# Reducing Water Conveyance Footprint through an Advanced Optimization Framework

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- A nonlinear chaotic honey badger algorithm, i.e., NCHBA, incorporating a nonlinear control parameter and a chaotic map to strike a balance between exploration and exploitation, is proposed. The efficiency of NCHBA is validated by solving a high-dimensional pump scheduling problem.
- A new multi-objective variant of NCHBA is proposed, and its performance is assessed using four ZDT benchmark functions.
- The proposed multi-objective algorithm is utilized to optimize the pump scheduling program of a large WDS to minimize the energy consumption and footprint of pumping stations, quality risk, and nodal pressure. The optimal compromise solution is determined through the TOPSIS method.

## 2. Materials and Methods

#### 2.1. Optimization Process and Problem Formulation

#### 2.1.1. Objective Functions

_{E}), which includes two parts: demand charge (C

_{D}) and consumption charge (Cc). Cc is the cost of electrical energy consumed during a time period. C

_{D}denotes the demand charge, which is the total cost associated with the maximum amount of power consumed (i.e., peak energy). Therefore, the total pumping energy cost is computed by Equation (1) as follows [36]:

_{D}, the model first finds the highest power required for each pump (${P}_{max}$) throughout the simulation and counts how often it happens. Then, it multiplies this number by a user-defined demand charge ($D{C}_{n}$) for each pump. Therefore, ${\alpha}_{n}$ is the product of the maximum power for a pump $n$, ${P}_{max}$, and the frequency of this power, ${n}_{{P}_{max}}$, i.e.,

#### 2.1.2. Constraints

#### 2.2. Optimization Model

#### 2.2.1. Honey Badger Algorithm

#### 2.2.2. Improved Honey Badger Algorithm

- Utilizing the chaotic maps instead of random numbers.
- Utilizing a nonlinear parameter to create a good balance between the exploitation and exploration phases.

#### 2.2.3. Multi-Objective NCHBA

## 3. Results and Implementation

#### 3.1. Model Validation

#### 3.2. Single Objective NCHBA for Energy Optimization

#### 3.3. Multi-Objective NCHBA for Benchmark Problems

_{1}to ZDT

_{4}and ZDT

_{6}. The optimal solution set of all three algorithms is uniformly converged to the actual Pareto front. By enlarging a part of the Pareto front, the uniformity of the solutions of the MONCHBA algorithm was much higher than the other two algorithms. At the same time, MOSMA has not been able to adapt effectively to the actual Pareto. The convergence of three algorithms for the ZDT

_{2}problem in Figure 6 shows the reasonable accuracy of MONCHBA compared to the other two algorithms. Hence, the solutions found by NSGA-II were far better than those found by MOSMA. However, the number of solutions obtained is less than the other two algorithms. This figure shows that the convergence of the proposed algorithm in solving the ZDT

_{3}function is also acceptable. The set of solutions provided by it is uniformly converged to the true Pareto front in the interaction diagram of the ZDT

_{4}and ZDT

_{6}examples presented in Figure 6, and it is the same as the algorithm MONCGWO has good accuracy and convergence. However, the noteworthy point is the poor performance of the MOSMA algorithm compared to the other two algorithms, which failed to converge accurately with the true solution. The optimal solutions were placed far from the true Pareto in almost all examples, meaning the weakness of this algorithm in escaping local optima and premature convergence.

#### 3.4. Multi-Objective NCHBA for Energy Optimization

_{2}based on the water level in the tank. Pumping stations S2 and S3 transferred water from the T2 tank to the tanks located at a higher altitude. Pumping stations S4 and S5 were responsible for transferring water from tank T1 to tanks T5, T6, and T7. Figure 7 shows the schematic of this network.

