# An Analytical Framework for Investigating Trade-Offs between Reservoir Power Generation and Flood Risk

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Study Area

^{9}m

^{3}, the flood-limited water level is 213.37 m, the flood-limited storage volume is 5.220 × 10

^{8}m

^{3}, the total installed capacity is 250 MW, the total overflow capacity of the turbine unit is 1270 m

^{3}/s, and the minimum downstream water demand (including industry, agriculture, urban life, ecology, etc.) is 200 m

^{3}/s.

^{9}m

^{3}) during the main stage of flood season (from 21 June to 25 August) and (2) normal storage volume (6.456 × 10

^{9}m

^{3}) during the early stage of flood season (from 1 June to 20 June) and the late stage of flood season (from 26 August to 30 September). Although the operation is easy, unnecessary spills during the flood season and energy reduction in both flood and non-flood seasons may be caused. Consequently, it is necessary to increase the storage volume of the main flood season for power generation improvement without adding extra flood risk.

## 3. Methods

#### 3.1. The Dynamic Control of Carryover Storage (DCCS)

#### 3.1.1. Two-Stage Model

_{0}and S

_{2}) and the inflow forecasting in two stages (i.e., I

_{1}and I

_{2}) are known in DCCS, while the release volume in two stages (i.e., R

_{1}and R

_{2}) and the carryover storage (S

_{1}) between the two stages are decision variables [34,35]. Only the release in Stage 1 (R

_{1}) and carryover storage (S

_{1}) is determined, and the window rolls over to the next period until the final scheduling period [28,29,36,37].

_{k}, I

_{k}, and R

_{k}are the release (in rate), inflow, and release in Stage k (k = 1, 2), respectively. S

_{2}is assumed to be equal to FLSV (S

^{L}) to avoid additional flood risk [27]. That is, all the excess storage above the primary FLSV should be released in Stage 2, namely,

_{2}does not have to be equal to S

^{L}in practice, because the decision is updated when the model rolls over to the next period.

#### 3.1.2. Forecast Errors and Flood Risk

_{2}and ${\overline{I}}_{2}$ represent actual and forecasted inflows. ${\overline{R}}_{2}$ denotes the expected release in Stage 2; τ (·) is the downstream flood risk, which indicates the probability of the actual release exceeding the downstream safety threshold Q

_{thres}. δ is the flood-safety margin, which is the difference between downstream safety threshold Q

_{thres}and expected release ${\overline{R}}_{2}$. H (∙) is the probability density function of ε.

#### 3.1.3. Power Generation in Two Stages

_{1}) and Stage 2 (E

_{2}),

_{k}is calculated as the product of hydro-turbine efficiency η, water head [(SSR(S

_{k}

_{−1}) + SSR(S

_{k}))/2 − SDR], and reservoir release R

_{k}[32,40]. When the release in two stages is replaced by carryover storage according to the water balance in Equations (1) and (2), power generation E

_{k}is the function of carryover storage S

_{1}, as expressed by Equation (7):

_{0}) and SSR (S

_{1}) represent the initial and ending water level of Stage 1, respectively, and SSR (S

^{L}) is the water level of S

^{L}. SDR is the downstream water level and can be simplified as a constant [32].

#### 3.2. Model Formulation

#### 3.2.1. Objective Function

_{1}and G

_{2}are power generation and flood risk objectives, respectively. E

^{max}is the maximum power generation in two stages without spilling and is equal to the product of installed capacity N

^{max}and two stages’ duration, i.e., E

^{max}= N

^{max}× T∙Δt.

_{1}to G

_{1}and δ to G

_{2}are given by

_{1}has been proven to be positive if there is no spilled water, i.e., increasing carryover storage generates a net gain in power generation in the two-period hydropower scheduling under no spilled water [32]. SSR′ (S

_{1}) is the first-order derivative of SSR (S

_{1}) and is positive, i.e., SSR′ (S

_{1}) > 0 [37,41]. The forecast error series (ε) is assumed to follow an unbiased Gaussian distribution, and the marginal contribution of δ to G

_{2}can be expressed as the second function of Equation (9) [27].

#### 3.2.2. Constraints

_{1}and δ, shown as follows:

_{thres}− δ denotes the expected volume of release in Stage 2, i.e., Q

_{thres}− δ = ${\overline{R}}_{2}$. ${S}_{1}^{C}$ is the carryover storage that Stage 1 releases water at the minimum downstream water demand, i.e., R

_{1}= ${R}_{1}^{\mathrm{m}\mathrm{i}\mathrm{n}}$. It is worth noting that the downstream water demand in Stage 2 (${R}_{2}^{\mathrm{m}\mathrm{i}\mathrm{n}}$) is regarded to be satisfied since all carryover storage should be released in Stage 2. The carryover storage should not be lower than S

^{L}to guarantee water supply. In the meantime, the thresholds for the tolerance of flood risk (τ ≤ τ

_{r}) should be met to guarantee downstream flood control [6,42]. That is, δ should be greater than the minimum safety margin δ

^{min}, and δ

^{min}is expressed as follows when the forecast error follows an unbiased Gaussian distribution [27]:

^{−1}(∙) is the inverse of the cumulative probability function with a standard normal distribution.

_{1}= ${E}_{1}^{\mathrm{m}\mathrm{a}\mathrm{x}}$) without spilling as dE

_{1}/dS

_{1}< 0, and ${S}_{1}^{B}$ is the maximum carryover storage that leads to the maximum power generation in Stage 2 without spilling (E

_{2}= ${E}_{2}^{\mathrm{m}\mathrm{a}\mathrm{x}}$) as dE

_{2}/dS

_{1}> 0. When the initial storage capacity, inflow in two stages, the installed capacity of a reservoir N

^{max}, and the length of two stages are given, then ${S}_{1}^{A}$ and ${S}_{1}^{B}$ can be calculated as:

_{1}, while the flood risk target is only related to δ according to Equation (9).

#### 3.2.3. Optimal Conditions

#### 3.3. Trade-Offs between Power Generation and Flood Risk

_{1}) and flood-safety margin (δ), which is determined by the relationships between the marginal utility of power generation (MUPG, f

_{1}) and that of flood risk (MUFR, f

_{2}). Therefore, in this section, the MUPG under various forecasted inflow and carryover storage is evaluated to obtain the allowable inflow range for hedging and trade-offs during DCCS. Besides, the effects of forecast uncertainty and risk tolerance on trade-offs and operation decisions are analyzed.

#### 3.3.1. Evaluation of the Marginal Utility of Power Generation

_{1}and ${\overline{I}}_{2}$) and the carryover storage (S

_{1}). To evaluate the MUPG, the three influencing factors should be discussed separately, where MUPG was proven to increase with S

_{1}by Zhao et al. [32], as shown by the solid blue line in Figure 3. The influence of I

_{1}is analyzed through two scenarios, that is, a relatively large I

_{1}resulting in ${S}_{1}^{A}$ ≥ S

^{L}and a small I

_{1}leading to ${S}_{1}^{A}$< S

^{L}. Then, this part focuses on the impacts of ${\overline{I}}_{2}$ under each scenario since ${\overline{I}}_{2}$> is directly engaged in the allocation of S

_{1}and δ.

