# Hybrid Cyber Petri net Modelling, Simulation and Analysis of Master-Slave Charging for Wireless Rechargeable Sensor Networks

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## Abstract

**:**

## 1. Introduction

- (1)
- Considering the disadvantages of single fixed or mobile charging, we propose a new charging paradigm. The number and location of fixed chargers are constrained by the growth rate of actual electric quantity, and the optimization goal is to maximize the total actual electric quantity of sensor nodes with fixed charging. Then, a mobile charger takes turns charging the remaining nodes with the optimization goal of minimizing the travelling distance.
- (2)
- Wireless charging for the WRSN is a hybrid system including discrete event decision and continuous energy transfer, in which there are energy flow, data flow and control flow relationships. In addition, a formal specification with graph-mathematical characterization ability is beneficial to the analysis and application of such system. As a result, the Hybrid Cyber Petri net System is proposed based on the classical Petri net to model the Master-Slave Charging.
- (3)
- Because Master-Slave Charging is NP-complete, it is difficult to solve using the traditional optimization method. A greedy-genetic algorithm is proposed by adding greedy crossover and mutation operators in a genetic algorithm, which enhances the local search capability and can be used for reference to solve the chargers’ scheduling problem.

## 2. Related Work

#### 2.1. Petri Net

#### 2.2. Wireless Charging

## 3. System Model

#### 3.1. Syetem Model of Wireless Charging

**Definition**

**1.**

**Actual Electric Quantity (AEQ).**To replenish full energy of${s}_{p}$, the least energy received is the sum of consumption energy ${E}_{c}\left({s}_{p}\right)$before deadline T and energy to be full${E}_{need}\left({s}_{p}\right)$. So AEQ is defined as the smaller value between the sum of${E}_{c}\left({s}_{p}\right)$, ${E}_{need}\left({s}_{p}\right)$and received energy${E}_{r}\left({s}_{p}\right)$. It reflects the actual valuable energy, and the CQ of${s}_{j}$is

**Definition**

**2.**

**Growth Rate of AEQ.**The definition refers to the growth rate obtained by adding a charger, that is

#### 3.2. Petri Net Model

**Definition 3.**

**Directed net**(referred to as net), if it satisfies the following conditions:

- (1)
- $S\cap T=\Phi \wedge S\cup T\ne \Phi $;
- (2)
- $F\subseteq S\times T\cup T\times S$;
- (3)
- $dom\left(F\right)\cup cod\left(F\right)=S\cup T$.

**Definition 4.**

**Petri net System**, if it satisfied the following conditions:

- (1)
- $N=\left(S,T;F\right)$is a directed net, which is named the basic net of$\sum $;
- (2)
- $K:S\to \left\{1,2,3,\cdots \right\}\cup \{\infty \}$is the capacity function of$N$;
- (3)
- $W:S\to \left\{1,2,3,\cdots \right\}$is the weight function of$N$;
- (4)
- $\forall s\in S,M\left(s\right)\le K\left(s\right)$,$M:S\to \left\{1,2,3,\cdots \right\}M:S\to \left\{1,2,3,\cdots \right\}$is the marking, and${M}_{0}$is the initial marking.

**Definition 5.**

**Cyber net System**, if it satisfied the following conditions:

- (1)
- $N=\left(S,T;F\right)$is a directed net, which is named the basic net of$\sum $;
- (2)
- $W:S\times T\cup T\times S\to \left\{0,1,2,\cdots \right\}\cup S$is the weight function of$\sum $,$W\left(x,y\right)\ne 0$, if and only if$\left(x,y\right)\in F$;
- (3)
- ${M}_{0}:S\to \left\{0,1,2,\cdots \right\}$is the initial marking.

**Definition 6.**

**incidence matrix**of a Petri net System, where${a}_{ij}=W\left({t}_{j},{s}_{i}\right)-W\left({s}_{i},{t}_{j}\right)$.

**Definition 7.**

**Generalized Cyber P/N System**if it satisfied the following conditions:

- (1)
- $N=\left(S,T;F\right)$is a directed net, which is named the basic net of$\sum $;
- (2)
- $K=\left\{{k}_{L},{k}_{H}\right\}$is the capacity function of$N$, and${k}_{L}:S\to \Re ;{k}_{H}:S\to \Re $,where$\Re $is a set of real numbers;${k}_{L}$and${k}_{H}$is lower and upper of$K$;
- (3)
- $W:F\to \Re \cup Exp\left(S\right)$is the weight function, where$Exp\left(S\right)$is the set of functional expressions for$S$;
- (4)
- $M:F\to \Re $is the marking, and${M}_{0}$is the initial marking.

