# Application of Empirical Mode Decomposition Method to Synthesize Flow Data: A Case Study of Hushan Reservoir in Taiwan

^{*}

## Abstract

**:**

^{3}/s); however, the flow distribution during wet season varies significantly (±0.63 m

^{3}/s). There are two analytic scenarios for demand. For Scenario I, without supporting industrial demand, the simulation results of the generation data of Method I and II show that both are more severe than the current condition, the shortage index of each method is between 0.67–1.96 but are acceptable. For Scenario II, no matter in which way the synthesis flow is simulated, supporting industrial demand will seriously affect the equity of domestic demand, the shortage index of each method is between 1.203 and 2.12.

## 1. Introduction

^{3}per day (up to 380,000 m

^{3}/day) and the reservoir utilization rate can reach 2.41 from 1.18 while Shortage Index (SI) = 1.

## 2. Materials and Method

#### 2.1. Study Area

#### 2.1.1. Hushan Reservoir

^{3}. The elevations of the water outlets are 165 m and 180 m respectively. The designed flow of the domestic water channel and the permanent river outlet are 12.27 m

^{3}/s and 1.00 m

^{3}/s, respectively. The watershed of Hushan reservoir is only 6.58 km

^{2}and leads to limited water resources. Therefore, the Tongtou weir, the intake of Hushan reservoir, on the Qingshui river needs to be guided through the diversion tunnel.

#### 2.1.2. Tongtou Weir

^{3}/s, with a gravity type.

#### 2.2. Research Method

#### 2.2.1. Empirical Mode Decomposition

- In the whole data series, the number of extrema and the number of zero-crossings must either equal or differ at most by 1;
- At any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero.

- Identify all the local maxima and minima of the original time series $\mathrm{x}\left(\mathrm{t}\right)$;
- Using the three-spline interpolation function to create the upper envelopes ${\mathrm{e}}_{\mathrm{up}}\left(\mathrm{t}\right)$ and the lower envelopes ${\mathrm{e}}_{\mathrm{low}}\left(\mathrm{t}\right)$ of the time series;
- Calculate mean value m(t) of the upper and lower envelopes $\left(\mathrm{m}\left(\mathrm{t}\right)=\left[{\mathrm{e}}_{\mathrm{up}}\left(\mathrm{t}\right)+{\mathrm{e}}_{\mathrm{low}}\left(\mathrm{t}\right)\right]/2\right)$;
- Calculate the difference value d(t) between time series x(t) and mean value m(t), $\left(\mathrm{d}\left(\mathrm{t}\right)=\mathrm{x}\left(\mathrm{t}\right)-\mathrm{m}\left(\mathrm{t}\right)\right)$;
- Check the difference value d(t): (a) if d(t) satisfies the two IMF conditions, then d(t) is defined as the ith IMF, the residue $\mathrm{r}\left(\mathrm{t}\right)=\mathrm{x}\left(\mathrm{t}\right)-\mathrm{d}\left(\mathrm{t}\right)$ replace the x(t). The ith IMF is denoted as ${\mathrm{c}}_{\mathrm{i}}\left(\mathrm{t}\right)$; (b) if d(t) is not an IMF, then d(t) replace the x(t).
- Repeat (1)–(5) until the residue item r(t) becomes a monotone function or the number of extrema is less than or equal to 1, so that the IMF component cannot be decomposed again.

#### 2.2.2. Ensemble Empirical Mode Decomposition

- Add white noise w(t) to the original time series x(t). The new time series can be defined as:$$\mathrm{X}\left(\mathrm{t}\right)=\text{}\mathrm{x}\left(\mathrm{t}\right)+\mathrm{w}\left(\mathrm{t}\right)$$
- Decompose the new time series into IMFs using EMD method;
- Repeat steps (1) and (2) with different white noises series each time;
- Obtain the mean of the ensemble corresponding IMFs of the decompositions as the final result.

