# Optimal Decomposition for the Monthly Contracted Electricity of Cascade Hydropower Plants Considering the Bidding Space in the Day-Ahead Spot Market

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

#### 2.1. Problem Description

#### 2.2. Uncertainty Treatment of Day-Ahead Market Clearing Price

- (1)
- It is assumed that the forecasted error of the clearing price series $\left\{{p}_{1}^{df},{p}_{2}^{df},\dots ,{p}_{T}^{df}\right\}$ obeys the normal distribution $N(\mu ,{\sigma}^{2})$ at any time period, where $\mu $ and $\sigma $ are the mean and standard deviation of the forecasted error, respectively, and they satisfy that $\mu =0,\sigma =0.1\times {p}_{t}^{df}$.
- (2)
- The Latin Hypercube Sampling (LHS) method [31] is adopted for generating the scenarios for the forecasted error of the day-ahead market price. In this method, the sampling probability distribution is stratified first, and then samples are randomly selected from each layer in turn, which can effectively improve the coverage degree of sampling samples to the distribution space of random variables.
- (3)
- To adequately reflect the stochastic characteristics of the day-ahead market clearing price, more price scenarios will be generated through LHS, which mainly aims to avoid a low calculation accuracy due to a small number of scenarios. In this paper, the fast backward/forward method [33] is adopted to balance the solving accuracy and efficiency, that is, to minimize the number of scenarios while maintaining the main characteristics of price scenarios.

#### 2.3. Objective Function

#### 2.4. Constraints

- (1)
- Water balance constraints

^{3}/s); ${Q}_{i,t}$ denotes the total water discharge of plant i in period t (in m

^{3}/s); $\Delta t$ is the duration of period t (in h); ${V}_{i,t}$ is the water storage at the end of period t (in m

^{3}).

- (2)
- Hydraulic connection constraints

^{3}/s); ${{\displaystyle Q}}_{i-1,t}^{g}$ and ${{\displaystyle Q}}_{i-1,t}^{s}$ denote the generating water flow and water spillage, respectively (in m

^{3}/s). Note that ${{\displaystyle Q}}_{i,t}^{s}$ is set to 0 in this paper because hydropower curtailment is generally not allowed according to China’s clean energy consumption policy.

- (3)
- Water level constraints

- (4)
- Water discharge constraints

^{3}/s).

- (5)
- Power output constraint

- (6)
- Water head constraints

- (7)
- Forebay water level–water storage relationship

- (8)
- Tailwater level–water discharge relationship

- (9)
- Constraints on trading electricity in the day-ahead market

- (10)
- Trading electricity constraints

## 3. Solving Technique

#### 3.1. Linear Approximation of the Objective Function

#### 3.2. Linear Approximation of the Power Generation Function

- (1)
- Set the convergence precision $\delta $ of the SA approach and let the index of iterations n = 1.
- (2)
- The initial solution has a great influence on the computational efficiency of the SA approach. Thus, to enhance the convergence speed, the initial water head of all the hydropower plants from upstream to downstream $\left\{{{\displaystyle H}}_{i,0}^{0},{{\displaystyle H}}_{i,1}^{0},\cdots ,{{\displaystyle H}}_{i,T}^{0}\right\}$ is generated using Equations (31)–(33). The initial output factor of each hydropower plant $\left\{{{\displaystyle k}}_{i,0}^{0},{{\displaystyle k}}_{i,1}^{0},\cdots ,{{\displaystyle k}}_{i,T}^{0}\right\}$ is then calculated using Equation (14).$$\Delta {W}_{i}={V}_{i,begin}-{V}_{i,end}+{\displaystyle \sum _{t=1}^{T}{Q}_{i,t}^{in}\times \Delta t}$$$$\overline{{Q}_{i}}=\Delta {W}_{i}/{\displaystyle \sum _{t=1}^{T}\Delta t}$$$$\overline{{H}_{i,t}}=({Z}_{i,begin}+{Z}_{i,end})/2-{f}_{i,zq}(\overline{{Q}_{i}})\forall t\in [1,T]$$
- (3)
- Based on the given water head and the output factor of each hydropower plant, the MILP based model for the optimal decomposition of monthly contracted electricity for cascade hydropower plants is established by using the linearization techniques presented in Section 3.1 and Ref. [30].
- (4)
- An efficient optimization solver is adopted to solve the MILP model, and the dispatching schemes, including the water discharge, forebay water level, and power output of each hydropower plant, can be obtained.
- (5)
- Calculate the new water head ${H}_{i,t}^{n}$ and corresponding output factor ${{\displaystyle k}}_{i,t}^{n}$ after the nth iteration using Equations (14)–(17). Judge if $max\left|{H}_{i,t}^{n}-{H}_{i,t}^{n-1}\right|/{H}_{i,t}^{n}\le \delta ,\forall i,\forall t$. If true, end the iteration and output the latest solution as the optimal dispatching scheme, otherwise let n = n + 1 and repeat steps (3)–(5).

