959 to 2014, iLSHADE can obtain better schedule results with lager generation benefits and better convergence property compared to LSHADE, JADE and CoDE. Above all, iLSHADE is a valid and reliable tool in solving the LSLCHS problem. Future research should consider the iLSHADE algorithm combined with other methods when solving multi objective scheduling problems in LSLCHS problem.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 8.**Historical runoff data from 1959 to 2014 for the annual power production increase that iLSHADE compares to LSHADE, JADE and CoDE.

Benchmark Function | Name | Domain | O-V | O-S |
---|---|---|---|---|

${f}_{1}={\displaystyle {\sum}_{i=1}^{n}{x}_{i}^{2}}$ | Sphere | ${\left[-100,100\right]}^{n}$ | 0 | $\left\{0,0,\cdots ,0\right\}$ |

${f}_{2}={\displaystyle {\sum}_{i=1}^{n}\left|{x}_{i}\right|+{\displaystyle {\prod}_{i}^{n}\left|{x}_{i}\right|}}$ | Schwefel (2.2) | ${\left[-100,100\right]}^{n}$ | 0 | $\left\{0,0,\cdots ,0\right\}$ |

${f}_{3}={{\displaystyle {\sum}_{i=1}^{n}\left({\displaystyle {\sum}_{j=1}^{i}{x}_{j}}\right)}}^{2}$ | Schwefel (1.2) | ${\left[-100,100\right]}^{n}$ | 0 | $\left\{0,0,\cdots ,0\right\}$ |

${f}_{4}={\displaystyle {\sum}_{i=1}^{n-1}\left[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}\right]}$ | Rosenbrock | ${\left[-30,30\right]}^{n}$ | 0 | $\left\{0,0,\cdots ,0\right\}$ |

${f}_{5}={{\displaystyle {\sum}_{i=1}^{n}\left(\lfloor {x}_{j}+0.5\rfloor \right)}}^{2}$ | Step | ${\left[-100,100\right]}^{n}$ | 0 | $\left\{0,0,\cdots ,0\right\}$ |

${f}_{6}={\displaystyle {\sum}_{i=1}^{n}i{x}_{i}^{4}}$ | Quartic | ${\left[-1.28,1.28\right]}^{n}$ | 0 | $\left\{0,0,\cdots ,0\right\}$ |

${f}_{7}={\displaystyle {\sum}_{i=1}^{n}-{x}_{i}\mathrm{sin}\left(\sqrt{\left|{x}_{i}\right|}\right)}$ | Schwefel (2.26) | ${\left[-500,500\right]}^{n}$ | −418.9n | * |

${f}_{8}={\displaystyle {\sum}_{i=1}^{n}\left[{x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)+10\right]}$ | Rastrigin | ${\left[-5.12,5.12\right]}^{n}$ | 0 | $\left\{0,0,\cdots ,0\right\}$ |

$\begin{array}{ll}\hfill {f}_{9}=& -20\mathrm{exp}\left(-0.2\sqrt{1/n{\displaystyle {\sum}_{i=1}^{n}{x}_{i}^{2}}}\right)\\ & -\mathrm{exp}\left(1/n\mathrm{cos}\left(2\pi {x}_{i}\right)\right)+20+e\end{array}$ | Ackley | ${\left[-32,32\right]}^{n}$ | 0 | $\left\{0,0,\cdots ,0\right\}$ |

${f}_{10}=1+{\displaystyle {\sum}_{i=1}^{n}\frac{{x}_{i}^{2}}{4000}+{\displaystyle {\prod}_{i}^{n}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)}}$ | Griewank | ${\left[-600,600\right]}^{n}$ | 0 | $\left\{0,0,\cdots ,0\right\}$ |

$\mathit{f}$ | LSHADE Mean (Std Dev) | JADE Mean (Std Dev) | CoDE Mean (Std Dev) | jDE Mean (Std Dev) | DE Mean (Std Dev) | iLSHADE Mean (Std Dev) |
---|---|---|---|---|---|---|

${f}_{1}$ | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 8.61 × 10^{−36}(7.18 × 10 ^{−36}) − | 5.93 × 10^{−38}(7.34 × 10 ^{−38}) − | 1.15 × 10^{−38}(1.18 × 10 ^{−38}) − | 2.73 × 10^{−46}(1.42 × 10 ^{−45}) − | 0.00 × 10^{0}(0.00 × 10 ^{0}) |

${f}_{2}$ | 6.49 × 10^{−49}(4.38 × 10 ^{−48}) − | 8.75 × 10^{−20}(4.68 × 10 ^{−20}) − | 7.88 × 10^{−22}(6.58 × 10 ^{−22}) − | 1.38 × 10^{−22}(1.01 × 10 ^{−22}) − | 8.03 × 10^{−25}(1.65 × 10 ^{−24}) − | 3.36 × 10^{−64}(2.05 × 10 ^{−63}) |