#### 2.2.3. Zero Up-Cross Method

#### 2.2.4. Data Synthesis

- Regression analysis is performed on the different residue of each group to fit a new residue to represent the residue value. By adding the results of the IMFs permutation and combination with the representative values of the residue, we can get 10-year new synthetic flow data for the ${\mathrm{n}}^{\mathrm{i}}$ groups.
- Permutation and combination of the different remainders of each group with IMFs, we can get 10-year new synthetic flow data for the ${\mathrm{n}}^{\mathrm{i}+1}$ groups.

#### 2.2.5. Water Supply System Simulation of Hushan Reservoir

^{5}m

^{3}water per day, and the deficit of demand was filled by pumping groundwater. After the completion of the Hushan reservoir, the situation of over pumping will be improved. The Hushan reservoir was originally intended to be used in conjunction with the Jiji weir to supply water for domestic demand in the Yunlin area. However, the conjunction operation of Hushan reservoir and Jiji weir will result in a low reservoir utilization rate of only 1.18 [25]. Therefore, we assume the Hushan reservoir operates independently in this study. The domestic demand in the Yunlin area is 2.8 × 10

^{5}/day, and the support demand for industrial is 3.0 × 10

^{5}/day. Based on the previous study, the operational regulations of reservoir were added [26], and the upper and lower rule curve is set at the water level of 190 m and 185 m for Hushan reservoir, respectively. In order to explore the situation of water resources utilization in the Yunlin area of the Hushan reservoir under different demand scenarios, the operation of reservoir should follow these principles:

- Supply of the Hushan reservoir must not be less than the projected demand if the water surface elevation exceeds the upper curve.
- Supply of the Hushan reservoir should satisfy 100% of the projected domestic demand, 90% of the projected industrial demand, and 50% of the projected irrigation demand if the water surface elevation is between the upper and lower curve.
- Supply of the Hushan reservoir should satisfy 80% of the planning domestic demand and 0% of the projected irrigation and industrial demand if the water surface elevation is under the lower curve.

#### 2.2.6. Indices for Impact and Risk Assessment

- Satisfaction:$$\mathrm{Satisfaction}=\frac{{\mathrm{Q}}_{\mathrm{sup}}}{{\mathrm{Q}}_{\mathrm{d}}}\times 100\%$$
- Reliability:$$\mathrm{Reliability}=\frac{{\mathrm{N}}_{\mathrm{sat}}}{\mathrm{N}}\times 100\%$$
- Shortage Index:$$\mathrm{SI}=\frac{100}{\mathrm{N}}{{\displaystyle \sum}}_{\mathrm{t}=1}^{\mathrm{N}}{\left(\frac{\left|{\mathrm{Q}}_{\mathrm{d}}-{\mathrm{Q}}_{\mathrm{sup}}\right|}{{\mathrm{Q}}_{\mathrm{d}}}\right)}^{2}$$
- Reservoir Efficiency:$$\mathrm{RE}=\frac{\mathrm{Annual}\text{}\mathrm{actual}\text{}\mathrm{water}\text{}\mathrm{supply}}{\mathrm{reservoir}\text{}\mathrm{effective}\text{}\mathrm{capacity}}$$
- The average duration of water shortage events:$${\overline{\mathrm{l}}}_{\mathrm{n}}=\frac{1}{\mathrm{M}}{\displaystyle \sum}_{\mathrm{j}=1}^{\mathrm{M}}{\mathrm{l}}_{\mathrm{j}}$$
- The average water shortage in the water shortage event (10
^{6}m^{3}):$${\overline{\mathrm{d}}}_{\mathrm{n}}=\frac{1}{\mathrm{M}}{\displaystyle \sum}_{\mathrm{j}=1}^{\mathrm{M}}{\mathrm{d}}_{\mathrm{j}}$$ - The daily average water shortage in the water shortage event (10
^{6}m^{3}/day):$$\text{}\frac{{\overline{\mathrm{d}}}_{\mathrm{n}}}{{\overline{\mathrm{l}}}_{\mathrm{n}}}$$

## 3. Results and Discussion

#### 3.1. 60-Year Flow Data Analysis with EEMD

_{1}–IMF

_{10}gradually decrease from ± 3 to ± 0.008, but the range of the residue gradually increased from 1.73 to 2.09 and then decreased to 1.75, which is a monotonic function. From the decomposition, we can find that the residue part plays an important role, which often affects the trend of whole time series.

