# Study on the Optimal Operation of a Hydropower Plant Group Based on the Stochastic Dynamic Programming with Consideration for Runoff Uncertainty

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

^{8}$\mathrm{k}\mathrm{W}\cdot \mathrm{h}$ and 306.91 × 10

^{8}$\mathrm{k}\mathrm{W}\cdot \mathrm{h}$), which proved that the operation diagram constructed in this study was reasonable.

## 1. Introduction

## 2. Study Area and Data

^{2}and an annual outbound water volume of 765 × 10

^{8}m

^{3}[27]. There are seven cascade hydropower plants on the Lancang River, including the Gongguoqiao (GGQ), Xiaowan (XW), Manwan (MW), Dachaoshan (DCS), Nuozhadu (NZD), Jinghong (JH), and Ganlanba (GLB) power plants. The lower reaches of the Lancang River include three hydropower plants: namely, NZD, JH, and GLB. The NZD hydropower plant has a multi-year regulation capacity, and its comprehensive utilization tasks are mainly for power generation, irrigation, flood control, shipping, ecology, tourism, etc. The JH hydropower plant is located 5 km away from the northern suburb of Jinghong city, and its main task is power generation, taking shipping, flood control, tourism, etc. into account. GLB is the reverse regulation power plant of the JH hydropower plant, and its main task is power generation, taking the needs of shipping and ecological water into account [28].

## 3. Methods

#### 3.1. Construction of Joint Distribution Model for Monthly Runoff

#### 3.1.1. Marginal Distribution Model of Incoming Runoff

#### 3.1.2. Joint Distribution Model of Adjacent Monthly Incoming Runoff

_{X}(x), F

_{Y}(y). According to Sklar’s theorem, the joint distribution of X, Y can be described by the 2-D copula function C:

#### 3.2. Stochastic Simulation Model of Runoff

_{0}, ${a}_{1}\in (0,1)$.

#### 3.3. Construction of Stochastic Operation Model and Solution

#### 3.3.1. Objective Function

#### 3.3.2. Constraint Condition

- Water balance constraint:$${V}_{t}={V}_{t-1}+({q}_{t}-{Q}_{t})\cdot \Delta t-{S}_{t}$$
- Discharge constraint:$${Q}_{t,min}\le {Q}_{t}\le {Q}_{t,max}$$
- Water level constraint:$${Z}_{t,min}\le {Z}_{t}\le {Z}_{t,max}$$
- Output constraint:$${N}_{min}\le {N}_{t}\le {N}_{max}$$
- The minimum ecological flow constraints:$${Q}_{t}\le bas{e}_{\left(i,t\right)}$$
- The minimum shipping flow constraints:

#### 3.3.3. Solution Algorithm

## 4. Results and Discussion

#### 4.1. Stochastic Simulation of Runoff

#### 4.1.1. Marginal Distribution for Incoming Runoff

#### 4.1.2. Copula Joint Distribution for Incoming Runoff

#### 4.1.3. Stochastic Simulation of Incoming Runoff

^{8}m

^{3}/s) was 0.7 10

^{8}m

^{3}/s less than the measured runoff (537.9 10

^{8}m

^{3}/s). The standard deviation of the simulated runoff (73.27 10

^{8}m

^{3}/s) was reduced by 4.71 10

^{8}m

^{3}/s compared with the standard deviation of the measured runoff (77.98 10

^{8}m

^{3}/s). According to the measured annual mean runoff of ±1 standard deviation as a reasonable range, the simulated annual runoff was statistically analyzed (in Figure 7). The proportion of the simulated annual runoff exceeding this range was 13.98%, and the proportion lower than this range was 14.9%. Most of the simulated runoff values were within this range, accounting for 71.12%, showing the uncertainty of the incoming water process.

#### 4.2. Process of Incoming Runoff

#### 4.2.1. Transfer Probability Matrix Based on Measured Runoff

#### 4.2.2. Transfer Probability Matrix based on Simulated Runoff

#### 4.3. Result of Hydropower Plants Operation

^{8}$\mathrm{k}\mathrm{W}\cdot \mathrm{h}$ and 306.91 × 10

^{8}$\mathrm{k}\mathrm{W}\cdot \mathrm{h}$, respectively. The total power generation and operation process of the SDP was same as those of the DP, which proved to be reliable for the SDP.

