Performance of Deterministic and Probabilistic Hydrological Forecasts for the ShortTerm Optimization of a Tropical Hydropower Reservoir
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Abstract
Hydropower is the most important source of electricity in Brazil. It is subject to the natural variability of water yield. One building block of the proper management of hydropower assets is the shortterm forecast of reservoir inflows as input for an online, eventbased optimization of its release strategy. While deterministic forecasts and optimization schemes are the established techniques for shortterm reservoir management, the use of probabilistic ensemble forecasts and multistage stochastic optimization techniques is receiving growing attention. The present work introduces a novel, mass conservative scenario tree reduction in combination with a detailed hindcasting and closedloop control experiments for a multipurpose hydropower reservoir in a tropical region in Brazil. The case study is the hydropower project Três Marias, which is operated with two main objectives: (i) hydroelectricity generation and (ii) flood control downstream. In the experiments, precipitation forecasts based on observed data, deterministic and probabilistic forecasts are used to generate streamflow forecasts in a hydrological model over a period of 2 years. Results for a perfect forecast show the potential benefit of the online optimization and indicate a desired forecast lead time of 30 days. In comparison, the use of actual forecasts of up to 15 days shows the practical benefit of operational forecasts, where stochastic optimization (15 days lead time) outperforms the deterministic version (10 days lead time) significantly. The range of the energy production rate between the different approaches is relatively small, between 78% and 80%, suggesting that the use of stochastic optimization combined with ensemble forecasts leads to a significantly higher level of flood protection without compromising the energy production.
Keywords
Hydrological forecasting Shortterm optimization Ensemble forecasting Realtime control Flood mitigation Três Marias Dam1 Introduction
Hydropower is the most important source of electricity in Brazil. During recent years, it has accounted for 60% to 70% of the total electrical power supply (EPE, 2014). The remaining electricity is provided mostly by thermal power plants using biomass, coal, natural gas, and nuclear power, but generally at higher costs. A nationwide transmission network allows for integrated management of the energy production. This management is done by a central organization called Operador Nacional do Sistema (ONS), whose objective is to optimize electricity production by increasing production at plants with lower operation costs and decreasing it at plants where these costs are higher. Operational costs of hydropower are lower than for thermal power plants; therefore, there is a strong economic reason to maximize the proportion of energy generated from hydropower (Hamlet et al. 2002). On the other hand, hydropower is dependent on weather and climate, which are naturally variable, leading to risks of power production shortage.
In such a complex system, benefits of reservoir inflow forecasts include the following: (1) spillage can be minimized; (2) reservoirs can operate with larger head for longer periods due to more reliable release schedules; (3) more energy can be generated at times when energy prices are higher; (4) more energy can be produced at hydropower plants that have the higher inflow forecasts (Faber and Stedinger 2001; Yeh et al., 1982; Hamlet et al. 2002; Maurer 2002). Forecast errors influence decisionmaking, leading to suboptimal operation when water is released unnecessarily from reservoirs, or when thermal power plants are activated needlessly.
ONS uses a chain of optimization models for the management of the Brazilian power system (Maceira and Damázio 2005; Pereira and Pinto 1991). Two of the optimization models are meant for longterm operational planning (5 to 10 years) and for seasonal operational planning (12 months). These optimization models use stochastically generated inflow scenarios, and no actual forecasts. The third optimization model used by ONS is applied to make operational decisions up to 14 days in advance, and uses forecasts of inflows to more than two hundred reservoirs. Until less than a decade ago, all of the inflow forecasts were provided by periodic autoregressive moving average (PARMA) models (Maceira and Damázio,). In 2005, ONS started to test inflow forecasting based on rainfallrunoff models with the input of quantitative precipitation forecasts. The forecasting models tested used precipitation forecasts generated by the Brazilian Weather Forecasting Center (Centro de Previsão de Tempo e Estudos Climáticos – CPTEC), primarily from the regional atmospheric Eta model (Chou et al. 2002). An assessment of results by Guilhon et al. (2007) showed that the new forecasting methods that included quantitative precipitation forecasts outperformed autoregressive forecasting models based on inflow time series input only. As a result, forecasting models and methods that use quantitative precipitation forecasts are gradually replacing PARMA models in the Brazilian electric energy optimization chain (Guilhon et al. 2007). However, until now, mostly medium range (14 days) forecasts are deterministic, i.e. generated considering only one possible future.
