# NN-Based Implicit Stochastic Optimization of Multi-Reservoir Systems Management

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Problem Definition

- Deterministic solution of many different open loop-problems each corresponding to one of the scenarios;
- Fitting of a suffcienctly general class of functions to reproduce as much as possible the behaviour of the open-loop solution;
- Testing of the identified rules with different scenarios on the complete system by new simulations.

#### 2.2. Deterministic Open-Loop Problem

- Allow solving fairly quickly nonlinear problems with lots of dimensions. They are faster than methods that discretize the decisions space and perform an exhaustive search;
- Guarantee a better performance than other random optimization methods, requiring an acceptable increasing of the computational effort;
- Behave better than gradient-based methods when the objective function has lots of local optima.

#### 2.3. Policy Identification

- radial basis functions, since it has been recently empirically proved that they can be more appropriate in some contexts similar to the one considered here [19].

#### 2.4. Rule Testing

## 3. Case Study Description

#### 3.1. The Nile River basin

#### 3.2. Model Formulation

#### 3.2.1. System Variables

#### (1) State Variables

#### (2) Input Variables

- Water demand in the respective five agricultural district, whose values are known at time t when the manager decides the control (green arrows in Figure 3):$${\mathbf{w}}_{t}=\left(\right)open="|"\; close="|">w{d}_{t}^{\mathrm{Tana}},w{d}_{t}^{\mathrm{Ros}},w{d}_{t}^{\mathrm{KeG}},w{d}_{t}^{\mathrm{Nas}},w{d}_{t}^{\mathrm{Egypt}}$$

- Random flows, whose subscripts t indicate that their values are known at the end of the time step $\left(\right)$.$${\mathbf{i}}_{t}=\left(\right)open="|"\; close="|">{n}_{t}^{\mathrm{Tana}},{e}_{t}^{\mathrm{Ros}},{e}_{t}^{\mathrm{KeG}},{e}_{t}^{\mathrm{Nas}},{l}_{t}^{\mathrm{Nas}},{i}_{t}^{\mathrm{Kes}},{i}_{t}^{\mathrm{Bor}},{i}_{t}^{\mathrm{Din}},{i}_{t}^{\mathrm{Rah}},{i}_{t}^{\mathrm{WN}},{i}_{t}^{\mathrm{u},\mathrm{At}},{i}_{t}^{\mathrm{l},\mathrm{At}}$$
- -
- ${n}_{t}^{\mathrm{Tana}}$ is the net inflow to Lake Tana;
- -
- ${e}_{t}^{\mathrm{Ros}}$, ${e}_{t}^{\mathrm{KeG}}$ and ${e}_{t}^{\mathrm{Nas}}$ are the evaporation rates of Roseires, Girba and Nasser reservoirs;
- -
- ${l}_{t}^{\mathrm{Nas}}$ is the seepage and bank loss at Nasser;
- -
- ${i}_{t}^{\mathrm{Kes}}$, ${i}_{t}^{\mathrm{Bor}}$, ${i}_{t}^{\mathrm{Din}}$, ${i}_{t}^{\mathrm{Rah}}$, ${i}_{t}^{\mathrm{u},\mathrm{At}}$ and ${i}_{t}^{\mathrm{l},\mathrm{At}}$ are the inflows to Kessie, Border, Dinder, Rahad, upper and lower Atbara;
- -
- ${i}_{t}^{\mathrm{WN}}$ represents the White Nile discharge at Khartoum.

#### (3) Decision Variables

#### (4) Internal Variables

- the natural outflow from Lake Tana at Bahir Dahr, ${q}_{t}^{\mathrm{Bahir}}$;
- the discharges downstream of Border, ${q}_{t}^{\mathrm{Bor}}$, and Dongola, ${q}_{t}^{\mathrm{Don}}$, which are the inflows to Roseires reservoir and Lake Nasser;
- others discharges at some significant points of the network such as Tamaniat (${q}_{t}^{\mathrm{Tam}}$) and Hassanab (${q}_{t}^{\mathrm{Has}}$) on the Main Nile, the discharge of the Blue Nile at Khartoum (${q}_{t}^{\mathrm{BN}}$) and of the river Atbara just before the junction with the Main Nile (${q}_{t}^{\mathrm{l},\mathrm{At}}$);
- the hydraulic head of the power plants, computed as the difference, if positive, between the water level and the tailwater: ${H}_{t}^{\mathrm{Tana}}$, ${H}_{t}^{\mathrm{Bahir}}$, ${H}_{t}^{\mathrm{Ros}}$, ${H}_{t}^{\mathrm{KeG}}$ and ${H}_{t}^{\mathrm{Nas}}$.