## 5. Conclusions

^{8}$\mathrm{k}\mathrm{W}\cdot \mathrm{h}$ and 306.91 × 10

^{8}$\mathrm{k}\mathrm{W}\cdot \mathrm{h}$), which proved that the operation diagram constructed in this study was reasonable.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

- Chang, J.; Meng, X.; Wang, Z.; Wang, X.; Huang, Q. Optimized cascade reservoir operation considering ice flood control and power generation. J. Hydrol.
**2014**, 519, 1042–1051. [Google Scholar] [CrossRef] - Chang, J.; Guo, A.; Du, H.; Wang, Y. Floodwater utilization for cascade reservoirs based on dynamic control of seasonal flood control limit levels. Environ. Earth Sci.
**2017**, 76, 260. [Google Scholar] [CrossRef] - Zhang, L.; Huang, Q.; Liu, D.; Deng, M.; Zhang, H.; Pan, B.; Zhang, H. Long-term and mid-term ecological operation of cascade hydropower plants considering ecological water demands in arid region. J. Clean. Prod.
**2021**, 279, 123599. [Google Scholar] [CrossRef] - Wang, Y.M.; Chang, J.; Huang, Q. Simulation with RBF Neural Network Model for Reservoir Operation Rules. Water Resour. Manag.
**2010**, 24, 2597–2610. [Google Scholar] [CrossRef] - Ponnambalam, K.; Vannelli, A.; Unny, T.E. An application of Karmarkar’s interior-point linear programming algorithm for multi-reservoir operations optimization. Stoch. Hydrol. Hydraul.
**1989**, 3, 17–29. [Google Scholar] [CrossRef] - Oliveira, R.; Loucks, D.P. Operating rules for multi-reservoir systems. Water Resour. Res.
**1997**, 33, 839–852. [Google Scholar] [CrossRef] - Chandramouli, V.; Raman, H. Multireservoir modeling with dynamic programming and neural networks. J. Water Resour. Plan. Manag.
**2001**, 127, 89–98. [Google Scholar] - Arunkumar, R.; Jothiprakash, V. Chaotic evolutionary algorithms for multi-reservoir optimization. Water Resour. Manag.
**2013**, 27, 5207–5222. [Google Scholar] [CrossRef] - Na, Y.; Ke, Z.; Yang, H.; Zhao, Q.; Huang, Q.; Xue, Y.; Xue, X.; Chen, S. Evaluation of the TRMM multisatellite precipitation analysis and its applicability in supporting reservoir operation and water resources management in Hanjiang basin, China. J. Hydrol.
**2017**, 549, 313–325. [Google Scholar] - Li, L.; Xu, H.; Chen, X.; Simonovic, S.P. Streamflow Forecast and Reservoir Operation Performance Assessment under Climate Change. Water Resour. Manag.
**2010**, 24, 83–104. [Google Scholar] [CrossRef] - Tegegne, G.; Kim, Y.O. Representing inflow uncertainty for the development of monthly reservoir operations using genetic algorithms. J. Hydrol.
**2020**, 586, 124876. [Google Scholar] [CrossRef] - Xu, B.; Huang, X.; Zhong, P.; Wu, Y. Two-phase risk hedging rules for informing conservation of flood resources in reservoir operation considering inflow forecast uncertainty. Water Resour. Manag.
**2020**, 34, 2731–2752. [Google Scholar] [CrossRef] - Galletti, A.; Avesani, D.; Bellin, A.; Majone, B. Detailed simulation of storage hydropower systems in large Alpine watersheds. J. Hydrol.
**2021**, 603 Pt D, 127125. [Google Scholar] [CrossRef] - Brekke, L.D.; Maurer, E.P.; Anderson, J.D.; Dettinger, M.D.; Townsley, E.S.; Harrison, A.; Pruitt, T. Assessing reservoir operations risk under climate change. Water Resour. Res.
**2009**, 45, 546–550. [Google Scholar] [CrossRef][Green Version] - Celeste, A.B.; Billib, M. Evaluation of stochastic reservoir operation optimization models. Adv. Water Resour.
**2009**, 32, 1429–1443. [Google Scholar] [CrossRef] - Archibald, T.W.; Buchanan, C.S.; Mckinnon, K.; Thomas, L. Nested Benders decomposition and dynamic programming for reservoir optimisation. J. Oper. Res. Soc.
**1999**, 50, 468–479. [Google Scholar] [CrossRef] - Kumar, D.N.; Baliarsingh, F. Folded dynamic programming for optimal operation of multireservoir system. Water Resour. Manag.
**2003**, 17, 337–353. [Google Scholar] [CrossRef] - Ganji, A.; Khalili, D.; Karamouz, M.; Ponnambalam, K.; Javan, M. A Fuzzy Stochastic Dynamic Nash Game Analysis of Policies for Managing Water Allocation in a Reservoir System. Water Resour. Manag.
**2008**, 22, 51–66. [Google Scholar] [CrossRef] - Nikoo, M.R.; Kerachian, R.; Karimi, A.; Azadnia, A.A.; Jafarzadegan, K. Optimal water and waste load allocation in reservoir–river systems: A case study. Environ. Earth Sci.
**2014**, 71, 4127–4142. [Google Scholar] [CrossRef] - Yang, Z.; Wang, Y.; Yang, K. The stochastic short-term hydropower generation scheduling considering uncertainty in load output forecasts. Energy
**2022**, 241, 122838. [Google Scholar] [CrossRef] - Yang, Z.; Yan, K.; Wang, Y.; Su, L.; Hu, H. Multi-objective short-term hydropower generation operation for cascade reservoirs and stochastic decision making under multiple uncertainties. J. Clean. Prod.