_{1}is relatively large and triggers the maximum power generation in the Stage 1 constraint, i.e., ${S}_{1}^{A}$ ≥ S

^{L}, the lower bound of carryover storage is ${S}_{1}^{A}$, as shown in Figure 3a. The bottom solid curve depicts the variation of MUPG with S

_{1}under the forecasting inflow ${\overline{I}}_{2}^{F}$, which equals downstream water demand (i.e., ${\overline{I}}_{2}^{F}$ = ${R}_{1}^{\mathrm{min}}$) and is the minimum allowable inflow for trade-offs. The small forecasting inflow in Stage 2 causes the larger ${S}_{1}^{B}$, i.e., ${S}_{1}^{B}$ > ${S}_{1}^{C}$, where the solid gray curve indicates the infeasibility domain of MUPG. When ${\overline{I}}_{2}$ increases from ${\overline{I}}_{2}^{F}$ to ${\overline{I}}_{2}^{D}$, the curve of MUPG moves upward, ${S}_{1}^{B}$ decreases, and the upper bound of the carryover storage is ${S}_{1}^{C}$. When Stage 1 releases available water to meet downstream water demand, the forecasting inflow ${\overline{I}}_{2}^{D}$ causes power generation in Stage 2 to peak, i.e., ${S}_{1}^{B}$ = ${S}_{1}^{C}$. When ${\overline{I}}_{2}$ exceeds ${\overline{I}}_{2}^{D}$, the upper bound of carryover storage is ${S}_{1}^{B}$, and the MUPG curve shortens with increasing ${\overline{I}}_{2}$. When ${\overline{I}}_{2}$ = ${\overline{I}}_{2}^{G}$, power generation in two stages reaches its maximum, i.e., ${S}_{1}^{B}$ = ${S}_{1}^{A}$, and the MUPG curve becomes a point. As a result, ${\overline{I}}_{2}^{G}$ is the maximum forecasting inflow allowed for trade-offs during DCCS, and the forecasting inflow in the Stage 2 within [${\overline{I}}_{2}^{F}$, ${\overline{I}}_{2}^{G}$] allows for trade-offs under relatively large I

_{1}.

_{1}is relatively small, leading to ${S}_{1}^{A}$ < S

^{L}, the lower bound of carryover storage is S

^{L}, as shown in Figure 3b. ${\overline{I}}_{2}^{H}$ is the forecasted inflow trigger of the maximum power generation of Stage 2 and FLSV constraints at the same time (i.e., ${S}_{1}^{B}$ = S

^{L}). Then, the forecasting inflow in Stage 2 within [${\overline{I}}_{2}^{F}$, ${\overline{I}}_{2}^{H}$] is allowed for trade-offs under a relatively small I

_{1}.

#### 3.3.2. The Starting and Ending Points of Hedging

^{SWA}represent the forecasted inflow and flood-safety margin at the starting point (SWA) of hedging, causing the MUPG (f

_{1}(${S}_{1}^{A}$)) at the lower bound of carryover storage (${S}_{1}^{A}$) to equal the marginal utility of flood risk (MUFR, f

_{2}(δ

^{SWA})), i.e., f

_{1}(${S}_{1}^{A}$) = f

_{2}(δ

^{SWA}). Equation (17) can be used to calculate ${\overline{I}}_{2}^{\mathit{SWA}}$ and δ

^{SWA}.

^{L}, the ${S}_{1}^{A}$ in Equation (17) needs to be replaced by S

^{L}. ${\overline{I}}_{2}^{\mathit{SWA}}$ and δ

^{SWA}under the lower bound of carryover storage S

^{L}are written as follows, illustrating that ${\overline{I}}_{2}^{\mathit{SWA}}$ and δ

^{SWA}vary with the inflow of Stage 1.

^{EWA}are the forecasted inflow and flood-safety margin at the ending point of hedging, which make the MUPG (f

_{1}(${S}_{1}^{C}$)) at the upper bound of carryover storage (${S}_{1}^{C}$) equal the marginal utility of flood risk (MUFR, f

_{2}(δ

^{EWA})), i.e., f

_{1}(${S}_{1}^{C}$) = f

_{2}(δ

^{EWA}), and they are obtained by Equation (19),

_{1}= ${S}_{1}^{B}$, the power generation of Stage 2 has its maximum value without abandoned water. If the forecasted inflow is smaller than the actual inflow, the reservoir fails to increase power generation, which is accompanied by abandoned water and increasing flood risk. Thus, the MUFR is larger than the MUPG in this situation. SWA and EWA points, as shown in Figure 3, divide the relationship between MUPG and MUFR into three categories: R.1, R.2, and R.3.

- R.1:
- When the forecasted inflow (${\overline{I}}_{2}$) ranges from ${\overline{I}}_{2}^{G}$ to ${\overline{I}}_{2}^{\mathit{SWA}}$ or from ${\overline{I}}_{2}^{H}$ to ${\overline{I}}_{2}^{\mathit{SWA}}$, the relatively large inflow illustrates a large flood risk, and the MUFR exceeds MUPG under optimal conditions in Appendix B1, i.e., f
_{2}(δ*) > f_{2}(δ^{SWA}) = f_{1}(${S}_{1}^{A}$) or f_{2}(δ*) > f_{2}(δ^{SWA}) = f_{1}(S^{L}). As a result, carryover storage should be kept at its lowest level in order to accommodate a large, forecasted inflow and to reduce the potential flood risk in Stage 2. Under the lower bound ${S}_{1}^{A}$, the optimal carryover storage and flood-safety margin are ${S}_{1}^{*}$ = ${S}_{1}^{A}$, δ* = Q_{thres}+ S^{L}− ${\overline{I}}_{2}$ − ${S}_{1}^{A}$. When the lower bound is S^{L}, they are written as ${S}_{1}^{*}$ = S^{L}, δ* = Q_{thres}− ${\overline{I}}_{2}$. - R.2.:
- When ${\overline{I}}_{2}$ ranges from ${\overline{I}}_{2}^{\mathit{SWA}}$ to ${\overline{I}}_{2}^{\mathit{EWA}}$, more carryover storage brings more power generation but also higher flood risk. The optimal carryover storage and flood-safety margin are to equalize the marginal utility, i.e., f
_{2}(δ*) = f_{1}(${S}_{1}^{*}$). - R.3.:
- When ${\overline{I}}_{2}$ ranges from ${\overline{I}}_{2}^{\mathit{EWA}}$ to ${\overline{I}}_{2}^{\mathit{F}}$, the relatively small inflow shows that floodwater can be carried over to the next stage as much as possible, and the MUPG is always greater than MUFR based on Equation (A9), i.e., f
_{1}(${S}_{1}^{C}$) > f_{2}(δ^{EWA}) = f_{2}(δ*). Therefore, the carryover storage should be kept at the upper bound of carryover storage, namely, Stage 1 releases the downstream water demand Rmin 1, and the remaining available water from Stage 1 is carried over to Stage 2 to increase power generation due to the relatively small forecasted inflow, i.e., ${S}_{1}^{*}$ = ${S}_{1}^{C}$, δ* = Q_{thres}+ S^{L}− ${\overline{I}}_{2}$ − ${S}_{1}^{C}$.

#### 3.3.3. Effects of Forecast Uncertainty and Risk Tolerance

^{SWA}and δ

^{EWA}) are related to the forecast uncertainty of inflow (σ) based on Equations (17)–(19), illustrating that the hedging range (the difference between δ

^{EWA}and δ

^{SWA}) varies with σ. However, extremely tiny or very big forecast uncertainties result in little or huge flood risk, and the specific target is power generation or flood risk. This section first derives the minimum and maximum forecast uncertainties to identify the forecast uncertainty range permitted for hedging during DCCS and then investigates the effects of forecast uncertainty and risk tolerance on the hedging range.

- The allowable forecast uncertainty

^{G}= Q

_{thres}+ S

^{L}− ${\overline{I}}_{2}^{G}$ − ${S}_{1}^{A}$, f

_{1}(${S}_{1}^{A}$) = f

_{2}(δ

^{G}) (equal marginal utility principle), and the forecast uncertainty at point G is deduced from Equation (16), denoted by σ

^{min},

^{min}, causing little flood risk, is the minimum forecast uncertainty for hedging, and power generation plays the dominant role within the allowable inflow range for DCCS application. There is no hedging between the two objectives until the forecasted inflow equals ${\overline{I}}_{2}^{G}$, and the power generation in two stages both reaches its maximum values (i.e., ${S}_{1}^{A}$ = ${S}_{1}^{B}$, ${\overline{I}}_{2}$ = ${\overline{I}}_{2}^{G}$). According to Equation (15), the unique hedging point G is as shown in Figure 5a, and the optimal carryover storage is always equal to the upper bounds because MUPG is greater than MUFR (i.e., f

_{1}> f

_{2}).