_{1}, f

_{2}, and f

_{3}). The answer series with the highest ${\mu}_{i}$ value is chosen as the final answer, indicating a consensus among all objectives.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Notation and List of Acronyms

${Q}_{ij}$ | Flow rate between nodes i and j | DMA | District mater area |

$NP\left(j\right)$ | Number of pipes meeting at node j | FSP | Fixed-speed pump |

${q}_{j}$ | Nodal demand at node j | VSP | Variable-speed pump |

GA | Genetic Algorithm | ||

$H{P}_{ij}$ | Head added by pumps in pipe j | LP | Linear programming |

$np\left(i\right)$ | Number of pipes included in loop i | NLP | Nonlinear programming |

${h}_{ij}$ | Head loss between node i and j | DP | Dynamic programming |

$L$ | Pipe length | ACO | Ant Colony Optimization |

$D$ | Pipe diameter | DA | Dragonfly Algorithm |

C | Hazen–Williams coefficient | NSGA-II | Non-Dominated Sorting Genetic Algorithm |

${Q}_{\left(n,t\right)}$ | Flow through pump during each time step t in pump n | HBA | Honey badger algorithm |

${H}_{\left(n,t\right)}$ | Total dynamic head during each time step t in pump n | WDS | Water distribution system |

$E{C}_{t}$ | Electricity tariff at time t (USD/kWh) | PDSM | Pressure-driven simulation method |

${b}_{nt}$ | Status of pump n as being off or on at time t | RUN | Runge–Kutta Optimization Algorithm |

$\u2206{t}_{t}$ | Length of a time interval t | NDS | Non-dominated sorting |

${\eta}_{\left(n,t\right)}$ | Efficiency of pump n during each time step t | CD | Crowding distance |

${Q}_{\left(n\right)}^{Max}$ | Peak discharge through the pump n | SMA | Slim Mould Algorithm |

ED | Demand charge (USD/kW) | AO | Aquila Optimizer |

${P}_{i,t}$ | Pressure at node i in time t | HGA | Hunger Games search |

${P}_{i}^{min}$ | Minimum required pressure at node i | EA | Evolutionary algorithm |

${Q}_{i,t}^{req}$ | Required demand for node i at time t | IGD NCHBA | Inverted generation distance Nonlinear chaotic honey badger algorithm |

${Q}_{i,t}^{avl}$ | Available discharge node i at time t | ||

FA | Firefly algorithm |

## Appendix A

#### Appendix A.1. EPANET Hydraulic Simulation Model

- Continuity at node j (j = 1 to N – 1)$$\sum}_{i=1}^{NP\left(j\right)}{Q}_{ij}-{q}_{j}=0$$
- Conservation of energy for loop i (i = 1 to NL)

^{2}); and $\theta $ is the rate of a constituent within pipe linking nodes i, j $\left(\frac{mass}{{m}^{3}/day}\right)$.

#### Appendix A.2. Modeling Variable-Speed Pumps

_{1}to N

_{2}, the new characteristic curve can be derived by substituting H1 and Q1 (head and flow at speed N

_{1}) with the formulas from the affinity laws, resulting in [36]:

_{1}and N

_{2}are two different pump speeds (N is the rotational speed in rpm, calculated by N = (ω/2π) × 60). The laws assume that the pump efficiency at the best efficiency point (BEP) does not change with the speed variation. The efficiency curve shifts to the left when the pump speed decreases or to the right when it increases.

## Appendix B

#### Developed Algorithm

Algorithm A1. The pseudo code of NCHBA |

Step 1: Initialize parameters (i.e., N, ${t}_{max}$, $\beta $, C)Step 2: Generate random solutionsStep 3: Evaluate the fitness of each search agent using objective function and save best solution $({x}_{prey}$ $\&{f}_{prey}$)while $t\le {t}_{max}$ doUpdate the decreasing factor $\alpha $ using (18). Generate Chaotic number. Calculate $w$ using Equation (24). for $i=1$ to
$N$doCalculate the intensity ${I}_{i}$ using Equation (17). if $rand<0.5$ thenUpdate the position ${x}_{new}$ using Equation (22). ElseUpdate the position ${x}_{new}$ using Equation (23). end ifEvaluate new position and assign to ${f}_{new}$. if ${f}_{new}\le {f}_{i}$ thenSet ${x}_{i}={x}_{new}$ and ${f}_{i}={f}_{new}$. end ifif ${f}_{new}={f}_{prey}$ thenSet ${x}_{prey}={x}_{new}$ and ${f}_{prey}={f}_{new}$. end ifend forend whileStop criteria satisfied. Return ${x}_{prey}$. |