**2020**, 276, 122995. [Google Scholar] [CrossRef] - Little, J.D.C. The Use of Storage Water in a Hydroelectric System. J. Oper. Res. Soc. Am.
**1955**, 3, 187–197. [Google Scholar] [CrossRef][Green Version] - Howard, R.A. Dynamic Programming and Markov Processes. Math. Gaz.
**1960**, 3, 120. [Google Scholar] - Rossman, L.A. Reliability constrained dynamic programing and randomized release rules in reservoir management. Water Resour. Res.
**1977**, 13, 247–255. [Google Scholar] [CrossRef] - Bras, R.L.; Buchanan, R.; Curry, K.C. Real-time adaptive closed loop control of reservoirs with the High Aswan Dam as a case study. Water Resour. Res.
**1983**, 19, 33–52. [Google Scholar] [CrossRef] - Saadat, M.; Asghari, K. Feasibility Improved Stochastic Dynamic Programming for Optimization of Reservoir Operation. Water Resour. Manag.
**2019**, 33, 3485–3498. [Google Scholar] [CrossRef] - Shi, W.; Yu, X.; Liao, W.; Wang, Y. Spatial and temporal variability of daily precipitation concentration in the Lancang River basin, China. J. Hydrol.
**2013**, 495, 197–207. [Google Scholar] [CrossRef] - Zhang, H.; Chang, J.; Gao, C.; Wu, H.; Wang, Y.; Lei, K.; Long, R.; Zhang, L. Cascade hydropower plants operation considering comprehensive ecological water demands. Energy Convers. Manag.
**2019**, 180, 119–133. [Google Scholar] [CrossRef] - Ogarekpe, N.M.; Tenebe, I.T.; Emenike, P.C.; Udodi, O.; Antigha, R.E.-E. Assessment of regional best-fit probability density function of annual maximum rainfall using CFSR precipitation data. J. Earth Syst. Sci.
**2020**, 129, 176. [Google Scholar] [CrossRef] - Hosseinifard, S.Z.; Abbasi, B.; Abdollahian, M. Performance Analysis in Non-Normal Linear Profiles Using Gamma Distribution. In Proceedings of the Eighth International Conference on Information Technology: New Generations, Las Vegas, NV, USA, 11–13 April 2011; IEEE: Piscataway, NJ, USA, 2011; pp. 603–607. [Google Scholar]
- Papalexiou, S.M.; Koutsoyiannis, D. A global survey on the seasonal variation of the marginal distribution of daily precipitation. Adv. Water Resour.
**2016**, 94, 131–145. [Google Scholar] - Zhang, L.; Singh, V.P. Bivariate Rainfall and Runoff Analysis Using Entropy and Copula Theories. Entropy
**2012**, 14, 1784–1812. [Google Scholar] [CrossRef] - Qian, L.; Wang, X.; Wang, Z. Modeling the dependence pattern between two precipitation variables using a coupled copula. Environ. Earth Sci.
**2020**, 79, 1–12. [Google Scholar] [CrossRef] - Abdollahi, S.; Akhoond-Ali, A.M.; Mirabbasi, R.; Adamowski, J.F. Probabilistic Event Based Rainfall-Runoff Modeling Using Copula Functions. Water Resour. Manag.
**2019**, 33, 3799–3814. [Google Scholar] [CrossRef] - Mirabbasi, R.; Fakheri-Fard, A.; Dinpashoh, Y. Bivariate drought frequency analysis using the copula method. Theor. Appl. Climatol.
**2012**, 108, 191–206. [Google Scholar] [CrossRef] - Tao, S.; Sheng, D.; Wang, N.; Soares, C.G. Estimating storm surge intensity with Poisson bivariate maximum entropy distributions based on copulas. Nat. Hazards
**2013**, 68, 791–807. [Google Scholar] [CrossRef] - Valle, L.D.; Giuli, M.D.; Tarantola, C.; Manelli, C. Default Probability Estimation via Pair Copula Constructions. Eur. J. Oper. Res.
**2016**, 249, 298–311. [Google Scholar] [CrossRef][Green Version] - Jackie, L. On the use of MCMC simulation for stochastic reserving. Aust. Actuar. J.
**2008**, 14, 227–271. [Google Scholar] - Ma, H.; Zhu, G.; Zhang, Y.; Pan, H.; Guo, H.; Jia, W.; Zhou, J.; Yong, L.; Wan, Q. The effects of runoff on Hydrochemistry in the Qilian Mountains: A case study of Xiying River Basin. Environ. Earth Sci.
**2019**, 78, 1866–6280. [Google Scholar] [CrossRef] - Nandalal, K.; Sakthivadivel, R. Planning and management of a complex water resource system: Case of Samanalawewa and Udawalawe reservoirs in the Walawe river, Sri Lanka. Agric. Water Manag.
**2002**, 57, 207–221. [Google Scholar] [CrossRef] - Jha, D.K.; Yorino, N.; Zoka, Y.; Sasaki, Y.; Hayashi, Y.; Iwata, K.; Oe, R. Backward search approach to incorporate excess stream inflows in SDP based reservoir scheduling of hydropower plants. In Proceedings of the 2009 IEEE/PES Power Systems Conference & Exposition, Seattle, WA, USA, 15–18 March 2009; IEEE: Piscataway, NJ, USA, 2009. [Google Scholar]
- Wu, X.; Cheng, C.; Lund, J.R.; Niu, W.; Miao, S. Stochastic dynamic programming for hydropower reservoir operations with multiple local optima. J. Hydrol.
**2018**, 564, 712–722. [Google Scholar] [CrossRef]

