# A Model for Assessing the Importance of Runoff Forecasts in Periodic Climate on Hydropower Production

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## Abstract

**:**

## 1. Introduction

## 2. Model Framework

#### 2.1. Assessment Model

#### 2.2. Ensemble Forecasting with Biennial Periodicity

_{H}, is shorter than the classified segments, there is a possibility of varying the start month. The time series for each subwatershed was abstracted from the 1961–2011 time series, and yearly time series were kept as a statistical repository for stochastic sampling. Each sampled yearly time series starts in December. The random samples of the forecasts and real runoff scenarios were taken from this statistical repository (see step 1 in Figure 1).

#### 2.3. Optimisation Model for Cascade Hydropower Stations

^{3}/s). $P$ (W) is the power from hydropower stations; ${E}_{s}$ (J) and $A$ (m

^{2}) are the energy stored and the surface water area of each reservoir, respectively; $\rho $ (kg/m

^{3}) is the density of water; $g$ (m/s

^{2}) is the acceleration because of gravity; and η is the generation efficiency of a hydropower plant, which is assumed to be constant over time. The hydraulic head of the stations, $\tilde{h}$ (m), is assumed to be constant in the optimisation problem but is updated after each decision in the updating time step ${t}_{u}$. Furthermore, $\underset{\_}{h}$ (m) is the minimum water level in each reservoir, the so-called dead water level, which cannot be used for regulation purposes. In the calculation of the energy of stored water, the routine subtracts the minimum water level from the actual water level and considers the remaining water non-usable. The potential production of water stored in a reservoir depends on the downstream fall height $hd$ (m), including the waterfall height at all downstream stations to the sea. The indices $j$, $n$, and $k$ are used to represent different indexes of time steps and forecasts in the programming; j is the index of the numerical time step; $J$ is the total number of $j$, i.e., $J={T}_{H}/\Delta t$; $\Delta t$(days) is the time length of one numerical time step, which is used to represent the flow dynamics; n is the index of the stochastic forecasts that are $N$ in total; and $i$ is the index of each of the multiple hydropower stations. $M$ is the total number of stations, including reservoirs and hydropower plants, and $NP$ is the total number of hydropower stations.

^{3}) is the water storage in the multireservoir system, the inflow ${q}_{r}$ (m

^{3}/s) is the runoff water from the subcatchment that connects directly as the inflow to one reservoir, and ${q}_{up}$ (m

^{3}/s) and ${s}_{up}$ (m

^{3}/s) are turbine discharges and water spillages, respectively, that come from the nearest upstream connected stations, which means that water released at one station is retained at the next reservoir regardless of the transport time along the river. The water spillage discharge is $s$ (m

^{3}/s). Water conservation (Equation (2)) indicates that the inflow to a specific reservoir comes from natural runoff as a result of precipitation and snowmelt and outflow from connected upstream multireservoirs. Note that the runoff ${q}_{r}$ is implemented both in forecasting step 1 as part of the above optimisation problem and in updating step 2 (Figure 1), in which we use the notations ${q}_{rf}$ for the forecasted runoff and ${q}_{ra}$ for the actual runoff applied in the management scenario. In the simulation routine (step 1 in Figure 1), the runoff discharge in Equation (2) is ${q}_{rf}$, while in the updating routine (step 2 in Figure 1), the runoff discharge is ${q}_{ra}$.

^{3}/s). To sustain the downstream water demand and meet environmental requirements to some extent, a lower limitation of spillage discharge $\underset{\_}{s}$ (m

^{3}/s) and an upper limitation $\overline{s}$ (m

^{3}/s) are given.

#### 2.4. Performance Indicators

_{sim}. The potential energy production in the above expression consists of both potential downstream production from runoff and energy stored in reservoirs. The potential downstream production indicates the estimated production generated at each station and the potential production from stations along the water path towards the sea. Energy stored in the water reservoir can be seen as the initial value of the scenario and can be used in production, which is the reason for the definition given by Equation (8). This study elucidates the relationship between forecasting error and production efficiency.