_{10}is NAN because the period of the sequence cannot get after the zero up cross analysis method.

_{4}, reaching 39% and the energy of IMF

_{4}is higher than that of other IMFs. Corresponding to IMF

_{4}in Figure 10, it can be found that the corresponding period is about 36 ten-day, which represents that the flow data has a strong annual (36 ten-day) period characteristic. In addition, the IMF

_{1}and IMF

_{2}still have a certain proportion of the energy percentage, which are 29% and 22% respectively, and their periods correspond to about 3 and 6 ten-day, respectively. Therefore, it can be seen that seasonal characteristics exists in the flow data.

#### 3.2. Analysis of 6 Groups of 10-Year Flow Data

_{1}–IMF

_{4}, which correspond to an average period from 2.96 to 37.03 ten-days. In addition, it can be found that the energy percentage of IMF

_{4}for the 6 groups has the most highest weight, with an average of about 20.5–44.5% compared with total weight of all IMFs, and it corresponds to an average period around 36 ten-days. Moreover, the minor IMFs, IMF

_{1}and IMF

_{2}, are have around 28–36% and 18–27% energy of the total respectively, and their corresponding periods are about 3 and 6 ten-days. These results are similar to the 60-year flow analysis, as shown in Table 4.

#### 3.3. Data Synthesis

^{i}(6

^{7}= 279,936) new IMF $\left({\mathrm{IMF}}_{1}+{\mathrm{IMF}}_{2}+\dots +{\mathrm{IMF}}_{7}\right)$ set. Where n is the number of groups and i is the number of IMFs each group. After permutations and combinations of the IMFs, add a representative value of the residue, which is the regression of six residues. The representative value of the residue is shown in Figure 13, the bold dashed line. Regression analysis fits the representative value of the residue, and then adds this residue to the IMF of the 279,936 group to obtain the synthesis flow data of the 279,936 group for 10 years. Method (II): all the IMFs and the residue in each group are permutations and combinations to obtain a total of n

^{i + 1}(6

^{7 + 1}= 1,679,616) new synthetic flow data. In the following, the new flow data synthesized from the two methods and historical flow data are compared with each other, and the differences are explored. In addition, this study will apply new flow data synthesized from two different methods to simulate Hushan reservoir water supply systems, and sequentially discuss the simulation results of different synthetic flow data in different water supply scenarios.

^{3}/s) and the lowest flow occurs in January (2.05 m

^{3}/s). Compared the historical with the synthetic flow, the monthly distribution is similar between both Method I and Method II. The synthetic flow is concentrated in May–October, and the highest flow always occurs in August, but the lowest flow occurs in the dry season in January and February. In terms of the overall flow during the wet season, the average synthesized by Method I is significantly less than the current situation; in contrast, the average synthesized by Method II is much higher. In the dry season, the average synthesized by Method I and Method II is not much different from the current.

^{3}/s, which is more than the annual average of Method I, 13.71 m

^{3}/s. The water supply system simulation will be based on these three flow sequences, historical data, Method I and Method II synthetic data, to simulate the water supply system of Hushan reservoir.

