Theoretical Analysis of Nonlinear Energy Harvesting From Wireless Mobile Nodes

This letter provides a theoretical analysis of the radio signal energy harvested from wireless mobile nodes. A nonlinear energy harvesting model based on the research results is proposed. This model is practical but tractable through stochastic geometry. A Gauss-Markov model is used as a mobility model, one that is versatile and can cover various mobile scenarios. The closed form formulas of the first and second moments of the total energy harvested are derived for a combination of these models.

SECTION I. Introduction

Energy harvesting (EH) is promising due to its potential for attaining fully wireless mobile communications [1]. Therefore, a huge number of papers have been published [2]. Such existing studies are based on a linear EH model; that is, the conversion efficiency from the radio frequency (RF) power to the direct current (DC) power is constant and independent of the RF power. However, some studies have shown that the conversion efficiency is not constant and that the harvested DC power becomes saturated as the input RF power increases [3], [4]. Two nonlinear EH models have been proposed on the basis of such studies. One is a sigmoid (logistic) function model [5], [6], [7], [8] and the other is a two-line-segment model [9]. These models revealed that the results on the basis of the conventional linear model can provide serious disagreement with real systems.

In addition to the linear model, the weakness of most existing studies is their focus on fixed nodes as energy sources. Simultaneous wireless information and power transfer (SWIPT) technology has been widely studied, and its progress shows that SWIPT has the potential to make wireless mobile nodes into RF energy sources. Wireless mobile nodes are promising as energy sources due to their rapid growth in number. Unfortunately, however, the performance analysis of EH from wireless mobile nodes is difficult because simulating EH from mobile nodes is time consuming. Thus, theoretical results such as basic statistics on the amount of EH are important. Using such statistics makes it more feasible to evaluate the conditions for certain applications of EH and the performance of EH control algorithms.

Several papers cover EH from/to mobile nodes, but all of them use a linear EH model. One is [10], which covers EH from mobile nodes moving on a straight line with a linear EH model. Another one is [11], where the optimal transmission through the Markov decision process is conducted for mobile nodes harvesting energy. The mobile nodes move from one discrete location to another. The third one is [12], where a single node moves on a straight line between two energy sources to find a better EH location. The reference [8] also investigates a single mobile node. This node is a relay node, and it moves along a predetermined route to offer wireless power transfer to two user nodes. The [13] also considers an optimal EH location from a single dedicated base station and shows that mobility can improve EH performance. The last one [14] assumes the following mobility scenario and analyzes the EH performance through stochastic geometry. When a device has depleted its power after a previous transmission, it checks whether or not its location is within an EH region. If not, the device keeps going straight until it reaches an EH region.

Stochastic geometric analysis for a nonlinear EH model has been conducted, although the wireless nodes are fixed in [15], [16], where the sigmoid nonlinear model was assumed. In [15], the moment generation function for the received RF has been derived. By using it and the gamma distribution approximation, the amount of energy harvested has been evaluated. In [16], a wireless power transfer using millimeter waves has been theoretically analyzed. The amount of energy harvested when the number of antennas becomes infinitely large has been evaluated.

This letter presents an analysis on the energy harvested from mobile nodes where a newly-proposed nonlinear EH model is used. The mobility model used here is a Gauss-Markov model, which is versatile and can cover various mobile scenarios. The nonlinear EH model is simple but well-fitted to measurement data. Most importantly, this model combined with the Gauss-Markov mobility model is tractable in stochastic geometry. Therefore, the closed form formulas of the first and second moment of the total amount of energy harvested can be obtained.