^{F}= Q

_{thres}+ S

^{L}− ${\overline{I}}_{2}^{F}$ − ${S}_{1}^{A}$, f

_{1}(${S}_{1}^{A}$) = f

_{2}(δ

^{F}) (equal marginal utility principle), and the forecast uncertainty at point F is expressed as σ

^{max},

^{max}represents the maximum forecast uncertainty allowed for hedging, illustrating a relatively high flood risk, and the carryover storage remains at the lower bound (i.e., ${S}_{1}^{*}$ = ${S}_{1}^{A}$) to reduce the flood risk. There is no hedging between two objectives until the forecasted inflow is as small as the downstream water demand ($\overline{I}$

_{2}= ${\overline{I}}_{2}^{F}$). Because MUPG is less than MUFR (i.e., f

_{1}< f

_{2}), when $\overline{I}$

_{2}> ${\overline{I}}_{2}^{F}$, there is only one hedging point F, as shown in Figure 5b.

^{L}, as shown in Figure 6, and they are similar to the points G and F, respectively. When point H is the starting point of hedging, we have ${S}_{1}^{*}$ = S

^{L}, δ

^{H}= Q

_{thres}− ${\overline{I}}_{2}^{H}$, f

_{1}(S

^{L}) = f

_{2}(δ

^{H}), and ${S}_{1}^{A}$ and ${\overline{I}}_{2}^{G}$ in Equation (20) for solving σ

^{min}are replaced by S

^{L}and ${\overline{I}}_{2}^{H}$, respectively. If point E is the starting point of hedging, we have ${S}_{1}^{*}$ = S

^{L}, δ

^{E}= Q

_{thres}− ${\overline{I}}_{2}^{F}$, f

_{1}(S

^{L}) = f

_{2}(δ

^{E}), and ${S}_{1}^{A}$ in Equation (21) for σ

^{max}becomes S

^{L}. Equation (22) expresses σ

^{min}and σ

^{max}under the lower bound S

^{L}:

^{min}≤ σ≤σ

^{max}. If σ < σ

^{min}, causing too little flood risk, the power generation objective takes precedence over the flood risk objective during DCCS, which requires carryover storage as high as its upper bounds, and the optimal carryover storage under different forecasted inflows is represented by black solid curves in Figure 5a and Figure 6a, S= min{${S}_{1}^{B}$, ${S}_{1}^{C}$}; if σ > σ

^{max}, resulting in a very large flood risk, then the reduced flood risk always dominates over the increasing power generation during DCCS, which calls for carryover storage as low as its lower bounds, and optimal solutions are represented by black solid curves in Figure 5b and Figure 6b, ${S}_{1}^{*}$ = max{${S}_{1}^{A}$, S

^{L}}.

- 2.
- Combined impacts of forecast uncertainty and risk tolerance

^{min}) was introduced by Ding et al. [23]. According to Equation (11), δ

^{min}decreases as the risk tolerance (τ

_{r}) increases and the forecast uncertainty (σ) decreases. Δ

^{SWA}and δ

^{EWA}, defined in Section 3.2, are the flood-safety margins at the start and end points of hedging, respectively, when the minimum flood-safety margin (δ

^{min}) constraint is unbinding. However, the flood-safety margin range for hedging may be altered when δ

^{min}is considered.

_{r}: Case 1 (δ

^{min}< δ

^{SWA}), Case 2 (δ

^{SWA}≤ δ

^{min}≤ δ

^{EWA}), and Case 3 (δ

^{min}> δ

^{EWA}), because τ

_{r}is only related to δ

^{min}but is not relevant to δ

^{SWA}and δ

^{EWA}. The flood-safety margin range for hedging in Case 1 is the same as the unbinding δ

^{min}; that range is from δ

^{min}to δ

^{EWA}in Case 2, and there is no hedging in Case 3. That is, as risk tolerance decreases, there are fewer opportunities for hedging between power generation and flood risk, and the flood risk objective becomes more important for reservoirs.

^{min}, δ

^{SWA}, and δ

^{EWA}for a given τ

_{r}. To evaluate the influence of σ, the trends of increasing δ

^{min}, δ

^{SWA}, and δ

^{EWA}as σ increases are deduced in Appendix A, i.e., dδ

^{min}/dσ = Փ

^{−1}(1-τ

_{r}) > 0, dδ

^{EWA}/dσ > dδ

^{SWA}/dσ > 0, and the increase rates of dδ

^{EWA}/dσ and dδ

^{SWA}/dσ as σ increase are derived to be small according to Equations (A3) and (A5). They are assumed to be zero here for simplicity (i.e., d

^{2}δ

^{SWA}/dσ

^{2}≈0, d

^{2}δ

^{EWA}/dσ

^{2}≈0). As a result, when the given τ

_{r}is very small (i.e., Փ

^{−1}(1 − τ

_{r}) > dδ

^{EWA}/dσ) or very large (i.e., Փ

^{−1}(1 − τ

_{r}) < dδ

^{SWA}/dσ), forecast uncertainty does not influence the relationships among δ

^{min}, δ

^{SWA}, and δ

^{EWA}within its allowable range [σ

^{min}, σ

^{max}], as illustrated in Figure 7a,b. That is, only Case 1 (i.e., δ

^{min}< δ

^{SWA}) exists under a large risk tolerance, whereas only Case 3 (δ

^{min}> δ

^{EWA}) occurs with a small risk tolerance. Furthermore, as seen in Figure 7c, increasing forecast uncertainty affects the starting of hedging when the risk tolerance is medium and satisfies dδ

^{SWA}/dσ ≤ Փ

^{−1}(1 − τ

_{r}) ≤ dδ

^{EWA}/dσ, and the range for hedging may be narrowed.

^{min}< δ

^{SWA}), Case 2 (δ

^{SWA}≤ δ

^{min}≤ δ

^{EWA}), and Case 3 (δ

^{min}> δ

^{EWA}).

#### 3.3.4. The Optimal Hedging Rules (OHR)

_{2}) and MUPG (f

_{1}) for the three cases under different lower bounds of carryover storage, where the left panel shows the variation of MUFR (f

_{2}) with the increasing δ

^{min}, and the right panel shows the variation of MUPG (f

_{1}) with the decreasing ${\overline{I}}_{2}$.

- Case 1: δ
^{min}< δ^{SWA}

^{min}< δ

^{SWA}, which has no influence on the start and end points of hedging. As a result, the optimal hedging rule under this case is the same as unbinding δ

^{min}. There are three relationships between MUPG and MUFR denoted as Case 1-R.1, Case 1-R.2, and Case 1-R.3 under different inflow conditions, which are the same as R.1, R.2, and R.2 in Section 3.2, respectively. The optimal solutions under different inflow conditions can be summarized as follows when the lower bound is ${S}_{1}^{A}$:

^{L}, as shown in Figure 9a, the maximum allowable inflow changes from ${\overline{I}}_{2}^{G}$ to ${\overline{I}}_{2}^{H}$, and the optimal carryover storage in the first subfunction in Equation (23) is S

^{L}, which is written as follows:

- 2.
- Case 2: δ
^{SWA}≤ δ^{min}≤ δ^{EWA}

^{SWA}≤ δ

^{min}≤ δ

^{EWA}. Point M is the new beginning point for hedging, where the flood-safety margin is δ

^{min}. The range for hedging is narrowed since δ

^{min}is greater than δ

^{SWA}, illustrating that much more flood-safety margin is required to reduce flood risk if the risk tolerance declines or the forecast uncertainty under a medium risk tolerance increase. The corresponding carryover storage ${S}_{1}^{\mathrm{M}}$ and forecasted inflow ${\overline{I}}_{2}^{M}$ to point M are acquired by the marginal utility principle, i.e., f

_{1}(${S}_{1}^{\mathrm{M}}$) = f

_{2}(δ

^{min}),

_{2}(δ

^{min}), i.e., f

_{1}(S

_{1}) > f

_{1}(${S}_{1}^{M}$) = f

_{2}(δ

^{min}), indicating that allocating available water after meeting the minimum safety margin requirement for carryover storage will increase power generation, i.e., δ* = δ

^{min}, ${S}_{1}^{*}$= Q

_{thres}+ S

^{L}− ${\overline{I}}_{2}$ − δ

^{min}.

_{1}(${S}_{1}^{*}$) = f

_{2}(δ*).

- 3.
- Case 3: δ
^{min}> δ^{EWA}

^{min}> δ

^{EWA}. ${\overline{I}}_{2}^{\mathit{CM}}$ is a special inflow that raises the carryover capacity to its upper bound while decreasing the optimal flood-safety margin to its lowest value, i.e., ${S}_{1}^{C}$ + δ

^{min}= Q

_{thres}+ S

^{L}− ${\overline{I}}_{2}^{\mathit{CM}}$.