## 4. Case Studies

^{4}CNY, including 286.62 × 10

^{4}CNY from the monthly contracted electricity revenue and 291.3 × 10

^{4}CNY from the day-ahead market trading electricity revenue. The calculation time of the model is 137 s, which fully meets the timeliness requirements of medium- and long-term scheduling of cascade hydropower plants, reflecting the high solving efficiency of the optimization model established in this paper.

^{4}kWh and 578.92 × 10

^{4}CNY, respectively, while under Model 2, the values are 2731 × 10

^{4}kWh and 567.49 × 10

^{4}CNY, respectively. Compared to the deterministic model, the total revenue of the proposed model increases by 2% when power generation is reduced. This shows that when making a monthly contract electricity decomposition plan, taking into account the uncertainty of the day-ahead market clearing price can significantly increase the expected benefits of the HGenCo.

## 5. Discussion

## 6. Conclusions

- (1)
- A scenario analysis technique and several effective linearization strategies are put forward to address the uncertain and nonlinear factors in the optimization model, including the uncertain day-ahead market clearing price, the nonlinear objective function, and the nonlinear power generation function of each hydropower plant. For such a complex research problem, the combination of the SA approach and MILP approach is computationally efficient with a calculation time of 137 s.
- (2)
- The total revenue obtained from the proposed stochastic optimization model is 578.92 × 10
^{4}CNY. Compared to the deterministic model, the total revenue of the proposed model increases by 2% when power generation is reduced. Furthermore, as the forecast errors of the day-ahead market clearing price are inevitable in actual operation, the proposed model can avoid solutions that imply small profits or major costs, hedging against risk and uncertainty. - (3)
- The penalty coefficient for imbalanced monthly contracted electricity (τ) is very important for the smooth settlement of the monthly contracted electricity. When τ is small (τ = 0.1 or 0.2), cascade hydropower plants will choose to violate the monthly electricity transaction contract and allow for more generation to participate in the day-ahead market transaction to obtain higher profits. While, when τ = 0.3 or 0.4, the cascade hydropower plants will fulfill the monthly contract. Therefore, market managers need to formulate a reasonable penalty coefficient to avoid a large number of defaults and ensure the long-term stable operation of the electricity market.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

