# Assessment of Water Resources Management Strategy Under Different Evolutionary Optimization Techniques

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Adopted Multi-Objective Optimization Approach

**F**(

**x**) = [f

_{1}(

**x**), ⋯, ⋯, ⋯f

_{M}(

**x**)]

^{T}

^{n}is the decision space, i.e. X = [

**x**

^{L},

**x**

^{U}] where

**x**= [x

_{1}, x

_{2}, …, x

_{n}]

^{T}is the decision variable vector of dimension n; and

**x**

^{L}and

**x**

^{U}are the vectors of the lower and upper bounds on

**x,**respectively.

**F**(

**x**) consists of M objective functions f

_{i}: X

`→`Z ∈ R

^{M}, where i = 1, …, M, and Z is the objective space’s feasible region containing all decision variables in X that satisfy all constraints. The ${g}_{i}\left(x\right)$ and ${h}_{j}\left(x\right)$ represents the i

^{th}of ${n}_{g}$ and j

^{th}for ${n}_{h}$ inequality and equality constraints, respectively. For unconstraint problems, ${n}_{g}={n}_{h}=\varnothing $, and Z = X [30].

- In a minimization problem, a vector
**u**= (u_{1}, . . . , u_{M})^{T}is said to dominate another vector**v**= (v_{1}, . . . , v_{M})^{T}if u_{i}≤ v_{i}for i = 1, . . . , M and u ≠ v. This property may be denoted as**u**≺**v**. - A feasible solution
**x**∈ X is called a Pareto-optimal solution, if there is no alternative solution**y**∈ X such that**F**(**y**) ≺**F**(**x**). - The Pareto-optimal set, PS, is the union of all Pareto-optimal solutions, and may be defined as PS = {
**x**∈ X :$\nexists $**y**∈ X,**F(y)**≺**F**(**x**)}. - The Pareto-optimal front, PF, is the set comprising the Pareto-optimal solutions in the objective space. It may be expressed as PF = {
**F(x)**|**x**∈ PS}.

#### 2.2. Details of Epsilon-Dominance-Driven Self-Adaptive Evolutionary Algorithm (ε-DSEA) Optimization Algorithm

- Diversity expansion to increase decision variables’ search space exploitation
- Self-adaptive operators’ parameters for parameters in process tuning
- Exploration extension for algorithm revival and stagnation coping
- Virtual dominance archive to improve diversity and convergence.

#### 2.2.1. Diversity Expansion

#### 2.2.2. Self-Adaptive Mechanism and Formulae

^{th}recombination operator, NDS

_{i}is the number of solutions in the archive contributed by the i

^{th}recombination operator, NRO is the number of recombination operators; The constant 1.0 is used to avoid probability values of zero.

#### 2.2.3. Exploration Extension Mechanism

_{r}such that ${N}_{r}$ ∈ ${\mathbb{N}}^{+}\in $ [1, 3]. When the algorithm starts, an N

_{r}value is selected at random and the maximum permissible number of function evaluations NFE

_{max}is divided by ${N}_{r}+1$ to determine the reset interval E

_{r}. For example, if NFE

_{max}= 300,000 and N

_{r}= 2, the reset occurs at every E

_{r}= 100,000 function evaluations. Hence, in this case, two resets occur during the entire optimization. Formally,

_{r}is the reset interval.

#### 2.2.4. Virtual Dominance Archive

#### 2.2.5. Constraint Handling Strategy

#### 2.3. Comparative Paradigms

- If there is no change in the archive size for a certain number of evaluations;
- If there is no improvement indicated by the $\u03f5$-progress indicator; and
- If the current population to archive ratio exceeds 1.25×γ

#### 2.4. Identification of a Real-World Experimental Test Problem

#### 2.4.1. Objectives Functions Formulae

_{W}, T

_{S}and T = winter, summer and total operation periods, respectively.

_{P}= penalty factor includes all the violations of the model, which could be expressed

^{th}) constraint

#### 2.4.2. Reservoir System Constraints

#### 2.5. Computational Properties

^{6}function evaluations and ε = 0.1 for three objectives, and ε = 0.5 for five objectives, with 10 and 20 runs for both algorithms. The minimum population size was 100 while the maximum was 1000. A Dell OptiPlex 780 computer was used (Core Duo 2 E8400, 2 × 3.0 GHz, 8.0 GB RAM, Ubuntu 16.04 operating system). Table 2 shows the parameter values used for both algorithms. A program (code) in C language was developed to build the current model.

## 3. Results

#### 3.1. Performance Achievement

#### 3.1.1. Algorithms’ Reliability

#### 3.1.2. Algorithms’ Robustness and Efficiency

^{5}function evaluation. The PCX operator is then involved by increasing the variation parameters (${\sigma}_{\eta}$ and ${\sigma}_{\zeta}$) to about 0.15. The SPX operator is also involved at the same time when its parameter (λ) changed to about 2.7. Both PCX and SPX operators compete to explore dominance solutions till the third resetting trigger, and after that the SPX operator starts to generate more dominance solutions in the dominance archive. Increasing PCX and SPX parameters will generate new offspring farther away from their original parents, which will increase algorithm exploration in the design search space.