SECTION II. Model

A wireless network has a set of nodes ${\mathcal N}\equiv \{X_{i}\}_{i>0}$ . Some nodes appear at ${t}$ when they are switched on, and their birth locations are modeled by a homogeneous Poisson point process (PPP) $\tilde \Phi (t)$ with intensity $\lambda$ . These nodes are assumed to move. $\mathbf {v}_{i}(t)$ denotes the movement of $X_{i}$ at ${t}$ and is assumed to follow the two-dimension version of the Gauss-Markov mobility model, which has been used in [17], [18], [19].

$$\begin{equation*} \mathbf {v}_{i}(t)= \sqrt {1-\zeta ^{2}}\omega (t-1)+\zeta \mathbf {v} _{i}(t-1)+(1-\zeta) \bar{\mathbf {v}}\tag{1}\end{equation*}$$ Here, $\omega (t-1)$ is an independent two-dimension Gaussian random variable with mean 0 and variance matrix $V_{v}$ , $\bar{\mathbf {v}}$ is an average movement vector, and $\zeta$ is a constant. At the birth of $X_{i}$ , assume that its initial value of $\mathbf {v}_{i}$ independently follows a known probabilistic distribution function (PDF). Here, assume that this PDF is the normal distribution with mean $\bar{\mathbf {v}}$ and variance matrix $V_{v}$ . For $X_{i}$ appearing as a part of $\tilde \Phi (t)$ , its location ${\mathbf {x}}_{i}(s)\in {\mathcal R}^{2}$ at ${s} > {t}$ is given as $$\begin{equation*} {\mathbf {x}}_{i}(s)= {\mathbf {x}}_{i}(t)+\sum _{\tau =t+1}^{s} \mathbf {v}_{i}(\tau).\tag{2}\end{equation*}$$ $X_{i}$ may disappear at each time slot with probability $\mu$ . Let $\Phi (t)\equiv \{ {\mathbf {x}}_{i}(t)\}_{i}$ be the set of locations of existing nodes at ${t}$ . Note that this is also a PPP. Define $\Phi [0,T]\equiv \cup _{s=0}^{T}\Phi (t)$ . (For simplicity, $[0, {T}]$ may be omitted in $\Phi [0,T]$ in the remainder of this letter.)

Let $X_{0}$ be a node harvesting energy from other nodes and continuously staying at the origin. Energy is harvested and stored in $X_{0}$ . Assume that the harvested RF power from $X_{i}$ is $g \tilde d^{-\alpha }$ , where $\tilde d_{i}(t)=d_{i}(t)+\epsilon$ and $d_{i}(t)\equiv \| {\mathbf {x}}_{i}(t)\|$ is the distance between $X_{0}$ and $X_{i}$ at ${t}$ , $\alpha >2$ is a path loss exponent, ${g}$ is the transmission power including gain at ${t}$ , and $\epsilon$ is a small positive constant. Also assume that $X_{0}$ ’s DC power converted from the RF power harvested from other nodes at time slot ${t}$ is given by the following model taking account of non-linear RF-to-DC conversion where ${y}$ (${t}$ ) is the output DC power, and $\xi$ is an RF-to-DC conversion function.

$$\begin{align*} y(t)=&\xi \left({\sum _{ {\mathbf {x}}_{i}(t)\in \Phi (t)} g \tilde d_{i}(t)^{-\alpha }}\right)\tag{3}\\ \xi (z)=&\gamma _{sat}-\gamma _{0}\exp (-\beta z)\tag{4}\end{align*}$$ Here, $\gamma _{sat}$ is a parameter denoting the harvested DC when the received RF power becomes infinitely large, and $\gamma _{0},\beta >0$ are constant parameters. Figure 1 demonstrates the validity of this proposed model. The total energy harvested during $[0, {T}]$ is $y[0,T]=\sum _{t=0}^{T} y(t)$ .

Fig. 1.

RF power $z$ vs. harvested DC power: proposed model $\xi$ and measurement data regenerated from [6, Fig. 8] for a practical harvesting circuit. $\gamma _{sat}$ , $\gamma _{0}$ , and $\beta$ used here are 0.1071, 0.14096, and 6.4087.

