# A Two-Step Approach for Analytical Optimal Hedging with Two Triggers

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Analytical Optimal Hedging Rule with Two Triggers (AOHR-TT)

_{1}and α

_{2}(0 < α

_{2}< α

_{1}< 1), are assigned to the expected demand, from the upper to the lower zones, when the initial and ending reservoir storage levels during the current period are located in the same zone. For levels that vary between zones, the rationing factor is between the corresponding values of the zones to keep the end-of-period storage level as close to the rule curves as possible. This policy, adopted in this paper, is illustrated in Figure 1.

**Figure 1.**The hedging policy for different reservoir zones based on the initial storage and ending storage levels.

#### 2.1.1. Mathematical Programming Model for the AOHR-TT

**Figure 2.**The penalty structure: (

**a**) penalty structure based on reservoir zone and (

**b**) penalty structure for water shortage based on demand during rationing.

#### 2.1.2. Model Solution and Analysis

_{2}proportion of the demand target, the proposed rule will be not applicable if the water availability is less than the minimum demand, i.e., $W{A}_{t}\text{\hspace{0.05em}}<\text{\hspace{0.05em}}{D}_{t}\text{\hspace{0.05em}}\cdot \text{\hspace{0.05em}}{\alpha}_{2}$, because the minimum water release constraint is not satisfied. When water availability exceeds the sum of the demand target and the carryover storage target, i.e., $W{A}_{t}\text{\hspace{0.05em}}>\text{\hspace{0.05em}}{D}_{t}\text{\hspace{0.05em}}+\text{\hspace{0.05em}}T{R}_{t}$, hedging is not necessary because the water stored in the reservoir is sufficient to satisfy both the water release and the carryover storage targets [5]. In these cases, the standard operating policy (SOP) is used to guide the reservoir operation. The level of water shortage located in Part 3 in Figure 2 means the implementation of SOP under which ${P}_{j}$ (j = 6) has no effect on hedging and can thus be set to a large value to punish the loss functions for avoiding a water shortage in Part 3.

- (1)
- Rule 1: ${S}_{t-1}$ is located in Zone 1, which corresponds to normal conditions, i.e., $\text{\hspace{0.05em}}T{R}_{t}\le {S}_{t-1}\le {K}_{t}$

- (2)
- Rule 2: ${S}_{t-1}$ is located in Zone 2, which corresponds to drought conditions, i.e., $\text{\hspace{0.05em}}F{R}_{t}\le {S}_{t-1}<T{R}_{t}$

- (3)
- Rule 3: ${S}_{t-1}$ is located in Zone 3, which corresponds to severe drought conditions, i.e., $0\le {S}_{t-1}<F{R}_{t}$

#### 2.2. Formulation of the Optimization-Simulation Model for the AOHR-TT

#### 2.2.1. Optimization Model

#### Optimization Objective

#### System Constraints

#### Method of Solution

#### 2.2.2. Simulation Model

- (1)
- According to the relationship between the initial storage level and the present level of the rule curves, the hedging sub-rule in the AOHR-TT is triggered first. Then, the current release is obtained based on the current water availability.
- (2)
- The ending reservoir storage level in the current period is then obtained by the water balance equation, and it is used to determine which hedging sub-rule is triggered in the next period. At the end of this step, the simulation procedure returns to step (1).

## 3. Study Area and Scenario Design

#### 3.1. Study Area

^{2}, and receives an annual rainfall of approximately 1032 mm. The maximum active storage and dead storage of the Xujiahe reservoir are 2.99 × 10

^{8}m

^{3}and 1.41 × 10

^{8}m

^{3}, respectively. The reservoir system is primarily designed to meet the water demands of agriculture in the irrigation district of Xujiahe.

**Figure 8.**The monthly average inflow into Xujiahe reservoir and the monthly average water demand of agriculture.

#### 3.2. Scenario Design

_{1}or β

_{2}) of the demand (where 0 < β

_{2}< β

_{1}< 1). In this paper, Equation (24) is used as the optimization objective for long-term periods in the conventional rule curves scenario. The rationing factors (β

_{1}and β

_{2}) and monthly rule curves are optimized as decision variables. As a discrete hedging rule, the operation results are compared with those of the continuous hedging rules.

_{t}) and monthly target rule curve, which have a significant impact on the operation benefits from long-term periods, are optimized as decision variables via IPSO. The results of Shiau’s method, a continuous hedging rule, are compared to those of the discrete hedging rule. This scenario and the proposed rule are both analytical continuous hedging rules. Thus, the results of this method can also be used for comparison to those of the proposed rule.