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**Figure 10.**Situation of C-Town WDS in terms of pressure and water age at 24:00: (

**a**) Scenario 2; (

**b**) Scenario 1.

**Figure 11.**Situation of C-Town WDS in terms of pressure and water age at 10:00: (

**a**) Scenario 2; (

**b**) Scenario 1.

**Figure 12.**Situation of C-Town WDS in terms of pressure and water age at 18:00: (

**a**) Scenario 2; (

**b**) Scenario 1.

Chebyshev | Circle | Gauss-Mouse | Iterative | Logistic | Sine | Singer | Sinusoidal | Tent | HBA | |
---|---|---|---|---|---|---|---|---|---|---|

F1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

F2 | 4.02 × 10^{−298} | 9.50 × 10^{−271} | 1.48 × 10^{−322} | 4.70 × 10^{−290} | 1.20 × 10^{−291} | 1.62 × 10^{−300} | 1.40 × 10^{275} | 1.17 × 10^{−247} | 3.70 × 10^{−273} | 4.18 × 10^{−169} |

F3 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 7.30 × 10^{−250} |

F4 | 4.55 × 10^{−295} | 2.12 × 10^{−271} | 2.09 × 10^{−320} | 3.49 × 10^{−291} | 3.35 × 10^{−296} | 1.50 × 10^{−299} | 3.80 × 10^{−276} | 2.89 × 10^{−244} | 7.26 × 10^{−265} | 5.32 × 10^{−143} |

F5 | −3.10 | −3.32 | −3.17 | −3.20 | −3.32 | −3.32 | −3.20 | −3.32 | −2.91 | −3.13 |

F6 | −1.02 × 10 | −1.02 × 10 | −4.82 × 10 | −1.02 × 10 | −1.02 × 10 | −1.01 × 10 | −1.02 × 10 | −1.02 × 10 | −8.75 | −1.02 × 10 |

F7 | 8.75 × 10^{−5} | 1.04 × 10^{−5} | 6.12 × 10^{−6} | 1.48 × 10^{−4} | 3.63 × 10^{−5} | 8.62 × 10^{−5} | 7.72 × 10^{−5} | 3.07 × 10^{−5} | 1.44 × 10^{−5} | 6.47 × 10^{−5} |

F8 | 3.90 × 10^{−1} | 3.90 × 10^{−1} | 3.90 × 10^{−1} | 3.90 × 10^{−1} | 3.90 × 10^{−1} | 3.90 × 10^{−1} | 3.90 × 10^{−1} | 3.90 × 10^{−1} | 3.90 × 10^{−1} | 3.90 × 10^{−1} |

F9 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

F10 | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} |

F11 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

F12 | −1.04 × 10 | −1.04 × 10 | −1.73 | −1.04 × 10 | −1.04 × 10 | −1.04 × 10 | −1.04 × 10 | −1.04 × 10 | −1.02 × 10 | −1.04 × 10 |

F13 | −1.05 × 10 | −1.05 × 10 | −1.05 × 10 | −1.05 × 10 | −1.05 × 10 | −1.05 × 10 | −1.05 × 10 | −1.05 × 10 | −1.05 × 10 | −1.05 × 10 |