SECTION III. Analysis

This section presents an analysis of the amount of the energy harvested by $X_{0}$ from other nodes during $[0, {T}]$ , and a derivation of its first and second moments (${E}[{y}[0, {T}]]$ and $E[y[0,T]^{2}]$ ). For simplicity, $i\in \Phi (t)$ denotes ${\mathbf {x}}_{i}(t)\in \Phi (t)$ in the remainder.

A. First Moment of Harvested Energy

Due to Eqs. (3) and (4),

$$\begin{equation*} E[y(t)]=\gamma _{sat}-\gamma _{0}E_{\Phi (t)}\left[{\prod _{i\in \Phi (t)}\exp (-\beta g \tilde d_{i}(t)^{-\alpha })}\right].\tag{5}\end{equation*}$$ Because $\Phi (t)$ is a PPP with intensity $\rho \equiv \lambda /\mu$ , use the moment generating functional (MGF) for a PPP (see Remark below). Then, we can obtain $$\begin{equation*} E[y[0,T]]=\sum _{t=0}^{T}E[y(t)]=(T+1)(\gamma _{sat}-\gamma _{0}A_{1}(\rho)),\tag{6}\end{equation*}$$ where $A_{k}(\rho)\equiv \exp \left\{{-2\pi \rho \int _{0}^{\infty } r (1-\exp (-k \beta g(\epsilon +r)^{-\alpha }))dr}\right\}$ . These equations show that ${E}[{y}[0, {T}]]$ is independent of $\bar{\mathbf {v}},V_{v},\zeta$ . This is because $\Phi (t)$ is a PPP for any ${t}$ . (This result is suggested by the displacement theorem for a PPP.) In addition, ${E}[{y}[0, {T}]]$ is independent of $\lambda,\mu$ when $\rho$ is fixed.

Remark:

The MGF for a homogeneous PPP $\{ {\mathbf {x}}_{i}\}_{i}$ of intensity $\rho$ in $\Omega$ is given as follows [20]. For a generic function ${h}$ (x),

$$\begin{equation*} E\left[{\prod _{i} h({\mathbf {x}}_{i})}\right]=\exp \left({-\rho \int _\Omega (1-h({\mathbf {x}}))d {\mathbf {x}}}\right).\tag{7}\end{equation*}$$

B. Second Moment of Harvested Energy

As a preliminary, analyze $\Pr \left({\sum _{\tau =t+1}^{s} \mathbf {v}_{i}(\tau)}\right)$ . By using that analysis result, the second moment of harvested energy can be analyzed.

1) Analysis of $\Pr\left({\sum_{\tau=t+1}^{s} \mathbf{v}_{i}(\tau)}\right)$ :

Because $\mathbf {v}_{i}(t_{0})$ follows ${\mathcal N}(\bar{\mathbf {v}},V_{v})$ at its birth epoch $t_{0}$ , $\Pr (\mathbf {v}_{i}(t_{0}+1))= {\mathcal N}(\bar{\mathbf {v}},V_{v})$ because of Eq. (1). Thus, for any $t\geq t_{0}$ , $\Pr (\mathbf {v}_{i}(t))= {\mathcal N}(\bar{\mathbf {v}},V_{v})$ .