_{thres}+ S

^{L}− ${\overline{I}}_{2}$ − ${S}_{1}^{C}$.

## 4. Application Results

#### 4.1. Inputs of Model

^{0.407}+ 184.10. The efficiency coefficient of the turbine η equaled 8.5, and the stage downstream water level SDR was 184.5 m based on the Nierji Reservoir Operation Manual issued in 2014.

_{thres}was 1600 m

^{3}/s according to the historical data. The downstream water demand, including municipal, industrial, agricultural, and environmental flow, was 200 m

^{3}/s.

#### 4.2. Optimal Hedging Rules Experiment

^{8}to 1.728 × 10

^{7}m

^{3}and 1.728 × 10

^{8}to 3.456 × 10

^{7}m

^{3}, respectively.

_{1}) increases, while ${S}_{1}^{B}$ falls as the forecasted inflow in Stage 2 (${\overline{I}}_{2}$) rises. For I

_{1}≤ 8.788 × 10

^{7}m

^{3}, the lower bound is S

^{L}(5.220 × 10

^{9}m

^{3}), and the maximum power generation of Stage 1 (${S}_{1}^{A}$) limits the lower bound when I

_{1}> 8.788 × 10

^{7}m

^{3}. The critical forecasted inflow (${\overline{I}}_{2}^{D}$) for the upper bound is 1.160 × 10

^{8}m

^{3}when I

_{1}= 7.540 × 10

^{7}m

^{3}. That is, ${S}_{1}^{C}$ is the upper bound (ranging from 5.237 × 10

^{9}to 5.279 × 10

^{9}m

^{3}) when 3.456 × 10

^{7}≤ I

_{1}≤ 7.540 × 10

^{7}m

^{3}and 3.456 × 10

^{7}≤ ${\overline{I}}_{2}$ ≤ 1.160 × 10

^{8}m

^{3}because of ${S}_{1}^{B}$ ≥ ${S}_{1}^{C}$. That bound shifts to ${S}_{1}^{B}$, which increases from 5.279 × 10

^{9}to 5.361 × 10

^{9}m

^{3}when 7.540 × 10

^{7}< I

_{1}≤ 9.957 × 10

^{7}m

^{3}and 1.160 × 10

^{8}< ${\overline{I}}_{2}$ ≤ 1.642 × 10

^{8}m

^{3}. In particular, ${S}_{1}^{A}$= ${S}_{1}^{B}$= 5.231 × 10

^{9}m

^{3}when I

_{1}= 9.957 × 10

^{7}m

^{3}and ${\overline{I}}_{2}$ = 1.642 × 10

^{8}m

^{3}, illustrating that the maximum forecasted inflow for hedging (${\overline{I}}_{2}^{G}$) is 1.642 × 10

^{8}m

^{3}. Besides, the minimum forecasted inflow is equivalent to 3.456 × 10

^{7}m

^{3}according to the downstream water demand.

^{7}, 5.450 × 10

^{7}].

^{6}, (σ = 2.368 × 10

^{7})

^{2}). Then, the flood-safety margins at the starting and ending points of hedging were calculated via Equations (17)–(19), i.e., δ

^{SWA}= 1.037 × 10

^{8}m

^{3}and δ

^{EWA}= 1.054 × 10

^{8}m

^{3}. The ranges of risk tolerance τ

_{r}corresponding to three cases (i.e., δ

^{min}< δ

^{SWA}, δ

^{SWA}≤ δ

^{min}≤ δ

^{EWA}, and δ

^{min}> δ

^{EWA}) were derived by δ

^{min}= μ + σ∙Φ

^{−1}(1 − τ

_{r}) and are listed in Table 2.

^{−3}, 4.50 × 10

^{−6}, 7.93 × 10

^{−7}) were chosen from its range obtained above to derive the forecast uncertainty ranges corresponding to three cases, and the results are shown in Table 2. When τ

_{r}= 5 × 10

^{−3}, δ

^{min}is always smaller than δ

^{SWA}at any forecast uncertainty within its allowable range for hedging. For example, if μ = 0, σ = 3.456 × 10

^{7}, then δ

^{min}< δ

^{SWA}can be known since δ

^{min}= 9.210 × 10

^{7}m

^{3}and δ

^{SWA}= 1.501 × 10

^{8}m

^{3}. When τ

_{r}= 4.50 × 10

^{−6}, Case 2 (δ

^{SWA}≤ δ

^{min}≤ δ

^{EWA}) occurs under forecast uncertainty ranging from 2.309 × 10

^{7}to 3.456 × 10

^{7}or within [3.974 × 10

^{7}, 5.450 × 10

^{7}], while Case 3 (δ

^{min}> δ

^{EWA}) happens at a relatively small range of forecast uncertainty, i.e., (3.456 × 10

^{7}, 3.974 × 10

^{7}). When τ

_{r}is 7.93 × 10

^{−7}, there is no hedging between power generation and flood risk within the allowable forecast uncertainty range.

_{r}= 5 × 10

^{−3}, 4.50 × 10

^{−6}, and 7.93 × 10

^{−7}, corresponding to Case 1, 2, and 3, respectively, and δ

^{min}takes the values of 6.000 × 10

^{7}m

^{3}, 1.047 × 10

^{8}m

^{3}, 1.082 × 10

^{8}m

^{3}for the three cases under current forecast level (σ = 2.368 × 10

^{7}). As τ

_{r}decreases in Figure 11, the hedging range is narrowed down, and there is no hedging when τ

_{r}is small enough.

^{min}under the same ${\overline{I}}_{2}$ within [1.613 × 10

^{8}m

^{3}to 1.401 × 10

^{8}m

^{3}], where they represent the inflows at the starting points of hedging for cases 1 and 2, respectively, namely, ${\overline{I}}_{2}^{\mathit{SWA}}$ = 1.613 × 10

^{8}m

^{3}and ${\overline{I}}_{2}^{M}$ = 1.401 × 10

^{8}m

^{3}. For example, when ${\overline{I}}_{2}$ = 1.613 × 10

^{8}m

^{3}, ${S}_{1}^{*}$ corresponds to three cases: 5.229 × 10

^{9}m

^{3}, 5.228 × 10

^{9}m

^{3}, and 5.224 × 10

^{9}m

^{3}, respectively. Under the same ${\overline{I}}_{2}$ ranging from 1.401 × 10

^{8}m

^{3}(${\overline{I}}_{2}^{M}$) to 1.140 × 10

^{8}m

^{3}(${\overline{I}}_{2}^{\mathit{EWA}}$), ${S}_{1}^{*}$ is the same for Cases 1 and 2 because hedging exists within the inflow range, and ${S}_{1}^{*}$ for Case 3 is less than for Cases 1 and 2. When ${\overline{I}}_{2}$ ranges from 1.140 × 10

^{8}m

^{3}(${\overline{I}}_{2}^{\mathit{EWA}}$) to 1.122 × 10

^{8}m

^{3}(${\overline{I}}_{2}^{\mathit{CM}}$), ${S}_{1}^{*}$ equals the upper bound of carryover storage for Cases 1 and 2, while ${S}_{1}^{*}$ for Case 3 is the carryover storage net of δ

^{min}and less than its upper bound. If the forecasted inflow is within [3.456 × 10

^{7}m

^{3}, 1.122 × 10

^{8}m

^{3}], then 3.456 × 10

^{7}m

^{3}, representing the minimum allowable inflow for hedging, i.e., ${\overline{I}}_{2}^{F}$, the three curves overlapping indicate that ${S}_{1}^{*}$ for three cases is equal. The tendency of optimal flood-safety margins (δ*) in three cases is the opposite to that of carryover storage, as shown in Figure 11b.

#### 4.3. Comparison with the Current Operation Rules

^{min}= 6.023 × 10

^{7}, thus, the optimal solutions are the same as in Case 1. The performance of OHR is compared to that of current operating rules (COR) in flood risk and power generation, as shown in Figure 12 and Table 3, respectively.