Best | 0.00 | 0.00 | 4.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

Algorithm | Parameter |
---|---|

AO | $\alpha =0.1$$,\delta =0.1$ |

HGA | $l=0.08$$,LH=100$ |

Run | $a=20$$,\mathrm{b}=12$ |

SMA | $vbandvc$ = [2 0] |

HBA | $\beta =6$$,C=2$ |

NCHBA | C (Nonlinear control parameter) $w=[20],\beta =6$$,C=2$ |

No. Run | AO | HGA | Run | SMA | HBA | NCHBA |
---|---|---|---|---|---|---|

1 | 360.627 | 310.873 | 299.894 | 300.080 | 281.745 | 261.891 |

2 | 361.164 | 303.752 | 300.279 | 300.087 | 301.290 | 249.798 |

3 | 360.191 | 301.465 | 301.207 | 299.416 | 284.834 | 266.977 |

5 | 350.543 | 306.884 | 291.234 | 297.725 | 300.661 | 259.558 |

6 | 359.868 | 298.520 | 295.697 | 299.458 | 316.968 | 260.232 |

7 | 359.938 | 302.123 | 290.906 | 284.270 | 295.006 | 259.945 |

8 | 380.171 | 308.970 | 304.920 | 299.252 | 308.427 | 263.015 |

9 | 358.250 | 299.807 | 290.167 | 299.743 | 300.812 | 260.346 |

10 | 361.700 | 310.898 | 308.929 | 300.279 | 324.785 | 264.362 |

Average | 361.384 | 304.810 | 298.137 | 297.812 | 301.614 | 260.680 |

Min | 350.543 | 298.52 | 290.167 | 284.27 | 281.745 | 249.798 |

Max | 380.171 | 310.898 | 308.929 | 300.279 | 324.785 | 266.977 |

Std | 7.801 | 4.737 | 6.605 | 5.134 | 13.878 | 4.752 |

Algorithm | Variables | Reference | Optimal Cost ($/day) |
---|---|---|---|

GA | Tank level controls (on/off) | [4] | 344.19 |

Hybrid GA | 344.19 | ||

EA | Tank level controls | [47] | 337.2 |

ABC | Tank level controls | [46] | 363.85 |

FF | 361.72 | ||

PSO | 363.44 | ||

ACO | Pump on/off | [48] | 388.04 |

Pump speed | 349.43 | ||

BDA | Pump on/off | [21] | 325.23 |

NCHBA | Pump speed | Current study | 249.79 |

IGD | SP | ||||||||
---|---|---|---|---|---|---|---|---|---|

Average | St.d | Best | Worst | Average | St.d | Best | Worst | ||

ZDT1 | MONCHBA | 4.63 × 10^{−3} | 1.95 × 10^{−4} | 4.43 × 10^{−3} | 4.93 × 10^{−3} | 5.85 × 10^{−3} | 3.51 × 10^{−3} | 5.50 × 10^{−3} | 6.39 × 10^{−3} |

NSGA-II | 6.23 × 10^{−2} | 7.39 × 10^{−2} | 2.97 × 10^{−1} | 1.97 × 10 | 6.23 × 10^{−2} | 4.51 × 10^{−2} | 9.47 × 10^{−3} | 1.09 × 10^{−1} | |

MOSMA | 1.21 × 10^{−2} | 7.52 × 10^{−2} | 1.48 × 10^{−2} | 3.33 × 10^{−2} | 1.21 × 10^{−2} | 4.75 × 10^{−2} | 8.96 × 10^{−3} | 1.62 × 10^{−2} | |

ZDT2 | MONCHBA | 4.54 × 10^{−3} | 2.58 × 10^{−4} | 4.32 × 10^{−3} | 4.96 × 10^{−3} | 5.23 × 10^{−3} | 3.95 × 10^{−4} | 4.92 × 10^{−3} | 5.98 × 10^{−3} |

NSGA-II | 1.05 | 6.33 × 10^{−1} | 8.45 × 10^{−3} | 1.61 | 2.46 × 10^{−2} | 1.49 × 10^{−2} | 4.65 × 10^{−2} | 1.23 × 10^{−2} | |

