Online joint power control for cooperative jamming systems with energy harvesting

Abstract

To maximize the long-term time-averaged secrecy rate of an energy harvesting wireless communication system, an online power control algorithm based on the Lyapunov optimization framework is proposed. The system is composed of a source node, a cooperative jamming node, and two destination nodes. The source node and the jamming node are powered by the energy harvesting device. Information sent to the two destination nodes is mutually confidential. Using the Lyapunov optimization framework, the original stochastic optimization problem is transformed into a per-time-slot optimization problem, and the power of the signal and that of the artificial noise are determined based on the current system state such as the power level of the batteries and channel coefficients. The fairness between the two destination nodes is considered too. Simulation results demonstrate that the proposed algorithm can effectively utilize the harvested energy and significantly improve the long-term averaged secrecy rate.

Introduction

With the rapid development of information technology, intelligent time is coming. To achieve intelligence, people have begun to explore in Internet of Things (IoT), mobile computing (MC), pervasive computing (PC), wireless sensor networks (WSNs) and cyber-physical systems (CPS). ¹ IoT connects different things by heterogeneous networks, which is a hot issue in academic and industrial researches. In recent years, IoT has been widely used in environmental monitoring, security surveillance, spatial crowdsourcing, crowd dynamics management, and smart cities. ² The sensor nodes in IoT that collect sensory data from environment connect with the network through a wireless link in most cases, making wireless communication technology one of the most important technologies of IoT.

The broadcast feature of wireless communications enables the signal sent by transmitter to be received by both the legitimate nodes and the eavesdroppers. Therefore, security of information is an important issue of IoT. Traditionally, information is encrypted at high level to guarantee its security, which relies on the high computational complexity in the decryption of the encrypted information without the key. The higher complexity of decryption is, the better security performance can be achieved, but it also makes the corresponding encryption and decryption with key more complex. Due to the limitations in the computing capability and energy supply of the nodes, the high complexity encryption algorithms cannot be used in IoT. Physical layer security (PLS) is another way to ensure the security of information. In 1975, Wyner ³ first proposed the eavesdropping channel model and pointed out that the security transmission of information can be achieved using physical layer technology when the legitimate channel is superior to the eavesdropping channel. The signal processing technology, which aims to create and increase the quality advantage of the legitimate channel relative to the eavesdropping channel, is a hot topic in the research of PLS. Commonly used technologies include multi-antenna technology, ⁴ artificial noise (AN) technology, and cooperative communication technology. ⁵ The nodes of IoT are limited in size and generally have single antenna. Due to the available resources of antenna, computing, and energy, PLS technology in IoT has its special features. Burg et al. ⁶ reviewed PLS technology in IoT.

As the scale of communication networks continues to expand, the energy consumption for information transmission and processing is increasing, and the supply and effective use of energy have gradually become an important topic. There are many nodes in IoT, which have a wide distribution range. In many cases, the nodes cannot be powered by the grid, but only by a battery. The replacement or charging for the battery is costly. Harvesting energy from the environment is a cost-effective energy supply solution. Energy in the environment comes from a wide range of sources, such as solar energy, thermal energy, environmental noise, and radio frequency signal. ⁷ In recent years, a lot of research has been conducted on the application of energy harvesting (EH) technology in communication systems. The main research topics include energy sources in the environment, EH communication model, and energy usage protocol. Ku et al. ⁸ provided a comprehensive review on the research and application of EH technology in communication systems. In addition, Zhao et al. ⁹ provided a comprehensive survey about the research on the utilization of interference for wireless EH systems.

In the wireless communication system powered by harvested energy, because of the random change of the amount of the harvested energy and the fading of channels, the control of the energy usage and the transmission rate is quite complicated. The power control algorithms in EH communication systems can be divided into two categories: offline power control algorithms and online power control algorithms, according to whether energy, channel state, and data arrival in the transmission process are available in advance. The offline control algorithms are applied in the case where the information of energy, channel state, and data arrival are known in advance. Some literature works have studied the offline power control algorithms under different system models. Tutuncuoglu and Yener ¹⁰ considered the discontinuous energy arrival model and presented an offline power control algorithm to maximize the short-term throughput, or minimize the transmission time of a certain amount of data. For the two-hop relay system in which the source node and the relay node are both powered by EH devices, Wu et al. ¹¹ decomposed an offline optimization problem of maximizing the end-to-end throughput into two sub-problems—the selection of forwarding relay and the control of transmission power. The optimal solution was obtained by solving the convex optimization problem. Ozel et al. ¹² aimed to maximize the transmission rate and minimize the transmission time for a point-to-point communication system. Under the premise that the process of EH and that of the channel fading were known, the water-filling algorithm was used to control the transmission power. In an actual system, the rate of EH, data flow, and channel fading are all randomly changing, and it is not possible to obtain the information in advance. So, they are online power control algorithms, not offline power control algorithms, which can be used in an actual system. The complexity of online algorithms is usually higher than that of offline algorithms. Modeling the power control process as a Markov decision process is a common method in online power control algorithms. For example, Sinha and Chaporkar ¹³ constructed the optimal transmission power control problem under a random fading channel to a Markov decision process. Under the condition that the statistical characteristics of system states such as energy, channel, and data arrival are available, the dynamic programming was used to solve the optimization problem and maximize the average transmission rate. The online power control algorithms based on the system’s statistical characteristics generally have high complexity. Furthermore, since the statistical information of the EH, channel fading, and data arriving is required, it is difficult to apply this category of algorithm in practice. Lyapunov framework ¹⁴ is a widely used control method in control engineering. It does not require the statistical information of the system and makes decisions based on the current system state to optimize the long-term time-averaged performance of the system. The Lyapunov framework is also a powerful tool to solve the online power control problem in EH communication systems. The basic task of Lyapunov optimization framework is to keep the queues and virtual queues stable in the long term. The constraints of the optimization are transformed to virtual queues. The optimization target is added to the drift of the queues (including virtual queues) as a penalty term, and the original optimization problem with the constraints is converted to the minimization of the drift-plus-penalty function. Lyapunov optimization transforms the long-term time-averaged optimization problem into an instantaneous optimization problem and simplifies the optimization problem. Some literature works have studied the optimization of EH communication systems based on Lyapunov framework. Qiu et al. ¹⁵ explored the long-term time-averaged throughput maximization problem of EH communication systems under the limitation of battery capacity and the requirement of bit error rate. The virtual queues of energy and bit error rate are constructed, and the transmission power and modulation scheme are jointly optimized. The Lyapunov optimization framework is used to optimize the performance for an EH point-to-point communication system by Amirnavaei and Dong. ¹⁶ The long-term time-averaged transmission rate is maximized by controlling the transmission power at each time slot (TS). The online joint power control for a two-hop amplifying-and-forwarding relay system is studied by Dong et al. ¹⁷ The relay is an EH node, and the harvested energy is used for the forwarding. Lyapunov framework was used to solve the joint online power control of the source node and the relay node for the maximization of the long-term time-averaged transmission rate.