_{1}to x

_{n}are the decision variables. Based on the best solution achieved, Figure 6b shows ${X}_{dv}$ convergence of both algorithms. Both achieved early convergence, but ε-DSEA converged faster, hence ε-DSEA’s efficiency was endorsed in the proposed test problem.

^{4}function evaluations for all iterations, and Borg MOEA converged at 25 × 10

^{4}. The ε-DSEA needs less execution time to achieve solutions. Where there are limited computational resources (e.g., CPU, Ram, etc.,) this achievement is significant. Furthermore, Borg MOEA suffered significant and interim stagnation in 7 trials (2, 3, 7, 9, and 4, 6, 10, respectively) in the early stage of evaluations. Only three out of 10 trials maintained dominance solutions improvement over the entire evaluation. The PCX operator’s adaptation with fixed parameters and recycling repetitively archive’s dominance solutions may restrict the extent of the algorithm’s exploration in the design search space. Conversely, only one trial (no. 9) suffered significate stagnation in ε-DSEA, however the expansion diversity and resetting techniques succeed in reviving the algorithm’s exploration to find new dominance solutions in the dominance archive. The robustness of ε-DSEA to escape from local optima are evident. Figure 7 shows trial no. 4 as a sample of convergence progress, since both algorithms achieved competitive solutions (based on Figure 5).

#### 3.1.3. Algorithms’ Effectiveness

#### 3.2. Strategic Achievement

^{2}, 51 and 44 km

^{2}achieved by ε-DSEA and Borg respectively. The small violation between these values indicates competitive results’ distribution, corollary more solutions greater than these values achieved by ε-DSEA. Hence, projects’ revenues could be improved even in such a critical scenario.

## 4. Discussion

#### 4.1. Algorithms’ Optimization Techniques

#### 4.2. Water Resources Management Case Study

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Illustrates operator’s parents’ selection from the entire population candidates after the initial random seeding at the beginning of the evaluation process.

**Figure 6.**(

**a**) Parameters self-adaptation of the most effective operators for the best solution achieved, and (

**b**) algorithm convergence to generate dominance solutions during the evaluation process.

**Figure 8.**Comparative graphs between achieved releases of both algorithms based on maximizing hydropower generation objective. (

**a**) and (

**b**) show detail results for 10 runs, while (

**c**) and (

**d**) illustrate historical and model releases.

**Figure 9.**Insight analysis of hydropower generation accomplished by both algorithms under flood risk management scenario (${f}_{summerD}$). (

**a**) and (

**b**) shows solutions repetition density, while (

**c**) and (

**d**) demonstrate head-discharge-hydropower solution space.

**Figure 10.**Reservoir’s surface area distribution achieved by both algorithms under ${f}_{summerD}$ scenario.

**Figure 11.**Demonstrates reservoir water level distribution achieved by both algorithms of 10 runs. (

**a**) and (

**b**) are the best solution to maximize hydropower generation; (

**c**) and (

**d**) the same as those to minimize summer storage.

**Table 1.**Parameters control formulae in epsilon-dominance-driven self-adaptive evolutionary algorithm (ε-DSEA).

Operator | Parameters | Domain | Adaptation Functions | Comments |
---|---|---|---|---|

SBX ^{1} | $\eta $ | [0, 100] | ${\mathcal{P}}_{i}^{NDS}\times 100$ | Distribution index |

DE ^{2} | CR F | [0.1,1.0] [0.5, 1.0] | $Max(0.1,{\mathcal{P}}_{i}^{NDS}{}_{i})$ $0.5+\left({\mathcal{P}}_{i}^{NDS}/2\right)$ | Crossover probability Step size |

SPX ^{3} | $\lambda $ | [2.5, 3.5] | $2.5+{\mathcal{P}}_{i}^{NDS}$ | Expansion rate |

PCX ^{4} | ${\sigma}_{\eta}$ ${\sigma}_{\zeta}$ | [0.1, 0.3] | $0.1+\left({\mathcal{P}}_{i}^{NDS}/5\right)$ $0.1+\left({\mathcal{P}}_{i}^{NDS}/5\right)$ | These parameters (standard deviations) control the spatial distribution of the offspring for PCX and UNDX |

UNDX ^{4} | ${\mathsf{\sigma}}_{\mathsf{\zeta}}$ ${\mathsf{\sigma}}_{\mathsf{\eta}}$ | [0.4, 0.6] [0.1, 0.35]$/\sqrt{L}$ | $0.4+\left({\mathcal{P}}_{\mathrm{i}}^{\mathrm{NDS}}/5\right)$ $\left[0.1+\left({\mathcal{P}}_{\mathrm{i}}^{\mathrm{NDS}}/3\right)\right]/\sqrt{L}$ |

^{1}Simulated Binary Crossover;

^{2}Differential Evolution;

^{3}Simplex Crossover;

^{4}Parent-Centric Crossover;

