# Research on Cascade Reservoirs’ Short-Term Optimal Operation under the Effect of Reverse Regulation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Coupling Model

#### 2.1. Objective Function

_{2,t}(q

_{2,t}, H

_{2,t}) is Xiaoxuan’s power output when its power generation flow and water head are q

_{2,t}and H

_{2,t}respectively in period t, unit: kW, and is obtained according to the power characteristic curve of its generator set; Δt is the length of the calculation period, unit: h, and Δt = 0.25 h; λ

_{1,t}is the energy efficiency coefficient of Pankou hydropower station in period t, unit: kWh/m

^{3}, and its physical meaning is the amount of energy contained in every cubic meter of Pankou reservoir’s water in period t; q

_{1,t}(N

_{1,t}, H

_{1,t}) is Pankou’s power generation discharge when its output and water head are N

_{1,t}and H

_{1,t}respectively in period t, unit: m

^{3}/s, and is also obtained according to the power characteristic curve of its generator set.

#### 2.2. Constraint Conditions

_{i}

_{,t}is the output of hydropower station i in period t, unit: kW (i = 1 for Pankou, i = 2 for Xiaoxuan and the same hereafter); N

_{t}is the output command from the superior dispatching center for Pankou hydropower station in period t, unit: kW.

_{i}

_{,t}is the water level of reservoir i at moment t, unit: m; Z

_{2,end}is Xiaoxuan reservoir’s controlled water level at the end of the operation period, unit: m.

_{i}

_{,t}is the power generation discharge of hydropower station i in period t, unit: m

^{3}/s, and in this paper, it equals to reservoir outflow because water abandonment is not taken into consideration; ${q}_{i}^{\mathrm{max}}$, ${q}_{i}^{\mathrm{min}}$ are the upper and lower bounds of power generation discharge of hydropower station i, respectively, unit: m

^{3}/s.

_{i}

_{,t}, V

_{i}

_{,t+1}are the storage capacity of reservoir i at respectively the beginning and the end of period t (i.e., moment t and moment t + 1), unit: m

^{3}; Q

_{i}

_{,t}is the inflow of reservoir i in period t, unit: m

^{3}/s.

_{i}

_{,j}is the output of generator unit j in hydropower station i, unit: kW; ${N}_{i,j}^{\mathrm{min}}$, ${N}_{i,j}^{\mathrm{max}}$ are the upper and lower bounds of the vibration zone of generator unit j in hydropower station i, unit: kW.

^{3}/s:

_{e}is the demand for downstream ecological flow, unit: m

^{3}/s, and q

_{e}= 16.7 m

^{3}/s.

_{s}is the auxiliary power need of the cascade hydropower stations, unit: kW, and N

_{s}= 2000 kW.

## 3. Calculation of the Model’s Key Variables

#### 3.1. The Downstream Reservoir’s Inflow Considering Water Flow Hysteresis and Interval Inflow

_{2,t}). As for the input layer, Pankou reservoir’s outflow (q

_{1,t}~q

_{1,t−4}) as well as the interval basin’s precipitation (P

_{t}~P

_{t}

_{−4}) in the current and the previous four periods are chosen as the primary data and screened by the correlation coefficient method. It is demonstrated in Table 1 that the correlation coefficient between Pankou reservoir’s outflow and Xiaoxuan reservoir’s inflow is smaller in the earlier period. For q

_{1,t}, q

_{1,t−1}and q

_{1,t−2}, their correlation coefficients with Q

_{2,t}are similar, whereas the correlation coefficients of q

_{1,t−3}and q

_{1,t−4}with Q

_{2,t}are noticeably smaller. Therefore, q

_{1,t}, q

_{1,t−1}and q

_{1,t−2}are selected as the input layer data. Compared with Pankou reservoir’s outflow, the interval basin’s precipitation has a much smaller correlation coefficient, which changes little over time. Considering that the interval inflow accounts for only a small portion of Xiaoxuan reservoir’s inflow, the precipitation P

_{t}

_{−1}, with the largest correlation coefficient with Q

_{2,t}in this regard, is selected as the input layer data. In 1989, Cybenko, G. and Hornik, K. proved that the three-layer network (with one input layer, one output layer, and one hidden layer) can simulate any complex nonlinear problems [33,34]. Therefore, one hidden layer is set up whose node numbers are determined according to Equation (11) [35]:

#### 3.2. The Upstream Reservoir’s Tail Water Level Considering the Influence of Dual Aftereffect Factors

_{1,t}) represents the output layer. As for the input layer data, aside from Pankou reservoir’s outflow in the current period (q

_{1,t}) and its tail water level in the previous period (DZ

_{1,t}

_{−1}), it is also needed to decide which periods of Xiaoxuan reservoir’s water level are concerned. Similarly, Xiaoxuan reservoir’s water levels in the current and the previous four periods (Z

_{2,t}~Z

_{2,t}

_{−4}) are chosen as the primary data to be filtered by the correlation coefficient method. From Table 2 it can be seen that the correlation coefficient between Xiaoxuan’s water level and Pankou’s tail water level gradually decreases with time. For Z

_{2,t}, Z

_{2,t}

_{−1}and Z

_{2,t}

_{−2}, their correlation coefficients have little difference, while for Z

_{2,t}

_{−3}and Z

_{2,t}

_{−4}their correlation coefficients are apparently smaller, which indicates that the time it takes for Xiaoxuan reservoir’s backwater to reach Pankou is between 0 and 2 periods. Thus, Z

_{2,t}, Z

_{2,t}

_{−1}and Z

_{2,t}

_{−2}are selected as the input layer data. The configuration of the hidden layer and the training parameters is similar to that described in Section 3.1. After the training is completed, the network is saved for later calculation of Pankou reservoir’s tail water level. Figure 5 shows the resulted tail water level variation of Pankou reservoir in the same power generation process mentioned before. Compared with the actual variation, the results of the BP neural network only produce a calculation error of 0.001 m, which is notably smaller than the calculation error (0.027 m) produced by the stage-discharge relation method, thus proving the advantage of the BP neural network.

## 4. Model Solution

^{N}

^{+1}combinations of Xiaoxuan’s water level and the total amount of calculation is T·M

^{N}

^{+1}with T times of multi-stage calculation. Since the value of T·M

^{N}

^{+1}grows exponentially with the increase in N, even in a one-dimensional circumstance the dimensionality curse still emerges due to the interconnection of N periods. As the number of dimensions increases to L (L > 1), the total amount of calculation will become T·(M

^{N}

^{+1})

^{L}with (M

^{N}

^{+1})

^{L}combinations of water level at each multistage, which is a more serious problem of dimensionality curse. In view of this, we come up with an improved POA (i.e., APOA) to solve the model of the cascade reservoirs’ short-term optimal operation with aftereffect.

#### 4.1. Basic Principle of APOA

^{N}

^{+1}produced by MSDP. As a result, APOA can effectively alleviate the problem of dimensionality curse.

#### 4.2. Calculation Procedure of APOA

**Step 1**: Determine the number of associated periods. Use the correlation coefficient method to calculate the number of periods connected with the current period in terms of water quantity and water head. If the first m periods are related to the current period in terms of Xiaoxuan reservoir’s inflow and the first n periods are related to the current period in terms of Pankou reservoir’s tail water level, then the number of the related periods can be calculated by Equation (12), where m = 2, n = 2 and N = 3 in this article:

**Step 2**: Train the BP neural networks. The BP neural networks are trained by inputting training data, including the type of training data, starting and ending time, as well as the structure and parameters of the neural networks, which are detailed in Section 3.1 and Section 3.2. After the training is completed, the neural network for calculating Xiaoxuan reservoir’s inflow (net

_{1}) and that for Pankou reservoir’s tail water level (net

_{2}) are obtained.

**Step 3**: Set the initial solution. The selection of initial solutions has a major influence on the convergence speed and computation time of APOA [39]. Long-term experience of the dispatchers is often embodied in the operation rule of the hydropower station, according to which a relatively optimal solution can be obtained. Thus, in this paper, such a solution is used as the initial solution. It should be noted that in the initial solution, the initial water level of Xiaoxuan reservoir (${Z}_{2}^{0}$ = {${Z}_{2,1}^{0}$, ${Z}_{2,2}^{0}$, …, ${Z}_{2,T+1}^{0}$}) is the most critical variable while that of Pankou reservoir (${Z}_{1}^{0}$ = {${Z}_{1,1}^{0}$, ${Z}_{1,2}^{0}$, …, ${Z}_{1,T+1}^{0}$}) is only the variable accompanying it.