HGenCo | Hydropower generation company |

NLP | Nonlinear programming |

MILP | Mixed integer linear programming |

DP | Dynamic programming |

SA | Successive approximation |

## References

- Yang, W.; Norrlund, P.; Saarinen, L.; Witt, A.; Smith, B.; Yang, J.; Lundin, U. Burden on hydropower units for short-term balancing of renewable power systems. Nat. Commun.
**2018**, 9, 2633. [Google Scholar] [CrossRef][Green Version] - Cheng, C.; Yan, L.; Mirchi, A.; Madani, K. China’s booming hydropower: Systems modeling challenges and opportunities. J. Water Resour. Plan. Manag.
**2017**, 143, 02516002. [Google Scholar] [CrossRef] - Llamosas, C.; Sovacool, B.K. The future of hydropower? A systematic review of the drivers, benefits and governance dynamics of transboundary dams. Renew. Sustain. Energy Rev.
**2021**, 137, 110495. [Google Scholar] [CrossRef] - Dukpa, R.D.; Joshi, D.; Boelens, R. Contesting hydropower dams in the Eastern Himalaya: The cultural politics of identity, territory and self-governance institutions in Sikkim, India. Water
**2019**, 11, 412. [Google Scholar] [CrossRef][Green Version] - Creţan, R.; Vesalon, L. The political economy of hydropower in the communist space: Iron Gates revisited. Tijdschr. Econ. Soc. Geogr.
**2017**, 108, 688–701. [Google Scholar] [CrossRef] - 2020 Hydropower Status Report. Available online: https://www.hydropower.org/publications/2020-hydropower-status-report (accessed on 10 May 2022).
- Statistics of the National Power Industry in 2020. Available online: http://www.nea.gov.cn/2021-01/20/c_139683739.htm (accessed on 10 May 2022).
- Zeng, M.; Yang, Y.; Wang, L.; Sun, J. The power industry reform in China 2015: Policies, evaluations and solutions. Renew. Sustain. Energy Rev.
**2016**, 57, 94–110. [Google Scholar] [CrossRef] - Guo, H.; Davidson, M.R.; Chen, Q.; Zhang, D.; Jiang, N.; Xia, Q.; Kang, C.; Zhang, X. Power market reform in China: Motivations, progress, and recommendations. Energy Policy
**2020**, 145, 111717. [Google Scholar] [CrossRef] - Bao, M.; Guo, C.; Wu, Z.; Wu, J.; Li, X.; Ding, Y. Review of electricity spot market reform in China: Current status and future development. In Proceedings of the 2019 IEEE Sustainable Power and Energy Conference (iSPEC), Beijing, China, 21–23 November 2019. [Google Scholar]
- Cheng, C.; Chen, F.; Li, G.; Ristic, B.; Mirchi, A.; Qiyu, T.; Madani, K. Reform and renewables in China: The architecture of Yunnan’s hydropower dominated electricity market. Renew. Sustain. Energy Rev.
**2018**, 94, 682–693. [Google Scholar] [CrossRef] - Chen, F.; Liu, B.; Cheng, C.; Mirchi, A. Simulation and regulation of market operation in hydro-dominated environment: The Yunnan case. Water
**2017**, 9, 623. [Google Scholar] [CrossRef][Green Version] - Kuriqi, A.; Pinheiro, A.N.; Sordo-Ward, A.; Garrote, L. Water-energy-ecosystem nexus: Balancing competing interests at a run-of-river hydropower plant coupling a hydrologic-ecohydraulic approach. Energy Convers. Manag.
**2020**, 223, 113267. [Google Scholar] [CrossRef] - Kuriqi, A.; Pinheiro, A.N.; Sordo-Ward, A.; Garrote, L. Influence of hydrologically based environmental flow methods on flow alteration and energy production in a run-of-river hydropower plant. J. Clean. Prod.
**2019**, 232, 1028–1042. [Google Scholar] [CrossRef] - Khaloie, H.; Abdollahi, A.; Shafie-khah, M.; Anvari-Moghaddam, A.; Nojavan, S.; Siano, P.; Catalao, J.P.S. Coordinated wind-thermal-energy storage offering strategy in energy and spinning reserve markets using a multi-stage model. Appl. Energy
**2020**, 259, 114168. [Google Scholar] [CrossRef] - Khaloie, H.; Mollahassani-Pour, M.; Anvari-Moghaddam, A. Optimal behavior of a hybrid power producer in day-ahead and intraday markets: A bi-objective CVaR-based approach. IEEE Trans. Sustain. Energy
**2021**, 12, 931–943. [Google Scholar] [CrossRef] - Sanchez de la Nieta, A.A.; Paterakis, N.G.; Gibescu, M. Participation of photovoltaic power producers in short-term electricity markets based on rescheduling and risk-hedging mapping. Appl. Energy
**2020**, 266, 114741. [Google Scholar] [CrossRef] - Lu, J.; Li, G.; Cheng, C.; Liu, B. A long-term intelligent operation and management model of cascade hydropower stations based on chance constrained programming under multi-market coupling. Environ. Res. Lett.
**2021**, 16, 055034. [Google Scholar] [CrossRef] - Li, G.; Lu, J.; Yang, R.; Cheng, C. IGDT-based medium-term optimal cascade hydropower operation in multimarket with hydrologic and economic uncertainties. J. Water Res. Plan. Manag.
**2021**, 147, 5021015. [Google Scholar] [CrossRef] - Shrestha, G.B.; Pokharel, B.K.; Lie, T.T.; Fleten, S.E. Medium term power planning with bilateral contracts. IEEE Trans. Power. Syst.
**2005**, 20, 627–633. [Google Scholar] [CrossRef] - Luo, B.; Miao, S.; Cheng, C.; Lei, Y.; Chen, G.; Gao, L. Long-term generation scheduling for cascade hydropower plants considering price correlation between multiple markets. Energy
**2019**, 12, 2239. [Google Scholar] [CrossRef][Green Version] - Yuan, X.; Wang, Y.; Xie, J.; Qi, X.; Nie, H.; Su, A. Optimal self-scheduling of hydro producer in the electricity market. Energy Convers. Manag.
