## 3 System Models
### Sensing Coverage Model
When \(\alpha=1.78\), IQPSO can achieve better global convergence; when \(\alpha<0.5\), IQPSO is difficult to converge. So \(\alpha\in[0.5,1.8]\) is choosen, the adaptive formula of \(\alpha\) with the number of iteration steps is

\[\alpha_{k+1}= \begin{cases}1.8-\omega(\alpha_{k}-0.5)\cos(k\pi/K),&if\quad \alpha_{k}>1.2\\ 0.5+\omega(\alpha_{k}-0.5)\cos(k\pi/K),&if\quad\alpha_{k}\leq 1.2\end{cases}  \tag{12}\]

where \(K\) is the maximum iterative step size, \(\omega\) is the adaptive adjustment coefficient, which is formulated as

\[\omega= \begin{cases}1,&if\quad\alpha_{k+1}>1.2\\ 0.88,&if\quad\alpha_{k+1}\leq 1.2\end{cases}  \tag{13}\]

In addition, different from the traditional way of closest integer and binary conversion, we use the swap operator in [31] to optimize the mobile charging sequence in this paper. The swap operator (\(SO\)) is defined as \(SO(m_{1},m_{2})\), \(m_{1},m_{2}\in[1,N]\), which can swap the node \(s_{m_{1}}\) and node \(s_{m_{2}}\) in \(\Phi\). So, \(\Phi^{\prime}=\Phi\oplus SO(m_{1},m_{2})\), \(\Phi^{\prime}\) is a new mobile charging sequence, \(\oplus\) is the function of changing \(\Phi\) to \(\Phi^{\prime}\) via \(SO\). Multiple \(SO\) compose a subset of swap operators (\(SSO\)), where \(SSO=(SO_{1},SO_{2},\ldots,SO_{w})\), \(w\) is the number of \(SOs\). If \(SSO1\) and \(SSO2\) act successively on the \(\Phi\) to get another \(\Phi^{\prime\prime}\), we can combine the SSO1 and SSO2 to get a new one, which is \(SSO^{\prime\prime}\)=\(SSO1\otimes SSO2\), where \(\otimes\) means merging two SSO. At this time, \(\Phi^{\prime\prime}=\Phi\oplus SSO^{\prime\prime}\). The subset with the least swap operators is named the subset of the basic swap operator (\(SBSO\)). Based on the above, the changes of \(SBSO\) can be extended. We assume \(SBSO=\Phi\odot\Phi^{\prime}\), the \(SBSO\) is expected to find, which can change charging sequence from \(\Phi\) to \(\Phi^{\prime}\). \(\odot\) means the function of finding \(SBSO\) from \(\Phi\) to \(\Phi^{\prime}\). Applying the \(SO\) algorithm to IQPSO, (10) is updated as

\[p_{i,j}(k)=\varphi_{j}(k)p_{i,j}(k)+(1-\varphi_{j}(k))\oplus G_{gj}(k)\pm \alpha\big{|}\frac{1}{\omega}\sum_{i=1}^{o}r_{j}(k)\odot X_{i,j}(k)|\ln[1/ \beta]. \tag{14}\]

For the MSSQ problem, the execution process of the IQPSO algorithm is as follows: (1) initialize parameters such as particle swarm \(o\), the maximum number of iterations \(K\) and minimum error \(e_{min}\) of algorithm termination; (2) calculate the \(TSCNS\) of each particle, and set it as the personal best value to find the current best position \(p_{i}(k)\) of each particle; (3) according to \(p_{i}(k)\) get the global best position \(G_{b}\); (4) update the current \(p_{i}(k)\) and \(G_{b}\) by comparing with the historical information; (5) update particles' positions; (6) repeat 2-5 until \(e_{min}\) is met or \(k=K\); (7) output global best \(TSCNS\) and \(G_{b}\). The pseudocode is shown in Algorithm 2. ```
0:\(\Phi(0)\), \(D\), \(E_{i}\), \(V_{cs}\), \(e_{m}\), \(v_{m}\), \(v_{c}\), \(o\), \(K\), \(e_{min}\)
0: optimal \(\Phi^{*}\) and \(TSCNS(\Phi^{*})\)
1:for\(k=1:K\)do
2:for\(i=1:o\)do
3: Initialize positions randomly \(X_{i}(k)\)
4: Update the \(p_{i}(k)\) according to the \(TSCNS(p_{i}(k))\) and \(TSCNS(X_{i}(k))\) via (8)
5:endfor
6: Update the global best position \(G_{b}(k)\) via (9)
7: To get the evolution equation of particles \(p_{i}(k)\) from (14)
8: From (12) to (13), adaptively choosing \(\alpha(k)\) according to the iteration steps
9: Calculate the new position of particles \(\tilde{X}(k+1)\) according to (11)
10:if\(e_{min}<=e_{min}\)then
11: Break
12:endif
13:endfor
14:\(\Phi^{*}=G_{b}(K)\)
```

**Algorithm 2** IQPSO algorithm for the MSSQ. ### IqPSO Convergence Analysis

The principle of the IQPSO algorithm is the same as that of QPSO; therefore, the proof of the convergence of QPSO is given in [31], and we will not repeat it here. In this part, we analyze the computational complexity of IQPSO.