Index | Distribution | January | February | March | April | May | June |
---|---|---|---|---|---|---|---|

K-S test | Weibull | 0.240 | 0.749 | 0.314 | 0.854 | 0.539 | 0.420 |

Logn | 0.449 | 0.895 | 0.369 | 0.774 | 0.954 | 0.975 | |

Gamma | 0.531 | 0.958 | 0.315 | 0.888 | 0.970 | 0.954 | |

Norm | 0 | 0 | 0 | 0 | 0 | 0 | |

Gev | 0.602 | 0.999 | 0.235 | 0.994 | 0.863 | 0.598 | |

AIC | Weibull | 296.555 | 257.53 | 275.016 | 303.779 | 397.945 | 493.377 |

Logn | 286.587 | 255.956 | 262.653 | 300.312 | 392.247 | 483.081 | |

Gamma | 286.931 | 255.069 | 263.75 | 299.536 | 391.855 | 484.338 | |

Norm | 288.843 | 254.317 | 266.92 | 299.446 | 393.985 | 490.282 | |

Gev | 288.81 | 256.575 | 263.048 | 301.98 | 394.265 | 485.293 | |

RMSE | Weibull | 0.062 | 0.035 | 0.054 | 0.029 | 0.038 | 0.044 |

Logn | 0.04 | 0.029 | 0.038 | 0.037 | 0.025 | 0.022 | |

Gamma | 0.041 | 0.025 | 0.039 | 0.032 | 0.021 | 0.025 | |

Norm | 0.046 | 0.02 | 0.045 | 0.024 | 0.025 | 0.038 | |

Gev | 0.306 | 0.316 | 0.302 | 0.302 | 0.275 | 0.262 | |

Index | Distribution | July | August | September | October | November | December |