**2010**, 51, 2523–2530. [Google Scholar] [CrossRef] - Conejo, A.; Arroyo, J.; Contreras, J.; Villamor, F. Self-scheduling of a hydro producer in a pool-based electricity market. IEEE Trans. Power Syst.
**2002**, 17, 1265–1272. [Google Scholar] [CrossRef] - Pousinho, H.; Contreras, J.; Catalão, J. Short-term optimal scheduling of a price-maker hydro producer in a pool-based day-ahead market. Iet. Gener. Transm. Dis.
**2012**, 6, 1243–1251. [Google Scholar] [CrossRef][Green Version] - Kongelf, H.; Overrein, K.; Klaeboe, G.; Fleten, S. Portfolio size’s effects on gains from coordinated bidding in electricity markets: A case study of a Norwegian hydropower producer. Energy Syst.
**2019**, 10, 567–591. [Google Scholar] [CrossRef] - 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] - Fu, X.; Li, A.; Wang, L.; Ji, C. Short-term scheduling of cascade reservoirs using an immune algorithm-based particle swarm optimization. Comput. Math. Appl.
**2011**, 62, 2463–2471. [Google Scholar] [CrossRef][Green Version] - Wang, J.; Cheng, C.; Shen, J.; Cao, R.; Yeh, W.W.G. Optimization of large-scale daily hydrothermal system operations with multiple objectives. Water Resour. Res.
**2018**, 54, 2834–2850. [Google Scholar] [CrossRef] - Catalao, J.P.S.; Mariano, S.J.P.S.; Mendes, V.M.F.; Ferreira, L.A.F.M. Scheduling of head-sensitive cascaded hydro systems: A nonlinear approach. IEEE Trans. Power Syst.
**2009**, 24, 337–346. [Google Scholar] [CrossRef] - Borghetti, A.; D’Ambrosio, C.; Lodi, A.; Martello, S. An MILP approach for short-term hydro scheduling and unit commitment with head-dependent reservoir. IEEE Trans. Power Syst.
**2008**, 23, 1115–1124. [Google Scholar] [CrossRef][Green Version] - Su, C.; Cheng, C.; Wang, P.; Shen, J.; Wu, X. Optimization model for long-distance integrated transmission of wind farms and pumped-storage hydropower plants. Appl. Energy
**2019**, 242, 285–293. [Google Scholar] [CrossRef] - Su, C.; Yuan, W.; Cheng, C.; Wang, P.; Sun, L.; Zhang, T. Short-term generation scheduling of cascade hydropower plants with strong hydraulic coupling and head-dependent prohibited operating zones. J. Hydrol.
**2020**, 591, 125556. [Google Scholar] [CrossRef] - Dupacova, J.; Growe-Kuska, N.; Romisch, W. Scenario reduction in stochastic programming: An approach using probability metrics. Math. Program
**2003**, 95, 493–511. [Google Scholar] [CrossRef] - Niu, W.; Feng, Z.; Cheng, C. Optimization of variable-head hydropower system operation considering power shortage aspect with quadratic programming and successive approximation. Energy
**2018**, 143, 1020–1028. [Google Scholar] [CrossRef] - Lu, P.; Zhou, J.; Wang, C.; Qiao, Q.; Mo, L. Short-term hydro generation scheduling of Xiluodu and Xiangjiaba cascade hydropower stations using improved binary-real coded bee colony optimization algorithm. Energy Convers. Manag.
**2015**, 91, 19–31. [Google Scholar] [CrossRef] - Ge, X.; Xia, S.; Lee, W.; Chung, C.Y. A successive approximation approach for short-term cascaded hydro scheduling with variable water flow delay. Electr. Power. Syst. Res.
**2018**, 154, 213–222. [Google Scholar] [CrossRef] - Zhou, Y.; Guo, S.; Chang, F.; Xu, C. Boosting hydropower output of mega cascade reservoirs using an evolutionary algorithm with successive approximation. Appl. Energy
**2018**, 228, 1726–1739. [Google Scholar] [CrossRef] - He, Z.; Wang, C.; Wang, Y.; Wei, B.; Zhou, J.; Zhang, H.; Qin, H. Dynamic programming with successive approximation and relaxation strategy for long-term joint power generation scheduling of large-scale hydropower station group. Energy
**2021**, 222, 119960. [Google Scholar] [CrossRef] - Pousinho, H.M.I.; Contreras, J.; Bakirtzis, A.G.; Catalão, J.P.S. Risk-constrained scheduling and offering strategies of a price-maker hydro producer under uncertainty. IEEE Trans. Power Syst.
**2013**, 28, 1879–1887. [Google Scholar] [CrossRef] - Yuan, W.; Wang, X.; Su, C.; Cheng, C.; Liu, Z.; Wu, Z. Stochastic optimization model for the short-term joint operation of photovoltaic power and hydropower plants based on chance constrained programming. Energy
**2021**, 222, 119996. [Google Scholar] [CrossRef] - Ming, B.; Liu, P.; Guo, S.; Cheng, L.; Zhou, Y.; Gao, S.; Li, H. Robust hydroelectric unit commitment considering integration of large-scale photovoltaic power: A case study in China. Appl. Energy
**2018**, 228, 1341–1352. [Google Scholar] [CrossRef] - Díaz, F.J.; Contreras, J.; Muñoz, J.I.; Pozo, D. Optimal scheduling of a price-taker cascaded reservoir system in a pool-based electricity market. IEEE Trans. Power Syst.
**2011**, 26, 604–615. [Google Scholar] [CrossRef] - Virtanen, M. Foreign direct investment and hydropower in Lao PDR: The Theun-Hinboun hydropower project. Corp. Soc. Responsib. Environ. Manag.
**2006**, 13, 183–193. [Google Scholar] [CrossRef] - Fan, J.; Liang, Y.; Tao, A.; Sheng, K.; Ma, H.; Xu, Y.; Wang, C.; Sun, W. Energy policies for sustainable livelihoods and sustainable development of poor areas in China. Energy Policy
**2011**, 39, 1200–1212. [Google Scholar] [CrossRef] - Méreiné-Berki, B.; Málovics, G.; Creţanc, R. You become one with the place: Social mixing, social capital, and the lived experience of urban desegregation in the Roma community. Cities
**2021**, 117, 103302. [Google Scholar] [CrossRef]

