An Analytical Framework for Investigating Trade-Offs between Reservoir Power Generation and Flood Risk
Abstract
:1. Introduction
2. Study Area
3. Methods
3.1. The Dynamic Control of Carryover Storage (DCCS)
3.1.1. Two-Stage Model
3.1.2. Forecast Errors and Flood Risk
3.1.3. Power Generation in Two Stages
3.2. Model Formulation
3.2.1. Objective Function
3.2.2. Constraints
3.2.3. Optimal Conditions
3.3. Trade-Offs between Power Generation and Flood Risk
3.3.1. Evaluation of the Marginal Utility of Power Generation
3.3.2. The Starting and Ending Points of Hedging
- R.1:
- When the forecasted inflow () ranges from to or from to , the relatively large inflow illustrates a large flood risk, and the MUFR exceeds MUPG under optimal conditions in Appendix B1, i.e., f2(δ*) > f2(δSWA) = f1() or f2(δ*) > f2(δSWA) = f1(SL). 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 , the optimal carryover storage and flood-safety margin are = , δ* = Qthres + SL − − . When the lower bound is SL, they are written as = SL, δ* = Qthres − .
- R.2.:
- When ranges from to , 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., f2(δ*) = f1().
- R.3.:
- When ranges from to , 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., f1() > f2(δEWA) = f2(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., = , δ* = Qthres + SL − − .
3.3.3. Effects of Forecast Uncertainty and Risk Tolerance
- The allowable forecast uncertainty
- 2.
- Combined impacts of forecast uncertainty and risk tolerance
3.3.4. The Optimal Hedging Rules (OHR)
- Case 1: δmin < δSWA
- 2.
- Case 2: δSWA ≤ δmin ≤ δEWA
- 3.
- Case 3: δmin > δEWA
4. Application Results
4.1. Inputs of Model
4.2. Optimal Hedging Rules Experiment
4.3. Comparison with the Current Operation Rules
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(SL) | Flood-limited storage volume. |
Δt | One period of two-stage operation. |
T | Forecast horizon. |
S0, Sk | Initial storage volume of Stage 1 and storage at the end of Stage k (k = 1, 2). |
Ik, Rk | Actual inflow and release volume in Stage k (k = 1, 2). |
, | Forecasted inflow and release volume in Stage 2. |
ε | Inflow forecasting error in Stage 2. |
Qthres | The threshold discharge capacity for the downstream safety. |
δ | Forecasted flood-safety margin. |
σ | Forecast uncertainty of inflow in Stage 2. |
Ek | Power generation in stage k (k = 1, 2). |
SSR(S0), SSR(Sk) | 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. |
G1, G2 | Power generation and flood risk objectives, respectively. |
, | Marginal utilities of power generation and flood risk, respectively. |
ω | The weight is designed for power generation. |
δmin | Minimum flood-safety margin required for flood risk in Stage 2. |
Maximum power capacity in Stage k (k = 1, 2). | |
The lower bound of carryover storage originated from maximum power generation of Stage 1. | |
The upper bound of carryover storage originated from maximum power generation of Stage 2. | |
Minimum downstream water demand. | |
The upper bound of carryover storage originated from downstream water demand of Stage 2. | |
τr | Risk tolerance. |
A specific forecasted inflow that triggers the maximum power generation in two stages without spilling when the lower bound of carryover storage is . | |
A specific forecasted inflow that triggers the maximum power generation in two stages without spilling when the lower bound of carryover storage is SL. | |
A specific forecasted inflow that triggers the maximum power generation in Stage 2 and downstream water demand constraints at the same time. | |
The minimum allowable inflow for DCCS application, which is equal to the minimum downstream water demand. | |
δSWA, | 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., f1() = f2(δSWA). |
δEWA, | Flood-safety margin and forecasted inflow at the ending hedging point that marginal utility of flood risk equals marginal utility of power generation (), i.e., f1() = f2(δEWA). |
MUPG(f1(S1)) | Marginal utility of power generation in Stage 1. |
MUFR (f2(δ)) | Marginal utility of flood risk in Stage 2. |
σmin, σmax | Minimum and maximum allowed forecast uncertainty for hedging, respectively. |
dδSWA/dσ | The trend of δSWA as σ increases. |
dδEWA/dσ | The trend of δEWA as σ increases. |
d2δSWA/dσ2 | The trend of dδSWA/ dσ as σ increases. |
d2δEWA/dσ2 | The trend of dδEWA/ dσ as σ increases. |
Optimal carryover storage from Stage 1 to Stage 2. | |
δ* | Optimal flood-safety margin in Stage 2. |
The carryover storage makes the marginal utility of power generation equal minimum marginal utility of flood risk, i.e., f2(δmin) = f1(). | |
The inflow for situation f2(δmin) = f1(). | |
The inflow for situation δ = δmin, S1 = . |
Appendix A
Appendix B
- Without considering δmin.
- 2.
- When < < or < < , f2(δ*) is larger than f2(δSWA) that exceeds MUHG, i.e., f2(δ*) > f2(δSWA) = f1() or f2(δ*) > f2(δSWA) = f1(SL), 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):
- 3.
- Case 1
- 4.
- Case 2
- 5.
- Case 3
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Cases | The Forecasted Inflow in Stage 2 for DCCS | The Optimal Solutions |
---|---|---|
Case 1: δmin < δSWA | R.1: < ≤ or < ≤ . | Little water or no water is stored, and carryover storage is kept at the lower bound of DCCS ( = or = SL). |
R.2: < ≤ . | Hydropower generation and flood risk are balanced (f1() = f2(δ*)). | |
R.3: < ≤ . | Carryover storage volume remains at the upper bound of DCCS ( = ). | |
Case 2: δSWA ≤ δmin ≤ δEWA | R.1: < ≤ or < ≤ . | After meeting δmin, the inflow from Stage 1 is carried over to Stage 2 ( = Qthres + SL − − δmin). |
R.2: < ≤ . | The optimal solutions are the same as in Case 1-R.2. | |
R.3: < ≤ . | The optimal solutions are the same as in Case 1-R.3. | |
Case 3: δmin > δEWA | R.1: < ≤ or < ≤ . | The optimal solutions are the same as in Case 2-R.1. |
R.2: < ≤ . | The optimal solutions are the same as in Case 1-R.3. |
Cases | Range of τr σ = 2.368 × 107 | Range of σ (×107) | ||
---|---|---|---|---|
τr = 5 × 10−3 | τr = 4.50 × 10−6 | τr = 7.93 × 10−7 | ||
Case 1: δmin < δSWA | (5.410 × 10−6, 0.500) | [2.309, 5.450] | -- | -- |
Case 2: δSWA ≤ δmin ≤δEWA | [3.730 × 10−6, 5.410 × 10−6] | -- | [2.309, 3.456], [3.974, 5.450] | -- |
Case 3: δmin > δEWA | (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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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
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 StyleZhang, 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
APA StyleZhang, L., Lund, J. R., Ding, W., Zhang, X., Jin, S., Wang, G., & Peng, Y. (2022). An Analytical Framework for Investigating Trade-Offs between Reservoir Power Generation and Flood Risk. Water, 14(23), 3841. https://doi.org/10.3390/w14233841