# Causal Inference of Optimal Control Water Level and Inflow in Reservoir Optimal Operation Using Fuzzy Cognitive Map

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- FCM-O is proposed to overcome the causal inference error caused by non-linear mapping of the activation function. In FCM-O, the activation function is not used, and the offset is introduced to better train directed weighted graphs to illustrate the specific relationship between any pair of elements.
- (2)
- The FCM-O of ROO for the Three Gorges Reservoir (TGR) is established. The causal relationships between optimal control water level and inflow are inferred using FCM-O, and they are presented as intuitive graphical forms. In addition, some relevant conclusions are obtained.

## 2. Problem Formulation

#### 2.1. Objective Function

#### 2.2. Constraints

## 3. Obtaining the Optimal Control Water Level of Reservoir Optimal Operation Using Dynamic Programming

_{begin}and V

_{end}of TGR are set as fixed values, which are usually specified by the dispatcher. At the first and last period, there is only one state different from other periods that the reservoir capacity V is discretized into n values: {${V}_{t,j}$}. At t period, ${V}_{t,j}$ will be selected from {${V}_{t,j}$} to obtain the optimal ${R}_{t}({V}_{t-1,i})$ for each ${V}_{t-1,i}$ in {${V}_{t-1,i}$}, and save the backtracking relationship between ${V}_{t-1,i}$ and ${V}_{t,j}$ After recursive computation from T − 1 to 1 period, the optimal benefit R

_{1}(V

_{begin}) and optimal state process {${V}_{t,j}^{*}$} can be obtained based on the backtracking relationship previously preserved. The reservoir capacity is selected as the state in the process of DP for ROO, and water level is often used as a control attribute in practical engineering applications. The water level and reservoir capacity can be queried by the relationship between water level and reservoir capacity as Equation (15), where $Czv()$ and $Cvz()$ respectively represent the functional relationship between reservoir capacity and water level.

Algorithm 1 DP for reservoir operation |

Input: |

1: set V_{begin} and V_{end}; select inflow series {I_{t}}. |

Initialization: |

1: the states (reservoir capacity) are discretized |

2: generate discrete set of states {${V}_{t,j}$} |

Calculation: |

1: for t = T to 1 |

2: for i = 1 to n select state ${V}_{t-1,i}$ from {${V}_{t-1,i}$} |

3: select optimal decision ${V}_{t,j}$ from {${V}_{t,j}$ } to obtain the optimal R_{t}(${V}_{t-1,i}$) |

4: save the backtracking relationship Backtracking(${V}_{t-1,i}$, t) = ${V}_{t,j}$ |

5: end for |

6: end for |

Output: |

the optimal benefit R_{1}(V_{begin}) and optimal state (decision) process {${V}_{t}$} |

## 4. Fuzzy Cognitive Map with Offset

#### 4.1. Fuzzy Cognitive Map

#### 4.2. Fuzzy Cognitive Map with Offset

#### 4.3. Algorithm for Learning the Structure of FCM: Differential Evolution Algorithm

## 5. Case Study

#### 5.1. Description of Research Area

#### 5.2. Dataset Acquisition and Preprocessing

- (1)
- For power generation, the water level of TGR should be higher than 145 m, which is the dead water level. In addition, TGR should keep lower than the normal water level 175 m.
- (2)
- From July to early September, the TGR runs according to the flood control mode.
- (3)
- TGR begins to store water in September, and reaches 175 m by late October. TGR had better fill up quickly to improve the efficiency of power generation.