#### 3.3. Evaluation Criteria

^{2}) and Nash-Sutcliffe coefficient (NSE) are employed to evaluate the similarities between DP and the other scenarios.

## 4. Results and Discussion

#### 4.1. Analysis of Rule Curves in the Proposed Rule

**Figure 11.**The triggering frequencies of the three hedging sub-rules of the proposed rule for each month during the 28 years from 1973 to 2000.

#### 4.2. Comparison and Analysis of Operation Scenarios

#### 4.2.1. Comparison and Analysis of the Operation Types

_{1}and α

_{2}) and unit penalties (i.e., P

_{1}, P

_{2}, P

_{3}, P

_{4}and P

_{5}), are listed in Table 1. The proposed rule can be obtained by substituting the level of the rule curves and parameters into Equations (11)–(23). As an example, the optimal policy for a representative month (July) is illustrated in Figure 12. The operation policies obtained using Shiau’s method and the conventional rule curves for the representative month are shown in Figure 13 and Figure 14. Figure 12, Figure 13 and Figure 14 clearly show that three hedging sub-rules are applied in the operation of the reservoir under the proposed rule, whereas only one rule is applied in the operation of the reservoir under the other methods. The triggering mechanism for hedging in the proposed rule is based not only on the current water availability but also on the relationship between the initial storage level and the levels of the rule curves (target rule curve and firm rule curve). This relationship determines whether the amount of stored water is increased. When the initial storage level is less than the levels of the rule curves at the end of month (i.e., $T{R}_{t}$ and $F{R}_{t}$), more water is stored. When the initial storage level is greater than the levels of the rule curves, less water is stored. Therefore, unlike the case of Rule 1 shown in Figure 12, when water availability is within the range of $290\times {10}^{6}{m}^{3}\text{\hspace{0.05em}}\le W{A}_{t}\le 293\text{\hspace{0.05em}}\times {10}^{6}{m}^{3}$, Rule 2 prioritizes the storage of water until the target storage level ($T{R}_{t}$) is reached (i.e., $W{A}_{t}=291\times {10}^{6}{m}^{3}$). A similar difference exists between Rule 2 and Rule 3. In contrast to the proposed rule, hedging is implemented only based on the current water availability or initial storage in the operation policies obtained using Shiau’s method and the conventional rule curves. Figure 12, Figure 13 and Figure 14 also show that the conventional rule curves produce a discrete form of the water supply curve (Figure 14), which leads to fixed rationing factors (0.8 and 0.9), whereas the proposed rule (Figure 12) and Shiau’s method (Figure 13) produce continuous forms with variable rationing factors. The amount of hedging is determined by the fixed rationing factors in the conventional rule curves scenario, whereas the amount of hedging is decided by the analytical expression of the water supply in the proposed rule and Shiau’s method. For each time step in the proposed rule, the linear hedging within the range bounded by SWA and EWA produces a step-wise hedging, which is obviously different from Shiau’s method. For Rule 1 in the propose rule, a large proportion of the water demand ($0.973{D}_{t}\text{\hspace{0.05em}}$, i.e., $\text{\hspace{0.05em}}50\text{\hspace{0.05em}}\times {10}^{6}{m}^{3}$ in July) is met within the range of $SW{A}_{2t}<W{A}_{t}\le EW{A}_{2t}$, which leads to a lower marginal benefit of release in this range compared with that in the range of $SW{A}_{3t}<W{A}_{t}\le EW{A}_{3t}$. Therefore, the slope of the linear hedging (0.552) in the range of $SW{A}_{2t}<W{A}_{t}\le EW{A}_{2t}$ is smaller than that (0.680) in the range of $SW{A}_{3t}<W{A}_{t}\le EW{A}_{3t}$. The operational form obtained by Taghian’s method is not discussed because it is built on the basis of an LP model run for each time step.

Parameters | ||||||
---|---|---|---|---|---|---|

α_{1} | α_{2} | P_{1} | P_{2} | P_{3} | P_{4} | P_{5} |

0.973 | 0.838 | 50.0 | 67.9 | 144.9 | 55.1 | 68.1 |

#### 4.2.2. Comparison and Analysis of Long-Term Operation Results

Indices | Scenarios | |||||
---|---|---|---|---|---|---|

SOP | Conventional Rule Curves | Shiau’s Method | Taghian’s Method | DP Model | Proposed Rule | |