In this article, we study the power control of a PLS transmission system. The system is composed of a source node, a friendly jamming node and two destination nodes. The source node and the jamming node are powered by the EH devices. Information sent to a destination node is required to keep secret to the other destination node. All nodes are equipped with single antenna. Without any prior information of channel fading process and EH process, an online algorithm to jointly control the power of the source node and that of the jamming node is designed based on Lyapunov framework. The Lyapunov optimization framework is used to transform the long-term time-averaged optimization problem into a single TS optimization problem. On the condition that only the current channel state and battery state are known, the transmission power of the information signal and AN are jointly controlled. Although the power control algorithm in this article is proposed for the single antenna system, it can be easily extended to the multi-antenna nodes scenario. The main contributions of this article are summarized as follows:

• We formulate the power control problem of the signal and AN as an optimization problem which aims to maximize the long-term time-averaged secrecy rate with long-term constrains of the batteries. We then transform the constrains of the batteries to the stability requirement of virtual queues.

• We use Lyapunov optimization framework to transform the long-term optimization problem into a per-TS minimization of the drift-plus-penalty function, and dynamically adjust the penalty weight to guarantee the fairness between the two destination nodes. However, the joint optimization of signal power and AN power is a non-convex problem. We use Karush-Kuhn-Tucker (KKT) condition to obtain all possible optimal power pairs and choose the pair that minimizes the drift-plus-penalty function to be the optimal power pair.

• We evaluate the performance of our proposed power control algorithm via simulations which show that the secrecy rate can be increased by the assistance of the cooperative jamming node. Compared with the algorithm that does not optimize the power, the proposed algorithm can utilize the harvested energy efficiently and achieve a higher secrecy rate.

The following parts of this article are arranged as follows: the second part introduces the system model, the third part presents the optimization problem, the fourth part solves the optimization problem using Lyapunov framework, the fifth part simulates the optimization algorithm, and sixth part summarizes the whole article.

System model

The system model is shown in Figure 1. The system is composed of a source node S, a cooperative jamming node (jammer) J, and two destination nodes D₁ and D₂. Each node is equipped with single antenna. The source node and the jammer are equipped with an EH device and a rechargeable battery. The EH equipment is used to harvest energy from the environment and convert it into electricity. The battery is used to store the harvested electricity for data transmission. During the transmission process, the arrival rate of the energy and the channel state change randomly. The source node sends information to the two destination nodes, and the information sent to a destination node is confidential to the other destination node. The source node chooses a destination node and sends its secrecy information to it at each TS according to the states of all channels and the power levels of the batteries. In order to ensure the secrecy of the transmission, the jammer uses harvested energy to send AN at the same time. The transmission power of the source node and that of the jammer is controlled jointly to obtain high energy efficiency.

Figure 1.

System model.

EH and using model

Assuming that the capacity of the battery at the source node is E_max (J). The energy stored in the battery at the beginning of TS t is E_sb(t) (J), and 0 ≤ E_sb(t) ≤ E_max. Denote the energy harvested by the EH device from the environment in TS t as E_sa(t) (J), and the electricity charged into the battery as E_ss(t) (J), which satisfies E_ss(t) ≤ E_sa(t). The transmission power is P_s(t) W in TS t, and the energy consumed in the TS is ΔtP_s(t) (J), where Δt is the duration of one TS. Limited by the storage capacity of the battery, the electricity charged in one TS does not exceed E_max − (E_sb(t) − ΔtP_s(t)). In addition, due to the battery’s physical feature, the charging rate is limited, and the maximum electricity charged into the battery in one TS is E_c,_max (J). Thus, the electricity charged into the battery can be written as

$$E_{\mathrm{ss}}(t)=\min \{E_{\max }-(E_{\mathrm{sb}}(t)-\mathrm{\Delta }t{P}_{\mathrm{s}}(t)),E_{\mathrm{sa}}(t),E_{\mathrm{c},\max }\}$$

(1)

In the formula, the first term in the minimum operation is the limitation of battery capacity, the second term is the amount of the harvested energy, and the third term is the limitation of the charging rate.

Similarly, for the jammer, we also assume that the capacity of the battery and the maximum energy charged into the battery in a TS as, respectively, E_max and E_c,_max (J) and denote the energy stored in the battery at the beginning of TS t as E_jb(t) J, and the energy harvested by the EH device and the energy charged into the battery and the transmission power in TS t, respectively, as E_ja(t) and E_js(t), and P_j(t). E_js(t) is determined by

$$E_{\mathrm{js}}(t)=\min \{E_{\max }-(E_{\mathrm{jb}}(t)-\mathrm{\Delta }t{P}_{\mathrm{j}}(t)),{E_{\mathrm{j}}}_{\mathrm{a}}(t),E_{\mathrm{c},\max }\}$$

(2)

The signal power P_s(t) and the AN power P_j(t) are constrained by the maximum discharge rate of the batteries P_max, so they are bounded by

$$\{\begin{array}{c}0\le P_{\mathrm{s}}(t)\le P_{\max }\hfill \\ \multicolumn{1}{c}{0\le P_{\mathrm{j}}(t)\le P_{\max }}\end{array}$$

(3)

The energy consumed in TS t cannot exceed the energy stored in the batteries at the begging of the TS, that is

$$\{\begin{array}{c}0\le \mathrm{\Delta }t{P}_{\mathrm{s}}(t)<E_{\mathrm{sb}}(t)\hfill \\ \multicolumn{1}{c}{0\le \mathrm{\Delta }t{P}_{\mathrm{j}}(t)<E_{\mathrm{jb}}(t)}\end{array}$$

(4)

Thus, after the charging and discharging process in TS t, the power levels at the beginning of next TS can be written as

$$\{\begin{array}{c}{E}_{\mathrm{sb}}(t+1)=E_{\mathrm{sb}}(t)-P_{\mathrm{s}}(t)\mathrm{\Delta }t+E_{\mathrm{ss}}(t)\hfill \\ \multicolumn{1}{c}{E_{\mathrm{jb}}(t+1)=E_{\mathrm{jb}}(t)-P_{\mathrm{j}}(t)\mathrm{\Delta }t+E_{\mathrm{js}}(t)}\end{array}$$

(5)