#### 5.3. Case Study and Discussion

^{3}/s, while ${I}_{10}^{}$ ranges in (11,000, 28,000) m

^{3}/s and ${I}_{11}^{}$ ranges in (8,000, 16,000) m

^{3}/s. These indicate the value of ${Z}_{9}^{*}$, ${Z}_{10}^{*}$ and ${Z}_{11}^{*}$ are more affected by upper boundaries of the water level constraint constraints than by inflow. However, the weighting coefficients of ${I}_{t,y}^{}$ to ${Z}_{9}^{*}$, ${Z}_{10}^{*}$, and ${Z}_{11}^{*}$ in FCM are 1 or −1 in Table 4, which shows that ${Z}_{9}^{*}$, ${Z}_{10}^{*}$ and ${Z}_{11}^{*}$ are greatly affected by ${I}_{t,y}^{}$. The conclusions obtained by FCM do not match the accepted conclusions that ${Z}_{9}^{*}$, ${Z}_{10}^{*}$ and ${Z}_{11}^{*}$ are more affected by constraints than by inflow, and the weighting coefficients of ${I}_{t,y}^{}$ to ${Z}_{9}^{*}$, ${Z}_{10}^{*}$ and ${Z}_{11}^{*}$ in FCM-O are 0, which seems reasonable. The reason for the above causal inference errors in FCM lies in the inability of the activation function $f(x)$ to deal with variables with offsets that are not 1 or −1. All of those shows that FCM-O is more competent than FCM in the causal relationship between optimal control water level and inflow in ROO both in min data error and reasonableness of results.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**An example of a fuzzy cognitive map (FCM) and its equivalent weight matrix. (

**a**) Graph Representation, (

**b**) Matrix Representation.

**Figure 3.**The function diagram of $f(x)$ with different ${\lambda}_{0}$; (

**a**) ${\lambda}_{0}$ = 1, 2, 3, 4, 5, 6, 8, 10, (

**b**) ${\lambda}_{0}$ = 10, 20, 30, 40, 50, 60, 80, 100.

**Figure 5.**The optimal control water level and inflow of the dataset from 1959 to 2014. (

**a**) optimal control water level {${Z}_{t}^{*}$}, (

**b**) inflow sequence {${I}_{t}$ }.

**Figure 6.**The schematic diagram of {${Z}_{t}^{*}$} and {${I}_{t}$} under the condition of a certain inflow.

**Figure 7.**The relationship between {${Z}_{t}^{*}$} and {${I}_{t}$} obtain by FCM-O. (

**a**) ${Z}_{9}^{*}$, (

**b**) ${Z}_{10}^{*}$, (

**c**) ${Z}_{11}^{*}$, (

**d**) ${Z}_{12}^{*}$, (

**e**) ${Z}_{1}^{*}$, (

**f**) ${Z}_{2}^{*}$, (

**g**) ${Z}_{3}^{*}$, (

**h**) ${Z}_{4}^{*}$.

Parameter | TGR |
---|---|

Adjustment ability | Season |

Total reservoir capacity (billion m^{3}) | 39.30 |

Regulating storage (billion m^{3}) | 16.50 |

Hydro plant discharge range(m^{3}/s) | (98, 800, 4500) |

Upriver water level range (m) | (175, 145) |

Installed capacity (MW) | 22,400 |

Normal water level (m) | 175 |

Maximum water level amplitude(m/d) | 0.6 |

t | t_{1} | ${t}_{2}$ | ${t}_{3}$ | ${t}_{4}$ | ${t}_{5}$ | ${t}_{6}$ | ${t}_{7}$ | ${t}_{8}$ | ${t}_{9}$ |

month | 9 | 10 | 11 | 12 | 1 | 2 | 3 | 4 | 5 |

Sept | Oct | Nov | Dec | Jan | Feb | Mar | Apr | May |

**Table 3.**The min data error of the fuzzy cognitive map (FCM) and fuzzy cognitive map with offset (FCM-O) for training and testing.

Method | Training | Testing | |
---|---|---|---|

min data error | FCM | 0.0045 | 0.0056 |

FCM-O | 0.0040 | 0.0052 | |

FCM-O vs. FCM in the reduction of min data error | 11.11% | 7.14% |

Z_{9} | Z_{10} | Z_{11} | Z_{12} | Z_{1} | Z_{2} | Z_{3} | Z_{4} | |
---|---|---|---|---|---|---|---|---|

I_{9} | −1.00 | 1.00 | 1.00 | 1.00 | 0.20 | −0.03 | −0.30 | −0.24 |

I_{10} | −1.00 | 1.00 | 1.00 | 1.00 | −0.09 | −0.13 | −0.07 | −0.16 |

I_{11} | −1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.72 | −0.16 |

I_{12} | −1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.62 | 0.79 | −0.40 |