In Brazil, the use of Ensemble Streamflow Forecasts (ESF) for both floods and reservoir inflow predictions is still starting (Fan et al. 2014). The aim of ESF is to account for uncertainties. They are usually produced by forcing hydrological models with meteorological forecasts which include multiple possible future trajectories of atmospheric variables (Cloke and Pappenberger 2009), although some others strategies to generate ESF such as the use of uncertainty of initial conditions and model parametrization are present in the literature (Demirel et al. 2013). The first examples of ESF for Brazilian cases are shown by Tucci et al. (2003 and 2008), who showed results of hindcasting experiments of seasonal streamflow forecasts for the rivers Uruguay and Grande. More recently, Calvetti and Pereira (2014), Collischonn et al. (2013) and Meller (2013) described experiments of short to mediumrange ensemble flood forecasting. Fernández Bou et al. (2015) showed the development of a methodology for 1month flood forecasting in the upper region of the Uruguay River basin, at the Itá Hydroelectric Power Plant (HPP) reservoir. Finally, Fan et al. (2014, 2015a b) showed the assessment of ESF within operational forecasting systems.
While ESF are not yet widely used operationally in Brazil for mediumrange hydropower optimization and planning (use of ensembles can be considered generally at a research level), a number of studies worldwide have shown the benefits of the use of ESF for reservoir operation. Selected recent works are cited as follows.
In the study of Zhao et al. (2011), synthetic experiments of ESF were applied to realtime reservoir operation and were compared with deterministic forecasts. Through the hypothetical example of a singleobjective realtime reservoir operation model, the results illustrated that forecast uncertainty exerts significant effects. Results showed that, in general, the benefit from the reservoir operation using ensembles was nearly as high as the one obtained with a perfect forecast. Zhao et al. (2012) demonstrated with synthetic tests that we may have an effective forecast horizon where the forecast is useful for operational purposes, which is the horizon where uncertainties do not excessively compromise the forecast quality.
Boucher et al. (2012) quantified the economic benefits of employing the forecasts in a stochastic decisionmaking assistance tool for hydroelectricity production during a flood event on the Gatineau River, Canada. According to the authors, this allowed for the comparison between different types of forecasts according to their value in terms of energy, spillage and storage in a reservoir. ESF exhibited excellent performances when compared to observations and were also satisfying when involved in the operational management of electricity production.
Liu et al. (2015) investigated how the sources of uncertainty impact the process of reservoir flood control based on forecasts, using a case study from the Ankang Reservoir, Han River, China. Based on the results of a series of synthetic tests, authors shed light on how different uncertainties can affect the operation of reservoirs while anticipating floods. Anvari et al. (2014) tested optimization models in the ZayandehRud reservoir system in Iran, aiming to investigate uncertaintybased optimal operation of this multipurpose water reservoir system. Authors also assessed the effect of Ensemble Streamflow Predictions (ESP) that were generated by artificial neural networks in the application. Results for the case study demonstrated that some of the specific approaches that were tested could be useful for realtime reservoir operation.
Ficchì (2015) presented a study where authors aimed to investigate the improvement of a fourreservoir operation system in the Seine River basin, France, using an ensemble of weather forecasts and a realtime control approach. In the proposed system, a model predictive control (MPC) optimization technique was used, and it was modified to a version that incorporated uncertainties in the form of a treebased model predictive control (TBMPC) approach in order to account for deterministic and ensemble forecasts, respectively. The management system was assessed by the simulation of historical events. Simulation results showed that the proposed realtime control system largely outperforms a noforecasts management strategy based on rule curves, and that explicitly considering forecast uncertainty through ensembles compensated for the loss in performance due to forecast inaccuracies.