#### 3.4. Application of Synthesis Data

^{6}m

^{3}and 0.27 × 10

^{6}m

^{3}, respectively. The water shortage index (SI) is only 0.002. While the water shortage occurring, the average daily water shortage (ADWS) is only 0.03 × 10

^{6}m

^{3}. The preceding indices show that if the Hushan reservoir only supplies water for domestic demand, the water supply situation will be very stable and shortage will not be easy to occur. In the case of supporting industrial water in Scenario II. If the daily support for industrial water is 300,000 m

^{3}, the annual water supply will reach 165.14 × 10

^{6}m

^{3}, and the annual water shortage will also increase significantly to 9.74 × 10

^{6}m

^{3}. Although the SI only increases to 0.309, and the ADWS only increases to 0.06 × 10

^{6}m

^{3}while the water shortage occurred. The reliability of the water supply reaches 0 already at the 9th and 11th ten-day, as depicted in Figure 17. The result show that the water supply risk in demand Scenario II is relatively severe than Scenario I.

^{3}/day of industrial water, and the annual water shortage has decreased to 2.18 × 10

^{6}m

^{3}. The number of average consecutive dry days (ACDD) for domestic has been greatly reduced to 45 days, and the reliability of the water supply is also raised to 0.4–0.5. In the following analyses, we recommend the daily supporting amount for industry from the Hushan reservoir should be 100,000 m

^{3}as basis.

^{6}m

^{3}, the annual water shortage is only 0.27 × 10

^{6}m

^{3}, and the SI is only 0.002. While the water shortage occurring, the ADWS is only 0.03 × 10

^{6}m

^{3}.

^{6}m

^{3}, the annual water shortage increased to 3.59 × 10

^{6}m

^{3}, and the SI increased significantly to 0.668. While the water shortage occurring, the ADWS is increased slightly to 0.05 × 10

^{6}m

^{3}. However, Figure 18 shows that whether the reliability is lower than the current simulation result, they are still above 0.5, the lowest satisfaction is 0.87, which is generally acceptable.

^{6}m

^{3}, the annual water shortage has increased significantly to 6.68 × 10

^{6}m

^{3}. The SI increased significantly to 1.96 and the ADWS increased to 0.07 × 10

^{6}m

^{3}as well. Figure 18 shows that the reliability is similar to the method I in dry season. However, Method II’s reliability is lower than method I value in the wet season. The satisfaction is generally lower than the current situation and the method I, but the minimum is still 0.84. The water supply situation of Method II is still acceptable. There is something odd: why the average annual flow of Method II is significantly more than the current situation and the average flow of Method I, but the simulation results are worse than the other cases? The main reason is that the extreme flow of the Method II often occurs; excessive flow cannot be used efficiently and there are many low flows during wet season. Therefore, the simulation result of Method II is the worst.

^{3}of industrial demand daily, the current annual water supply is 128.30 × 10

^{6}m

^{3}, the annual water shortage is 2.18 × 10

^{6}m

^{3}and the SI is 0.1. While water shortage occurs, the average daily water shortage is 0.05 × 10

^{6}m

^{3}.

^{6}m

^{3}, the annual water shortage increased to 11 × 10

^{6}m

^{3}, and the SI increased significantly to 1.203. While the water shortage occurring, the ADWS is still maintained at 0.05 × 10

^{6}m

^{3}. Figure 19 shows that the reliability decrease more earlier than the current situation, the lowest monthly reliability reaches 0.24 in March, and recoveries in May. The lowest fulfillment of demand still occurs in March (0.74). In terms of dry season, the risk of water supply shortage is much worse than the current situation.

^{6}m

^{3}, the annual water shortage increased sharply to 13.46 × 10

^{6}m

^{3}, and the SI increased significantly to 2.12. While occurs the water shortage event, the ADWS also increased to 0.07 × 10

^{6}m

^{3}. Figure 19 depicts that the reliability is similar to but lower than Method I in the dry season. Compared with the current situation, the water shortage occurs earlier, but it is lower than the Method I in the wet season. The satisfaction is generally lower than the current situation and the Method I, but similar to Method I. The satisfaction rises rapidly in May, which is higher than the current simulation results. However, the satisfaction is decreasing from August to October in wet season, it seems worse than the others.