^{5}Unimodal Normal Distribution Crossover

Parameters | Borg | ε-DSEA^{a} | Parameters | Borg | ε-DSEA |
---|---|---|---|---|---|

Initial population size | 100 | 100 | SPX parents | 10 | 3 |

Tournament selection size | 2 | 2 | SPX offspring | 2 | 2 |

SBX crossover rate | 1.0 | 1.0 | SPX expansion rate λ | 3 | [2.5, 3.5] |

SBX distribution index η | 15.0 | [0, 100] | UNDX parents | 10 | 10 |

DE crossover rate CR | 0.1 | [0.1, 1.0] | UNDX offspring | 2 | 2 |

DE step size F | 0.5 | [0.5, 1.0] | UNDX σ_{ζ} | 0.5 | [0.4, 0.6] |

PCX parents | 10 | 10 | UNDX σ_{η} | 0.35/$\sqrt{L}$ | [0.1, 0.35]/$\sqrt{L}$ |

PCX offspring | 2 | 2 | UM mutation rate | 1/L | 1/L |

PCX σ_{η} | 0.1 | [0.1, 0.3] | PM mutation rate | 1/L | 1/L |

PCX σ_{ζ} | 0.1 | [0.1, 0.3] | PM distribution index η_{m} | 20 | 20 |

_{η}and σ

_{ζ}are defined in Table 1.

^{a}The initial values of dynamic parameters used in ε-DSEA are as shown for Borg MOEA.

**Table 3.**Results’ analysis of water resources management strategy of Derbendikhan dam achieved by both algorithms based on maximising hydropower generation.

Borg MOEA | ε-DSEA | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Area ^{1} | Head ^{2} | Power ^{3} | Storage ^{4} | Releases ^{5} | Area | Head | Power | Storage | Releases | |

3 Objective problem | ||||||||||

Min. | 19.18 | 437.79 | 24.50 | 433.55 | 129.94 | 17.32 | 434.73 | 24.83 | 373.82 | 129.75 |

Max. | 122.79 | 485.97 | 249.00 | 2565.84 | 866.25 | 121.80 | 485.86 | 246.37 | 2551.05 | 877.20 |

Mean | 74.37 | 474.71 | 94.76 | 1732.33 | 336.06 | 76.39 | 475.53 | 94.77 | 1775.22 | 336.17 |

Median | 72.03 | 477.12 | 83.00 | 1743.17 | 297.19 | 78.92 | 478.95 | 84.37 | 1867.41 | 316.10 |

St.^{7} | 27.19 | 10.11 | 50.88 | 523.86 | 174.84 | 24.99 | 10.42 | 51.88 | 496.30 | 183.22 |

Gross^{6} | 37.52 | 686.00 | 133.08 | 37.53 | 702.99 | 133.12 | ||||

5 Objective problem | ||||||||||

Min. | 16.94 | 434.09 | 23.44 | 361.60 | 130.72 | 19.10 | 437.67 | 24.09 | 431.16 | 130.45 |

Max. | 122.97 | 485.98 | 249.00 | 2568.50 | 866.08 | 123.14 | 486.00 | 249.00 | 2570.98 | 797.97 |

Mean | 66.09 | 470.67 | 90.46 | 1555.39 | 337.72 | 71.90 | 472.95 | 91.77 | 1672.36 | 334.80 |

Median | 61.55 | 473.63 | 82.14 | 1540.19 | 316.81 | 71.78 | 477.05 | 82.00 | 1738.53 | 298.79 |

St. | 29.89 | 12.83 | 45.64 | 597.36 | 162.74 | 28.94 | 12.58 | 47.18 | 583.53 | 169.11 |

Gross | 35.82 | 615.94 | 133.74 | 36.34 | 662.25 | 132.58 |

^{1}Surface area of reservoir in km

^{2};

^{2}Head of water in m.a.s.l;

^{3}Hydropower generation in MW;

^{4}Reservoir storage in m

^{3}×10

^{6});

^{5}Reservoir releases in m

^{3}/month ×10

^{6};

^{6}Gross sum units of: Power in GW; Storage in m

^{3}×10

^{9}); Releases in m

^{3}/month ×10

^{9}.

^{7}St. for Standard Deviation.

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**MDPI and ACS Style**

Y. Al-Jawad, J.; M. Kalin, R. Assessment of Water Resources Management Strategy Under Different Evolutionary Optimization Techniques. *Water* **2019**, *11*, 2021.
https://doi.org/10.3390/w11102021

**AMA Style**

Y. Al-Jawad J, M. Kalin R. Assessment of Water Resources Management Strategy Under Different Evolutionary Optimization Techniques. *Water*. 2019; 11(10):2021.
https://doi.org/10.3390/w11102021

**Chicago/Turabian Style**

Y. Al-Jawad, Jafar, and Robert M. Kalin. 2019. "Assessment of Water Resources Management Strategy Under Different Evolutionary Optimization Techniques" *Water* 11, no. 10: 2021.
https://doi.org/10.3390/w11102021