^{9}m

^{3}at the end of the main flood season in 1985, despite being at the flood recession stage, because the forecasted inflow at the end of the main flood season continued to exceed its maximum allowable inflow for DCCS. The storage volume in 1994 and 2005 was raised using OHR almost the whole main flood season, but the raised storage volume at the end of the main flood season in 1994 was greater than that in 2005 (5.282 × 10

^{9}m

^{3}versus 5.265 × 10

^{9}m

^{3}) since the inflow at the end of the main flood season in 2005 was less than the downstream water demand. Over these three years, OHR performance was beneficial in increasing storage capacity and limiting energy losses due to inflow constraints.

## 5. Conclusions

- (1)
- Hedging and trade-offs between power generation and flood risk exist during DCCS only when the forecasted inflow is greater than the minimum downstream water demand and less than the inflow that allows power generation in two stages to reach its peak without spilling.
- (2)
- We identified the forecast uncertainty range that allows for hedging between two objectives by calculating the minimum and maximum forecast uncertainties. If the forecast uncertainty is greater than its maximum, reducing flood risk is the unique objective considered by decision-makers. If the forecast uncertainty is less than its minimum, the carryover storage keeping at its upper bounds meets the current flood control standard.
- (3)
- Compared to forecast uncertainty, downstream risk tolerance plays a more important role in determining which case of the OHR is adopted in real-world operations.
- (4)
- In the real-world application, compared with COR, OHR had an excellent performance in power generation improvement in dry years, indicating that OHR can alleviate the energy crisis during dry years.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

OHR | The optimal hedging rules. |

DCCS | Dynamic control of carryover storage. |

FLSV(S^{L}) | Flood-limited storage volume. |

Δt | One period of two-stage operation. |

T | Forecast horizon. |

S_{0}, S_{k} | Initial storage volume of Stage 1 and storage at the end of Stage k (k = 1, 2). |

I_{k}, R_{k} | Actual inflow and release volume in Stage k (k = 1, 2). |

${\overline{I}}_{2}$, ${\overline{R}}_{2}$ | Forecasted inflow and release volume in Stage 2. |

ε | Inflow forecasting error in Stage 2. |

Q_{thres} | The threshold discharge capacity for the downstream safety. |

δ | Forecasted flood-safety margin. |

σ | Forecast uncertainty of inflow in Stage 2. |

E_{k} | Power generation in stage k (k = 1, 2). |

SSR(S_{0}), SSR(S_{k}) | The initial stage-storage relationship water level in Stage 1 and stage-storage relationship at the end of Stage k (k = 1, 2). |

SDR | Downstream water level. |

G_{1}, G_{2} | Power generation and flood risk objectives, respectively. |

${G}_{1}^{\prime}$, ${G}_{2}^{\prime}$ | Marginal utilities of power generation and flood risk, respectively. |

ω | The weight is designed for power generation. |

δ^{min} | Minimum flood-safety margin required for flood risk in Stage 2. |

${E}_{k}^{\mathrm{max}}$ | Maximum power capacity in Stage k (k = 1, 2). |

${S}_{1}^{A}$ | The lower bound of carryover storage originated from maximum power generation of Stage 1. |

${S}_{1}^{B}$ | The upper bound of carryover storage originated from maximum power generation of Stage 2. |

${R}_{1}^{\mathrm{min}}$ | Minimum downstream water demand. |

${S}_{1}^{C}$ | The upper bound of carryover storage originated from downstream water demand of Stage 2. |

τ_{r} | Risk tolerance. |

${\overline{I}}_{2}^{G}$ | A specific forecasted inflow that triggers the maximum power generation in two stages without spilling when the lower bound of carryover storage is ${S}_{1}^{A}$. |

${\overline{I}}_{2}^{H}$ | A specific forecasted inflow that triggers the maximum power generation in two stages without spilling when the lower bound of carryover storage is S^{L}. |

${\overline{I}}_{2}^{D}$ | A specific forecasted inflow that triggers the maximum power generation in Stage 2 and downstream water demand constraints at the same time. |

${\overline{I}}_{2}^{F}$ | The minimum allowable inflow for DCCS application, which is equal to the minimum downstream water demand. |

δ^{SWA}, ${\overline{I}}_{2}^{\mathit{SWA}}$ | Flood-safety margin and forecasted inflow at the starting hedging point that marginal utility of power generation equals marginal utility of flood risk, i.e., f_{1}(${S}_{1}^{A}$) = f_{2}(δ^{SWA}). |

δ^{EWA}, ${\overline{I}}_{2}^{\mathit{EWA}}$ | Flood-safety margin and forecasted inflow at the ending hedging point that marginal utility of flood risk equals marginal utility of power generation (${S}_{1}^{C}$), i.e., f_{1}(${S}_{1}^{C}$) = f_{2}(δ^{EWA}). |

MUPG(f_{1}(S_{1})) | Marginal utility of power generation in Stage 1. |

MUFR (f_{2}(δ)) | Marginal utility of flood risk in Stage 2. |

σ^{min}, σ^{max} | Minimum and maximum allowed forecast uncertainty for hedging, respectively. |

dδ^{SWA}/dσ | The trend of δ^{SWA} as σ increases. |

dδ^{EWA}/dσ | The trend of δ^{EWA} as σ increases. |

d^{2}δ^{SWA}/dσ^{2} | The trend of dδ^{SWA}/ dσ as σ increases. |

d^{2}δ^{EWA}/dσ^{2} | The trend of dδ^{EWA}/ dσ as σ increases. |

${S}_{*}^{1}$ | Optimal carryover storage from Stage 1 to Stage 2. |

δ* | Optimal flood-safety margin in Stage 2. |

${S}_{1}^{M}$ | The carryover storage makes the marginal utility of power generation equal minimum marginal utility of flood risk, i.e., f_{2}(δ^{min}) = f_{1}(${S}_{1}^{M}$). |

${\overline{I}}_{2}^{M}$ | The inflow for situation f_{2}(δ^{min}) = f_{1}(${S}_{1}^{M}$). |

${\overline{I}}_{2}^{\mathit{CM}}$ | The inflow for situation δ = δ^{min}, S_{1} = ${S}_{1}^{C}$. |

## Appendix A

^{min}, the trend of δ

^{min}as σ increases (dδ

^{min}/dσ) can be expressed as:

^{min}/dσ is positive and monotonically increases because Փ

^{−1}(1 − τ

_{r}) is positive.

^{SWA}/dσ illustrates the trend of δ

^{SWA}as σ increases:

^{SWA}/σ

^{2}− 1/(I

_{1}+ ${\overline{I}}_{2}^{\mathit{SWA}}$) is smaller than δ

^{SWA}/σ

^{2}; therefore, dδ

^{SWA}/dσ is greater than δ

^{SWA}/σ − σ/δ

^{SWA}. δ

^{SWA}is defined as the difference between the threshold for downstream levees Q

_{thres}and the release at point SWA (i.e., δ

^{SWA}= Q

_{thres}− ${R}_{2}^{SWA}$), which is larger than the difference between Q

_{thres}and the threshold of turbine release capacity (${R}_{2}^{\mathrm{max}}$) since the maximum power generation binding lead to ${R}_{2}^{SWA}$ ≤ R

^{max}. To ensure downstream safety, the threshold of turbine release capacity was designed far lower than the downstream levee threshold [19], i.e., Q

_{thres}− Rmax 2 > σ. As a result, δ

^{SWA}= Q

_{thres}-${R}_{2}^{SWA}$> σ and dδ

^{SWA}/dσ > δ

^{SWA}/σ − σ/δ

^{SWA}> 0.

^{2}δ

^{SWA}/dσ

^{2}illustrates the trend of dδ

^{SWA}/dσ as σ increases:

^{2}δ

^{SWA}/dσ

^{2}for simplicity, assume that δ

^{SWA}= σ and 1/(I

_{1}+ ${\overline{I}}_{2}^{\mathit{SWA}}$) = 0. Then, the actual value of d

^{2}δ

^{SWA}/dσ

^{2}is close to (−2/σ) and can be derived since σ < δ

^{SWA}< σ

^{2}, that is, the denominator is one order of σ higher than the numerator.