By repeatedly applying Eq. (1), for any $\tau \geq t$ ,

$$\begin{equation*} \mathbf {v}_{i}(\tau)=\sqrt {1-\zeta ^{2}}\sum _{k=t}^{\tau -1}\zeta ^{\tau -1-k} \omega _{i}(k)+(1-\zeta ^{\tau -t}) \bar{\mathbf {v}} +\zeta ^{\tau -t} \mathbf {v}_{i}(t).\tag{8}\end{equation*}$$ By using Eq. (8) with $\tau =t+1,\ldots, s$ , for $0\leq \zeta < 1$ , $$\begin{align*}&\sum _{\tau =t+1}^{s} \mathbf {v}_{i}(\tau)=\sqrt {1-\zeta ^{2}}\sum _{k=t}^{s-1}\frac {1-\zeta ^{s-k}}{1-\zeta }\omega _{i}(k) \\&\quad +\,\,\left({s-t-\frac {\zeta (1-\zeta ^{s-t})}{1-\zeta }}\right) \bar{\mathbf {v}} +\frac {\zeta (1-\zeta ^{s-t})}{1-\zeta } \mathbf {v}_{i}(t).\tag{9}\end{align*}$$ By using Eq. (8) for $\zeta =1$ , $$\begin{equation*} \sum _{\tau =t+1}^{s} \mathbf {v}_{i}(\tau)=(s-t) \mathbf {v}_{i}(t).\tag{10}\end{equation*}$$ Because $\Pr (\mathbf {v}_{i}(t))= {\mathcal N}(\bar{\mathbf {v}},V_{v})$ , $\Pr \left({\sum _{\tau =t+1}^{s} \mathbf {v}_{i}(\tau)}\right)$ follows ${\mathcal N}((s-t) \bar{\mathbf {v}}, \phi _{i}(s-t)V_{v})$ , where $\phi _{i}(s-t)\equiv \frac {1}{(1-\zeta)^{2}}\{(1-\zeta ^{2})(s-t)-2\zeta (1-\zeta ^{s-t})\}$ for $0\leq \zeta < 1$ and $(s-t)^{2}$ for $\zeta =1$ .

2) Analysis of Second Moment:

Note $E[y[0,T]^{2}]=\sum _{t=0,\ldots, T}E[y(t)^{2}]+2\sum _{s>t}E[y(s)y(t)]$ . By using the MGF for a PPP,

$$\begin{align*} E[y(t)^{2}]=&\gamma _{sat}^{2}-2\gamma _{0}\gamma _{sat}E_{\Phi (t)}\left[{\prod _{i\in \Phi (t)} \exp (-\beta g \tilde d_{i}(t)^{-\alpha })}\right] \\&+\,\,\gamma _{0}^{2}E_{\Phi (t)}\left[{\prod _{i\in \Phi (t)} \exp (-2\beta g \tilde d_{i}(t)^{-\alpha })}\right] \\=&\gamma _{sat}^{2}-2\gamma _{0}\gamma _{sat}A_{1}(\rho)+\gamma _{0}^{2}A_{2}(\rho).\tag{11}\end{align*}$$ For $s>t$ , $$\begin{align*}&E[y(s)y(t)] \\&\;= \gamma _{sat}^{2}-2\gamma _{0}\gamma _{sat}A_{1}(\rho)+\gamma _{0}^{2}E_{\Phi (t),\Phi (s)}\big[\prod _{i\in \Phi (t), j\in \Phi (s)} \\&\qquad \exp \left(-\beta g\left(\tilde d_{i}(t)^{-\alpha }+ \tilde d_{j}(s)^{-\alpha }\right)\right)\big].\tag{12}\end{align*}$$ Because $d_{i}(t)$ and $d_{j}(s)$ (${i}~\neq ~{j}$ ) are independently distributed, $$\begin{align*}&E_{\Phi (t),\Phi (s)}\left[{\prod _{i\in \Phi (t), j\in \Phi (s)} \exp (-\beta g(\tilde d_{i}(t)^{-\alpha }+ \tilde d_{j}(s)^{-\alpha }))}\right] \\&\;= E_{\Phi (t)\cap \Phi (s)}\left[{\prod _{i\in \Phi (t)\cap \Phi (s)} \exp (-\beta g(\tilde d_{i}(t)^{-\alpha }+ \tilde d_{i}(s)^{-\alpha }))}\right] \\&E_{\Phi (t)/\Phi (s)}\left[{\prod _{i\in \Phi (t)}\exp (-\beta g \tilde d_{i}(t)^{-\alpha })}\right] \\&E_{\Phi (s)/\Phi (t)}\left[{\prod _{j\in \Phi (s)}\exp (-\beta g \tilde d_{j}(s)^{-\alpha })}\right].\tag{13}\end{align*}$$ Because $\Phi (t)/\Phi (s)$ and $\Phi (s)/\Phi (t)$ are also PPPs with intensity $\rho (1-(1-\mu)^{s-t})$ , both of the last two terms of Eq. (13) are $A_{1}(\rho (1-(1-\mu)^{s-t)}))$ .