I_{1} | −1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | −1.00 |

I_{2} | −1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.36 |

I_{3} | −1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | −1.00 |

I_{4} | −1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.09 |

I_{5} | −1.00 | 1.00 | 1.00 | 1.00 | 0.01 | −0.29 | 0.10 | −0.34 |

λ_{0} | 5.00 | 5.00 | 5.00 | 5.00 | 5.00 | 5.00 | 5.00 | 5.00 |

Z_{9} | Z_{10} | Z_{11} | Z_{12} | Z_{1} | Z_{2} | Z_{3} | Z_{4} | |
---|---|---|---|---|---|---|---|---|

I_{9} | 0.00 | 0.00 | 0.00 | 0.00 | 0.01 | −0.08 | −0.05 | −0.03 |

I_{10} | 0.00 | 0.00 | 0.00 | 0.00 | −0.08 | −0.01 | −0.03 | −0.01 |

I_{11} | 0.00 | 0.00 | 0.00 | 0.00 | 0.18 | 0.51 | 0.71 | 0.01 |

I_{12} | 0.00 | 0.00 | 0.00 | 0.02 | 0.71 | 0.15 | 0.42 | 0.11 |

I_{1} | 0.00 | 0.00 | 0.00 | −0.01 | 1.00 | 1.00 | 1.00 | 0.14 |

I_{2} | 0.00 | 0.00 | 0.00 | 0.02 | 1.00 | 1.00 | 1.00 | 0.16 |

I_{3} | 0.00 | 0.00 | 0.00 | 0.00 | 0.50 | 1.00 | 1.00 | −0.16 |

I_{4} | 0.00 | 0.00 | 0.00 | 0.00 | 0.03 | 0.38 | 1.00 | 0.12 |

I_{5} | 0.00 | 0.00 | 0.00 | 0.00 | −0.18 | −0.18 | 0.15 | 0.02 |

$Offse{t}_{tZ}$ | 0.00 | 1.00 | 1.00 | 1.00 | 0.84 | 0.67 | 0.56 | 0.11 |

In Degree | In Element | $\mathit{O}\mathit{f}\mathit{f}\mathit{s}\mathit{e}{\mathit{t}}_{\mathit{t}\mathit{Z}}$ | |
---|---|---|---|

Z_{9} | 0 | 0 | |

Z_{10} | 0 | 1 | |

Z_{11} | 0 | 1 | |

Z_{12} | 3 | I_{12}, I_{1}, I_{2} | 1 |

Z_{1} | 9 | I_{9}, I_{10}, I_{11}, I_{12},I _{1}, I_{2}, I_{3}, I_{4}, I_{5} | 0.84 |

Z_{2} | 9 | I_{9}, I_{10}, I_{11}, I_{12},I _{1}, I_{2}, I_{3}, I_{4}, I_{5} | 0.67 |

Z_{3} | 9 | I_{9}, I_{10}, I_{11}, I_{12},I _{1}, I_{2}, I_{3}, I_{4}, I_{5} | 0.56 |

Z_{4} | 9 | I_{9}, I_{10}, I_{11}, I_{12},I _{1}, I_{2}, I_{3}, I_{4}, I_{5} | 0.11 |

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## Share and Cite

**MDPI and ACS Style**

Liu, Y.; Zhou, J.; He, Z.; Lu, C.; Jia, B.; Qin, H.; Feng, K.; He, F.; Liu, G. Causal Inference of Optimal Control Water Level and Inflow in Reservoir Optimal Operation Using Fuzzy Cognitive Map. *Water* **2019**, *11*, 2147.
https://doi.org/10.3390/w11102147

**AMA Style**

Liu Y, Zhou J, He Z, Lu C, Jia B, Qin H, Feng K, He F, Liu G. Causal Inference of Optimal Control Water Level and Inflow in Reservoir Optimal Operation Using Fuzzy Cognitive Map. *Water*. 2019; 11(10):2147.
https://doi.org/10.3390/w11102147

**Chicago/Turabian Style**

Liu, Yi, Jianzhong Zhou, Zhongzheng He, Chengwei Lu, Benjun Jia, Hui Qin, Kuaile Feng, Feifei He, and Guangbiao Liu. 2019. "Causal Inference of Optimal Control Water Level and Inflow in Reservoir Optimal Operation Using Fuzzy Cognitive Map" *Water* 11, no. 10: 2147.
https://doi.org/10.3390/w11102147