# Real-Time Flood Control by Tree-Based Model Predictive Control Including Forecast Uncertainty: A Case Study Reservoir in Turkey

^{1}

^{2}

^{3}

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^{*}

## Abstract

**:**

## 1. Introduction

^{2}and located in Turkey, owing to its challenging operation due to downstream flow constraints. The reservoir serves as a main water supply for Kocaeli province. First, a hydro-meteorological rule based decision support system is developed for daily and hourly operation [38]. Later, it is shown that underestimated daily inflow forecasts during a flood event operation in critical reservoir level may result in underestimated spillage releases when using deterministic MPC [39]. Therefore, the case establishes a precedent for similar relatively small reservoirs with multi-purpose operational characteristics. At this point, this paper complements deterministic methods by PSF integrated TB-MPC including forecast uncertainty.

## 2. Materials and Methods

- Perfect Hindcast Experiments: The flood hydrograph corresponding to a 100 years return period (Q
_{100}) of the basin is utilized as perfect forecast data in deterministic MPC. This represents the best solution since it exhibits the optimized releases and forebay elevations under perfect knowledge of the future inflows, and it is evaluated as reference case for comparative analyses with forecast based models. - Deterministic Hindcast Experiments: This represents the skill of single value DSF evaluation by deterministic MPC. Randomly perturbed inflows in each receding horizon are employed in deterministic MPC mode. The random perturbation having a forecast horizon up to 48 h is updated in each hour.
- Probabilistic Hindcast Experiments: This represents the skill of ensemble PSF evaluation by multi-stage stochastic TB-MPC. Synthetically generated ensemble inflows in each receding horizon are employed in stochastic MPC mode (Exp-A). This provides forecast uncertainty consideration in the application. Moreover, features of TB-MPC are investigated with selection of different forecast horizons, tree branching numbers, and different inflow conditions (Exp-B).

#### 2.1. Deterministic and Probabilistic Synthetic Streamflow Generation

- In the initial time step, PSF (${\tilde{q}}_{j}^{1}$) members are generated as:$${\tilde{q}}_{j}^{k}=N~\left({\widehat{q}}^{k},{\widehat{q}}^{k}\ast {\widehat{\sigma}}^{k}\right),\text{}k=1\text{}\text{}j=1,2,\dots ,M$$
- The PSF should be correlated with previous members, therefore, the differences between successive DSF values (referred to $k$ time instants for the same DSF sequence) are calculated, then normally distributed errors having mean and standard error from the previous time step are generated and added to the differences. Maximum function is added in order to eliminate negative values, and the remaining PSF members ($\tilde{q}$) are formulated as:$$\begin{array}{c}{\tilde{q}}_{j}^{k}=\left({\widehat{q}}^{k}-{\widehat{q}}^{k-1}\right)+\mathrm{max}\left(N~\left({\widehat{q}}^{k-1},{\widehat{q}}^{k-1}\ast {\widehat{\sigma}}^{k}\right),0\right)\\ k=2,\dots ,N\text{}\text{}j=1,2,\dots ,M\end{array}$$

#### 2.2. Deterministic Model Predictive Control (MPC)

#### 2.3. Multi-Stage Stochastic Tree-Based MPC (TB-MPC)

## 3. A Real World Test Case and Model Set-Up

#### 3.1. Study Area

^{2}catchment area, located in the east part of Marmara Region, Turkey (Figure 3). The earth-filled dam was constructed in 1999 for the water supply of Kocaeli city. An annual 142 hm

^{3}of drinking, domestic and industrial water for 1.5 million inhabitants should be supplied. The relatively limited reservoir has an active storage capacity of approximately 51.2 hm

^{3}at maximum operating level of 169.3 m whereas 169.8 m is the maximum water level. The spillway is controlled by four radial gates. It should be noted that while a volume of 36.60 hm

^{3}is stored between minimum operation level of 112.50 m and spillway crest elevation of 159.95 m, a volume of 14.60 hm

^{3}is kept above spillway crest elevation and behind the radial gates. On the other hand, a Dam Management System (DMS) which has been developed as a part of a Supervisory Control and Data Acquisition (SCADA) system by the reservoir operators provides data collection and transmission from automatic gauges.

#### 3.2. MPC Inputs

^{3}/s by the regional water authority taking the drainage discharge conditions of the downstream canal and lateral flows into consideration. The main reason for that is a 12 km long downstream reach that passes from a narrow valley near a rural district and flows into the Marmara Sea after a sharp curvature by a manmade channel next to industrial and urban areas. The dam is built to protect the downstream region against extreme flood events, and the maximum spillway release is set to 200 m

^{3}/s during hourly flood control. This value is the maximum allowable flood limit without severe damage in the downstream. The operation of the dam is multi-purpose subject to two main objectives: (i) water supply and (ii) flood control.

