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

^{1}

^{2}

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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 |

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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