## 4. Conclusions

_{1}and IMF

_{4}, which IMF

_{4}has the largest weight (39%), and the corresponding period is about 36 ten-day periods (1 year). However, IMF

_{1}still has a certain proportion of energy percentage, the corresponding period is about 3 ten-day periods (1 month). This reflects that the flow of Tongtou weir is not only affected by a 1-year periodic sequence, but influenced by the monthly rainfall event. Comparing the current situation with the new data generated by EEMD, the generated data is similar to the current flow distribution, but the flow distribution during the wet season is significantly different (±0.63 m

^{3}/s), and the flow difference during the dry season is insignificant (±0.78 m

^{3}/s).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Show of the zero up-cross method [30]. H, The wave high; n

_{max}: The average value of the highest point (peak).

**Figure 13.**Flow data residues and their representative equation. CMS, m

^{3}/s; r*(t): representative values of the residue.

**Figure 17.**Ten-day reliability and satisfaction of the water supply in present each scenario. CMD, m

^{3}/day.

Water Level (m) | Area (ha) | Volume (10^{4} m^{3}) |
---|---|---|

Dead water level: 165 | 12 | 36 |

170 | 32 | 147 |

175 | 76 | 418 |

180 | 99 | 857 |

185 | 107 | 1371 |

190 | 132 | 1969 |

195 | 142 | 2653 |

200 | 170 | 3432 |

205 | 181 | 4310 |

210 | 210 | 5288 |

Full water level: 211.5 | 214 | 5612 |

Scenario | I | II |
---|---|---|

Water supply range | Supply the domestic water | Support the industrial water |

The operational regulations | The upper and lower rule was set at the water level of 190 and 185 m in Hushan reservoir, respectively. |

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

Energy (%) | 29% | 22% | 10% | 39% | 1% | 0% | 0% | 0% | 0% | 0% |

Period (ten-day) | 3.06 | 6.10 | 12.51 | 36.15 | 68.15 | 122.69 | 219.52 | 506.72 | 1262.74 | N/A |

Energy and Period of Each IMFs Title | 1956–1965 | 1966–1975 | 1976–1985 | 1986–1995 | 1996–2005 | 2006–2015 | |
---|---|---|---|---|---|---|---|

IMF_{1} | Energy (%) | 29.21 | 29.80 | 30.01 | 32.27 | 36.61 | 28.63 |

Period (ten-day) | 3.34 | 3.39 | 3.04 | 2.96 | 3.05 | 3.18 | |

IMF_{2} | Energy (%) | 17.89 | 21.43 | 23.75 | 27.34 | 24.22 | 16.89 |

Period (ten-day) | 5.89 | 6.38 | 6.79 | 6.29 | 6.17 | 5.72 | |

IMF_{3} | Energy (%) | 7.94 | 7.57 | 11.98 | 8.09 | 17.82 | 9.12 |

Period (ten-day) | 11.80 | 11.48 | 14.16 | 12.28 | 15.97 | 11.87 | |

IMF_{4} | Energy (%) | 44.32 | 40.85 | 30.22 | 31.93 | 20.54 | 44.46 |

Period (ten-day) | 37.03 | 35.97 | 33.00 | 36.98 | 36.78 | 36.90 | |

IMF_{5} | Energy (%) | 0.60 | 0.31 | 3.47 | 0.28 | 0.72 | 0.59 |

Period (ten-day) | 68.73 | 83.47 | 63.53 | 61.08 | 75.53 | 78.63 | |

IMF_{6} | Energy (%) | 0.04 | 0.04 | 0.56 | 0.10 | 0.09 | 0.31 |

Period (ten-day) | 93.06 | 155.07 | 95.69 | 114.80 | 129.14 | 173.41 | |

IMF_{7} | Energy (%) | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

Period (ten-day) | - | - | - | - | - | - |

Month | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Average | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Flow (cms) | ||||||||||||||