Focusing more on operational aspects (not using forecasts), Che and Mays (2015) developed and tested a methodology for determining reservoir release schedules before, during, and after an extreme flood event in real time. The problem was formulated as a realtime optimal control problem in which reservoir releases represent the decision variables, similar to the methodology adopted in the present work.
The present paper presents one of the first hindcasting experiments with multistage stochastic reservoir optimization on the basis of mediumrange ESF for a tropical study case in Brazil. It is a followup of a previous study presented by Schwanenberg et al. (2015a) which used a similar approach, but restricted for a singleevent simulation. The novel components of the present paper are: (i) the closed loop experiment of the stochastic optimization in application to a realworld case study and (ii) a novel, mass conservative scenario tree generation with a detailed comparison of the forecast skill of the ensemble against the reduced scenario tree.
2 Methodology
The proposed methodology here does not intend to reduce forecast uncertainties, but, instead, is focused in the usage of these uncertainties for better decision making by a shortterm reservoir optimization approach.
where J( ) is a cost function associated with a state transition and h( ) includes hard constraints on control variables and states. The notation x * refers to a subset of state variables which is used in the hard constraints and which therefore become independent optimization variables. In this case, the related process model becomes an equality constraint of the optimization problem.
where p _{ j } is the probability of the ensemble member j and m are the number of ensembles.
The features of the stochastic optimization are determined by the choice of u _{ j } ^{ k } . In its simplest form, the control trajectory is identical in all ensemble members according to u _{ j } ^{ k } = u ^{ k }. In this case, the optimization tries to find a trajectory which minimizes the objective function on average over all ensemble members. If we assume that the reservoir release is the typical control variable in a reservoir management application, this means that the variability in the reservoir inflow recurs as a variation of the forebay elevation of the reservoir. This can lead to infeasibility if the inflow variability is larger than the reservoir storage over the forecast horizon.
The key to the method is the derivation of the scenario tree from an ensemble by a scenario tree reduction. We revised existing tree reduction techniques and found that most existing methods (Heitsch & Rőmisch 2003; Raso et al. 2013 among others) do not preserve the probabilityweighted sum of a quantity of the original ensemble ∑ _{ j = 0} ^{ m } p _{ j } u _{ j } ^{ k } due to a complete deletion of ensembles and an allocation of their probability to the closest remaining one. This is a drawback for reservoir management applications, since the probabilityweighted inflow volumes change with the tree reduction techniques and their parameters.
The purpose in this study is to define an efficient and robust scenario tree reduction technique which i) keeps probabilityweighted quantities constant in the tree reduction process; ii) leads to a constant number of branches in the tree (to keep the optimization effort roughly the same); and iii) preserves other properties of the original ensemble as much as possible.
 1.
The number of ensemble members is reduced to 2^{ x }, e.g. 2, 4, 8, 16, by deleting m − 2^{ x } members with the closest distance to remaining members by means of the distance matrix dis(i, j) = ∑ _{ k = 1} ^{ n } ‖d _{ i } ^{ k } − d _{ j } ^{ k } ‖. The probability of a deleted member is added to the closest remaining one. Next, the remaining member is replaced by the probabilityweighted sum of the deleted and remaining members.
 2.
A number of x branching points are distributed in equidistant steps along the forecast horizon.
 3.
All ensemble members up to the first branching point get aggregated into a single branch. From this point on, the ensemble members get classified into two groups according to their remaining inflow volume.
 4.
Step 3 is repeated at all other branching points for each of the previously generated subsets and split into two again.
 5.
Optional smoothing is applied over a number of time steps after the branching point to reduce the negative impact of large, unphysical inflow steps on the optimization.