Because the set of node locations $\Phi (t)\cap \Phi (s)$ is also a PPP and because its intensity is $\rho (1-\mu)^{s-t}$ ,

$$\begin{align*}&E_{\Phi (t)\cap \Phi (s)}\left[{\prod _{i\in \Phi (t)\cap \Phi (s)} \exp (-\beta g(\tilde d_{i}(t)^{-\alpha }+ \tilde d_{i}(s)^{-\alpha }))}\right] \\&\;= E_{\Phi (t),\sum _{\tau =t+1}^{s} \mathbf {v}_{i}(\tau), \mathbf {v}_{i}(t)}\bigg[\prod _{i\in \Phi (t)\cap \Phi (s)} \\&\qquad \bigg[\exp \bigg(-\beta g\bigg((\epsilon +| {\mathbf {x}}_{i}(t)|)^{-\alpha } \\&\qquad \qquad +\,\,\left({\epsilon +| {\mathbf {x}}_{i}(t)+\sum _{\tau =t+1}^{s} \mathbf {v}_{i}(\tau)|}\right)^{-\alpha }\bigg)\bigg)\bigg] \\&\;= \int \Psi ((s-t) \bar{\mathbf {v}}, \phi _{i}(s-t)V_{v})\tilde B_{s-t}(\mathbf {u})d \mathbf {u} \\&\;\equiv B_{s-t}.\tag{14}\end{align*}$$ Here, $\Psi ((s-t) \bar{\mathbf {v}}, \phi _{i}(s-t)V_{v})$ is the PDF of ${\mathcal N}((s-t) \bar{\mathbf {v}}, \phi _{i}(s-t)V_{v})$ , and $$\begin{align*}&\tilde B_{s-t}(\mathbf {u})\equiv \exp \bigg\{-\rho (1-\mu)^{s-t}\int _{0}^\infty \int _{0}^{2\pi } \\&\qquad r(1-\exp \bigg(-\beta g\bigg(\bigg(\epsilon +r)^{-\alpha } \\&\qquad +\,\,(\epsilon +|(r\cos \psi,r\sin \psi)+ \mathbf {u}|)^{-\alpha }\bigg)\bigg)\bigg) dr\,d\psi \bigg\}.\tag{15}\end{align*}$$ Thus, $$\begin{align*}&E[y[0,T]^{2}] \\&\;= (T+1)(\gamma _{sat}^{2}-2\gamma _{0}\gamma _{sat}A_{1}(\rho)+\gamma _{0}^{2}A_{2}(\rho)) \\&\quad +\,\,T(T+1)(\gamma _{sat}^{2}-2\gamma _{0}\gamma _{sat}A_{1}(\rho)) \\&\quad +\,\,2\gamma _{0}^{2}\sum _{s>t}B_{s-t}A_{1}(\rho (1-(1-\mu)^{s-t}))^{2}.\tag{16}\end{align*}$$ Because $B_{s-t}$ depends on ${s}\,\,-\,\,{t}$ but not on ${s}$ or ${t}$ , the variance $\sigma (y[0,T])^{2}\equiv E[y[0,T]^{2}]-E[y[0,T]]^{2}$ of ${y}[0, {T}]$ is given as follows. $$\begin{align*} \sigma (y[0,T])^{2}=&\gamma _{0}^{2}(-(T+1)^{2}A_{1}(\rho)^{2}+(T+1)A_{2}(\rho) \\&+2\sum _{k=1}^{T}(T-k+1)B_{k}A_{1}(\rho (1-(1-\mu)^{k}))^{2}) \\\tag{17}\end{align*}$$