^{EWA}/dσ measures the trend of δ

^{EWA}as σ increases:

^{EWA}/dσ is calculated in the same way as the process in Equation (A2), and δ

^{EWA}is larger than δ

^{SWA}based on its definition; hence, dδ

^{EWA}/dσ > dδ

^{SWA}/dσ > 0.

^{2}δ

^{EWA}/dσ

^{2}measures the trend of dδ

^{EWA}/dσ as σ increases:

^{2}δ

^{SWA}/dσ

^{2}, −1 < d

^{2}δ

^{SWA}/dσ

^{2}< 0 can be derived.

## Appendix B

- Without considering δ
^{min}.

- 2.
- When ${\overline{I}}_{2}^{\mathit{SWA}}$ < ${\overline{I}}_{2}$ < ${\overline{I}}_{2}^{G}$ or ${\overline{I}}_{2}^{\mathit{SWA}}$ < ${\overline{I}}_{2}$ < ${\overline{I}}_{2}^{H}$, f
_{2}(δ*) is larger than f_{2}(δ^{SWA}) that exceeds MUHG, i.e., f_{2}(δ*) > f_{2}(δ^{SWA}) = f_{1}(${S}_{1}^{A}$) or f_{2}(δ*) > f_{2}(δ^{SWA}) = f_{1}(S^{L}), allocating as much space as possible or all of the space above FLSV, which depends on the lower bound of carryover storage, to accommodate the relative larger inflows to reduce flood risk. The optimal conditions of R.1 under different lower bounds of carryover storage can be written as Equations (A6) and (A7):$$\{\begin{array}{l}{S}_{1}^{\ast}={S}_{1}^{A}\\ {\delta}^{\ast}={Q}_{thres}+{S}^{L}-{\overline{I}}_{2}-{S}_{1}^{A}\\ {f}_{2}({\delta}^{*})=\lambda \\ {f}_{2}({\delta}^{*})-{f}_{1}({S}_{1}^{A})={\mu}_{1}^{sl}{}^{{}_{1}}>0\\ {\mu}_{1}^{s{u}_{1}}={\mu}_{1}^{s{u}_{2}}={\mu}_{1}^{s{l}_{2}}={\mu}_{2}^{\delta}=0\end{array}$$$$\{\begin{array}{l}{S}_{1}^{\ast}={S}^{L}\\ {\delta}^{\ast}={Q}_{thres}-{\overline{I}}_{2}\\ {f}_{2}({\delta}^{*})=\lambda \\ {f}_{2}({\delta}^{*})-{f}_{1}({S}^{L})={\mu}_{1}^{s{l}_{2}}>0\\ {\mu}_{1}^{s{u}_{1}}={\mu}_{1}^{s{u}_{2}}={\mu}_{1}^{s{l}_{1}}={\mu}_{2}^{\delta}=0\end{array}$$

_{1}(${S}_{1}^{*}$) = f

_{2}(δ*), and the optimality conditions of R.2 are:

_{1}(${S}_{1}^{C}$) = f

_{2}(δ

^{SWA}) > f

_{2}(δ*), the shadow price of the upper bound of carryover storage is positive (i.e., ${\mu}_{1}^{s{u}_{1}}$ > 0), and the optimality conditions of R.3 are:

- 3.
- Case 1

^{min}is smaller than δ

^{SWA}, the optimal conditions are the same as the optimal conditions without considering δ

^{min}since it will not alter the space of hedging and decision making.

- 4.
- Case 2

^{SWA}≤ δ

^{min}≤ δ

^{EWA}and the forecasted inflow satisfies ${\overline{I}}_{2}^{M}$ < ${\overline{I}}_{2}$ ≤ ${\overline{I}}_{2}^{G}$ or ${\overline{I}}_{2}^{M}$ < ${\overline{I}}_{2}$ ≤ ${\overline{I}}_{2}^{H}$, the MUPG exceeds MUFR, i.e., f

_{1}(${S}_{1}^{*}$) > f

_{1}(${S}_{1}^{M}$) = f

_{2}(δ

^{min}), implying that floodwater should be stored after satisfying the minimum flood-safety margin. Under this situation, the shadow price of the minimum safety margin constraint is positive, i.e., ${\mu}_{2}^{\delta}$ > 0, and the optimality conditions are:

- 5.
- Case 3

^{min}is larger than δ

^{EWA}, illustrating a very small flood risk, and the optimal conditions of Case 3-R.1 and Case 3-R.2 are the same as Equations (A9) and (A10).