Mean Absolute Error (MAE): this metric is the absolute difference between two values, in this case, between observations and forecasts at each lead time. In the case of ensemble forecasts. The MAE of a perfect model would be zero.

Mean Continuous Ranked Probability Score (CRPS): the CRPS is a score that summarizes the quality of a probability forecast into a number by comparing the integrated square difference between the cumulative distribution function of forecasts and observations. Lower CRPS values correspond to more accurate results.

Brier Score: the BS measures the average square error of a probability forecast for a dichotomous event, defined, for example, by a flow threshold exceedance. A perfectly sharp set of forecasts would show a BS value of zero.
3 Test Case Description
Três Marias HPP is operated by CEMIG (Companhia Elétrica de Minas Gerais). During low and medium flows, CEMIG follows rules defined by ONS to operate the HPP. However, during flood events, CEMIG operators take over the control of the reservoir turbines and spillways to mitigate downstream floods and without compromising future energy generation.
This case has already been the object of several recent studies. In the research of Fan et al. (2014), a first version of an ensemble forecasting system based on the MGBIPH (Collischonn et al. 2007) hydrological model and on GEFS v2 meteorological forecasts (Hamill et al. 2013) was shown. Subsequently, Fan et al. (2015a) presented a second version of the ensemble forecasting system coupled with DelftFEWS software shell (Werner et al. 2013) and tested it using multiple ensemble rainfall forecasts from the TIGGE database (Bougeault et al. 2010) and in comparison to other Brazilian basins. Finally, Schwanenberg et al. (2015a) developed a reservoir optimization system by coupling the MGBIPH hydrological model, ECMWF rainfall forecasts (Buizza et al. 2007) and the RTCTools (RealTime Control Tools) software (Schwanenberg et al. 2015b) to suggest optimum operations of the Três Marias HPP dam.
In the tests presented by Schwanenberg et al. (2015a) that developed system were applied to the reservoir’s operation during a single flood event, comparing results from perfect forecasts (e.g. observations), deterministic forecasts, and ensemble forecasts (stochastic optimization). Results showed that ensemble forecasts coupled with stochastic optimization were as good as perfect forecasts to the operation of the reservoir for the tested single event. The present paper extends the analysis to a more representative period of two consecutive years using optimum decisions to control the reservoir’s release. This means that the impact of mediumrange forecasts to the longterm operation of the reservoir is also evaluated.
Hydrological forecasts are produced using the MGBIPH model (Collischonn et al. 2007; Paz et al. 2007; Fan et al. 2014), forced by historical rainfall forecasts from the ECMWF global meteorological Bougeault et al. 2010). Both deterministic (fc) and probabilistic ensemble forecasts (pf) are used. The system runs in the DelftFEWS software shell (Werner et al. 2013). This hydrological forecasting framework is the same one presented by Fan et al. (2015a), and forecasts used in the experiments shown here are the same as presented by Fan et al. (2015a).
 1.
The minimum flow requirement is 460m^{3}/s;
 2.
Reservoir elevations must be between seasonallydependent lower and upper bounds;
 3.
The forebay elevation should be close to a requested forebay elevation at each timestep given by a guide cuve;
 4.
Spillage must be avoided as much as possible;
 5.
A discharge of 2000m^{3}/s at the city of Pirapora must be avoided as much as possible to prevent inundating small islands along the river;
 6.
If it is not possible to accomplish constraint 5, a discharge of 3800m^{3}/s at the city of Pirapora must be avoided as much as possible to prevent severe inundations; and
 7.
Changes in outflow should be as smooth as possible.
All of these constraints are included in the RTCTools model to optimize the operation of Três Marias HPP for the test period. Constraints 1 and 2 are considered hard constraints. The others are implemented as soft constraints by penalizing their violation in the objective function.
Weighting factors are adjusted under the supervision of CEMIG to represent the current operation of the Três Marias HPP. Under low flow conditions, operation of the Três Marias HPP must follow a guideline curve defined by ONS. Under high flow conditions, operation is focused on avoiding floods at the city of Pirapora while also protecting the dam’s integrity.

Configuration 01: perfect knowledge of the future (using observed discharges), at different total lead times (2, 3, 5, 7, 10, 15, 30, and 60 days);

Configuration 02: perfect knowledge of future rainfall is used to generate discharges of the hydrological model with different total lead times (2, 3, 5, 7, 10, 15, 30, and 60 days);

Configuration 03: deterministic future rainfalls obtained from ECMWF (ECMWFfc) are used to generate discharges with the hydrological model, at different lead times (2, 3, 5, 7, and 10 days);

Configuration 04: ensemble mean of future rainfalls obtained from ECMWF (ECMWFpf mean) are used to generate discharges within the hydrological model for different lead times (2, 3, 5, 7, 10, and 15days); and