_{100}flood hydrograph) in the main test application. The flood peak-occurrence time from the beginning is 6 h (i.e., the response time of the catchment to the rainfall event) and the peak flow corresponds to 597 m

^{3}/s. The total flood volume equals to 17.1 hm

^{3}. Being an extreme case, the peak flow of the selected flood hydrograph is three times greater than the downstream channel capacity (200 m

^{3}/s). The hindcasting experiments were conducted in an arbitrary year during a critical operation period (May) when the initial forebay elevation was high. The whole closed-loop hindcasting period covers 96 h, from [1-May-2012 00:00:00] to [4-May-2012 23:00:00]. It is assumed that a 24-h long flood hydrograph occurs between [3-May-2012 00:00:00] and [3-May-2012 23:00:00]. Hourly forecast data are produced for 48 h lead-time. This means that in each 1 h, 48-h long DSF data (with 1 member) and 48-h long PSF data (with M = 50 ensemble members) are generated throughout the whole hindcasting period. Given the lack of probabilistic hydrological forecasts for this case study, we decided to recreate the stochasticity by assuming a normally distributed noise around the deterministic forecast. This in fact becomes an innovation on its own in the paper, since an objective approach has not been reported in the literature before. Although it was desired to increase the number of PSF members in order to cover much more possibilities, this ended up with the reverse situation in the optimization model due to the high-dimensional data space. Different ensemble sizes were also tried in the control model, and it was observed that increasing member number estimates higher uncertainty range, 50 members are considered to provide enough spread to capture the major uncertainties in the forecasts.

_{25}and Q

_{50}(from Table 1) are also tested under Section 4.3.3. Characteristics of the PSF scenarios are presented with the performance assessment by using a mean Continuous Ranked Probability Score (CRPS) which generalizes the Mean Absolute Error (MAE) in the case of probabilistic forecasts (Supplementary Figure S1). Mean CRPS 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 [24]. According to that, mean CRPS increases with forecast lead-time, while each scenario shows a different performance. The scenario number is not critical in the study because the focal idea is to develop an objective approach for stochasticity of the flows, to use them in stochastic optimization set-up and to compare the results with a deterministic equivalent. Thus, scenarios can be deliberated as different source based forecast data sets.

- Forebay elevation at the end of a flood event: Forebay elevation should be same at the end of a flood event in order to provide long term water supply targets,
- Flooding threshold value: Spillway discharges should be less than channel capacity, thus this is considered as the flooding threshold value and the maximum discharge at the dam outlet is checked,
- Total flood volume at the downstream area: The cumulative volume of the released flood water (only above the maximum flood limit of 200 m
^{3}/s) should be zero for the best flood management, - Flood storage index (FSI): It is essential to have enough flood pool in the reservoir to attenuate the hourly extremes. To measure this, FSI is defined by the ratio of the total effective flood storage over the total volume of storage corresponding to Flood Control Levels (FCLs) [4] as shown in Equations (14) and (15) . The reservoir level should be kept at FCL as suggested to reserve space for flood attenuation. FSI ranges between zero and one. While zero indicates the reservoir level is always kept above FCL, one indicates operation is totally based on FCL. Higher FSI ensures more reliable flood operation (under forecast uncertainty) by having a high empty reservoir volume against flood risk.$$FS{I}_{f}=\frac{{{\displaystyle \sum}}_{k=1}^{N}{v}_{f}^{k}}{{{\displaystyle \sum}}_{k=1}^{N}{v}_{FCL}^{k}}$$$${v}_{f}^{k}=\{\begin{array}{c}{v}_{act}^{k}\text{}if\text{}{v}^{k}\le {v}_{FCL}^{k}\\ {v}_{FCL}^{k}\text{}if\text{}{v}^{k}{v}_{FCL}^{k}\end{array}for\text{}k=1,2,\dots N$$

#### 3.3. Model Set-Up

^{3}), $\Delta t$ is the time difference between ${k}^{th}$ and ${\left(k-1\right)}^{th}$ time steps, ${Q}_{I}$, $Qs$ , and ${Q}_{WS}$ are the reservoir inflows, spillway flow and water supply (m

^{3}/s), respectively. Also, forebay elevation, $fb$ could be computed by:

#### 3.4. Physical and Operational Constraints

#### 3.5. Objective Function

- Differences between optimized forebay elevation and maximum operating elevation are minimized in Equation (22),
- Spillway discharges are minimized in Equation (23),
- Spillway releases above a specified discharge (200 m
^{3}/s) are constrained in Equation (24). This has a high weight in order to prevent damage in the downstream, - The case of a set point for forebay elevation is given i.e., a variable guide curve (it is the same for each hour in a day) for long term targets [39], this term stands to minimize deviations from it in Equation (25),
- Spillway discharges between two consecutive time steps are constrained by consideration of the mechanical gate operation efficiency (against wear and tear) in Equation (26).$$J1\left(fb\right)={w}_{1}{\displaystyle \sum}_{k=1}^{N}\left({f}_{max}-f{b}^{k}\right)$$$$J2\left(Qs\right)={w}_{2}{\displaystyle \sum}_{k=1}^{N}\left(Q{s}^{k}\right)$$$$J3\left(Qs\right)={w}_{3}{\displaystyle \sum}_{k=1}^{N}\mathrm{max}{\left(Q{s}^{k}-{Q}_{set},0\right)}^{2}$$$$J4\left(fb\right)={w}_{4}{\displaystyle \sum}_{k=1}^{N}\mathrm{max}{\left(f{b}^{k}-f{b}_{set},0\right)}^{2}$$$$J5\left(Qs\right)={w}_{5}{\displaystyle \sum}_{k=1}^{N}{\left(Q{s}^{k+1}-Q{s}^{k}\right)}^{2}$$

## 4. Numerical Experiments and Results

#### 4.1. Deterministic MPC Hindcasts Using Perfect Forecasts

^{3}/s limit at the outlet. Longer forecast horizons than 18 h e.g., 24, 36, and 48 h, result in a similar response. Therefore, the experiment results of PER24, PER36, and PER48 overlap in the figure. According to the results, one can note that the mitigation of a major flood even with maximum operating levels and 200 m

^{3}/s downstream channel constraint can be achieved under perfect future knowledge of flood inflows at least 18 h beforehand.

#### 4.2. Deterministic MPC Hindcasts Using DSFs

^{3}/s and create flooding in the downstream. This is considered as lower reliability compared to perfect data based experiments, and mainly attributed to the forecast disturbance which introduces 30% bias to the control strategy. Longer forecast horizons (such as 18, 24, 36, 48 h) perform better and releases are shifted to earlier time steps. However, it is not possible to mitigate the flood event even for forecast horizons longer than 18 h due to the given bias in the inflows and the lack of uncertainty in the system optimization. Compared to perfect forecasts based MPC, the variations in spillages are higher due to updated information for each receding horizon.

#### 4.3. Multi-Stage Stochastic TB-MPC Hindcasts Using PSFs

#### 4.3.1. TB-MPC Hindcasts Considering a Different Number of Tree Branches

^{x}branches, e.g., 1, 2, 4, 8,… etc. [24]. Therefore, in this study 50 PSF ensembles were reduced to six different branches (1, 2, 4, 8, 16 and 32) and tested in a hindcast test. The experiments were done using Sce-Q100a PSF as input forecast data and comparison of the optimization results from different tree branches numbers is given in Figure 10 in terms of spillway flows and forebay elevation. This experiment shows the effects of the resolution of tree and correspondingly capturing forecast uncertainty in stochastic optimization. If the forecast horizon is set to 48 h, we can get optimum results after 16 branches as shown (Figure 10a) in terms of the spillway discharge. Since the results for 16–32 trees are exactly the same, they overlap with each other in the same figure. Although higher resolution overestimates the inflows which increases the pre-releases, it is still able to restore the forebay elevation target at the end of the flood event (Figure 10b).

#### 4.3.2. TB-MPC Hindcasts Considering Different Forecast Horizons

#### 4.3.3. Assessment of the Approach for Different Inflow Conditions and Scenarios

_{25}and Q

_{50}scenarios, respectively. On the other hand, deterministic (perfect and DSF) and stochastic (PSF) closed-loop MPC results from Sce-Q100a, Sce-Q100b and Sce-Q100c are shown in Figure 12. It is notable that the stochastic set-up always provides pre-releases and takes precautions against flood event. The deterministic model only takes actions over several hours which is similar to the perfect data based reference model, but generates much more spillage above the flood threshold compared to stochastic TM-MPC.

^{3}/s for each condition, they are not shown in the table. According to results, there is always an improvement in spillway discharges for different flows conditions of Q

_{25}, Q

_{50}and Q

_{100}.