Observation | 2.05 | 2.56 | 3.39 | 5.74 | 16.85 | 45.82 | 41.02 | 62.09 | 36.43 | 10.67 | 3.88 | 2.55 | 19.24 | |

Synthesis (I) | 1.91 | 1.98 | 2.28 | 3.44 | 12.17 | 33.89 | 27.69 | 41.37 | 25.33 | 8.59 | 3.54 | 2.33 | 13.71 | |

Synthesis (II) | 3.29 | 3.10 | 3.70 | 5.67 | 20.70 | 57.92 | 55.11 | 80.93 | 40.40 | 12.68 | 5.08 | 3.54 | 24.34 |

Simulation Results | ||||
---|---|---|---|---|

Unit: Quantity of Water in 10^{6} m^{3}/yr | ||||

Present | ||||

Scenario | No Support Industrial Water | Support Industrial Water (30 × 10^{4} CMD) | Support Industrial Water (10 × 10^{4} CMD) | |

Project | ||||

Reservoir supply | 101.93 | 165.14 | 128.30 | |

Domestic shortage | 0.27 | 9.74 | 2.18 | |

Reservoir inflow | 101.90 | 162.80 | 128.05 | |

Domestic SI | 0.002 | 0.309 | 0.10 | |

Reservoir efficiency | 1.91 | 3.09 | 2.40 | |

Public ACDD | 9.00 | 135.00 | 45.00 | |

ADWS during water shortage events | 0.03 | 0.06 | 0.05 |

^{3}/day; ACDD, average consecutive dry days; ADWS, average daily water shortage.

Simulation Results | ||||
---|---|---|---|---|

Unit: Quantity of Water in 10^{6} m^{3}/yr | ||||

No Support Industrial Water | ||||

Scenario | Present | Method I | Method II | |

Project | ||||

Reservoir supply | 101.93 | 98.25 | 95.52 | |

Domestic shortage | 0.27 | 3.95 | 6.68 | |

Reservoir inflow | 101.90 | 94.72 | 92.07 | |

Domestic SI | 0.002 | 0.668 | 1.96 | |

Reservoir efficiency | 1.91 | 1.84 | 1.79 | |

Domestic ACDD | 9.00 | 51.2 | 59.28 | |

ADWS during water shortage events | 0.03 | 0.05 | 0.07 |

Simulation Results | ||||
---|---|---|---|---|

Unit: Quantity of Water in 10^{6} m^{3}/yr; | ||||

Support Industrial Water (10 × 10^{4} m^{3}/s) | ||||

Scenario | Present | Method I | Method II | |

Project | ||||

Reservoir supply | 128.30 | 116.92 | 113.82 | |

Public shortage | 2.18 | 11.00 | 13.46 | |

Reservoir inflow | 128.05 | 112.37 | 109.43 | |

Public SI | 0.10 | 1.203 | 2.12 | |

Reservoir efficiency | 2.40 | 2.19 | 2.13 | |

Public ACDD | 45.0 | 111.0 | 113.82 | |

ADWS during water shortage events | 0.05 | 0.05 | 0.07 |

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## Share and Cite

**MDPI and ACS Style**

Chu, T.-Y.; Huang, W.-C.
Application of Empirical Mode Decomposition Method to Synthesize Flow Data: A Case Study of Hushan Reservoir in Taiwan. *Water* **2020**, *12*, 927.
https://doi.org/10.3390/w12040927

**AMA Style**

Chu T-Y, Huang W-C.
Application of Empirical Mode Decomposition Method to Synthesize Flow Data: A Case Study of Hushan Reservoir in Taiwan. *Water*. 2020; 12(4):927.
https://doi.org/10.3390/w12040927

**Chicago/Turabian Style**

Chu, Tai-Yi, and Wen-Cheng Huang.
2020. "Application of Empirical Mode Decomposition Method to Synthesize Flow Data: A Case Study of Hushan Reservoir in Taiwan" *Water* 12, no. 4: 927.
https://doi.org/10.3390/w12040927