Secrecy transmission model

The channels from the source node and the jamming node to the two destination nodes are time-varying fading channel, and their coefficients are denoted as h_sd1(t), h_sd2(t), h_jd1(t), and h_jd2(t) respectively, which remain unchanged within a TS. The noise of each channel is additive white Gaussian noise (AWGN) with mean zero and variance ${\sigma }_{\mathrm{n}}^2$ . The instantaneous channel capacities of legitimate channel and eavesdropping channel are

$$\{\begin{array}{c}{C}_{\mathrm{d}}(t)={\log }_2(1+\frac{P_{\mathrm{s}}(t){|h_{\mathrm{sd}i}(t)|}^2}{{\sigma }_{\mathrm{n}}^2+P_{\mathrm{j}}(t){|h_{\mathrm{jd}i}(t)|}^2})\hfill \\ C_{\mathrm{e}}(t)={\log }_2(1+\frac{P_{\mathrm{s}}(t){|h_{\mathrm{sd}\hat{i}}(t)|}^2}{{\sigma }_{\mathrm{n}}^2+P_{\mathrm{j}}(t){|h_{\mathrm{jd}\hat{i}}(t)|}^2})\hfill \end{array}$$

(6)

where i = 1 and $\hat{i}=2$ when the information is sent to D₁, otherwise i = 2 and $\hat{i}=1$ . According to the theory of PLS, when the capacity of legitimate channel is larger than that of eavesdropping channel, the legitimate receiver can correctly decode the confidential information while the eavesdropper cannot obtain any meaningful information if the rate of the confidential information is not higher than the achievable secrecy rate of the system. The achievable secrecy rate of a system is defined as the difference between the capacity of the legitimate channel and that of the eavesdropping channel

$$R_{\mathrm{s}}(t)=[C_{\mathrm{d}}(t)-C_{\mathrm{e}}(t){]}^{+}$$

(7)

where [x]⁺ = max{0, x}. For the model in this article, the destination of the information transmission is determined by the states of all channels, and the battery levels of the source node and the jammer. Only the destination node with non-zero achievable secrecy rate can be taken as a legitimate receiver. Therefore, [x]⁺ in equation (7) can be omitted, and the achievable secrecy rate can be written as

$$\begin{array}{cc}\multicolumn{1}{c}{R_{\mathrm{s}}(t)=}& {\log }_2(1+\frac{P_{\mathrm{s}}(t){|h_{\mathrm{sd}i}(t)|}^2}{{\sigma }_{\mathrm{n}}^2+P_{\mathrm{j}}(t){|h_{\mathrm{jd}i}(t)|}^2})\hfill \\ \multicolumn{1}{c}{}& -{\log }_2(1+\frac{P_{\mathrm{s}}(t){|h_{\mathrm{sd}\hat{i}}(t)|}^2}{{\sigma }_{\mathrm{n}}^2+P_{\mathrm{j}}(t){|h_{\mathrm{jd}\hat{i}}(t)|}^2})\hfill \end{array}$$

(8)

Optimization algorithm based on Lyapunov framework

Optimization problem

For each TS, in order to utilize the harvested energy efficiently, it is necessary to choose the legitimate receiver and determine the power of signal and AN based on the current channel state and the power levels of the batteries at the source node and the jammer. The target of the power control strategy is to maximize the long-term time-averaged achievable secrecy rate under the constraints of the batteries’ performance and the energy amount stored in the batteries. The optimization problem for all t is illustrated as

$$\begin{array}{c}\mathrm{P}1:\underset{\{P_{\mathrm{s}}(t),P_{\mathrm{j}}(t)\}}{\max }\underset{T\to \infty }{\lim }\frac{1}{T}\sum _{t=0}^{T-1}E[R_{\mathrm{s}}(t)]\hfill \\ \mathrm{s}.\mathrm{t}.\mathrm{equations}(3)(4)(5)\hfill \end{array}$$

(9)

where E[x] represents expectation operation.

Rewrite the first formula of equation (5) as

$$E_{\mathrm{sb}}(t+1)-E_{\mathrm{sb}}(t)=E_{\mathrm{ss}}(t)-\mathrm{\Delta }t{P}_{\mathrm{s}}(t)$$

(10)

From TS 0 to TS T, it is easy to get

$$\begin{array}{c}{E}_{\mathrm{sb}}(T)-E_{\mathrm{sb}}(T-1)=E_{\mathrm{ss}}(T-1)-\mathrm{\Delta }t{P}_{\mathrm{s}}(T-1)\hfill \\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{E_{\mathrm{sb}}(2)-E_{\mathrm{sb}}(1)=E_{\mathrm{ss}}(1)-\mathrm{\Delta }t{P}_{\mathrm{s}}(1)}\\ \multicolumn{1}{c}{E_{\mathrm{sb}}(1)-E_{\mathrm{sb}}(0)=E_{\mathrm{ss}}(0)-\mathrm{\Delta }t{P}_{\mathrm{s}}(0)}\end{array}$$

(11)

Calculate the expectation for the left-hand side and the right-hand side of equation (11), respectively, and superimposing all formulae, we can obtain

$$E[E_{\mathrm{sb}}(T)]-E[E_{\mathrm{sb}}(0)]=\sum _{t=0}^{T-1}E[E_{\mathrm{ss}}(t)-\mathrm{\Delta }t{P}_{\mathrm{s}}(t)]$$

(12)

Dividing both sides of the above equation by T and letting T → ∞, we can get

$$\begin{array}{c}{\lim }_{T\to \infty }\frac{1}{T}E[E_{\mathrm{sb}}(T)]-\underset{T\to \infty }{\lim }\frac{1}{T}E[E_{\mathrm{sb}}(0)]\hfill \\ =\underset{T\to \infty }{\lim }\frac{1}{T}\sum _{t=0}^{T-1}E[E_{\mathrm{ss}}(t)-\mathrm{\Delta }t{P}_{\mathrm{s}}(t)]\hfill \end{array}$$

(13)

Due to the finite capacity of the battery, E_sb(0) and E_sb(T) must be limited, so the left-hand side of equation (13) is 0. Denoting $\underset{T\to \infty }{\lim }(1/T)E[E_{\mathrm{ss}}(t)]$ as ${\overline{E}}_{\mathrm{ss}}$ and $\underset{T\to \infty }{\lim }(1/T)E[P_{\mathrm{s}}(t)]$ as ${\overline{P}}_{\mathrm{s}}$ , we get ${\overline{E}}_{\mathrm{ss}}-\mathrm{\Delta }t{\overline{P}}_{\mathrm{s}}=0$ . Similarly, ${\overline{E}}_{\mathrm{js}}-\mathrm{\Delta }t{\overline{P}}_{\mathrm{j}}=0$ can be obtained where ${\overline{E}}_{\mathrm{js}}=\underset{T\to \infty }{\lim }(1/T)E[E_{\mathrm{js}}(t)]$ and ${\overline{P}}_{\mathrm{j}}=\underset{T\to \infty }{\lim }(1/T)E[P_{\mathrm{j}}(t)]$ . So, the constraints about the operations of the two batteries (equation (5)) are transformed to

$$\{\begin{array}{c}{\overline{E}}_{\mathrm{ss}}-\mathrm{\Delta }t{\overline{P}}_{\mathrm{s}}=0\hfill \\ {\overline{E}}_{\mathrm{js}}-\mathrm{\Delta }t{\overline{P}}_{\mathrm{j}}=0\hfill \end{array}$$

(14)

Equation (14) indicates that, from a long-term perspective, all energy harvested by the EH device of the source node should be used to transmit information, and all energy harvested by the EH device of the jammer is used to transmit AN.

