# Petri-Net Based Multi-Objective Optimization in Multi-UAV Aided Large-Scale Wireless Power and Information Transfer Networks

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## Abstract

**:**

## 1. Introduction

- The HCCPNS (hybrid colored cyber Petri net system) is proposed for the first time to model the multi-UAV aided wireless power and information transfer system. The place represents the status of the UAV or SD, where the continuous part is the energy, and the discrete part is information. The variation of a marking or a token corresponds to the continuous transition and the discrete transition, respectively. To the best of our knowledge, this is the first time that Petri net is employed to express the energy flow, control flow, and information flow simultaneously.
- The multi-UAV aided wireless power and information transfer is constructed as a multi-objective optimization problem. On the one hand, we hoped that the UAV can replenish more energy for SDs, thus improving energy efficiency. Since the wireless charging power is constant, this inevitably results in a longer hovering time. On the other hand, when an SD sends out the request for information transmission, it expects a UAV to arrive at the corresponding position to receive data as soon as possible, that is, the time delay of information collection should be minimal. It is not difficult to see that the two targets are in conflict and a trade-off needs to be found.
- Under the premise of one-to-one service, the strategy of trajectory assignment and hover of multiple UAVs is designed. The MAC-NSGA II is proposed in order to optimize the energy utilization and average delay of information simultaneously based on the optimal trajectory of multiple UAVs. Numerical simulation results demonstrate that the proposed algorithm has excellent performance, especially for large scale networks.

## 2. Related Work

## 3. System Model and Problem Formulation

#### 3.1. UAV-Aided Wireless Power and Information Transfer Model

#### 3.2. The Specification of Petri Net

**Definition**

**1.**

- (1)
- $N=\left(S,T;F\right)$is a directed net, which is named the basic net of$\sum$;
- (2)
- C is the vector of colors with red and green,$C=\left\{{c}_{1},{c}_{2}\right\}$;
- (3)
- The place, transition, and weight function of directed arc all contain two colors; the red one represents the continuous part and the green one represents the discrete part, i.e.,$M\left(S\right)=\left[{M}_{C}\left(S\right),{M}_{D}\left(S\right)\right]$,$T=\left[{T}_{C},{T}_{D}\right]$,$W=\left[{W}_{C},{W}_{D}\right]$;
- (4)
- $F\subseteq S\times T\cup T\times S$,$Rd\subseteq S\times T$,$Wr\subseteq T\times S$and$F\cap Rd\cap Wr=\Phi $;$F$,$Rd$,$Wr$,$Inhibit$and$Permit$is the flow, read, write, inhibitor, and permission relationship;
- (5)
- $K=\left\{{k}_{L},{k}_{H}\right\}$is the lower and upper capacity function of$N$, and${k}_{L}:S\to \mathbb{R};{k}_{H}:S\to \mathbb{R}$,where$\mathbb{R}$is a set of real numbers;
- (6)
- $W:F\to \mathbb{R}\cup Exp\left(S,\tau \right)$is the weight function, where$Exp\left(S,\tau \right)$is the set of functional expressions for$\tau $;
- (7)
- $M:F\to \mathbb{R}$is the marking, and${M}_{0}$is the initial marking.

**Definition**

**2.**

**Definition**

**3.**

## 4. Multiple Ant Colony-Nondominated Sorting Genetic Algorithm II

#### 4.1. Ant State Transition Strategy

Algorithm 1 Pseudo-code of the multiple ant colony-nondominated sorting genetic algorithm II |

Input: Number of UAVs and SDs, Location of the depot and SDs, Residual energy and power consumption of SDs |

Output: Optimal trajectory of UAVs, the set of hovering |

1: Initialize all parameters and pheromone trails; 2: $iter\leftarrow 0$; |

3: while $iter\le ite{r}_{max}$ do |

4: for each UAV5: for each SD6: generate initial solution with pheromone concentration and distances between SDs; 7: end for8: construct solution following (20); 9: update the pheromone following (21) and according to the residual pheromone and the length of path; |