_{100}is used for Q

_{100}operation assessment whereas FCL Q

_{50}is used for the Q

_{50}based flood case. According to this, TB-MPC stands as the more confident by higher FSI but also still can provide a high reservoir level at the end of the event without compromising water supply targets. For an uncertain future, a higher FSI is more reliable and preferable with lower risk for water supply as well.

_{25}, Q

_{50}, Q

_{100}with a forecast horizon up to 48 h with respect to (i) maximum discharge at the dam outlet (compared to flooding threshold value), (ii) total flood volume at the downstream area and (iii) FSI, while keeping the forebay elevation at the desired level for water supply at the end of the flood event. This shows the added value of the approach and provides reasonable outputs compared to the deterministic counterpart. The developed framework also indicates robust solutions against forecast uncertainty along with a different independent hindcasting experiment assessment.

## 5. Conclusions and Outlook

- Forecast uncertainty is indispensable especially for flood management. It is critical for those cases in which wrong or poor decisions may result with loss of life and property. At this point, considering uncertainty provides better management in terms of flood metrics without discarding water supply purposes.
- Independent closed-loop hindcasting experiment scenarios demonstrate the robustness of the system developed against biased information (disturbances).
- Probabilistic data represent forecast uncertainty in comparison to deterministic equivalents. In this study, a new synthetic streamflow generation method is proposed to represent forecast uncertainty for reservoir optimization.
- The synthetic PSF generation model that considers the dynamic evolution of uncertainties is valuable if hydrological model outputs driven by a rainfall and temperature forecast ensemble are not available. This method is very advantageous from the operational standpoint, since it does not require complex computations and is easy to implement while considering conditional (flow dependent) increasing uncertainty through time. It is simple to formulate, comprehend, and easy to repeat.
- Besides the ensemble generation, tree reduction parameters should be carefully investigated in the problem definition phase. In the case of selecting a lower branch and forecast horizon than required, TB-MPC results may converge to deterministic MPC results.
- The system was also tested against different inflow conditions which have greater flood value than the downstream channel capacity. According to the results, the method provides reliable results against different high flood conditions in the hindcasting experiments.

## Supplementary Materials

_{25}: (a) Sce-Q25a; (b) Sce-Q25b; (c) Sce-Q25c, Figure S3: Comparison of deterministic (perfect and DSF) and stochastic (PSF) closed-loop MPC results with different forecast scenarios for Q

_{50}: (a) Sce-Q50a; (b) Sce-Q50b; (c) Sce-Q50c.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**The general framework of the experiments. DSFs stands for Deterministic Streamflow Forecasts. PSFs stands for Probabilistic Streamflow Forecasts. MPC stands for Model Predictive Control.

**Figure 2.**Schematic of single time-step streamflow forecast uncertainty: (

**a**) DSF schematization; (

**b**) PSF schematization. ${q}^{k}$ stands for observed inflow. $\epsilon $ stands for relative inflow forecasting error. ${\widehat{q}}^{k}$ stands for DSF member. ${\tilde{q}}_{j}^{k}$ stands for ${j}^{th}$ PSF member. $k$ stands for time index.

**Figure 4.**Graphical representation of the scenarios (Sce-Q100a and Sce-Q100b) for: (

**a**) T0: 01-May-2012 12:00:00; (

**b**) T0: 01-May-2012 13:00:00.

**Figure 7.**Comparison of closed-loop MPC forecast horizon performance using perfect streamflow forecasts: (

**a**) Spillway discharge (m

^{3}/s); (

**b**) forebay elevation (m).

**Figure 8.**Comparison of closed-loop MPC forecast horizon performances with DSF (Sce-Q100a): (

**a**) Spillway discharge (m

^{3}/s); (

**b**) forebay elevation (m).

**Figure 9.**Open-loop optimization results of multi-stage stochastic optimization (from Sce-Q100a): (

**a**) Spillway discharge trees (m

^{3}/s); (

**b**) Forebay elevation trees (m).

**Figure 10.**Comparison of closed-loop MPC with different tree reduction branches for 48 h forecast horizon (Sce-Q100a): (

**a**) Spillway discharge (m

^{3}/s); (

**b**) forebay elevation (m).

**Figure 11.**Comparison of deterministic (perfect and DSF) and stochastic (PSF) closed-loop MPC results with different forecast horizons (Sce-Q100a): (

**a**) 18 h; (

**b**) 24 h; (

**c**) 36 h; (

**d**) 48 h.

**Figure 12.**Comparison of deterministic (perfect and DSF) and stochastic (PSF) closed-loop MPC results with different forecast scenarios for Q

_{100}: (

**a**) Sce-Q100a; (

**b**) Q100-Sceb; (

**c**) Q100-Scec.

Return Periods (Years) | Project Value (m^{3}/s) |
---|---|

5 | 208 |

10 | 297 |

25 | 410 |

50 | 506 |

100 | 597 |

**Table 2.**Computation times in MPC hindcasting experiments. MPC stands for Model Predictive Control. DSF stands for Deterministic Streamflow Forecast. TB-MPC stands for Tree-based MPC.