Define the two virtual queues, respectively, for the power levels of the batteries of the source node and the jammer as

$$\{\begin{array}{c}{X}_{\mathrm{s}}(t)\stackrel{\mathrm{\Delta }}{=}{E}_{\mathrm{sb}}(t)-{\delta }_{\mathrm{s}}\hfill \\ X_{\mathrm{j}}(t)\stackrel{\mathrm{\Delta }}{=}{E}_{\mathrm{jb}}(t)-{\delta }_{\mathrm{j}}\hfill \end{array}$$

(15)

where δ_s and δ_j are constants. Keeping the two queues stable in the long term is equivalent to satisfying the constraint (14), that is, all harvested energy runs out. By adding an offset to the energy queue, the battery level will fluctuate around the offset. By choosing appropriate constants, enough energy is stored in the batteries for the transmission while there is enough free capacity to store the harvested energy in each TS.

The dynamics of the virtual queues from one TS to the next TS can be formulated as

$$\{\begin{array}{c}{X}_{\mathrm{s}}(t+1)=X_{\mathrm{s}}(t)-\mathrm{\Delta }t{P}_{\mathrm{s}}(t)+E_{\mathrm{ss}}(t)\hfill \\ X_{\mathrm{j}}(t+1)=X_{\mathrm{j}}(t)-\mathrm{\Delta }t{P}_{\mathrm{j}}(t)+E_{\mathrm{js}}(t)\hfill \end{array}$$

(16)

If the virtual queues are stable in the long term, that is

$$\{\begin{array}{c}\multicolumn{1}{c}{\underset{t\to \infty }{\lim }\frac{E[X_{\mathrm{s}}(t)]}{t}=0}\\ \multicolumn{1}{c}{\underset{t\to \infty }{\lim }\frac{E[X_{\mathrm{j}}(t)]}{t}=0}\end{array}$$

(17)

the constraint (14) is satisfied.

Due to the random variation of the amount of the harvested energy and the channel states, the virtual queues X_s(t) and X_j(t) will fluctuate up and down around 0, so the values of X_s(t) and X_j(t) can be either positive or negative. Replacing the constraint (5) in P1 with equation (17), we can convert the optimization problem into

$$\begin{array}{c}\mathrm{P}2:\underset{\{P_{\mathrm{s}}(t),P_{\mathrm{j}}(t)\}}{\max }\underset{T\to \infty }{\lim }\frac{1}{T}\sum _{t=0}^{T-1}E[R_{\mathrm{s}}(t)]\hfill \\ \mathrm{s}.\mathrm{t}.\mathrm{equations}(3)(4)(17)\hfill \end{array}$$

(18)

Transformation of the optimization problem

To solve the above optimization problem, some traditional offline power control strategies, such as the water-filling algorithm, can be used, but the complete information of the channel states and the energy arrival amounts need to be known in advance. In an actual system, this information is not easy to get. We use Lyapunov framework to solve the optimization problem based on the current system states.

The optimization goal of power control is to maximize the long-term time-averaged secrecy rate while the power level of the two batteries (i.e. the two energy virtual queues) is kept stable. The optimizing objects are the transmission power of the source node and that of the jammer. The optimization goal can be achieved by exploiting the minimization of the drift-plus-penalty of Lyapunov optimization framework.

Let $\mathbf{X}(t)\stackrel{\mathrm{\Delta }}{=}[X_{\mathrm{s}}(t),X_{\mathrm{j}}(t)]$ . Define the quadratic Lyapunov function as

$$L(\mathbf{X}(t))\stackrel{\mathrm{\Delta }}{=}\frac{X_{\mathrm{s}}^2(t)+X_{\mathrm{j}}^2(t)}{2}$$

(19)

Define the Lyapunov drift as

$$\mathrm{\Delta }\mathbf{X}(t)\stackrel{\mathrm{\Delta }}{=}E[L(\mathbf{X}(t+1))-L(\mathbf{X}(t))|\mathbf{X}(t)]$$

(20)

The smaller the drift is, the more stable the queues are. In order to maximize the secrecy rate while ensuring the stability of queues, the negative of the mean value of the secrecy rate is used as the penalty and we construct the drift-plus-penalty function as

$$\mathrm{\Delta }\mathbf{X}(t)-\mathrm{VE}[R_{\mathrm{s}}(t)|\mathbf{X}(t)]$$

(21)

Now optimization problem P2 has been transformed into the minimization of the drift-plus-penalty equation (21). V in equation (21) is the penalty weight, which is a positive constant and used to trade between the rate maximization and queues stability in the optimization. It is not easy to minimize the drift-plus-penalty directly. However, it has an upper bound as follows, and the minimization of the drift-plus-penalty can be replaced by the minimization of its upper bound.

Lemma 1

Equation (21) has an upper bound

$$\begin{array}{c}\mathrm{\Delta }\mathbf{X}(t)-\mathrm{VE}[R_{\mathrm{s}}(t)|\mathbf{X}(t)]\le B+X_{\mathrm{s}}(t)E[E_{\mathrm{ss}}(t)-\mathrm{\Delta }t{P}_{\mathrm{s}}(t)|\mathbf{X}(t)]\hfill \\ +X_{\mathrm{j}}(t)E[E_{\mathrm{js}}(t)-\mathrm{\Delta }t{P}_{\mathrm{j}}(t)|\mathbf{X}(t)]-\mathrm{VE}[R_{\mathrm{s}}(t)|\mathbf{X}(t)]\hfill \end{array}$$

(22)

where B is a constant not smaller than $(1/2)E[{(E_{\mathrm{ss}}(t)-\mathrm{\Delta }t{P}_{\mathrm{s}}(t))}^2+{(E_{\mathrm{js}}(t)-\mathrm{\Delta }t{P}_{\mathrm{j}}(t))}^2|\mathbf{X}(t)]$ .