When $B_{s-t}A_{1}(\rho (1-(1-\mu)^{s-t)}))^{2}=A_{1}(\rho)^{2}$ for all ${s}\,\,-\,\,{t}$ , $\sigma (y[0,T])^{2}=\gamma _{0}^{2}(T+1)(A_{2}(\rho)-A_{1}(\rho)^{2})=\sum _{t=0}^{T}\sigma (y(t))^{2}$ . That is, $\triangle \sigma (y[0,T])^{2}\equiv \sigma (y[0,T])^{2}-\sum _{t=0}^{T}\sigma (y(t))^{2}=2\gamma _{0}^{2} \sum _{k=1}^{T}(T-k+1)(B_{s-t}A_{1}(\rho (1-(1-\mu)^{s-t)}))^{2}-A_{1}(\rho)^{2})$ is the effect of the correlations of mobile node locations at different times.

C. Approximation of $B_{s-t}$

Although Eq. (17) provides $\sigma (y[0,T])^{2}$ , it contains a time-consuming computation for Eq. (14). According to an intensive simulation, $\sigma (y[0,T])$ is almost insensitive to $V_{v}$ . Thus, $B_{s-t}$ is approximately equal to that with $V_{v}=0$ , and the computation of $B_{s-t}$ with $V_{v}=0$ becomes much easier than that with $V_{v} \neq 0$ . This means that the straight-line mobility model can approximately be used to derive $B_{s-t}$ . An intuitive explanation is as follows. $B_{s-t}$ is determined by the distance from the origin to the location of $X_{i}$ at ${t}$ and that at ${s}$ . $X_{i}$ may become closer to the origin during $[{s}, {t}]$ , and $X_{j}$ may become further away. As a whole, $B_{s-t}$ is almost insensitive to the direction of the movement because $\{X_{i}\}_{i}$ is a PPP. The validity of this approximation is demonstrated using numerical examples (see the Appendix).

Because $B_{s-t}$ with $V_{v}=0$ is $\tilde B_{s-t}((s-t) \bar{\mathbf {v}})$ ,

$$\begin{align*}&\sigma (y[0,T])^{2} \approx \gamma _{0}^{2}(-(T+1)^{2}A_{1}(\rho)^{2}+(T+1)A_{2}(\rho) \\&\;+2\sum _{k=1}^{T}(T-k+1)\tilde B_{k}(k \bar{\mathbf {v}})A_{1}(\rho (1-(1-\mu)^{k}))^{2}.)\tag{18}\end{align*}$$

SECTION IV. Numerical Examples

A. Default Conditions of Numerical Examples

If not explicitly mentioned otherwise, the following conditions were used: $\lambda =10^{-4}$ (1/m)², $\mu =0.01$ , ${g}\,\,=$ 0.1 W, $\zeta =0.9$ , $\bar{\mathbf {v}} =$ (0.5, 0.5) m/s (walking speed), ${V}_{v}=\textit {diag}$ (0.1, 0.1) (m/s)², ${T}\,\,=$ 10, and the time slot length is 1 s. In addition, values of $\gamma _{sat},\gamma _{0},\beta$ in Fig. 1 were used. $\sigma (y[0,T])$ , which was calculated theoretically in this section, used the approximation Eq. (18).

Remark:

To harvest energy, a minimum power requirement must be satisfied [2]. Here, assume that no EH was obtained in the simulation if the total RF power at each time was less than 0.1 (mW).

B. $[{y}[0, {T}]]$ and $\sigma(y[0,T])$ vs. $\rho$

Figure 2 plots ${E}[{y}[0, {T}]]$ against $\rho$ , where the simulation results assuming Eq. (4) were with 95% confidential intervals. It clearly demonstrates that ${E}[{y}[0, {T}]]$ becomes saturated as $\rho$ increases, particularly when $\rho >0.02$ . The theoretical results and simulation results were in good agreement. However, for a very small $\rho$ , Eq. (6) overestimated the simulation results. This may have been because Eq. (6) does not take account of the minimum power requirement.