#### 3.2.2. Operating Objectives

#### (1) Hydropower Production

- $\eta $ is the overall efficiency of the power plant;
- $\gamma $ is the specific weight of water, equal to $9810\phantom{\rule{0.277778em}{0ex}}\frac{N}{{m}^{3}}$;
- ${H}_{t}^{\mathrm{i}}$ is the hydraulic head, expressed in m;
- the fourth factor represents the flow rate that will be used for power production, expressed in $\left(\right)$, computed as the minimum between the release from the reservoir and the maximum flow that can go through the turbines (${mtf}^{\mathrm{i}}$);
- $\psi $ is a coefficient of dimensional conversion, which contains the duration of the time step, 1 month, and whose value is $\frac{1\mathrm{month}}{3600\phantom{\rule{0.277778em}{0ex}}\mathrm{s}/\mathrm{h}}$ in order to obtain an energy value expressed in $\left[\mathrm{GWh}\right]$.

#### (2) Water Supply to Agricultural Districts

#### (3) Objective Space Definition

#### 3.2.3. Constraints

#### (1) White Nile

#### (2) Blue Nile

#### (3) Atbara

#### (4) Main Nile

#### 3.3. Deterministic Open-Loop Problem

## 4. Results

#### 4.1. Control Laws

#### 4.2. System Performances

- a trivial policy, which releases the maximum feasible flow at each time step;
- a closed-loop linear policy, which takes as input the same variables of the neural policy. Since we considered four different cases depending on the available information (DI, DF, CI, and CF), four different linear policies have been identified;
- the open-loop control sequence computed with the GA. The performance obtained using this control sequence can be considered as a utopia point, in the sense that it is the target to which we wish to get close. This point represents the best performance which is possible to obtain when all the future inflows are deterministically known.

## 5. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

CF | Centralized-full hydrological |

CI | Centralized-incomplete hydrological |

DEM | Digital elevation model |

DF | Decentralized-full hydrological |

DI | Decentralized-incomplete hydrological |

ESO | Explicit stochastic optimization |

GA | Genetic Algorithm |

ISO | Imolicit stochastic optimization |

Nile-DST | Nile decision support tool |

NN | Neural network |

Probability density function | |

ReLU | Rectified Linear Unit |

SDP | Stochastic dynamic programming |

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**Figure 1.**Schematic representation of the ISO procedure (modified from [14]). First, the optimal open-loop decisions are computed. Then, the obtained dataset is used for the identification of the closed-loop policy.

**Figure 3.**Schematic representation of the Nile River system. White Nile upstream Khartoum and Atbara before Khasm el Girba Dam have been considered as unregulated flows.

**Figure 5.**Objective space for a dry (

**a**) and a wet (

**b**) scenario. Averages of inflows for the considered scenarios are reported in Table 1.

**Figure 7.**Error distribution for the identified control laws. Errors are computed as the difference between targets and model outputs and normalized between $-1$ and 1.

**Figure 9.**Control laws for Lake Nasser with decentralized and incomplete information: the release is expressed as function of the storage.

**Table 1.**Yearly average of inflows (${\mathrm{km}}^{3}/\mathrm{month}$) and objectives improvement with respect to the trivial policy, obtained using the optimal open-loop control sequence for two synthetic scenarios.

Inflows | Dry | Wet |
---|---|---|

${n}_{t}^{\mathrm{Tana}}$ | $0.809$ | $0.830$ |

${i}_{t}^{\mathrm{Kes}}$ | $1.008$ | $1.173$ |

${i}_{t}^{\mathrm{Bor}}$ | $2.504$ | $2.972$ |

${i}_{t}^{\mathrm{Din}}$ | $0.177$ | $0.215$ |

${i}_{t}^{\mathrm{Rah}}$ | $0.090$ | $0.095$ |

${i}_{t}^{\mathrm{WN}}$ | $2.214$ | $2.356$ |

${i}_{t}^{\mathrm{u},\mathrm{At}}$ | $0.849$ | $1.063$ |

${i}_{t}^{\mathrm{l},\mathrm{At}}$ | $0.036$ | $0.020$ |

$\sum \mathrm{inflows}$ | $7.687$ | $8.724$ |

Energy increase $\left[\%\right]$ | $8.31$ | $11.48$ |

Water deficit decrease $\left[\%\right]$ | $56.14$ | $51.22$ |

**Table 2.**Variables given as input to the control laws for the four considered cases: centralized-full (CF), centralized-incomplete (CI), decentralized-full (DF), and decentralized-incomplete (DI).