Proof

$$\begin{array}{c}\mathrm{\Delta }\mathbf{X}(t)=E[L(\mathbf{X}(t+1))-L(\mathbf{X}(t))|\mathbf{X}(t)]\hfill \\ =\frac{1}{2}E[(X_{\mathrm{s}}^2(t+1)-X_{\mathrm{s}}^2(t))+(X_{\mathrm{j}}^2(t+1)-X_{\mathrm{j}}^2(t))|\mathbf{X}(t)]\hfill \\ =E[X_{\mathrm{s}}(t)(E_{\mathrm{ss}}(t)-\mathrm{\Delta }t{P}_{\mathrm{s}}(t))+X_{\mathrm{j}}(t)(E_{\mathrm{js}}(t)-\mathrm{\Delta }t{P}_{\mathrm{j}}(t))|\mathbf{X}(t)]\hfill \\ +\frac{1}{2}E[{(E_{\mathrm{ss}}(t)-\mathrm{\Delta }t{P}_{\mathrm{s}}(t))}^2+{(E_{\mathrm{js}}(t)-\mathrm{\Delta }t{P}_{\mathrm{j}}(t))}^2|\mathbf{X}(t)]\hfill \end{array}$$

Since E_ss(t), E_js(t), P_s(t), and P_j(t) are all finite and $(1/2)E[(E_{\mathrm{ss}}(t)-\mathrm{\Delta }t{P}_{\mathrm{s}}(t){)}^2+(E_{\mathrm{js}}(t)-\mathrm{\Delta }t{P}_{\mathrm{j}}(t){)}^2\mathbf{X}(t)]$ are always bigger than or equal to 0, there must be a constant B which is bigger than or equal to 0 and meets

$$B\ge \frac{1}{2}E[{(E_{\mathrm{ss}}(t)-\mathrm{\Delta }t{P}_{\mathrm{s}}(t))}^2+{(E_{\mathrm{js}}(t)-\mathrm{\Delta }t{P}_{\mathrm{j}}(t))}^2|\mathbf{X}(t)]$$

In summary

$$\begin{array}{cc}\multicolumn{1}{c}{\mathrm{\Delta }\mathbf{X}(t)}& \le B+E[X_{\mathrm{s}}(t)(E_{\mathrm{ss}}(t)-\mathrm{\Delta }t{P}_{\mathrm{s}}(t))|\mathbf{X}(t)]\hfill \\ \multicolumn{1}{c}{}& +[X_{\mathrm{j}}(t)(E_{\mathrm{js}}(t)-\mathrm{\Delta }t{P}_{\mathrm{j}}(t))|\mathbf{X}(t)]\hfill \end{array}$$

and the drift-plus-penalty equation (21) has an upper bound as

$$\begin{array}{cc}\multicolumn{1}{c}{\mathrm{\Delta }\mathbf{X}(t)-\mathrm{VE}[R_{\mathrm{s}}(t)|\mathbf{X}(t)]}& \le B+X(t)E[E_{\mathrm{ss}}(t)-\mathrm{\Delta }t{P}_{\mathrm{s}}(t)|\mathbf{X}(t)]\hfill \\ \multicolumn{1}{c}{}& +X_{\mathrm{j}}(t)E[E_{\mathrm{js}}(t)-\mathrm{\Delta }t{P}_{\mathrm{j}}(t)|\mathbf{X}(t)]\hfill \\ \multicolumn{1}{c}{}& -\mathrm{VE}[R_{\mathrm{s}}(t)|\mathbf{X}(t)]\hfill \end{array}$$

The proof is completed.

Under the condition that the current states of the channels and the energy virtual queues are known, equation (21) can be changed to minimize its upper bound in per-TS fashion. By removing the expectation and constant B in the drift-plus-penalty equation (22), an equivalent per-TS optimization problem of P2 can be obtained as

$$\begin{array}{c}\underset{\{P_{\mathrm{s}}(t),P_{\mathrm{j}}(t)\}}{\min }{X}_{\mathrm{s}}(t)(E_{\mathrm{ss}}(t)-\mathrm{\Delta }t{P}_{\mathrm{s}}(t))+X_{\mathrm{j}}(t)(E_{\mathrm{js}}(t)-\mathrm{\Delta }t{P}_{\mathrm{j}}(t))-V{R}_{\mathrm{s}}(t)\hfill \\ \mathrm{s}.\mathrm{t}.\mathrm{equations}(3)\mathrm{and}(4)\hfill \end{array}$$

(23)

In the transmission process, due to the random changing of the channels and the uncertainty of the harvested energy, there may be a significant difference between the information throughputs of the two destination nodes, so fairness should be considered in the power control. The idea of maintaining fairness between the information transmissions to the two destination nodes in the power control is as follows. Assume that the average secrecy rate in the past TS before TS t of destination nodes D₁ and D₂ are ${\overline{R}}_1(t)$ and ${\overline{R}}_2(t)$ , respectively. When the source node transmits secret data to destination node D₁, if ${\overline{R}}_1(t)>{\overline{R}}_2(t)$ , which means that the throughput to D₁ in the past is larger than that to D₂, it is necessary to appropriately reduce the transmission power and reserve more power for subsequent TS to transmit secrecy data to D₂ so that the difference between ${\overline{R}}_1(t)$ and ${\overline{R}}_2(t)$ can be reduced. For this purpose, the weight of penalty V in the drift-plus-penalty should be appropriately reduced, and the optimization focuses more on keeping the energy virtual queues stable. Conversely, if ${\overline{R}}_1(t)<{\overline{R}}_2(t)$ , the weight V should be increased. The same operation is conducted when the source nodes transmits confidential information to destination node D₂. Based on the idea, the weighting factor V in equation (23) is adjusted according to the difference between the average secrecy rates of the two destination nodes, and the optimization problem is modified to

$$\begin{array}{c}\mathrm{P}3:\underset{\{P_{\mathrm{s}}(t),P_{\mathrm{j}}(t)\}}{\min }{X}_{\mathrm{s}}(t)(E_{\mathrm{ss}}(t)-\mathrm{\Delta }t{P}_{\mathrm{s}}(t))+X_{\mathrm{j}}(t)(E_{\mathrm{js}}(t)-\mathrm{\Delta }t{P}_{\mathrm{j}}(t))-\tilde{V}(t)R_{\mathrm{s}}(t)\hfill \\ \mathrm{s}.\mathrm{t}.\mathrm{equations}(3)\mathrm{and}(4)\hfill \end{array}$$

(24)

where $\tilde{V}(t)$ is the adjusted weight

$$\tilde{V}(t)=\{\begin{array}{cc}{V}^{*}(t),\hfill & V_{\min }<V^{*}(t)<V_{\max }\hfill \\ \multicolumn{1}{c}{V_{\max },}& V^{*}(t)\ge V_{\max }\hfill \\ \multicolumn{1}{c}{V_{\min },}& V^{*}(t)\le V_{\min }\hfill \end{array}$$

(25)

and

$$V^{*}(t)=\{\begin{array}{cc}V+U({\overline{R}}_2(t)-{\overline{R}}_1(t)),\hfill & \mathrm{Send}\mathrm{confidential}\mathrm{data}\mathrm{to}{\mathrm{D}}_1\hfill \\ \multicolumn{1}{c}{V+U({\overline{R}}_1(t)-{\overline{R}}_2(t)),}& \mathrm{Send}\mathrm{confidential}\mathrm{data}\mathrm{to}{\mathrm{D}}_2\hfill \end{array}$$

(26)

In the above formula, positive constant U is the adjustment factor, and V_max and V_min are the maximum and minimum values of the weight which are used to avoid extreme large or small weight after the adjustment and maintain the necessary balance between the batteries’ power stability and the maximization of secrecy rate in the optimization.