Fig. 2.

${E}[{y}[0, {T}]]$ vs. $\rho$ .

Figure 3 plots $\sigma (y[0,T])$ against $\rho$ . $\sigma (y[0,T])$ showed a sharp decrease as $\rho$ increased. This was because ${y}[0, {T}]$ becomes almost constant due to saturation as $\rho$ increases. $\sigma (y[0,T])$ increased as $\mu$ decreased (that is, the mean lifetime increased). This was because the number of mobile nodes in $\Phi (t)\cap \Phi (s)$ increases as their lifetimes increase. Thus, the correlations of ${y}$ (${t}$ ) and ${y}$ (${s}$ ) increase. However, $\sigma (y[0,T])$ is not so sensitive to $\mu$ . The theoretical results and simulation results were in good agreement except for a very small $\rho$ .

Fig. 3.

$\sigma (y[0,T])$ vs. $\rho$ .

C. Effect of Mobility on $\sigma(y[0,T])$

Equation (6) shows that mobility has no effect on ${E}[{y}[0, {T}]]$ . However, mobility affects $\sigma (y[0,T])$ . This subsection presents a quantitative evaluation of its effect.

Figures 4 and 5 demonstrate the results. The ${x}$ -axis of Fig. 4 denotes ${c}$ , where $\bar{\mathbf {v}} = {c}(0.5, 0.5)$ . That is, ${c}$ is the relative speed against the default $\bar{\mathbf {v}}$ . This figure shows that $\sigma (y[0,T])$ decreased as $\bar{\mathbf {v}}$ increased. This is because ${\mathbf {x}}_{i}(s)$ was less affected by ${\mathbf {x}}_{i}(t)$ as $\bar{\mathbf {v}}$ increased. The theoretical results overestimated the simulation when the relative speed was large with $\lambda =10^{-4}$ .

Fig. 4.

$\sigma (y[0,T])$ vs. relative speed.

Fig. 5.

$\sigma (y[0,T])$ vs. $\zeta$ .

Figure 5 shows that $\sigma (y[0,T])$ is almost insensitive to $\zeta$ . This is similar to the fact that $\sigma (y[0,T])$ is almost insensitive to $V_{v}$ .

SECTION V. Conclusion

This letter provided the closed form formulas for the average and standard deviation of the energy harvested from mobile nodes, where the nonlinear effect of RF-to-DC conversion is taken into account. The mobility model used is a Gauss-Markov model, which is versatile, and stochastic geometry was used as a theoretical tool. The obtained results can contribute to evaluating the conditions for certain applications of EH and the performance of EH control algorithms.

The current model ($\xi$ defined by Eq. (4)) can be extended to cover a wider range of the RF power by introducing multiple exponential functions.

$$\begin{equation*} \xi (z)=\gamma _{sat}-\sum _{k}\gamma _{k}\exp (-\beta _{k} z)\tag{19}\end{equation*}$$ This extension could resolve the problem that the current model provides a negative value of the DC power for a small RF power. The extension of the analysis results in Section III is straightforward for the extended model.

Appendix

This Appendix compares the intensive simulation results regarding $\sigma (y[0,T])$ and theoretical $\sigma (y[0,T])$ on the basis of the approximation (Eq. (18)) for various $V_{v}$ . ($V_{v}$ is $\begin{aligned} c \begin{pmatrix}0.1&~0\\ 0&~0.1\end{pmatrix} \end{aligned}$ or $\begin{aligned} c \begin{pmatrix}0.1&~0.01\\ 0.01&~0.05\end{pmatrix} \end{aligned}$ , where ${c}$ is an ${x}$ -axis in Fig. 6.) As Fig. 6 demonstrates, $\sigma (y[0,T])$ is almost insensitive to $V_{v}$ . Thus, the approximation Eq. (18) works.

Fig. 6.

Comparison of simulation and theory: $\sigma (y[0,T])$ vs $V_{v}$ .