C | D | F | I | Variables Composing ${\mathit{I}}_{\mathit{t}}$ |
---|---|---|---|---|

• | • | Tana: ${s}_{t}^{\mathrm{Tana}}$, $w{d}_{t}^{\mathrm{Tana}}$ | ||

Roseires: ${s}_{t}^{\mathrm{Ros}}$, $w{d}_{t}^{\mathrm{Ros}}$ | ||||

Girba: ${s}_{t}^{\mathrm{KeG}}$, $w{d}_{t}^{\mathrm{KeG}}$ | ||||

Nasser: ${s}_{t}^{\mathrm{Nas}}$, $w{d}_{t}^{\mathrm{Egypt}}$ | ||||

• | • | Tana: ${s}_{t}^{\mathrm{Tana}}$, $w{d}_{t}^{\mathrm{Tana}}$, ${n}_{t}^{\mathrm{Tana}}$ | ||

Roseires: ${s}_{t}^{\mathrm{Ros}}$, $w{d}_{t}^{\mathrm{Ros}}$, ${e}_{t}^{\mathrm{Ros}}$, ${i}_{t}^{\mathrm{Ros}}$ | ||||

Girba: ${s}_{t}^{\mathrm{KeG}}$, $w{d}_{t}^{\mathrm{KeG}}$, ${e}_{t}^{\mathrm{KeG}}$, ${i}_{t}^{\mathrm{KeG}}$ | ||||

Nasser: ${s}_{t}^{\mathrm{Nas}}$, $w{d}_{t}^{\mathrm{Egypt}}$, $w{d}_{t}^{\mathrm{Nas}}$, ${e}_{t}^{\mathrm{Nas}}$, ${l}_{t}^{\mathrm{Nas}}$, ${i}_{t}^{\mathrm{Nas}}$ | ||||

• | • | ${s}_{t}^{\mathrm{Tana}}$, ${s}_{t}^{\mathrm{Ros}}$, ${s}_{t}^{\mathrm{KeG}}$, ${s}_{t}^{\mathrm{Nas}}$, $w{d}_{t}^{\mathrm{Tana}}$, $w{d}_{t}^{\mathrm{Ros}}$, $w{d}_{t}^{\mathrm{KeG}}$, $w{d}_{t}^{\mathrm{Egypt}}$ | ||

• | • | ${s}_{t}^{\mathrm{Tana}}$, ${s}_{t}^{\mathrm{Ros}}$, ${s}_{t}^{\mathrm{KeG}}$, ${s}_{t}^{\mathrm{Nas}}$, $w{d}_{t}^{\mathrm{Tana}}$, $w{d}_{t}^{\mathrm{Ros}}$, $w{d}_{t}^{\mathrm{KeG}}$, $w{d}_{t}^{\mathrm{Egypt}}$, $w{d}_{t}^{\mathrm{Nas}}$ | ||

${n}_{t}^{\mathrm{Tana}}$, ${e}_{t}^{\mathrm{Ros}}$, ${i}_{t}^{\mathrm{Ros}}$, ${e}_{t}^{\mathrm{KeG}}$, ${i}_{t}^{\mathrm{KeG}}$, ${e}_{t}^{\mathrm{Nas}}$, ${l}_{t}^{\mathrm{Nas}}$, ${i}_{t}^{\mathrm{Nas}}$ |

Reservoir | Function | Case | Training | Validation | Testing | All |
---|---|---|---|---|---|---|