Solution of the optimization problem

The achievable secrecy rate is jointly determined by the channel states h_sd1(t), h_sd2(t), h_jd1(t), and h_jd2(t); the signal power P_s(t); and AN power P_j(t). Under the premise that secrecy rate is positive, the source node can send secrecy data to destination node D₁ or D₂, which needs a further discussion.

Lemma 2

Denote $|h_i^2(t)|={\gamma }_i(t)(i\in \{\mathrm{sd}1,\mathrm{sd}2,\mathrm{jd}1,\mathrm{jd}2\})$ , $A_1={\gamma }_{\mathrm{sd}2}(t){\gamma }_{\mathrm{jd}1}(t)-{\gamma }_{\mathrm{sd}1}(t){\gamma }_{\mathrm{jd}2}(t)$ , $A_2={\gamma }_{\mathrm{sd}1}(t){\gamma }_{\mathrm{jd}2}(t)-{\gamma }_{\mathrm{sd}2}(t){\gamma }_{\mathrm{jd}1}(t)$ , $B_1=({\gamma }_{\mathrm{sd}1}(t)-{\gamma }_{\mathrm{sd}2}(t)){\sigma }_{\mathrm{n}}^2/A_1$ , and $B_2=({\gamma }_{\mathrm{sd}2}(t)-{\gamma }_{\mathrm{sd}1}(t)){\sigma }_{\mathrm{n}}^2/A_2$ . There are several possible cases in the transmission for each TS:

• γ _sd1(t) > γ_sd2(t):

1. If γ_jd1(t) < γ_jd2(t): only the confidential data of D₁ can be sent and $P_{\mathrm{j}}(t)\in [0,\min (E_{\mathrm{jb}}(t)/\mathrm{\Delta }t,P_{\max })]$ .

2. If γ_jd1(t) > γ_jd2(t):

3. In other cases, both the confidential data of D₁ and D₂ can be sent. When the data of D₁ are sent, P_j(t) = 0; otherwise, $P_{\mathrm{j}}(t)\in [B_2,\min (E_{\mathrm{jb}}(t)/\mathrm{\Delta }t,P_{\max })]$ .

• γ _sd1(t) < γ_sd2(t):

1. If γ_jd1(t) > γ_jd2(t): only the confidential data of D₂ can be sent and $P_{\mathrm{j}}(t)\in [0,\min (E_{\mathrm{jb}}(t)/\mathrm{\Delta }t,P_{\max })]$ .

2. If γ_jd1(t) < γ_jd2(t):

3. In other cases, both the confidential data of D₁ and D₂ can be sent. When the data of D₂ are sent, P_j(t) = 0; otherwise, $P_{\mathrm{j}}(t)\in [B_1,\min (E_{\mathrm{jb}}(t)/\mathrm{\Delta }t,P_{\max })]$ .

For Proof, see Appendix 1.

The optimization problem P3 includes two constraints on the transmission power of the source node and that of the jammer, which can be solved by the KKT condition. ¹⁸ Since the KKT condition is a necessary condition for the optimal solution of nonlinear programming, the solution must satisfy KKT condition. The optimal pair of the signal power and AN power must be one of the solutions which satisfy the KKT condition.

It can be seen from Lemma 2 that the power pair of the signal and AN must belong to one of the following three cases when transmitting confidential data of D _i (i = 1, 2):

• Case 1: $P_{\mathrm{s}}(t)\in [0,\min (E_{\mathrm{sb}}(t)/\mathrm{\Delta }t,P_{\max }]$ and $P_{\mathrm{j}}(t)\in [0,\min (E_{\mathrm{jb}}(t)/\mathrm{\Delta }t,P_{\max })]$ ;

• Case 2: $P_{\mathrm{s}}(t)\in [0,\min (E_{\mathrm{sb}}(t)/\mathrm{\Delta }t,P_{\max })]$ and $P_{\mathrm{j}}(t)=0$ ;

• Case 3: $P_{\mathrm{s}}(t)\in [0,\min (E_{\mathrm{sb}}(t)/\mathrm{\Delta }t),P_{\max })]$ and $P_{\mathrm{j}}(t)\in [B_i(i=1,2),\min (E_{\mathrm{jb}}(t)/\mathrm{\Delta }t,P_{\max })]$ .

Now, we analyze the solution of the optimization, respectively, for the three cases.

Case 1

In Case 1, the optimization problem (24) can be rewritten as

$$\begin{array}{c}\underset{\{P_{\mathrm{s}}(t),P_{\mathrm{j}}(t)\}}{\min }J(P_{\mathrm{s}}(t),P_{\mathrm{j}}(t))\hfill \\ \mathrm{s}.\mathrm{t}.0\le P_{\mathrm{s}}(t)\le \min (\frac{E_{\mathrm{sb}}(t)}{\mathrm{\Delta }t},P_{\max })\hfill \\ 0\le P_{\mathrm{j}}(t)\le \min (\frac{E_{\mathrm{jb}}(t)}{\mathrm{\Delta }t},P_{\max })\hfill \end{array}$$

(27)

where

$$\begin{array}{cc}\multicolumn{1}{c}{J(P_{\mathrm{s}}(t),P_{\mathrm{j}}(t))}& =X_{\mathrm{s}}(t)(E_{\mathrm{ss}}(t)-\mathrm{\Delta }t{P}_{\mathrm{s}}(t))\hfill \\ \multicolumn{1}{c}{}& +X_{\mathrm{j}}(t)(E_{\mathrm{js}}(t)-\mathrm{\Delta }t{P}_{\mathrm{j}}(t))-\tilde{V}(t)R_{\mathrm{s}}(t)\hfill \end{array}$$

(28)