Configuration 05: ensemble of future rainfalls obtained from ECMWF (ECMWFpf) are used to generate discharges within the hydrological model, with a lead time of 15 days.
Results of these experiments are showed and discussed in the following section.
4 Results and Discussions
4.1 Evaluation of Hydrological Forecasts and Derived Scenario Trees
MAE metric results (Fig 4) indicate that deterministic forecasts (ECMWFfc) and ensemble forecasts (ECMWFpf) perform very similarly until a lead time of 96 h, with errors increasing from 60 m^{3}/s (0 h) to 170 m^{3}/s (96 h). For higher lead times, the ensemble forecasts show smaller errors (198 m^{3}/s at 240 h) than the deterministic ones (220 m^{3}/s at 240 h). Figure 4 also demonstrates that the computed MAE of the decision trees 8B and 32B have the same value than the full ensemble. This is expected since the scenario tree reduction generation process supposes that the mean value of the distribution must be maintained. Hence, the test confirms the expected feature of the tree reduction.
CRPS metric results (Fig 5) indicate a similar performance to the deterministic forecasts (ECMWFfc) and the ensemble forecasts (ECMWFpf) until a lead time 48 h, with errors varying from 60 m^{3}/s (0 h) to 115 m^{3}/s (96 h). For higher lead times, the ensemble forecasts have smaller errors (151 m^{3}/s at 240 h) than the deterministic ones (220 m^{3}/s at 240 h). The CRPS of the scenario trees are between the deterministic forecasts and the ensemble. The tree with 8 branches (8B) has slightly higher error values than the 32B tree.
Another important feature noticed on Fig 5 is that for decision trees the mean CRPS values have “steps” along lead times. This happens due to the branching process. The trees have a lower number of branches for shorter lead times, whereas the number of branches increases for larger lead times. This property makes the decision tree more similar to a deterministic forecast at smaller lead times and more alike the probabilistic forecast for higher lead times. As such, the CRPS of the trees propagates from the deterministic forecasts to the probabilistic forecasts.
BS metric results (Fig 6) indicate that the performance of the deterministic forecasts (ECMWFfc) is similar to the ensemble forecasts’ (ECMWFpf) performance until a lead time of 120 h, with BS values varying from 0.01 (0 h) to 0.04 (120 h). For higher lead times, the ensemble forecasts show smaller BS values (0.035 at 240 h) than the deterministic forecasts (0.048 at 240 h).
As it was also observed in the CRPS analysis, the performance of the scenario trees is between the ones of the deterministic forecasts and the ensemble. Trees with 8 branches have slighter higher errors than trees with 32 branches. Both trees almost converge to the values of the full ensemble for high lead times, with a slightly better performance of the larger (32B) tree.
For MAE, CRPS and BS metrics it is visible that the lead time 100 h has critical importance. This lead time is when the uncertainties related to meteorological forecasts start to be more expressive in the basin and results start to differ more substantially, what is accused by the visible “split” between metrics results between different forecasting techniques (deterministic, full ensemble, or ensemble trees).
The metrics assessment results indicate two main conclusions. The first is that the ensemble forecasts have a better performance than the deterministic ones. This has been already reported in the broader analysis of Fan et al. (2015a) and recurs in the optimization results. Second, the generated scenario trees reveal a performance in between the deterministic forecast and the full ensemble. Usually, results are more similar to the ensemble at greater lead times when decision trees have more branches. At initial lead times, trees are more similar to the deterministic forecasts due to the smaller number of branches. This last fact is not a problem, because deterministic and probabilistic forecasts are usually close for smaller lead times; thus, the overall performance loss of the trees is small.
From the exclusive perspective of the tree metrics, the best option for the optimization is the use of a tree with 32 branches. However, a preassessment of the CPU time of the optimization shows an increase by a factor of 7 when increasing the number of branches from 8 to 32 in the multistage stochastic optimization. From this perspective, the metrics differences between these two trees are considered too small to justify the much longer processing time in an operational setup of the daily operation of the Três Marias HPP. Therefore, further analysis is based on the 8B tree only.
4.2 Optimization Results
The “ForecastOBS” configuration uses perfect rainfall observations as forcing of the hyrological model. Once again, the performance increases with longer forecast horizons. However, the decrease in the peak flow is significantly less than the one in the “Perfect” configuration. This reflects the negative impact of the relatively sparse gauging network as well as the error introduced by the hydrological model.
The results obtained with the deterministic forecasts (ECMWFfc) and the ensemble mean (ECMWFpf mean) are very similar until forecast horizons of 10 days. Peak flows decrease with longer horizons, but stay above 3800 m^{3}/s in all cases. For the horizon of 15 days, the remaining optimization with the ensemble mean (ECMWFpf mean) shows a peak flow of 3500 m^{3}/s and performs similarly to the ForecastOBS result. This shows the added value of the extra lead time of the ensemble mean (15 days) over the deterministic forecast (10 days).
The use of the 8B scenario tree in the multistage stochastic optimization significantly improve the performance at a horizon of 15 days by reducing the peak flow from 3500 m^{3}/s (ECMWFpf mean) to 2900 m^{3}/s (ECMWFpf). This clearly shows the benefit of the full tree in the stochastic setup versus the averaging of the ensemble and application of a deterministic optimization.
At Fig. 7 there are no values reported below the lower threshold. This happens because there is no motivation to have values below the lower threshold, once the objective of the optimization is to keep values discharges equal or below 2000m^{3}/s., and values equals to 2000m^{3}/s are the ones that guarantee the maximum flood protection without breaking the constrain.
A general analysis of Fig. 9 indicates that for all studied circumstances increasing lead times in the optimization procedure results in decreasing volumes above 2000m^{3}/s threshold. The lowest observed values were obtained for perfect inflows, culminating in zero values for horizons of 30 days and greater. The performance of the configurations of practical relevance decrease from 1450 hm^{3} for the deterministic ECMWFfc run of 10 days to a value of 1250 hm^{3} for the run of 15 days with the ensemble mean and the deterministic optimization (ECMWFpf mean) and to 650 hm^{3} for the stochastic optimization with the 8B tree (ECMWFpf). The latter shows the same performance as the deterministic run with the perfect inflow forecast (Perfect).
The analysis of Fig. 10 indicates that the fluctuation in the total energy generated is relatively small among the different configurations and for different forecast horizons; it is between 78% and 80% for the forecast horizon of 15 days. Longer lead times do not necessarily lead to a higher energy production. This is due to the fact that the optimization becomes more floodaware with longer horizons and probably deploys lower hydropower generation to achieve better flood mitigation. The cases of practical relevance are again the runs ECMWFfc (10 days), ECMWFpf mean (15 days) and ECMWFpf (15 days). In all of these runs, the energy generation is between 79.5% and 80%, which can be considered as an almost identical performance. This means that the added value in the flood mitigation objectives does not jeopardize the generation efficiency.
5 Conclusions
In this study, deterministic and probabilistic forecasts force a shortterm optimization model to operate the Três Marias HPP reservoir over a 2years test period. Main focus is the assessment of the added value of probabilistic forecast, the novel massconservative scenario tree reduction technique and multistagestochastic optimization in comparison to their deterministic counterparts.