- (1)
- The weight is changed to the function of $S$;
- (2)
- The capacity of place is limited;
- (3)
- The marking and weight can be real numbers.

**Definition 8.**

**reading arc**(

**writing arc**) in the Generalized Cyber P/N System. As shown in Figure 2, if arc$\left(s,t\right)$is a reading arc, incidence matrix$A\left(s,t\right)=0$; and if arc$\left(t,s\right)$is a writing arc,$A\left(s,t\right)=W\left(t,s\right)-s$.

**Definition 9.**

**Generalized Continuous Cyber Net System**if it satisfied the following conditions:

- (1)
- $N=\left(S,T;F\right)$is a directed net, which is named the basic net of$\sum $;
- (2)
- $K=\left\{{k}_{L},{k}_{H}\right\}$is the capacity function of$N$;
- (3)
- $W:F\to \Re \cup Exp\left(S,t\right)$is the weight function, where$Exp\left(S,t\right)$is the set of functional expressions for$S$and$t$;
- (4)
- $M:F\to \Re $is the marking, and${M}_{0}$is the initial marking.

**Definition 10.**

**Hybrid Cyber Petri net System**(HCPNS, if it satisfied the following conditions:

- (1)
- $N=\left(S,T;F\right)$is a directed net, which is named the basic net of$\sum $;
- (2)
- $S={S}_{D}\cup {S}_{C}$,${S}_{D}$and${S}_{C}$is discrete place and continuous place, respectively;
- (3)
- $T={T}_{D}\cup {T}_{C}$,${T}_{D}$and${T}_{C}$is discrete transition and continuous transition, respectively;
- (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 \Re ;{k}_{H}:S\to \Re $, where$\Re $is a set of real numbers;
- (6)
- $W:F\to \Re \cup Exp\left(S,\tau \right)$is the weight function, where$Exp\left(S,\tau \right)$is the set of functional expressions for$\tau $;
- (7)
- $M:F\to \Re $is the marking, and${M}_{0}$is the initial marking.

**Definition 11.**

**The condition that transition t will be fired:**

**Definition**

**12.**

**Result of transition firing.**After transition t is fired, the original marking$M$will be changed to${M}^{\prime}$, then

## 4. HCPNS Modelling Approach of a Master-Slave Charging System

#### 4.1. Fixed Charging Subnet

#### 4.2. Mobile Charging Subnet

## 5. Simulation and Numeric Results

#### 5.1. Simulation Setup and Environment Parameters

- (1)
- Total AEQ: The actual electric quantity received by fixed charging. As the objective of master mode, this metric implies the efficiency of our algorithm.
- (2)
- Proportion of fully-charged sensors: The ratio of fully-charged sensors to all the sensors.
- (3)
- Exceeded energy: Additional energy that the chargers can provide when nodes are fully charged. Since the mobile charger will leave once the node is fully charged, there is no exceeded energy in slave mode.
- (4)
- Traveling distance of the mobile charger: The total distance of the mobile charger from and back to the base station.
- (5)
- Number of dead nodes: The number of nodes running out of energy. This metric implied the effect of AEQ growth threshold on network performance.

#### 5.2. Performance

- (1)
- Total AEQ: The total AEQ on both data instances are evaluated by setting the threshold of AEQ growth rate between 2% and 12%. Each different setup is run 10 times, and the average of them will be taken.

- (2)
- Effectiveness of applying AEQ: The effectiveness of AEQ is evaluated in terms of the proportion of fully charged and exceeded energy in large instances.

- (3)
- Traveling distance and dead nodes: The traveling distance of the mobile charger and number of dead nodes are evaluated in large data instances. As shown in Figure 12 and Figure 13, when the threshold of AEQ growth rate is 10%, the number of dead nodes is 8 and the traveling distance of the mobile charger is 4300 m. When the threshold of AEQ growth rate is 8%, the traveling distance of the mobile charger is 3900 m and the number of dead nodes is 5. This comparison demonstrates that both the traveling distance and number of dead nodes are turning down with the decrement of the threshold of AEQ growth rate. This is because the lower threshold of the AEQ growth rate allows more fixed chargers to be placed, reducing the number of nodes with mobile charging, thereby easing the burden on the mobile charger and decreasing the waiting time of nodes.