## References

- Panwar, N.L.; Kaushik, S.C.; Kothari, S. Role of renewable energy sources in environmental protection: A review. Renew. Sust. Energ. Rev.
**2011**, 15, 1513–1524. [Google Scholar] [CrossRef] - Fallah-Mehdipour, E.; Bozorg Haddad, O.; Mariño, M.A. Real-time operation of reservoir system by genetic programming. Water Resour. Manag.
**2012**, 26, 4091–4103. [Google Scholar] [CrossRef] - Gupta, P.; Singh, S.P.; Jangid, A.; Kumar, R. Characterization of black carbon in the ambient air of Agra, India: Seasonal variation and meteorological influence. Adv. Atmos. Sci.
**2017**, 34, 1082–1094. [Google Scholar] [CrossRef] - Kuriqi, A.; Pinheiro, A.N.; Sordo-Ward, A.; Bejarano, M.D.; Garrote, L. Ecological impacts of run-of-river hydropower plants—Current status and prospects on the brink of energy transition. Renew. Sustain. Energy Rev.
**2021**, 142, 110833. [Google Scholar] [CrossRef] - Oliveira, R.; Loucks, D.P. Operating rules for multireservoir systems. Water Resour. Res.
**1997**, 33, 839–852. [Google Scholar] [CrossRef] - Zhao, J.; Cai, X.; Wang, Z. Optimality conditions for a two-stage reservoir operation problem. Water Resour. Res.
**2011**, 47, W08503. [Google Scholar] [CrossRef] - Ortiz-Partida, J.P.; Kahil, T.; Ermolieva, T.; Ermoliev, Y.; Lane, B.; Sandoval-Solis, S.; Wada, Y. A two-stage stochastic optimization for robust operation of multipurpose reservoirs. Water Resour. Manag.
**2019**, 33, 3815–3830. [Google Scholar] [CrossRef][Green Version] - Kelman, J.; Damazio, J.M.; Marien, J.L.; Da Costa, J.P. The determination of flood control volumes in a multireservoir system. Water Resour. Res.
**1989**, 25, 337–344. [Google Scholar] [CrossRef] - Shah MA, R.; Rahman, A.; Chowdhury, S.H. Challenges for achieving sustainable flood risk management. J. Flood Risk Manag.
**2018**, 11, S352–S358. [Google Scholar] [CrossRef][Green Version] - Yao, H.; Dong, Z.; Jia, W.; Ni, X.; Chen, M.; Zhu, C.; Li, D. Competitive relationship between flood control and power generation with flood season division: A case study in downstream Jinsha River Cascade Reservoirs. Water
**2019**, 11, 2401. [Google Scholar] - Mu, J.; Ma, C.; Zhao, J.; Lian, J. Optimal operation rules of Three-gorge and Gezhouba cascade hydropower stations in flood season. Energy Conv. Manag.
**2015**, 96, 159–174. [Google Scholar] [CrossRef] - Shang, Y.; Lu, S.; Ye, Y.; Liu, R.; Shang, L.; Liu, C.; Meng, X.; Li, X.; Fan, Q. China’ energy-water nexus: Hydropower generation potential of joint operation of the Three Gorges and Qingjiang cascade reservoirs. Energy
**2018**, 142, 14–32. [Google Scholar] [CrossRef] - Xie, A.; Liu, P.; Guo, S.; Zhang, X.; Jiang, H.; Yang, G. Optimal design of seasonal flood limited water levels by jointing operation of the reservoir and floodplains. Water Resour. Manag.
**2018**, 32, 179–193. [Google Scholar] [CrossRef] - Chen, J.; Guo, S.; Li, Y.; Liu, P.; Zhou, Y. Joint operation and dynamic control of flood limiting water levels for cascade reservoirs. Water Resour. Manag.
**2013**, 27, 749–763. [Google Scholar] [CrossRef] - Zhou, Y.; Guo, S.; Liu, P.; Xu, C. Joint operation and dynamic control of flood limiting water levels for mixed cascade reservoir systems. J. Hydrol.
**2014**, 519, 248–257. [Google Scholar] [CrossRef] - Li, X.; Guo, S.; Liu, P.; Chen, G. Dynamic control of flood limited water level for reservoir operation by considering inflow uncertainty. J. Hydrol.
**2010**, 391, 124–132. [Google Scholar] [CrossRef] - Jiang, Z.; Sun, P.; Ji, C.; Zhou, J. Credibility theory based dynamic control bound optimization for reservoir flood limited water level. J. Hydrol.
**2015**, 529, 928–939. [Google Scholar] [CrossRef] - Liu, G.; Qin, H.; Shen, Q.; Tian, R.; Liu, Y. Multi-objective optimal scheduling model of dynamic control of flood limit water level for cascade reservoirs. Water
**2019**, 11, 1836. [Google Scholar] [CrossRef][Green Version] - Huang, X.; Xu, B.; Zhong, P.-A.; Yao, H.; Yue, H.; Zhu, F.; Lu, Q.; Sun, Y.; Mo, R.; Li, Z.; et al. Robust multiobjective reservoir operation and risk decision-making model for real-time flood control coping with forecast uncertainty. J. Hydrol.
**2022**, 605, 127334. [Google Scholar] [CrossRef] - Gong, Y.; Liu, P.; Cheng, L.; Chen, G.; Zhou, Y.; Zhang, X.; Xu, W. Determining dynamic water level control boundaries for a multi-reservoir system during flood seasons with considering channel storage. J. Flood Risk Manag.
**2020**, 13, e12586. [Google Scholar] [CrossRef] - Chen, L.; Singh, V.P.; Lu, W.; Zhang, J.; Zhou, J.; Guo, S. Streamflow forecast uncertainty evolution and its effect on real-time reservoir operation. J. Hydrol.
**2016**, 540, 712–726. [Google Scholar] [CrossRef] - Huang, K.; Ye, L.; Chen, L.; Wang, Q.; Dai, L.; Zhou, J.; Singh, V.P.; Huang, M.; Zhang, J. Risk analysis of flood control reservoir operation considering multiple uncertainties. J. Hydrol.
**2018**, 565, 672–684. [Google Scholar] [CrossRef] - Nayak, M.A.; Herman, J.D.; Steinschneider, S. Balancing flood risk and water supply in California: Policy search integrating short-term forecast ensembles with conjunctive use. Water Resour. Res.
**2018**, 54, 7557–7576. [Google Scholar] [CrossRef] - Lu, J.; Li, G.; Cheng, C.; Liu, B. A long-term intelligent operation and management model of cascade hydropower stations based on chance constrained programming under multi-market coupling. Environ. Res. Lett.
**2021**, 16, 055034. [Google Scholar] [CrossRef] - Ouarda, T.B.; Labadie, J.W. Chance-constrained optimal control for multireservoir system optimization and risk analysis. Stoch. Environ. Res. Risk Assess.
**2001**, 15, 185–204. [Google Scholar] [CrossRef] - Simonovic, S. Influence of different downstream users on single multipurpose reservoir operation by chance constraints. Can. J. Civ. Eng.
**1988**, 15, 596–600. [Google Scholar] [CrossRef] - Ding, W.; Zhang, C.; Peng, Y.; Zeng, R.; Zhou, H.; Cai, X. An analytical framework for flood water conservation considering forecast uncertainty and acceptable risk. Water Resour. Res.
**2015**, 51, 4702–4726. [Google Scholar] [CrossRef] - Draper, A.J.; Lund, J.R. Optimal hedging and carryover storage value. J. Water Resour. Plann. Manag.
**2004**, 130, 83–87. [Google Scholar] [CrossRef][Green Version] - You, J.Y.; Cai, X. Hedging rule for reservoir operations: 1. A theoretical analysis. Water Resour. Res.
**2008**, 44, W01415. [Google Scholar] [CrossRef] - You, J.Y.; Cai, X. Hedging rule for reservoir operations: 2. A numerical model. Water Resour. Res.
**2008**, 44, W01416. [Google Scholar] [CrossRef] - Zhao, T.; Zhao, J.; Liu, P.; Lei, X. Evaluating the marginal utility principle for long-term hydropower scheduling. Energy Conv. Manag.
**2015**, 106, 213–223. [Google Scholar] [CrossRef] - Tan, Q.F.; Wen, X.; Fang, G.H.; Wang, Y.Q.; Qin, G.H.; Li, H.M. Long-term optimal operation of cascade hydropower stations based on the utility function of the carryover potential energy. J. Hydrol.
**2020**, 580, 124359. [Google Scholar] [CrossRef] - Li, H.; Liu, P.; Guo, S.; Ming, B.; Cheng, L.; Zhou, Y. Hybrid two-stage stochastic methods using scenario-based forecasts for reservoir refill operations. J. Water Resour. Plann. Manag.
**2018**, 144, 04018080. [Google Scholar] [CrossRef] - Wan, W.; Zhao, J.; Lund, J.R.; Zhao, T.; Lei, X.; Wang, H. Optimal hedging rule for reservoir refill. J. Water Resour. Plann. Manag.
**2016**, 142, 04016051. [Google Scholar] [CrossRef] - Zhang, X.; Liu, P.; Xu, C.-Y.; Gong, Y.; Cheng, L.; He, S. Real-time reservoir flood control operation for cascade reservoirs using a two-stage flood risk analysis method. J. Hydrol.
**2019**, 577, 123954. [Google Scholar] [CrossRef] - Quinn, J.D.; Reed, P.M.; Giuliani, M.; Castelletti, A. Rival framings: A framework for discovering how problem formulation uncertainties shape risk management trade-offs in water resources systems. Water Resour. Res.
**2017**, 53, 7208–7233. [Google Scholar] [CrossRef] - Jain, S.K. Investigating the behavior of statistical indices for performance assessment of a reservoir. J. Hydrol.
**2010**, 391, 90–96. [Google Scholar] [CrossRef] - Kim, T.H.; Kim, B.; Han, K.Y. Application of fuzzy TOPSIS to flood hazard mapping for levee failure. Water
**2019**, 11, 592. [Google Scholar] [CrossRef][Green Version] - Valipour, M.; Banihabib, M.E.; Behbahani, S.M.R. Comparison of the ARMA, ARIMA, and the autoregressive artificial neural network models in forecasting the monthly inflow of Dez dam reservoir. J. Hydrol.
**2013**, 476, 433–441. [Google Scholar] [CrossRef] - Pérez-Díaz, J.I.; Wilhelmi, J.R. Assessment of the economic impact of environmental constraints on short-term hydropower plant operation. Energy Policy
**2010**, 38, 7960–7970. [Google Scholar] [CrossRef] - Georgakakos, K.P.; Graham, N.E. Potential benefits of seasonal inflow prediction uncertainty for reservoir release decisions. J. Appl. Meteorol. Climatol.
**2008**, 47, 1297–1321. [Google Scholar] [CrossRef] - Jane, J.Y. Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl.
**2005**, 307, 350–369. [Google Scholar] - Kyparisis, J. On uniqueness of Kuhn-Tucker multipliers in nonlinear programming. Math. Program.
**1985**, 32, 242–246. [Google Scholar] [CrossRef] - Herskovits, J. Feasible direction interior-point technique for nonlinear optimization. J. Optim. Theory Appl.
**1998**, 99, 121–146. [Google Scholar] [CrossRef] - Zeng, X.; Lund, J.R.; Cai, X. Linear Versus Nonlinear (Convex and Concave) Hedging Rules for Reservoir Optimization Operation. Water Resour. Res.
**2021**, 57, e2020WR029160. [Google Scholar] [CrossRef]

**Figure 3.**The variation of MUPG under different lower bounds as carryover storage increases when the forecasted inflow ranges from ${\overline{I}}_{2}^{F}$ to ${\overline{I}}_{2}^{G}$>: (

**a**) lower bound is ${S}_{1}^{A}$ (${S}_{1}^{A}$ ≥ S

^{L}); (

**b**) lower bound is S

^{L}(${S}_{1}^{A}$ < S

^{L}).