K-S test | Weibull | 0.364 | 0.229 | 0.471 | 0.373 | 0.15 | 0.104 |

Logn | 0.535 | 0.915 | 0.697 | 0.977 | 0.85 | 0.67 | |

Gamma | 0.431 | 0.722 | 0.624 | 0.907 | 0.606 | 0.498 | |

Norm | 0 | 0 | 0 | 0 | 0 | 0 | |

Gev | 0.291 | 0.293 | 0.547 | 0.517 | 0.216 | 0.23 | |

AIC | Weibull | 531.949 | 588.492 | 560.242 | 516.981 | 472.988 | 371.753 |

Logn | 523.338 | 578.995 | 547.004 | 505.39 | 451.806 | 351.651 | |

Gamma | 524.208 | 580.171 | 548.8 | 506.957 | 455.697 | 354.275 | |

Norm | 527.805 | 586.518 | 555.915 | 513.035 | 468.245 | 361.636 | |

Gev | 525.192 | 581.272 | 548.725 | 507.028 | 451.248 | 350.342 | |

RMSE | Weibull | 0.052 | 0.05 | 0.051 | 0.056 | 0.074 | 0.075 |

Logn | 0.038 | 0.021 | 0.027 | 0.024 | 0.031 | 0.033 | |

Gamma | 0.041 | 0.028 | 0.033 | 0.032 | 0.041 | 0.04 | |

Norm | 0.049 | 0.047 | 0.047 | 0.049 | 0.064 | 0.056 | |

Gev | 0.281 | 0.255 | 0.266 | 0.269 | 0.265 | 0.29 |

Monthly | Parameters | Copula | ||
---|---|---|---|---|

Clayton | Frank | Gumbel | ||

January–February | θ | 2.84 | 12.11 | 2.99 |

AIC | −74.30 | −94.26 | −85.33 | |

February–March | θ | 1.04 | 4.95 | 1.68 |

AIC | −24.75 | −29.79 | −25.47 | |

March–April | θ | 0.94 | 4.93 | 1.78 |

AIC | −19.82 | −29.76 | −32.66 | |

April–May | θ | 0.72 | 3.54 | 1.45 |

AIC | −15.08 | −15.07 | −11.72 | |

May–June | θ | 0.60 | 2.07 | 1.29 |

AIC | −6.74 | −4.90 | −7.35 | |

June–July | θ | 0.74 | 2.92 | 1.40 |

AIC | −10.76 | −12.24 | −12.72 | |

July–August | θ | 0.78 | 2.61 | 1.37 |

AIC | −7.86 | −9.88 | −11.78 | |

August–September | θ | 1.09 | 3.60 | 1.50 |

AIC | −20.51 | −16.76 | −18.56 | |

September–October | θ | 0.87 | 2.94 | 1.34 |

AIC | −13.81 | −11.10 | −9.92 | |

October–November | θ | 1.65 | 3.78 | 1.26 |

AIC | −34.69 | −17.03 | −3.12 | |

November–December | θ | 5.13 | 12.59 | 2.49 |

AIC | −109.38 | −91.74 | −62.23 | |

December–January | θ | 0.13 | 0.55 | 1.04 |

AIC | 1.43 | 1.55 | 1.62 |

**Table 3.**Representative runoff in each month for measured runoff of the NZD hydropower plant (10

^{8}m

^{3}).

Month | Representative Runoff QR | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

June | 67.2 | 60.9 | 53.1 | 51.5 | 48.1 | 44.9 | 41.5 | 37.3 | 34.1 | 27.6 |

July | 112.9 | 102.4 | 95.3 | 88.0 | 82.7 | 75.9 | 74.3 | 70.1 | 64.9 | 56.2 |

August | 141.6 | 122.4 | 112.9 | 104.5 | 96.9 | 91.4 | 84.6 | 75.6 | 71.2 | 53.3 |

September | 119.0 | 106.1 | 94.6 | 89.0 | 83.8 | 77.0 | 71.4 | 68.0 | 60.4 | 49.6 |