Hindcasting Experiment | Total CPU Time (s) |
---|---|

MPC with DSF | 151 |

TB-MPC with 1 tree branch | 491 |

TB-MPC with 2 tree branches | 551 |

TB-MPC with 4 tree branches | 633 |

TB-MPC with 8 tree branches | 677 |

TB-MPC with 16 tree branches | 867 |

TB-MPC with 32 tree branches | 1354 |

**Table 3.**Peakflow assessment of deterministic and stochastic closed-loop MPC results for different inflow conditions with forecast horizons of 48 h.

Flood Hydrograph | Scenarios | Peakflow at Yuvacik Outlet (m^{3}/s) | |
---|---|---|---|

Deterministic MPC | Stochastic MPC | ||

Q_{25} | Sce-Q25a | 243 | 231 |

Sce-Q25b | 255 | 243 | |

Sce-Q25c | 248 | 243 | |

Q_{50} | Sce-Q50a | 241 | 211 |

Sce-Q50b | 245 | 200 | |

Sce-Q50c | 246 | 200 | |

Q_{100} | Sce-Q100a | 242 | 200 |

Sce-Q100b | 269 | 235 | |

Sce-Q100c | 278 | 233 |

**Table 4.**Flood volume assessment of deterministic and stochastic closed-loop MPC for different inflow conditions with forecast horizon of 48 h.

Flood Condition | Scenarios | Total Flood Volume (1 × 10^{6} m^{3}) | |
---|---|---|---|

Deterministic MPC | Stochastic MPC | ||

Q_{25} | Sce-Q25a | 0.507 | 0.302 |

Sce-Q25b | 0.549 | 0.254 | |

Sce-Q25c | 0.438 | 0.271 | |

Q_{50} | Sce-Q50a | 0.666 | 0.062 |

Sce-Q50b | 0.471 | 0.004 | |

Sce-Q50c | 0.331 | 0.004 | |

Q_{100} | Sce-Q100a | 0.690 | 0.004 |

Sce-Q100b | 1.256 | 0.184 | |

Sce-Q100c | 1.018 | 0.127 |

**Table 5.**FSI value assessment of deterministic and stochastic closed-loop MPC according to Flood Control Levels (FCLs) for different inflow conditions with forecast horizon of 48 h.

Flood Condition | Scenarios | Flood Storage Index (FSI) | |
---|---|---|---|

Deterministic MPC | Stochastic MPC | ||

Q_{25} | Sce-Q25a | 0.652 | 0.800 |

Sce-Q25b | 0.659 | 0.990 | |

Sce-Q25c | 0.659 | 0.796 | |

Q_{50} | Sce-Q50a | 0.566 | 0.723 |

Sce-Q50b | 0.598 | 0.770 | |

Sce-Q50c | 0.606 | 0.758 | |

Q_{100} | Sce-Q100a | 0.457 | 0.650 |

Sce-Q100b | 0.463 | 0.645 | |

Sce-Q100c | 0.456 | 0.645 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Uysal, G.; Alvarado-Montero, R.; Schwanenberg, D.; Şensoy, A.
Real-Time Flood Control by Tree-Based Model Predictive Control Including Forecast Uncertainty: A Case Study Reservoir in Turkey. *Water* **2018**, *10*, 340.
https://doi.org/10.3390/w10030340

**AMA Style**

Uysal G, Alvarado-Montero R, Schwanenberg D, Şensoy A.
Real-Time Flood Control by Tree-Based Model Predictive Control Including Forecast Uncertainty: A Case Study Reservoir in Turkey. *Water*. 2018; 10(3):340.
https://doi.org/10.3390/w10030340

**Chicago/Turabian Style**

Uysal, Gökçen, Rodolfo Alvarado-Montero, Dirk Schwanenberg, and Aynur Şensoy.
2018. "Real-Time Flood Control by Tree-Based Model Predictive Control Including Forecast Uncertainty: A Case Study Reservoir in Turkey" *Water* 10, no. 3: 340.
https://doi.org/10.3390/w10030340