**Figure 4.**The relationship between MUFR (

**left**, f

_{2}) and MUPG (

**right**, f

_{1}) when the inflow is relatively large and the lower bound is ${S}_{1}^{A}$ without considering minimum flood-safety margin constraint.

**Figure 8.**The relationship between marginal utilities of (left) flood risk (f

_{2}) and (right) power generation (f

_{1}) with under three values of δ

^{min}when the lower bound of carryover storage is ${S}_{1}^{A}$: (

**a**) Case 1: δ

^{min}< δ

^{SWA}, (

**b**) Case 2: δ

^{SWA}≤ δ

^{min}≤ δ

^{EWA}, (

**c**) Case 3: δ

^{EWA}< δ

^{min}.

**Figure 9.**The relationship between marginal utilities of (left) flood risk (f

_{2}) and (right) power generation (f

_{1}) under three values of δ

^{min}when the lower bound of carryover storage is S

^{L}: (

**a**) Case 1: δ

^{min}< δ

^{SWA}, (

**b**) Case 2: δ

^{SWA}≤ δ

^{min}≤ δ

^{EWA}, (

**c**) Case 3: δ

^{min}> δ

^{EWA}.

**Figure 5.**Relationship between MUFR (

**left**, f

_{2}) and MUPG (

**right**, f

_{1}) under minimum and maximum forecast uncertainties when the lower bound is ${S}_{1}^{A}$: (

**a**) σ = σ

^{min}; (

**b**) σ = σ

^{max}.

**Figure 6.**The relationship between MUFR (

**left**, f

_{2}) and MUPG (

**right**, f

_{1}) under minimum and maximum forecast uncertainties when the lower bound is S

^{L}: (

**a**) σ = σ

^{min}; (

**b**) σ = σ

^{max}.

**Figure 7.**The minimum flood-safety margin (δ

^{min}) changes with forecast uncertainty (σ) and risk tolerance (τ

_{r}), and flood-safety margins the at start and end of hedging (δ

^{SWA}and δ

^{EWA}) vary with different σ.

**Figure 10.**The bounds of carryover storage under different inflows: (

**a**) ${S}_{1}^{A}$ and ${S}_{1}^{C}$ vary with increasing I

_{1}(

**left**); (

**b**) ${S}_{1}^{B}$ varies with increasing ${\overline{I}}_{2}$ (

**right**).

**Figure 11.**The OHR for three cases caused by three risk tolerances τ

_{r}when confronting different inflows: (

**a**) the optimal carryover storage (${S}_{1}^{*}$) and (

**b**) the optimal flood-safety margin (δ*).

**Figure 13.**Comparison of storage volume for OHR and COR under a: (

**a**) wet year; (

**b**) normal year; (

**c**) dry year.

Cases | The Forecasted Inflow in Stage 2 for DCCS | The Optimal Solutions |
---|---|---|

Case 1: δ^{min} < δ^{SWA} | R.1: ${\overline{I}}_{2}^{\mathit{SWA}}$ < ${\overline{I}}_{2}$ ≤ ${\overline{I}}_{2}^{\mathit{G}}$ or ${\overline{I}}_{2}^{\mathit{SWA}}$ < ${\overline{I}}_{2}$ ≤ ${\overline{I}}_{2}^{H}$. | Little water or no water is stored, and carryover storage is kept at the lower bound of DCCS (${S}_{1}^{*}$ = ${S}_{1}^{A}$ or ${S}_{1}^{*}$ = S^{L}). |

R.2: ${\overline{I}}_{2}^{\mathit{EWA}}$ < ${\overline{I}}_{2}$ ≤ ${\overline{I}}_{2}^{\mathit{EWA}}$. | Hydropower generation and flood risk are balanced (f_{1}(${S}_{1}^{*}$) = f_{2}(δ*)). | |

R.3: ${\overline{I}}_{2}^{\mathit{F}}$ < ${\overline{I}}_{2}$ ≤ ${\overline{I}}_{2}^{\mathit{EWA}}$. | Carryover storage volume remains at the upper bound of DCCS (${S}_{1}^{*}$ = ${S}_{1}^{C}$). | |

Case 2: δ^{SWA} ≤ δ^{min} ≤ δ^{EWA} | R.1: ${\overline{I}}_{2}^{M}$ < ${\overline{I}}_{2}$ ≤ ${\overline{I}}_{2}^{\mathit{G}}$ or ${\overline{I}}_{2}^{\mathit{CM}}$ < ${\overline{I}}_{2}$ ≤ ${\overline{I}}_{2}^{H}$. | After meeting δ^{min}, the inflow from Stage 1 is carried over to Stage 2 (${S}_{1}^{*}$ = Q_{thres} + S^{L} − ${\overline{I}}_{2}$ − δ^{min}). |

R.2: ${\overline{I}}_{2}^{C}$ < ${\overline{I}}_{2}$ ≤ ${\overline{I}}_{2}^{M}$. | The optimal solutions are the same as in Case 1-R.2. | |

R.3: ${\overline{I}}_{2}^{F}$ < ${\overline{I}}_{2}$ ≤ ${\overline{I}}_{2}^{C}$. | The optimal solutions are the same as in Case 1-R.3. | |

Case 3: δ^{min} > δ^{EWA} | R.1: ${\overline{I}}_{2}^{\mathit{CM}}$ < ${\overline{I}}_{2}$ ≤ ${\overline{I}}_{2}^{G}$ or ${\overline{I}}_{2}^{\mathit{CM}}$ < ${\overline{I}}_{2}$ ≤ ${\overline{I}}_{2}^{\mathit{H}}$. | The optimal solutions are the same as in Case 2-R.1. |

R.2: ${\overline{I}}_{2}^{\mathit{F}}$ < ${\overline{I}}_{2}$ ≤ ${\overline{I}}_{2}^{\mathit{CM}}$. | The optimal solutions are the same as in Case 1-R.3. |

Cases | Range of τ_{r} σ = 2.368 × 10^{7} | Range of σ (×10^{7}) | ||
---|---|---|---|---|

τ_{r} = 5 × 10^{−3} | τ_{r} = 4.50 × 10^{−6} | τ_{r} = 7.93 × 10^{−7} | ||

Case 1: δ^{min} < δ^{SWA} | (5.410 × 10^{−6}, 0.500) | [2.309, 5.450] | -- | -- |

Case 2: δ^{SWA} ≤ δ^{min} ≤δ^{EWA} | [3.730 × 10^{−6}, 5.410 × 10^{−6}] | -- | [2.309, 3.456], [3.974, 5.450] | -- |

Case 3: δ^{min} > δ^{EWA} | (0.000, 5.410 × 10^{−6}) | -- | (3.456, 3.974) | [2.309, 5.450] |

Methods | Annual Average | Flood Season | Non-Flood Season | Wet Year | Normal Year | Dry Year |
---|---|---|---|---|---|---|

COR | 6.36 | 3.23 | 3.13 | 7.93 | 6.57 | 4.63 |

OHR | 6.62 | 3.34 | 3.28 | 8.15 | 6.74 | 4.99 |

Change | 0.26 | 0.11 | 0.15 | 0.22 | 0.17 | 0.33 |

Rate | 4.09% | 4.02% | 4.79% | 2.77% | 2.59% | 7.13% |

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## Share and Cite

**MDPI and ACS Style**

Zhang, L.; Lund, J.R.; Ding, W.; Zhang, X.; Jin, S.; Wang, G.; Peng, Y. An Analytical Framework for Investigating Trade-Offs between Reservoir Power Generation and Flood Risk. *Water* **2022**, *14*, 3841.
https://doi.org/10.3390/w14233841

**AMA Style**

Zhang L, Lund JR, Ding W, Zhang X, Jin S, Wang G, Peng Y. An Analytical Framework for Investigating Trade-Offs between Reservoir Power Generation and Flood Risk. *Water*. 2022; 14(23):3841.
https://doi.org/10.3390/w14233841

**Chicago/Turabian Style**

Zhang, Lin, Jay R. Lund, Wei Ding, Xiaoli Zhang, Sifan Jin, Guoli Wang, and Yong Peng. 2022. "An Analytical Framework for Investigating Trade-Offs between Reservoir Power Generation and Flood Risk" *Water* 14, no. 23: 3841.
https://doi.org/10.3390/w14233841