Define Lagrangian function as

$$\begin{array}{c}L(P_{\mathrm{s}}(t),P_{\mathrm{j}}(t),{\lambda }_1,{\lambda }_2,{\mu }_1,{\mu }_2)\hfill \\ =J(P_{\mathrm{s}}(t),P_{\mathrm{j}}(t))-{\lambda }_1{P}_{\mathrm{s}}(t)-{\lambda }_2{P}_{\mathrm{j}}(t)\hfill \\ +{\mu }_1(P_{\mathrm{s}}(t)-\min (\frac{E_{\mathrm{sb}}(t)}{\mathrm{\Delta }t},P_{\max }))\hfill \\ +{\mu }_2(P_{\mathrm{j}}(t)-\min (\frac{E_{\mathrm{jb}}(t)}{\mathrm{\Delta }t},P_{\max }))\hfill \end{array}$$

The KKT condition is

$$\{\begin{array}{c}\frac{\partial J(P_{\mathrm{s}}(t),P_{\mathrm{j}}(t))}{\partial P_{\mathrm{s}}(t)}-{\lambda }_1+{\mu }_1=0\hfill \\ \frac{\partial J(P_{\mathrm{s}}(t),P_{\mathrm{j}}(t))}{\partial P_{\mathrm{j}}(t)}-{\lambda }_2+{\mu }_2=0\hfill \\ {\lambda }_1=0\mathrm{or}{P}_{\mathrm{s}}(t)=0\hfill \\ {\lambda }_2=0\mathrm{or}{P}_{\mathrm{j}}(t)=0\hfill \\ {\mu }_1=0\mathrm{or}{P}_{\mathrm{s}}(t)=\min (\frac{E_{\mathrm{sb}}(t)}{\mathrm{\Delta }t},P_{\max })\hfill \\ {\mu }_2=0\mathrm{or}{P}_{\mathrm{j}}(t)=\min (\frac{E_{\mathrm{jb}}(t)}{\mathrm{\Delta }t},P_{\max })\hfill \end{array}$$

(29)

First, we analyze the solutions of P_s(t) which meet the KKT condition. (1) If λ₁ = 0 and μ₁ = 0, the solution of P_s(t) is the root of $\partial J(P_{\mathrm{s}}(t),P_{\mathrm{j}}(t))/\partial P_{\mathrm{s}}(t)=0$ . Denote the root of this equation as $P_{\mathrm{s}}^{*}(t)$ . (2) If λ₁ = 0 and μ₁ ≠ 0, $P_{\mathrm{s}}(t)=\min (E_{\mathrm{sb}}(t)/\mathrm{\Delta }t,P_{\max })$ . (3) If λ₁ ≠ 0 and μ₁ = 0, P_s(t) = 0. (4) If λ₁ ≠ 0 and μ₁ ≠ 0, P_s(t) = 0 and $P_{\mathrm{s}}(t)=\min (E_{\mathrm{sb}}(t)/\mathrm{\Delta }t,P_{\max })$ needs to be met at the same time, so the solution does not exist.

Similarly, there are three possible solutions to P_j(t), which are the root of $\partial J(P_{\mathrm{s}}(t),P_{\mathrm{j}}(t))/\partial P_{\mathrm{j}}(t)=0$ (the root is denoted as $P_{\mathrm{j}}^{*}(t)$ ), 0, and $\min (E_{\mathrm{jb}}(t)/\mathrm{\Delta }t,P_{\max })$ .

By combining the three possible values of P_s(t) and P_j(t), there are nine possible solutions of power pair of the signal and AN which meet the KKT condition. When the power of signal P_s(t) is zero, it is not necessary to transmit AN. So, the two pairs of P_s(t) = 0 and P_j(t) ≠ 0 cannot be the solution of the optimization and are removed. The optimal pair of the signal power and AN power $({P_{\mathrm{s}}}^{\mathrm{opt}},{P_{\mathrm{j}}}^{\mathrm{opt}})$ must be one of the following seven pairs: (0, 0), $(\min (E_{\mathrm{sb}}(t)/\mathrm{\Delta }t,P_{\max }),0)$ , $(\min (E_{\mathrm{sb}}(t)/\mathrm{\Delta }t,P_{\max }),\min (E_{\mathrm{jb}}(t)/\mathrm{\Delta }t,P_{\max }))$ , $(P_{\mathrm{s}}^{*}(t),0)$ , $(P_{\mathrm{s}}^{*}(t),\min (E_{\mathrm{jb}}(t)/\mathrm{\Delta }t,P_{\max }))$ , $(P_{\mathrm{s}}^{*}(t),P_{\mathrm{j}}^{*}(t))$ , and $(\min (E_{\mathrm{sb}}(t)/\mathrm{\Delta }t,P_{\max }),P_{\mathrm{j}}^{*}(t))$ .

Next, we give the expressions of $\partial J(P_{\mathrm{s}}(t),P_{\mathrm{j}}(t))/\partial P_{\mathrm{s}}(t)=0$ and $\partial J(P_{\mathrm{s}}(t),P_{\mathrm{j}}(t))/\partial P_{\mathrm{j}}(t)=0$ . Taking the first derivative of $J(P_{\mathrm{s}}(t),P_{\mathrm{j}}(t))$ with respect to P_s(t) and making it equal to 0, we get

$$\begin{array}{c}\multicolumn{1}{c}{{\gamma }_{11}}\end{array}{X}_{\mathrm{s}}(t)P_{\mathrm{s}}^2(t)+X_{\mathrm{s}}(t)({\gamma }_{12}+{\gamma }_{13}{P}_{\mathrm{j}}(t))P_{\mathrm{s}}(t)+X_{\mathrm{s}}(t)({({\sigma }_{\mathrm{n}}^2)}^2+{\gamma }_{14}{P}_{\mathrm{j}}(t)+{\gamma }_{15}{P}_{\mathrm{j}}^2(t))-\frac{\tilde{V}(t)}{\ln 2}({\gamma }_{16}+{\gamma }_{17}{P}_{\mathrm{j}}(t))=0$$