10: end for |

11: end while |

12: Obtain the optimal flying trajectory; 13: Taking the hovering time above every SD as independent variable, initialize the population ${P}_{t}$; 14: Combine parent and offspring population ${R}_{t}={P}_{t}\cup {Q}_{t}$; 15: all nondominated of ${R}_{t}$: $F=fast-non-dominated-sort\left({R}_{t}\right)$; 16: ${P}_{t+1}=\varnothing $ and $i=1$; 17: while $\left|{P}_{t+1}\right|+\left|{F}_{i}\right|\le N$ do18: calculate crowding-distance in ${F}_{i}$; 19: ${P}_{t+1}={P}_{t+1}\cup {F}_{i}$; 20: $i=i+1$; 21: $Sort\left({F}_{i},{\prec}_{n}\right)$; 22: Choose the first $\left(N-{P}_{t+1}\right)$ element of ${F}_{i}$,${P}_{t+1}={P}_{t+1}\cup {F}_{i}\left[1:\left(N-{P}_{t+1}\right)\right]$; 23: Use selection, crossover and mutation to create a new population ${Q}_{t+1}=make-new-pop\left({P}_{t+1}\right)$; 24: $t=t+1$; 25: end while26: The pareto set of hovering time. |

- The greater the pheromone concentration from one location to the next, the more likely the ant is to choose that path.
- The shorter the distance between the current position and the next position the ant traverses, the greater the probability that the ant chooses the path.

#### 4.2. Pheromone Update Strategies

- (1)
- Pheromones $\lambda $ that are positively correlated with the superiority of feasible solutions are uniformly added to all subpath.
- (2)
- If the length of the subpath is less than the mean of all the subpaths, $\lambda $ is reduced to $\left(0.7~0.95\right)\lambda $. On the contrary, $\lambda $ is reduced to $\left(0.1~0.5\right)\lambda $.
- (3)
- The pheromone of the shortest subpath and the longest subpath is reduced to$0.1\lambda $, so that they can be recombined to form a better feasible solution.

## 5. Simulation and Numeric Results

#### 5.1. Simulation Setup and Environment Parameters

- (1)
- UAV’s trajectory. This indicator reflects the influence of the number of UAVs on the trajectory length.
- (2)
- UAV’s energy efficiency. It represents the proportion of the energy received by SDs in the total energy consumption of UAVs, which is affected by different network sizes and the number of UAVs.
- (3)
- Average time delay of information collection. This index reflects the timeliness of gathering information perceived by SDs.

#### 5.2. Performance

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**A simple HCCPNS model. (

**a**) Folded model; (

**b**) unfolded model: continuous part; (

**c**) unfolded model: discrete part.

Notation | Description |
---|---|

$M$ | number of UAVs |

$N$ | number of SDs |

${P}_{mov}$ | moving power of the UAV |

${P}_{hov}$ | hovering power of the UAV |

${P}_{tra}$ | transmitting power of the UAV |

${E}_{j,rec}^{i}$ | received energy of the $j$th SD from the $i$th UAV |

${E}_{mov}^{i}$ | moving energy of the $i$th UAV |

${E}_{hov}^{i}$ | hovring energy of the $i$th UAV |

${E}_{tra}^{i}$ | transmitting energy of the $i$th UAV |

Name | Function |
---|---|

${U}_{j}^{i}$ | Place when the $i$-th UAV hovers directly over the $j$-th SD |

${I}_{j}$ | Place of the $j$-th SD |

$T{\left(i\right)}_{j+1}^{j}$ | Transition that the $i$-th UAV flies from the $j$-th to ($j+1$)th SD |

$T{h}_{j}^{i}$ | Transition that the $i$-th UAV hovers directly over the $j$-th SD |

$T{d}_{j}^{i}$ | Transition that the $i$-th UAV collects information from the $j$-th SD |