October | 89.8 | 74.6 | 69.3 | 65.1 | 62.0 | 57.5 | 54.1 | 49.6 | 45.7 | 35.5 |

November | 54.1 | 43.6 | 40.4 | 38.1 | 37.3 | 34.7 | 31.0 | 29.2 | 27.6 | 20.4 |

December | 28.9 | 26.5 | 24.2 | 23.2 | 22.6 | 21.5 | 20.8 | 19.5 | 18.4 | 16.0 |

January | 21.1 | 19.3 | 18.1 | 17.4 | 16.9 | 16.8 | 16.4 | 15.1 | 14.2 | 12.0 |

February | 17.4 | 16.0 | 15.4 | 14.9 | 14.5 | 14.2 | 13.7 | 13.0 | 11.9 | 10.5 |

March | 16.8 | 16.2 | 15.0 | 14.4 | 14.0 | 12.9 | 12.6 | 12.1 | 11.7 | 10.8 |

April | 20.7 | 19.7 | 18.8 | 18.3 | 17.5 | 17.0 | 16.1 | 15.3 | 13.5 | 11.8 |

May | 35.5 | 30.7 | 29.7 | 27.6 | 26.5 | 25.3 | 23.5 | 21.6 | 19.0 | 15.3 |

**Table 4.**Representative flow in each month for simulated runoff of the NZD hydropower plant (10

^{8}m

^{3}).

Month | Representative Runoff QR | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

June | 65.9 | 59.5 | 54.5 | 51.0 | 47.9 | 44.8 | 41.7 | 38.6 | 34.4 | 20.0 |

July | 108.7 | 99.3 | 92.9 | 87.8 | 83.3 | 79.0 | 75.1 | 70.5 | 64.1 | 39.8 |

August | 139.0 | 122.4 | 112.7 | 104.9 | 97.7 | 91.1 | 84.7 | 77.7 | 68.7 | 36.2 |

September | 114.6 | 103.0 | 94.9 | 88.4 | 82.8 | 77.8 | 72.9 | 67.5 | 60.8 | 36.6 |

October | 83.7 | 75.4 | 70.0 | 65.5 | 61.6 | 57.9 | 54.4 | 50.3 | 45.5 | 28.3 |

November | 51.4 | 45.5 | 42.1 | 39.1 | 36.7 | 34.4 | 32.0 | 29.4 | 26.4 | 15.9 |

December | 29.0 | 26.7 | 25.1 | 23.9 | 22.9 | 21.9 | 20.8 | 19.6 | 18.0 | 12.6 |

January | 20.6 | 19.4 | 18.5 | 17.9 | 17.2 | 16.6 | 16.0 | 15.2 | 14.3 | 10.0 |

February | 17.3 | 16.3 | 15.6 | 15.0 | 14.5 | 14.0 | 13.5 | 13.0 | 12.2 | 8.4 |

March | 16.8 | 15.8 | 15.0 | 14.5 | 13.9 | 13.4 | 12.9 | 12.3 | 11.5 | 8.1 |

April | 21.1 | 19.6 | 18.7 | 17.9 | 17.2 | 16.5 | 15.8 | 15.1 | 14.1 | 9.6 |

May | 34.9 | 31.6 | 29.5 | 27.8 | 26.2 | 24.8 | 23.4 | 21.7 | 19.6 | 10.2 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, H.; Zhang, L.; Chang, J.; Li, Y.; Long, R.; Xing, Z.
Study on the Optimal Operation of a Hydropower Plant Group Based on the Stochastic Dynamic Programming with Consideration for Runoff Uncertainty. *Water* **2022**, *14*, 220.
https://doi.org/10.3390/w14020220

**AMA Style**

Zhang H, Zhang L, Chang J, Li Y, Long R, Xing Z.
Study on the Optimal Operation of a Hydropower Plant Group Based on the Stochastic Dynamic Programming with Consideration for Runoff Uncertainty. *Water*. 2022; 14(2):220.
https://doi.org/10.3390/w14020220

**Chicago/Turabian Style**

Zhang, Hongxue, Lianpeng Zhang, Jianxia Chang, Yunyun Li, Ruihao Long, and Zhenxiang Xing.
2022. "Study on the Optimal Operation of a Hydropower Plant Group Based on the Stochastic Dynamic Programming with Consideration for Runoff Uncertainty" *Water* 14, no. 2: 220.
https://doi.org/10.3390/w14020220