MSI | 0.5340 | 0.2470 | 0.0840 | 0.1563 | 0.0533 | 0.0695 |

MSR | 100.00% | 20.00% | 19.94% | 20.00% | 19.68% | 16.22% |

Reliability | 98.50% | 82.44% | 82.74% | 87.20% | 93.40% | 81.85% |

Scenarios | Rationing Factors (RF) of the Scenarios | ||||
---|---|---|---|---|---|

RF = 0.8 | 0.8 < RF < 0.9 | RF = 0.9 | 0.9 < RF < 1 | RF = 1 | |

Conventional Rule Curves | 2.38% | 0 | 15.18% | 0 | 82.44% |

Shiau’s Method | 0 | 2.68% | 0 | 14.58% | 82.74% |

Taghian’s Method | 1.49% | 0 | 0 | 11.31% | 87.20% |

Proposed Rule | 0 | 2.08% | 0 | 15.77% | 81.85% |

#### 4.2.3. Comparison and Analysis of Critical-Period Operation Results

Years | Indices | Scenarios | ||||
---|---|---|---|---|---|---|

SOP | Conventional Rule Curves | Shiau’s Method | Taghian’s Method | Proposed Rule | ||

1977 | MSI | 0.0000 | 0.1667 | 0.0014 | 0.1464 | 0.0125 |

MSR | 0.00% | 10.00% | 1.29% | 9.37% | 2.74% | |

Reliability | 100.00% | 83.33% | 91.67% | 83.33% | 83.33% | |

1978 | MSI | 0.0000 | 1.0833 | 0.4698 | 0.4393 | 0.4591 |

MSR | 0.00% | 20.00% | 19.94% | 9.37% | 16.04% | |

Reliability | 100% | 41.67% | 50.00% | 50.00% | 41.67% | |

1979 | MSI | 14.9586 | 2.1667 | 0.9523 | 1.8459 | 0.7874 |

MSR | 100.00% | 20.00% | 19.80% | 20.00% | 16.22% | |

Reliability | 58.33% | 8.33% | 8.33% | 8.33% | 16.67% | |

1980 | MSI | 0.0000 | 0.5833 | 0.0375 | 0.5530 | 0.0188 |

MSR | 0.00% | 20.00% | 6.20% | 20.00% | 2.74% | |

Reliability | 100.00% | 66.67% | 66.67% | 66.67% | 75.00% | |

1981 | MSI | 0.0000 | 0.0833 | 0.0431 | 0.0732 | 0.0313 |

MSR | 0.00% | 10.00% | 7.17% | 9.37% | 4.74% | |

Reliability | 100.00% | 91.67% | 75.00% | 91.67% | 75% | |

5 years | MSI | 2.9917 | 0.8167 | 0.3008 | 0.6116 | 0.2618 |

MSR | 100.00% | 20.00% | 19.94% | 20.00% | 16.22% | |

Reliability | 91.67% | 58.33% | 58.33% | 60.00% | 58.33% |

^{2}and NSE values, produced by the proposed rule, indicate the greatest similarity to the DP model. Therefore, the reservoir storage process obtained using the proposed rule achieves a high level of agreement with the optimal reservoir storage process. Thus, the proposed rule in this paper is reasonable and effective during critical periods.

**Figure 15.**The end-of-month reservoir storage levels obtained using dynamic programming (DP) and the other scenarios during the critical periods from 1977 to 1981.

**Table 5.**The similarity comparison of storage process produced by dynamic programming (DP) and the other scenarios during the critical periods.

Evaluation Criteria | Scenarios | ||||
---|---|---|---|---|---|

SOP | Conventional Rule Curves | Shiau’s Method | Taghian’s Method | Proposed Rule | |

R^{2} | 0.974 | 0.985 | 0.987 | 0.984 | 0.990 |

NSE | 0.917 | 0.960 | 0.967 | 0.959 | 0.982 |

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Hu, T.; Zhang, X.-Z.; Zeng, X.; Wang, J.
A Two-Step Approach for Analytical Optimal Hedging with Two Triggers. *Water* **2016**, *8*, 52.
https://doi.org/10.3390/w8020052

**AMA Style**

Hu T, Zhang X-Z, Zeng X, Wang J.
A Two-Step Approach for Analytical Optimal Hedging with Two Triggers. *Water*. 2016; 8(2):52.
https://doi.org/10.3390/w8020052

**Chicago/Turabian Style**

Hu, Tiesong, Xu-Zhao Zhang, Xiang Zeng, and Jing Wang.
2016. "A Two-Step Approach for Analytical Optimal Hedging with Two Triggers" *Water* 8, no. 2: 52.
https://doi.org/10.3390/w8020052