${T}_{se{n}_{j}}$ | Transition that the $j$-th SD consumes energy |

$Teme\left(i\right)$ | Transition that the $i$-th UAV returns to the depot emergently |

Arc | Weight |
---|---|

${W}_{C}\left({U}_{j}^{i},T{\left(i\right)}_{C}{}_{j+1}^{j}\right)$ | ${M}_{C}\left({U}_{j}^{i}\_{t}_{j}^{+}\right)$ |

${W}_{D}\left({U}_{j}^{i},T{\left(i\right)}_{D}{}_{j+1}^{j}\right)$ | ${M}_{D}\left({U}_{j}^{i}\_{t}_{j}^{+}\right)$ |

$writ{e}_{D}\left(T{\left(i\right)}_{D}{}_{j+1}^{j},{U}_{j+1}^{i}\right)$ | ${M}_{D}\left({U}_{j}^{i}\_{t}_{j}^{+}\right)$ |

$writ{e}_{C}\left(T{\left(i\right)}_{C}{}_{j+1}^{j},{U}_{j+1}^{i}\right)$ | ${M}_{C}\left({U}_{j}^{i}\_{t}_{j}^{+}\right)$-${P}_{mov}\times {t}_{j,j+1}$ |

$W\left({U}_{j}^{i},T{h}_{j}^{i}\right)$ | ${P}_{hov}$ |

$W\left({U}_{j}^{i},T{c}_{j}^{i}\right)$ | ${P}_{tra}$ |

$W\left(T{c}_{j}^{i},{I}_{j}\right)$ | ${P}_{rec}$ |

$W\left({I}_{j},{T}_{se{n}_{j}}\right)$ | ${P}_{sen}^{j}$ |

$W\left({I}_{j},T{d}_{j}^{i}\right)$ | 1 |

$write\left(T{d}_{j}^{i},{U}_{j}^{i}\right)$ | 1 |

$inhibit\left({I}_{j},T{c}_{j}^{i}\right)$,$permit\left({I}_{j},T{\left(i\right)}_{j+1}^{j}\right)$, | ${M}_{C}\left({I}_{j}\right)={k}_{H}\left({I}_{j}\right)$ |

$inhibit\left({U}_{j}^{i},T{c}_{j}^{i}\right)$,$inhibit\left({U}_{j}^{i},T{h}_{j}^{i}\right)$, $inhibit\left({U}_{j}^{i},T{\left(i\right)}_{j+1}^{j}\right)$ | ${M}_{C}\left({U}_{j}^{i}\right)={k}_{L}\left({U}_{j}^{i}\right)$ |

$permit\left({U}_{j}^{i},Teme\left(i\right)\right)$ | ${M}_{C}\left({U}_{j}^{i}\right)={k}_{L}\left({U}_{j}^{i}\right)$ |

Parameter | Value | Parameter | Value |
---|---|---|---|

$N$ | $\left\{100,200,\cdots ,1000\right\}$ | ${h}_{0}$ | $\left(200,200\right)$ |

$\mathrm{A}({\mathrm{m}}^{2}$) | 0.503 | ${v}_{0}$ | 4.03 |

${U}_{tip}\left(\mathrm{m}/\mathrm{s}\right)$ | 120 | ${d}_{0}$ | 0.6 |

$s$ | 0.05 | $\rho \left(\mathrm{kg}/{\mathrm{m}}^{3}\right)$ | 1.225 |

$H\left(\mathrm{m}\right)$ | 20 | ${P}_{hov}\left(\mathrm{W}\right)$ | 13.7 |

${V}_{e}\left(\mathrm{m}/\mathrm{s}\right)$ | 10 | ${P}_{mov}\left(\mathrm{W}\right)$ | 30 |

${\eta}^{\prime}$ | 90% | ${P}_{tra}\left(\mathrm{W}\right)$ | 3 |

$M\left({\mathrm{m}}^{2}\right)$ | $400\times 400$ | ${\rho}_{0}\left(\mathrm{dB}\right)$ | −60 |