**Step 4**: Optimize progressively. Carry out the optimization step by step from period 1 to period T. When the calculation progresses to period t and period t + 1 (accompanied by the two related periods of t − 2 and t − 1), fix Xiaoxuan reservoir’s water levels at all moments except that at moment t + 1 and calculate all the discrete points of its water level at that moment. Xiaoxuan reservoir’s inflow and Pankou reservoir’s tail water level are calculated by the BP neural networks, as shown in Equation (13), with Pankou reservoir’s water level acting as the variable accompanying Xiaoxuan’s water level. After running through all the discrete points, select the optimal water level of Xiaoxuan reservoir (${Z}_{2,t+1}^{*}$) and correspondingly the optimal water level of Pankou (${Z}_{1,t+1}^{*}$) at moment t + 1. Make ${Z}_{2,t+1}^{1}$ = ${Z}_{2,t+1}^{*}$ and ${Z}_{1,t+1}^{1}$ = ${Z}_{1,t+1}^{*}$.

**Step 5**: Iterative calculation. After Step 4 is executed, the first iteration completes and produces a new solution ${Z}_{2}^{1}$ = {${Z}_{2,1}^{1}$, ${Z}_{2,2}^{1}$, …, ${Z}_{2,T+1}^{1}$} accompanied by ${Z}_{1}^{1}$ = {${Z}_{1,1}^{1}$, ${Z}_{1,2}^{1}$, …, ${Z}_{1,T+1}^{1}$}. Compare ${Z}_{2}^{1}$ with ${Z}_{2}^{0}$ in terms of the value of the objective function. If the difference between their values is less than the calculation precision (ε), then the convergence condition is satisfied and ${Z}_{2}^{1}$ (accompanied by ${Z}_{1}^{1}$) is the final solution. Otherwise replace the initial solution with ${Z}_{2}^{1}$ and repeat Step 4 until the convergence condition is satisfied.

## 5. Case Study

#### 5.1. Comparative Analysis of Model Solving Methods

#### 5.1.1. Comparative Analysis of the Objective Function Value of Each Method

#### 5.1.2. Analysis of Computation Time

^{2}for DP and T·M

^{N}

^{+1}for MSDP with the aftereffect factors taken into consideration. With the number of discrete points M = 560 and the number of related periods N = 3, the amount of calculation of MSDP is 313,600 times that of DP. However, Table 4 shows that the computation time of MSDP is about 1400 times that of DP, far fewer than 313,600 times. This is because MSDP imposes constraint on the feasible domain and leaves out those discrete points that do not satisfy the constraints, saving a great deal of calculation time. Nevertheless, its computation time of 262 h still cannot meet the efficiency requirement in practical production.

#### 5.1.3. Analysis of Calculation Accuracy

#### 5.2. Analysis of the Reverse Regulation Rule

^{3}/s, Xiaoxuan sees an increase of 0.0198 in power generation efficiency, much greater than the reduction of 0.0063 that Pankou suffers. Therefore, to raise Xiaoxuan’s water level by 2 m means an increase of 0.0021 in the overall power generation efficiency.

## 6. Conclusions

- On the basis of considering Xiaoxuan reservoir’s regulation on both water quantity and water head of Pankou reservoir, the model takes into account both Pankou’s power generation efficiency and Xiaoxuan’s generated energy to seek the maximum of overall power generation benefits from the angle of the cascade hydropower stations’ total energy, which fits the requirements of actual production. The calculation results show that the model can effectively enhance power generation benefits of the cascade hydropower stations, which also verifies the model’s validity.
- The BP neural network has excellent performance in exploring water flow hysteresis and the aftereffect of tail water level variation, so that the accurate values of downstream reservoir’s inflow and upstream reservoir’s tail water level can be obtained, which significantly improves the coupling model’s accuracy. The proposed APOA can efficiently work out the short-term optimal operation model of cascade reservoirs with aftereffect. With the merits and accuracy of its calculation results demonstrated, APOA is proved to meet the demand of actual production.
- As for the rule of reverse regulation, from the aspect of water quantity regulation, Xiaoxuan reservoir should strategically store and discharge the inflow from Pankou reservoir and try to discharge flow in the mode where its generator units are in the high-efficiency zone, so that this portion of water can be utilized more efficiently; from the aspect of water head regulation, the increase in Xiaoxuan’s generated energy brought by raising its operation water level is greater than Pankou’s hydroenergy loss caused by the fall in its power generation efficiency. Therefore, to raise Xiaoxuan’s operation water level is beneficial to power generation of the whole cascade hydropower stations.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 8.**Comparison of Pankou’s calculated results between POA and APOA. (