The skill of the scenario trees, measured by several performance metrics, tends to decrease with a decreasing number of branches, while the computational performance increases with a decreasing number of branches.

A larger lead time improves the skill of optimized reservoir operation in terms of flood protection. If forecasts were perfect, the required lead time to accomplish the best operation in terms of flood mitigation would be approximately 30 days.

In forcing the deterministic and probabilistic ECMWF forecasts into the deterministic optimization (by using the ensemble average), the main benefit is achieved by the longer lead time of the probabilistic forecast. Little benefit can be explained by the higher skill of the probabilistic forecast average.

The use of probabilistic forecasts in combination with the stochastic optimization (15 days lead time) outperforms the deterministic version (10 days lead time) significantly.

The range of the energy production rate between the different approaches is relatively small, between 78% and 80%. This means that the flood protection improvement of the probabilistic / stochastic approach is achieved without compromising the energy generation.
Since early 2016, the presented approach is being used in the operational flow forecasting and decision support system of CEMIG for the operation of six single reservoirs at different locations and climates. Future studies are intended to address the extension of this approach to reservoir systems, to analyze alternate scenario tree reduction techniques and to assess additional meteorological and hydrological models.
Notes
Acknowledgments
The first author would like to acknowledge the Brazilian funding agency CAPES for the scholarship received during the work development period. Also, authors would like to thank Sheila Ball from Deltares, the Netherlands, for English language review.
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