Number of SDs | Number of UAVs | MAC-NSGA II | NSGAIII | MOEA/D | |||||
---|---|---|---|---|---|---|---|---|---|

Optimal | Worst | Average | Median | Optimal | Median | Optimal | Median | ||

200 | 6 | 2.13% | 1.08% | 1.64% | 1.69% | 2.02% | 1.59% | 1.93% | 1.54% |

9 | 2.15% | 1.13% | 1.69% | 1.72% | 2.06% | 1.62% | 1.97% | 1.57% | |

12 | 2.16% | 1.17% | 1.75% | 1.82% | 2.09% | 1.72% | 1.97% | 1.69% | |

15 | 2.21% | 1.32% | 1.84% | 1.88% | 2.12% | 1.79% | 2.00% | 1.70% | |

400 | 6 | 1.90% | 0.66% | 1.32% | 1.33% | 1.80% | 1.27% | 1.73% | 1.25% |

9 | 1.92% | 0.69% | 1.33% | 1.35% | 1.83% | 1.30% | 1.75% | 1.27% | |

12 | 2.00% | 0.80% | 1.42% | 1.39% | 1.92% | 1.34% | 1.82% | 1.30% | |

15 | 2.09% | 0.98% | 1.58% | 1.61% | 1.97% | 1.51% | 1.84% | 1.47% | |

600 | 6 | 1.58% | 0.39% | 0.93% | 0.90% | 1.49% | 0.84% | 1.43% | 0.79% |

9 | 1.72% | 0.51% | 1.09% | 1.03% | 1.61% | 0.97% | 1.55% | 0.92% | |

12 | 1.74% | 0.51% | 1.17% | 1.19% | 1.67% | 1.12% | 1.57% | 1.07% | |

15 | 1.88% | 0.64% | 1.28% | 1.30% | 1.77% | 1.22% | 1.71% | 1.14% | |

800 | 6 | 1.48% | 0.33% | 0.83% | 0.84% | 1.39% | 0.80% | 1.35% | 0.73% |

9 | 1.51% | 0.35% | 0.86% | 0.87% | 1.44% | 0.77% | 1.37% | 0.75% | |

12 | 1.70% | 0.47% | 1.03% | 1.00% | 1.62% | 0.94% | 1.57% | 0.89% | |

15 | 1.77% | 0.53% | 1.12% | 1.13% | 1.67% | 1.07% | 1.63% | 1.04% | |

1000 | 6 | 1.20% | 0.22% | 0.62% | 0.58% | 1.13% | 0.54% | 1.09% | 0.51% |

9 | 1.49% | 0.34% | 0.86% | 0.83% | 1.42% | 0.79% | 1.35% | 0.74% | |

12 | 1.54% | 0.36% | 0.90% | 0.87% | 1.47% | 0.82% | 1.39% | 0.78% | |

15 | 1.60% | 0.40% | 0.95% | 0.95% | 1.51% | 0.89% | 1.46% | 0.84% |

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**MDPI and ACS Style**

Qin, H.; Zhao, B.; Xu, L.; Bai, X.
Petri-Net Based Multi-Objective Optimization in Multi-UAV Aided Large-Scale Wireless Power and Information Transfer Networks. *Remote Sens.* **2021**, *13*, 2611.
https://doi.org/10.3390/rs13132611

**AMA Style**

Qin H, Zhao B, Xu L, Bai X.
Petri-Net Based Multi-Objective Optimization in Multi-UAV Aided Large-Scale Wireless Power and Information Transfer Networks. *Remote Sensing*. 2021; 13(13):2611.
https://doi.org/10.3390/rs13132611

**Chicago/Turabian Style**

Qin, Huaiyu, Buhui Zhao, Leijun Xu, and Xue Bai.
2021. "Petri-Net Based Multi-Objective Optimization in Multi-UAV Aided Large-Scale Wireless Power and Information Transfer Networks" *Remote Sensing* 13, no. 13: 2611.
https://doi.org/10.3390/rs13132611