Plant | Regulation Performance | Normal Water Level/m | Dead Water Level/m | Installed Capacity /MW | Maximum Generating Water Flow/(m^{3}/s) | Minimum Total Water Discharge /(m^{3}/s) |
---|---|---|---|---|---|---|

A | Seasonal | 835 | 818 | 2 × 60 | 260 | 5 |

B | Weekly | 756 | 740 | 2 × 65 | 209 | 5 |

Plant | Monthly Contract Electricity/kWh | Contract Price/(CNY/kWh) | Water Level at the Beginning of Month/m | Control Water Level at the End of Month/m |
---|---|---|---|---|

A | 673 × 104 | 0.19062 | 822.29 | 821.86 |

B | 832 × 104 | 0.19039 | 751.04 | 752.76 |

Model | ${\mathit{F}}_{1}/\times {10}^{4}\mathbf{CNY}$ | ${\mathit{F}}_{2}/\times {10}^{4}\mathbf{CNY}$ | ${\mathit{F}}_{3}/\times {10}^{4}\mathbf{CNY}$ | ${\mathit{F}}_{4}/\times {10}^{4}\mathbf{CNY}$ | $\mathit{F}/\times {10}^{4}\mathbf{CNY}$ |
---|---|---|---|---|---|

Model 1 | 286.62 | 0 | 0 | 292.3 | 578.92 |

Model 2 | 286.62 | 0 | 0 | 280.87 | 567.49 |

$\mathit{\tau}$ | Completed Monthly Contracted Electricity/×10^{4} kWh | Day-Ahead Market Trading Electricity/×10^{4} kWh | Total Power Generation /×10^{4} kWh | Total Revenue/×10^{4} CNY |
---|---|---|---|---|

0.1 | 0 | 2711 | 2711 | 609.45 |

0.2 | 282 | 2430 | 2712 | 581.90 |

0.3 | 1505 | 1215 | 2720 | 578.92 |

0.4 | 1505 | 1215 | 2720 | 578.92 |

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**MDPI and ACS Style**

Wu, Y.; Su, C.; Liu, S.; Guo, H.; Sun, Y.; Jiang, Y.; Shao, Q. Optimal Decomposition for the Monthly Contracted Electricity of Cascade Hydropower Plants Considering the Bidding Space in the Day-Ahead Spot Market. *Water* **2022**, *14*, 2347.
https://doi.org/10.3390/w14152347

**AMA Style**

Wu Y, Su C, Liu S, Guo H, Sun Y, Jiang Y, Shao Q. Optimal Decomposition for the Monthly Contracted Electricity of Cascade Hydropower Plants Considering the Bidding Space in the Day-Ahead Spot Market. *Water*. 2022; 14(15):2347.
https://doi.org/10.3390/w14152347

**Chicago/Turabian Style**

Wu, Yang, Chengguo Su, Shuangquan Liu, Hangtian Guo, Yingyi Sun, Yan Jiang, and Qizhuan Shao. 2022. "Optimal Decomposition for the Monthly Contracted Electricity of Cascade Hydropower Plants Considering the Bidding Space in the Day-Ahead Spot Market" *Water* 14, no. 15: 2347.
https://doi.org/10.3390/w14152347