## 3. Case Study

## 4. Results

#### 4.1. Assessment of the Effects of Forecast Error on Production Efficiency in the Dalälven River Basin

#### 4.2. Start-Month Impact on the Biennial Periodicity

^{3}/s) of 64 subcatchments from the Dalälven River basin. Runoff data were deduced for every set of odd (blue dots) and even (red dots) years using different start months from January to December in the time series that covers daily data from 1961 to 2011. The green line is the average daily runoff of the 64 subcatchments over 51 years. The graph clearly shows that the biennial classification results in differences in the daily mean runoff from the Dalälven River basin, but the pattern depends on the start month of the segments. Note that the odd–even classification used in this figure denotes the year of the start month, but the yearly data selected for a start month later than January cover both odd and even years. The two curves, odd year (blue line) and even year (black line), in Figure 6 highlight an interchange of the implication of odd and even for the dry–wet year characteristic. At the crossing of the curves in Figure 6, both odd and even years are statistically equally wet, whereas the difference becomes larger when the curves are farther apart. The vertical bars show a 95% confidence interval using a t-distribution and indicate somewhat weak significance in the differences between the curves, which can be explained by the prevalence of other periodic climatic phenomena and some degree of randomness. Nevertheless, the results show the systematic biennial hydrological periodicity of the annual mean runoff in the Dalälven River basin, which varies with the start month of the selection and is most vital with a December start month. This circumstance implies that the long-term forecasts used in hydropower production management in December are more sensitive to recognising biennial periodicity compared with planning conducted in a summer month. Compared with the mean runoff of all years (green line), odd years present a higher mean daily runoff than the average, while even years show a relatively low mean daily runoff regardless of the start month.

## 5. Conclusions and Discussions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Ensemble classifications for runoff forecasting in wet years (odd years) and dry years (even years).

**Figure 4.**Comparison of scenario 1 (forecasts of wet years) and scenario 2 (forecasts of dry years), with both scenarios using simulated real runoff from a wet year.

**Figure 6.**Averaged daily runoff (m

^{3}/s) over 1961–2011 for the 64 subcatchments of the Dalälven River basin in units of m

^{3}/s using different start months. Each start month provides up to 25 yearly samples in odd or even years, and the bars indicate the 95% confidence interval of the mean value estimate (using a t-distribution).

Parameter | Definition | Parameter Value in the Example Application |
---|---|---|

${T}_{H}$ | Time horizon of optimisation: the duration of the forecasted time series placed into one optimisation procedure. | ${T}_{H}$ = 90 (days) |

${T}_{sim}$ | Period of simulation: the maximum shift in time of the horizon in the receding horizon approach; ${T}_{sim}=K\times {t}_{u}$. | ${T}_{sim}$ = 90 (days) |

${t}_{u}$ | Updating period: the time during which the decided turbine discharges are applied, whereafter the reservoir levels are updated and new decisions are taken; ${t}_{u}={T}_{sim}/K$. | ${t}_{u}$ = 2 (days) |

$\Delta t$ | Numerical time step used to represent the watershed dynamics and to move between the states used in the optimization. | $\Delta t$ = 0.5 (days) |

$j$ | $j=1:J$. Index for the numerical time step for water dynamics; $J={T}_{H}/\Delta t$. | $J$ = 180 |

$i$ | $i=1:M$ Index for the reservoirs. | $M$ = 49 |

$n$ | $n=1:N$. Index for the repetition number of one updating period simulation with different stochastic runoff forecasts, which was used to make the average decision. | $N$ = 10 |

$k$ | $k=1:K$. Index for the simulation time step in order to progress over the simulation period ${T}_{sim}$. The number of updating time steps is $K={T}_{sim}/{t}_{u}$. | $K$ = 45 |

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**MDPI and ACS Style**

Hao, S.; Wörman, A.; Riml, J.; Bottacin-Busolin, A.
A Model for Assessing the Importance of Runoff Forecasts in Periodic Climate on Hydropower Production. *Water* **2023**, *15*, 1559.
https://doi.org/10.3390/w15081559

**AMA Style**

Hao S, Wörman A, Riml J, Bottacin-Busolin A.
A Model for Assessing the Importance of Runoff Forecasts in Periodic Climate on Hydropower Production. *Water*. 2023; 15(8):1559.
https://doi.org/10.3390/w15081559

**Chicago/Turabian Style**

Hao, Shuang, Anders Wörman, Joakim Riml, and Andrea Bottacin-Busolin.
2023. "A Model for Assessing the Importance of Runoff Forecasts in Periodic Climate on Hydropower Production" *Water* 15, no. 8: 1559.
https://doi.org/10.3390/w15081559