**a**) Pankou’s outflow process; (

**b**) Pankou’s tail water level process.

**Figure 9.**Comparison of Xiaoxuan’s calculated results between POA and APOA. (

**a**) Xiaoxuan’s inflow process; (

**b**) Xiaoxuan’s water level process; (

**c**) Xiaoxuan’s output process.

**Figure 10.**Comparison between the optimal and the actual operation schemes. (

**a**) Pankou; (

**b**) Xiaoxuan.

**Figure 11.**Comparison between optimal and actual operation schemes of Xiaoxuan, (

**a**) 1 July; (

**b**) 12 June.

Input Layer Data | q_{1,t} | q_{1,t−1} | q_{1,t−2} | q_{1,t−3} | q_{1,t−4} | P_{t} | P_{t}_{−1} | P_{t}_{−2} | P_{t}_{−3} | P_{t}_{−4} |
---|---|---|---|---|---|---|---|---|---|---|

Correlation Coefficient | 0.91 | 0.89 | 0.88 | 0.83 | 0.79 | 0.19 | 0.20 | 0.19 | 0.19 | 0.18 |

Input layer Data | Z_{2,t} | Z_{2,t−1} | Z_{2,t−2} | Z_{2,t−3} | Z_{2,t−4} |
---|---|---|---|---|---|

Correlation Coefficient | 0.74 | 0.73 | 0.73 | 0.70 | 0.67 |

Items | Unit | Pankou | Xiaoxuan |
---|---|---|---|

Normal water level | M | 355 | 264 |

Dead water level | M | 330 | 261.3 |

Regulation volume | 10^{8} m^{3} | 11.2 | 0.0678 |

Regulation performance | - | annual regulation | daily regulation |

Installed capacity | MW | 500 | 50 |

Operation mode | - | ‘electricity to water’ | ‘water to electricity’ |

Items | Unit | Actual | DP | POA | MSDP | APOA |
---|---|---|---|---|---|---|

Pankou’s hydroenergy consumption | 10^{3} kWh | 1518.5 | 1519.4 | 1519.4 | 1518.9 | 1518.9 |

Xiaoxuan’s generated energy | 10^{3} kWh | 267.7 | 297.5 | 297.5 | 307.4 | 307.4 |

Objective function | 10^{3} kWh | −1250.8 | −1221.9 | −1221.9 | −1211.5 | −1211.5 |

Optimization margin | % | - | 1.61 | 1.61 | 2.19 | 2.19 |

Computation time | s | - | 673.10 | 86.56 | 9.43×10^{5} | 156.37 |

Items | Unit | Scheme 1 | Scheme 2 | Scheme 3 |
---|---|---|---|---|

H_{1} | m | 80 | 78 | 78.8 |

H_{2} | m | 12 | 14 | 14 |

q_{1} | m^{3}/s | 100 | 100 | 100 |

q_{2} | m^{3}/s | 100 | 100 | 100 |

η_{1} | - | 0.6467 | 0.6404 | 0.6429 |

η_{2} | - | 0.8503 | 0.8701 | 0.8701 |

η | - | 0.6732 | 0.6753 | 0.6771 |

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## Share and Cite

**MDPI and ACS Style**

Ji, C.; Yu, H.; Wu, J.; Yan, X.; Li, R.
Research on Cascade Reservoirs’ Short-Term Optimal Operation under the Effect of Reverse Regulation. *Water* **2018**, *10*, 808.
https://doi.org/10.3390/w10060808

**AMA Style**

Ji C, Yu H, Wu J, Yan X, Li R.
Research on Cascade Reservoirs’ Short-Term Optimal Operation under the Effect of Reverse Regulation. *Water*. 2018; 10(6):808.
https://doi.org/10.3390/w10060808

**Chicago/Turabian Style**

Ji, Changming, Hongjie Yu, Jiajie Wu, Xiaoran Yan, and Rui Li.
2018. "Research on Cascade Reservoirs’ Short-Term Optimal Operation under the Effect of Reverse Regulation" *Water* 10, no. 6: 808.
https://doi.org/10.3390/w10060808