${f}_{3}$ | 1.10 × 10^{−91}(5.90 × 10 ^{−91}) − | 2.81 × 10^{−35}(2.75 × 10 ^{−35}) − | 2.43 × 10^{−39}(4.02 × 10 ^{−39}) − | 1.24 × 10^{−40}(1.79 × 10 ^{−40}) − | 4.58 × 10^{−45}(2.03 × 10 ^{−44}) − | 0.00 × 10^{0}(0.00 × 10 ^{0}) |

${f}_{4}$ | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 1.47 × 10^{−05}(1.96 × 10 ^{−05}) − | 7.39 × 10^{−03}(9.11 × 10 ^{−03}) − | 7.82 × 10^{−02}(5.53 × 10 ^{−01}) − | 0.00 × 10^{0}(0.00 × 10 ^{0}) |

${f}_{5}$ | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 0.00 × 10^{0}(0.00 × 10 ^{0})≈ | 0.00 × 10^{0}(0.00 × 10 ^{0}) |

${f}_{6}$ | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 1.40 × 10^{−71}(3.27 × 10 ^{−71}) − | 6.47 × 10^{−71}(1.52 × 10 ^{−70}) − | 1.40 × 10^{−72}(3.66 × 10 ^{−72}) − | 8.39 × 10^{−88}(4.17 × 10 ^{−87}) − | 0.00 × 10^{0}(0.00 × 10 ^{0}) |

${f}_{7}$ | −4189.83 (2.73 × 10 ^{−12}) ≈ | −4189.83 (2.73 × 10 ^{−12}) ≈ | −4189.83 (2.73 × 10 ^{−12}) ≈ | −4189.83 (2.73 × 10 ^{−12}) ≈ | −4189.83 (2.73 × 10 ^{−12}) ≈ | −4189.83 (2.73 × 10 ^{−12}) |

${f}_{8}$ | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 3.06 × 10^{0}(2.47 × 10 ^{0}) − | 0.00 × 10^{0}(0.00 × 10 ^{0}) |

${f}_{9}$ | 3.72 × 10^{−15}(9.55 × 10 ^{−16}) ≈ | 3.86 × 10^{−15}(6.90 × 10 ^{−16}) − | 4.00 × 10^{−15}(2.37 × 10 ^{−30}) − | 3.93 × 10^{−15}(4.93 × 10 ^{−16}) − | 3.93 × 10^{−15}(4.93 × 10 ^{−16}) − | 3.72 × 10^{−15}(9.55 × 10 ^{−16}) |

${f}_{10}$ | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 2.42 × 10^{−12}(6.51 × 10 ^{−12}) − | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 3.22 × 10^{−04}(2.28 × 10 ^{−03}) − | 8.58 × 10^{−02}(6.17 × 10 ^{−02}) − | 0.00 × 10^{0}(0.00 × 10 ^{0}) |

− | 2 | 6 | 6 | 7 | 8 | |

+ | 0 | 0 | 0 | 0 | 0 | |

≈ | 8 | 4 | 4 | 3 | 2 |

$\mathit{f}$ | LSHADE Mean (Std Dev) | JADE Mean (Std Dev) | CoDE Mean (Std Dev) | jDE Mean (Std Dev) | DE Mean (Std Dev) | iLSHAD EMean (Std Dev) |
---|---|---|---|---|---|---|

${f}_{1}$ | 1.12 × 10^{−90}(6.44 × 10 ^{−90}) − | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 9.85 × 10^{−19}(7.04 × 10 ^{−19}) − | 4.31 × 10^{−41}(4.35 × 10 ^{−41}) − | 9.34 × 10^{−44}(2.74 × 10 ^{−43}) − | 0.00 × 10^{0}(0.00 × 10 ^{0}) |

${f}_{2}$ | 2.09 × 10^{−42}(1.03 × 10 ^{−41}) − | 4.11 × 10^{−27}(4.89 × 10 ^{−27}) − | 4.01 × 10^{−12}(1.34 × 10 ^{−12}) − | 3.48 × 10^{−24}(1.96 × 10 ^{−24}) − | 1.16 × 10^{−05}(8.17 × 10 ^{−05}) − | 4.88 × 10^{−58}(1.55 × 10 ^{−57}) |

${f}_{3}$ | 3.85 × 10^{−81}(1.74 × 10 ^{−80}) − | 3.58 × 10^{−49}(7.53 × 10 ^{−49}) − | 4.22 × 10^{−19}(3.41 × 10 ^{−19}) − | 7.27 × 10^{−43}(1.05 × 10 ^{−42}) − | 1.16 × 10^{−46}(6.85 × 10 ^{−46}) − | 0.00 × 10^{0}(0.00 × 10 ^{0}) |

${f}_{4}$ | 1.40 × 10^{−25}(9.70 × 10 ^{−25}) − | 1.85 × 10^{+01}(1.01 × 10 ^{+01}) − | 1.82 × 10^{+01}(3.29 × 10 ^{0}) − | 1.15 × 10^{+01}(8.23 × 10 ^{0}) − | 2.62 × 10^{0}(2.60 × 10 ^{0}) − | 0.00 × 10^{0}(0.00 × 10 ^{0}) |

${f}_{5}$ | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 1.96 × 10^{−02}(1.39 × 10 ^{−01}) − | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 4.22 × 10^{0}(7.61 × 10 ^{0}) − | 0.00 × 10^{0}(0.00 × 10 ^{0}) |