**Review 1:**Compared with the traditional genetic algorithm, the greedy crossover and greedy mutation operator are introduced in this paper, which can improve the defects of weak local search ability and precocity easily. Moreover, the proposed method not only has strong optimization ability, but also the convergence speed is obviously faster than the traditional genetic algorithm, which has a good reference for solving the chargers’ scheduling problem.

**Review 2:**The growth rate of actual electric quantity is introduced to regulate the number of fixed chargers, and the locations of them are determined on the basis of the nodes’ state including the remaining energy, power consumption and position. This scheme not only expands the coverage of fixed chargers, but also reduces the burden of slave charging and ultimately maintains the energy balance of the network.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Continuous transition. (

**a**) Discrete transition (

**b**) timing sequence (

**c**) Continuous transition

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

n | number of sensor nodes |

m | number of chargers |

${p}_{f}$ | source power of the charger |

${r}_{p}$ | power consumption rate of sensor node ${s}_{p}$ |

${p}_{r}\left({s}_{p},{c}_{i}\right)$ | received power of ${s}_{p}$ from ${c}_{i}$ |

${E}_{c}\left({s}_{p}\right)$ | consumed energy of ${s}_{p}$ |

E_{r}(${s}_{p}$) | energy received by ${s}_{p}$ |

${E}_{need}\left({s}_{p}\right)$ | energy needed to be full of ${s}_{p}$ |

$Q\left({s}_{p}\right)$ | actual electric quantity of ${s}_{p}$ |

$\delta $ | growth rate of actual electric quantity |

${v}_{m}$ | energy consumption per time unit of travelling by ${c}_{m}$ |

$T$ | charging period of fixed chargers |

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

${c}_{i}$ | Represents the charger ${c}_{i}$. |

$s{c}_{i}^{p}$ | Specify whether the sensor node ${s}_{p}$ is within the radiation of ${c}_{i}$. It is the case when $M\left(s{c}_{i}^{p}\right)=1$. |

${s}_{p}$ | Represents the node ${s}_{p}$. Its marking $M\left({s}_{p}\right)$ corresponds to the current energy. |

${t}_{p}^{i}$ | Used to model the charging for ${s}_{p}$ by the charger ${c}_{i}$. |

$t{c}_{{s}_{p}}$ | Used to model the consumption of the sensor node ${s}_{p}$. |

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

$A{c}_{m}^{n}$ | Specify whether the mobile charger ${c}_{m}$ reach the node ${s}_{n}$. It is the case when $M\left(A{c}_{m}^{n}\right)=1$. |

${c}_{m}$ | Represents the mobile charger ${c}_{m}$. Its marking $M\left({c}_{m}\right)$ corresponds to the current energy. |

$D{c}_{m}^{n}$ | Specify whether the mobile charger ${c}_{m}$ can leave the node ${s}_{n}$. It is the case when $M\left(D{c}_{m}^{n}\right)=1$. |

${t}_{n}^{m}$ | Used to model the charging for ${s}_{n}$ by the mobile charger ${c}_{m}$. |

$t{r}_{n-1}^{n}$ | Used to model the mobile charger’s travelling from the nth to the n−1th node. |

$t{d}_{n-1}$ | Used to model the leaving from the n−1th node. |

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## Share and Cite

**MDPI and ACS Style**

Qin, H.; Zhao, B.; Xu, L.; Bai, X.
Hybrid Cyber Petri net Modelling, Simulation and Analysis of Master-Slave Charging for Wireless Rechargeable Sensor Networks. *Sensors* **2021**, *21*, 551.
https://doi.org/10.3390/s21020551

**AMA Style**

Qin H, Zhao B, Xu L, Bai X.
Hybrid Cyber Petri net Modelling, Simulation and Analysis of Master-Slave Charging for Wireless Rechargeable Sensor Networks. *Sensors*. 2021; 21(2):551.
https://doi.org/10.3390/s21020551

**Chicago/Turabian Style**

Qin, Huaiyu, Buhui Zhao, Leijun Xu, and Xue Bai.
2021. "Hybrid Cyber Petri net Modelling, Simulation and Analysis of Master-Slave Charging for Wireless Rechargeable Sensor Networks" *Sensors* 21, no. 2: 551.
https://doi.org/10.3390/s21020551