MOSMA | 2.98 × 10^{−1} | 2.90 × 10^{−1} | 2.46 × 10^{−2} | 7.72 × 10^{−1} | 1.21 × 10^{−2} | 4.75 × 10^{−2} | 8.96 × 10^{−3} | 1.62 × 10^{−2} | |

ZDT3 | MONCHBA | 6.12 × 10^{−3} | 1.24 × 10^{−3} | 4.68 × 10^{−3} | 7.11 × 10^{−3} | 6.28 × 10^{−3} | 8.63 × 10^{−4} | 5.60 × 10^{−3} | 7.31 × 10^{−3} |

NSGA-II | 7.68 × 10^{−2} | 1.18 × 10^{−1} | 8.72 × 10^{−3} | 2.86 × 10^{−1} | 1.21 × 10^{−1} | 1.75 × 10^{−1} | 2.34 × 10^{−2} | 4.32 × 10^{−1} | |

MOSMA | 6.05 × 10^{−2} | 1.72 × 10^{−2} | 8.76 × 10^{−2} | 4.13 × 10^{−2} | 3.35 × 10^{−2} | 4.25 × 10^{−2} | 1.91 × 10^{−2} | 4.65 × 10^{−2} | |

ZDT4 | MONCHBA | 4.80 × 10^{−3} | 2.70 × 10^{−4} | 4.47 × 10^{−3} | 5.22 × 10^{−3} | 5.87 × 10^{−3} | 5.49 × 10^{−3} | 5.18 × 10^{−3} | 6.36 × 10^{−3} |

NSGA-II | 3.71 | 2.33 | 7.77 × 10^{−1} | 6.45 | 6.50 × 10^{−1} | 3.73 × 10^{−1} | 1.90 × 10^{−1} | 1.08 | |

MOSMA | 4.75 × 10 | 2.43 × 10 | 1.94 × 10 | 7.45 × 10 | 2.64 | 1.12 × 10^{−2} | 1.23 × 10^{−1} | 5.15 | |

ZDT6 | MONCHBA | 3.33 × 10^{−3} | 4.37 × 10^{−4} | 2.69 × 10^{−3} | 3.82 × 10^{−3} | 5.05 × 10^{−3} | 4.34 × 10^{−4} | 4.60 × 10^{−3} | 5.73 × 10^{−3} |

NSGA-II | 5.69 × 10^{−2} | 2.02 × 10^{−2} | 3.55 × 10^{−2} | 8.47 × 10^{−2} | 5.99 × 10^{−2} | 4.75 × 10^{−2} | 1.34 × 10^{−1} | 2.28 × 10^{−2} | |

MOSMA | 5.16 × 10^{−1} | 1.14 | 3.89 × 10^{−3} | 2.55 | 8.72 × 10^{−2} | 3.33 × 10^{−1} | 1.86 × 10^{−2} | 2.22 × 10^{−1} |

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## Share and Cite

**MDPI and ACS Style**

Jafari-Asl, J.; Hashemi Monfared, S.A.; Abolfathi, S.
Reducing Water Conveyance Footprint through an Advanced Optimization Framework. *Water* **2024**, *16*, 874.
https://doi.org/10.3390/w16060874

**AMA Style**

Jafari-Asl J, Hashemi Monfared SA, Abolfathi S.
Reducing Water Conveyance Footprint through an Advanced Optimization Framework. *Water*. 2024; 16(6):874.
https://doi.org/10.3390/w16060874

**Chicago/Turabian Style**

Jafari-Asl, Jafar, Seyed Arman Hashemi Monfared, and Soroush Abolfathi.
2024. "Reducing Water Conveyance Footprint through an Advanced Optimization Framework" *Water* 16, no. 6: 874.
https://doi.org/10.3390/w16060874