${f}_{6}$ | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 2.35 × 10^{−33}(2.93 × 10 ^{−33}) − | 1.79 × 10^{−69}(3.86 × 10 ^{−69}) − | 1.74 × 10^{−59}(1.10 × 10 ^{−58}) − | 0.00 × 10^{0}(0.00 × 10 ^{0}) |

${f}_{7}$ | −12,569.49 (1.82 × 10 ^{−12}) ≈ | −12,567.16 (1.64 × 10 ^{+01}) − | −12,569.49 (1.82 × 10 ^{−12}) ≈ | −12,569.49 (1.82 × 10 ^{−12}) ≈ | −11552.14 (3.68 × 10 ^{+02}) − | −12,569.49 (1.88 × 10 ^{−05}) |

${f}_{8}$ | 1.74 × 10^{−16}(6.35 × 10 ^{−16}) + | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 8.38 × 10^{−12}(9.18 × 10 ^{−12}) + | 4.83 × 10^{0}(3.86 × 10 ^{0}) − | 3.62 × 10^{+01}(1.44 × 10 ^{+01}) − | 3.16 × 10^{−11}(1.66 × 10 ^{−10}) |

${f}_{9}$ | 4.00 × 10^{−15}(2.37 × 10 ^{−30}) ≈ | 4.76 × 10^{−15}(1.46 × 10 ^{−15}) − | 2.74 × 10^{−10}(1.08 × 10 ^{−10}) − | 5.60 × 10^{−15}(1.77 × 10 ^{−15}) − | 2.64 × 10^{−01}(5.21 × 10 ^{−01}) − | 4.00 × 10^{−15}(2.37 × 10 ^{−30}) |

${f}_{10}$ | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 1.55 × 10^{−03}(3.81 × 10 ^{−03}) − | 3.05 × 10^{−17}(1.08 × 10 ^{−16}) − | 0.00 × 10^{0}(0.00 × 10 ^{0}) ≈ | 7.99 × 10^{−03}(1.46 × 10 ^{−02}) − | 0.00 × 10^{0}(0.00 × 10 ^{0}) |

− | 4 | 6 | 7 | 7 | 10 | |

+ | 1 | 1 | 1 | 0 | 0 | |

≈ | 5 | 3 | 2 | 3 | 0 |

Parameter | Wudongde | Baihetan | Xiluodu | Xiangjiaba |
---|---|---|---|---|

Adjustment ability | Season | Annual | Annual | Season |

Regulating storage (billion m^{3}) | 2.60 | 10.40 | 6.46 | 0.90 |

Hydro plant discharge range (m^{3}/s) | [49,400, 906] | [49,700, 905] | [43,700, 1500] | [49,800, 1500] |

Upriver water level range (m) | [975, 945] | [825, 765] | [600, 540] | [380, 370] |

Installed capacity (MW) | 12000 | 16000 | 13860 | 6400 |

Normal water level (m) | 975 | 825 | 600 | 380 |

Method | Wet Year (1999) | Normal Year (2008) | Dry Year (1969) | ||||||
---|---|---|---|---|---|---|---|---|---|

Max | Mean | Std | Max | Mean | Std | Max | Mean | Std | |

iLSHADE | 2425.03 | 2425.01 | 0.02 | 2268.13 | 2268.11 | 0.01 | 1814.36 | 1814.35 | 0.01 |

LSHADE | 2423.86 | 2422.99 | 0.48 | 2266.88 | 2265.08 | 0.94 | 1812.64 | 1810.87 | 0.99 |

Diff | 1.17 | 2.02 | 1.25 | 3.03 | 1.72 | 3.48 | |||

JADE | 2424.43 | 2420.97 | 1.32 | 2267.51 | 2260.74 | 2.43 | 1814.00 | 1808.67 | 2.234 |

Diff | 0.6 | 4.04 | 0.62 | 7.37 | 0.36 | 5.68 | |||

CoDE | 2423.39 | 2422.62 | 0.30 | 2265.62 | 2264.35 | 0.64 | 1813.08 | 1812.39 | 0.39 |

Diff | 1.64 | 2.39 | 2.51 | 3.76 | 1.28 | 1.96 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wen, X.; Zhou, J.; He, Z.; Wang, C.
Long-Term Scheduling of Large-Scale Cascade Hydropower Stations Using Improved Differential Evolution Algorithm. *Water* **2018**, *10*, 383.
https://doi.org/10.3390/w10040383

**AMA Style**

Wen X, Zhou J, He Z, Wang C.
Long-Term Scheduling of Large-Scale Cascade Hydropower Stations Using Improved Differential Evolution Algorithm. *Water*. 2018; 10(4):383.
https://doi.org/10.3390/w10040383

**Chicago/Turabian Style**

Wen, Xiaohao, Jianzhong Zhou, Zhongzheng He, and Chao Wang.
2018. "Long-Term Scheduling of Large-Scale Cascade Hydropower Stations Using Improved Differential Evolution Algorithm" *Water* 10, no. 4: 383.
https://doi.org/10.3390/w10040383