Ros | Sigmoid | DI | $89.1$ | $89.0$ | $88.7$ | $89.0$ |

DF | $90.5$ | $90.4$ | $90.1$ | $90.4$ | ||

CI | $89.7$ | $89.5$ | $89.3$ | $89.6$ | ||

CF | $90.7$ | $90.5$ | $90.3$ | $90.6$ | ||

Radial Basis | DI | $89.1$ | $89.0$ | $88.7$ | $89.0$ | |

DF | $90.5$ | $90.4$ | $90.1$ | $90.4$ | ||

CI | $89.7$ | $89.6$ | $89.2$ | $89.6$ | ||

CF | $90.8$ | $90.6$ | $90.3$ | $90.6$ | ||

KeG | Sigmoid | DI | $90.1$ | $89.7$ | $89.3$ | $89.9$ |

DF | $91.8$ | $91.4$ | $91.3$ | $91.6$ | ||

CI | $91.0$ | $90.6$ | $90.3$ | $90.8$ | ||

CF | $92.4$ | $92.1$ | $91.9$ | $92.2$ | ||

Radial Basis | DI | $90.1$ | $89.7$ | $89.4$ | $89.9$ | |

DF | $91.7$ | $91.4$ | $91.2$ | $91.6$ | ||

CI | $91.0$ | $90.5$ | $90.3$ | $90.8$ | ||

CF | $92.4$ | $92.0$ | $91.9$ | $92.2$ | ||

Nas | Sigmoid | DI | $78.1$ | $77.3$ | $77.4$ | $77.8$ |

DF | $91.2$ | $90.7$ | $90.8$ | $91.0$ | ||

CI | $78.7$ | $77.8$ | $78.0$ | $78.4$ | ||

CF | $91.2$ | $90.7$ | $90.7$ | $91.0$ | ||

Radial Basis | DI | $78.1$ | $77.3$ | $77.4$ | $77.8$ | |

DF | $91.2$ | $90.7$ | $90.8$ | $91.0$ | ||

CI | $78.8$ | $77.9$ | $78.0$ | $78.4$ | ||

CF | $91.2$ | $90.7$ | $90.7$ | $91.0$ |

**Table 4.**Improvements over the solution obtained with the trivial closed-loop policy for all the considered cases, expressed as percentage of the distance between this solution and the optimal open-loop release sequence.

Control | Case | Distance from the Trivial Policy [%] | |
---|---|---|---|

Hydropower Producers | Agricultural Districts | ||

Trivial policy | $0.00$ | $0.00$ | |

Linear policy | DI | $-\phantom{\rule{0.277778em}{0ex}}13.73$ | $+\phantom{\rule{0.277778em}{0ex}}61.28$ |

DF | $+\phantom{\rule{0.277778em}{0ex}}7.95$ | $+\phantom{\rule{0.277778em}{0ex}}67.97$ | |

CI | $+\phantom{\rule{0.277778em}{0ex}}19.41$ | $+\phantom{\rule{0.277778em}{0ex}}65.99$ | |

CF | $+\phantom{\rule{0.277778em}{0ex}}48.82$ | $+\phantom{\rule{0.277778em}{0ex}}93.60$ | |

Neural policy | DI | $+\phantom{\rule{0.277778em}{0ex}}51.52$ | $+\phantom{\rule{0.277778em}{0ex}}91.98$ |

DF | $+\phantom{\rule{0.277778em}{0ex}}52.46$ | $+\phantom{\rule{0.277778em}{0ex}}93.79$ | |

CI | $+\phantom{\rule{0.277778em}{0ex}}50.64$ | $+\phantom{\rule{0.277778em}{0ex}}94.44$ | |

CF | $+\phantom{\rule{0.277778em}{0ex}}56.47$ | $+\phantom{\rule{0.277778em}{0ex}}96.13$ | |

OL sequence | $+\phantom{\rule{0.277778em}{0ex}}100.00$ | $+\phantom{\rule{0.277778em}{0ex}}100.00$ |

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**MDPI and ACS Style**

Sangiorgio, M.; Guariso, G.
NN-Based Implicit Stochastic Optimization of Multi-Reservoir Systems Management. *Water* **2018**, *10*, 303.
https://doi.org/10.3390/w10030303

**AMA Style**

Sangiorgio M, Guariso G.
NN-Based Implicit Stochastic Optimization of Multi-Reservoir Systems Management. *Water*. 2018; 10(3):303.
https://doi.org/10.3390/w10030303

**Chicago/Turabian Style**

Sangiorgio, Matteo, and Giorgio Guariso.
2018. "NN-Based Implicit Stochastic Optimization of Multi-Reservoir Systems Management" *Water* 10, no. 3: 303.
https://doi.org/10.3390/w10030303