(30)

where ${\gamma }_{11}={\gamma }_{\mathrm{sd}1}(t){\gamma }_{\mathrm{sd}2}(t)$ , ${\gamma }_{12}=({\gamma }_{\mathrm{sd}1}(t)+{\gamma }_{\mathrm{sd}2}(t)){\sigma }_{\mathrm{n}}^2$ , ${\gamma }_{13}={\gamma }_{\mathrm{sd}1}(t){\gamma }_{\mathrm{jd}2}(t)+{\gamma }_{\mathrm{sd}2}(t){\gamma }_{\mathrm{jd}1}(t)$ , ${\gamma }_{14}=({\gamma }_{\mathrm{jd}1}(t)+{\gamma }_{\mathrm{jd}2}(t)){\sigma }_{\mathrm{n}}^2$ , and ${\gamma }_{15}={\gamma }_{\mathrm{jd}1}(t){\gamma }_{\mathrm{jd}2}(t)$ . When the confidential data of D₁ is sent, ${\gamma }_{16}=({\gamma }_{\mathrm{sd}2}(t)-{\gamma }_{\mathrm{sd}1}(t)){\sigma }_{\mathrm{n}}^2$ and ${\gamma }_{17}={\gamma }_{\mathrm{sd}1}(t){\gamma }_{\mathrm{jd}2}(t)-{\gamma }_{\mathrm{sd}2}(t){\gamma }_{\mathrm{jd}1}(t)$ ; otherwise, when the confidential data of D₂ is sent, ${\gamma }_{16}=({\gamma }_{\mathrm{sd}1}(t)-{\gamma }_{\mathrm{sd}2}(t)){\sigma }_{\mathrm{n}}^2$ and ${\gamma }_{17}={\gamma }_{\mathrm{sd}2}(t){\gamma }_{\mathrm{jd}1}(t)-{\gamma }_{\mathrm{sd}1}(t){\gamma }_{\mathrm{jd}2}(t)$ .

Taking the first derivative of $J(P_{\mathrm{s}}(t),P_{\mathrm{j}}(t))$ with respect to P_j(t) and making it equal to 0, we get

$$\begin{array}{c}{X}_{\mathrm{j}}(t){\gamma }_{21}{P}_{\mathrm{j}}^4(t)+X_{\mathrm{j}}(t)({\gamma }_{22}+{\gamma }_{23}{P}_{\mathrm{s}}(t))P_{\mathrm{j}}^3(t)\hfill \\ +(X_{\mathrm{j}}(t)({\gamma }_{24}+{\gamma }_{25}{P}_{\mathrm{s}}(t)+{\gamma }_{26}{P}_{\mathrm{s}}^2(t))-\frac{\tilde{V}(t)}{\ln 2}{\gamma }_{27}{P}_{\mathrm{s}}(t))P_{\mathrm{j}}^2(t)\hfill \\ +(X_{\mathrm{j}}(t)({\gamma }_{28}+{\gamma }_{29}{P}_{\mathrm{s}}(t)+{\gamma }_{210}{P}_{\mathrm{s}}^2(t))-\frac{\tilde{V}(t)}{\ln 2}{\gamma }_{211}{P}_{\mathrm{s}}(t))\hfill \\ P_{\mathrm{j}}(t)+X_{\mathrm{j}}(t)({({\sigma }_{\mathrm{n}}^2)}^4+{\gamma }_{212}{P}_{\mathrm{s}}(t)+{\gamma }_{213}{P}_{\mathrm{s}}^2(t))\hfill \\ -\frac{\tilde{V}(t)}{\ln 2}({\gamma }_{214}{P}_{\mathrm{s}}(t)+{\gamma }_{215}{P}_{\mathrm{s}}^2(t))=0\hfill \end{array}$$

(31)

where ${\gamma }_{21}={\gamma }_{\mathrm{jd}1}^2(t){\gamma }_{\mathrm{jd}2}^2(t)$ , ${\gamma }_{22}=2{\sigma }_{\mathrm{n}}^2({\gamma }_{\mathrm{jd}1}^2(t){\gamma }_{\mathrm{jd}2}(t)+{\gamma }_{\mathrm{jd}2}^2(t){\gamma }_{\mathrm{jd}1}(t))$ , ${\gamma }_{23}=({\gamma }_{\mathrm{jd}1}(t){\gamma }_{\mathrm{sd}2}(t)+{\gamma }_{\mathrm{jd}2}(t){\gamma }_{\mathrm{sd}1}(t)){\gamma }_{\mathrm{jd}1}(t){\gamma }_{\mathrm{jd}2}(t)$ , ${\gamma }_{24}={\gamma }_{\mathrm{jd}1}^2(t)+{\gamma }_{\mathrm{jd}2}^2(t)+4{\gamma }_{\mathrm{jd}1}(t){\gamma }_{\mathrm{jd}2}(t){\sigma }_{\mathrm{n}}^2$ , ${\gamma }_{25}=2{\gamma }_{\mathrm{jd}1}(t){\gamma }_{\mathrm{jd}2}(t)({\gamma }_{\mathrm{sd}2}(t)+{\gamma }_{\mathrm{sd}1}(t))+{\gamma }_{\mathrm{jd}1}^2(t){\gamma }_{\mathrm{sd}2}(t)+{\gamma }_{\mathrm{jd}2}^2(t){\gamma }_{\mathrm{sd}1}(t)$ , ${\gamma }_{26}={\gamma }_{\mathrm{jd}1}(t){\gamma }_{\mathrm{jd}2}(t){\gamma }_{\mathrm{sd}2}(t){\gamma }_{\mathrm{sd}1}(t)$ , ${\gamma }_{28}=2({\sigma }_{\mathrm{n}}^2{)}^3({\gamma }_{\mathrm{jd}1}(t)+{\gamma }_{\mathrm{jd}2}(t))$ , ${\gamma }_{29}={\gamma }_{\mathrm{jd}1}(t)({\sigma }_{\mathrm{n}}^2{)}^2·({\gamma }_{\mathrm{sd}1}(t)+2{\gamma }_{\mathrm{sd}2}(t))+{\gamma }_{\mathrm{jd}2}(t)({\sigma }_{\mathrm{n}}^2{)}^2({\gamma }_{\mathrm{sd}2}(t)+2{\gamma }_{\mathrm{sd}1}(t))$ , ${\gamma }_{210}={\gamma }_{\mathrm{sd}1}(t){\gamma }_{\mathrm{sd}2}(t)({\gamma }_{\mathrm{jd}1}(t)+{\gamma }_{\mathrm{jd}2}(t)){\sigma }_{\mathrm{n}}^2$ , ${\gamma }_{212}=({\sigma }_{\mathrm{n}}^2{)}^3·({\gamma }_{\mathrm{sd}1}(t)+{\gamma }_{\mathrm{sd}2}(t))$ , and ${\gamma }_{213}={\gamma }_{\mathrm{sd}2}(t){\gamma }_{\mathrm{sd}1}(t)({\sigma }_{\mathrm{n}}^2{)}^2$ . When the confidential data of D₁ are sent, ${\gamma }_{27}={\gamma }_{\mathrm{jd}1}(t)·{\gamma }_{\mathrm{jd}2}(t)({\gamma }_{\mathrm{jd}2}(t){\gamma }_{\mathrm{sd}1}(t)-{\gamma }_{\mathrm{jd}1}(t){\gamma }_{\mathrm{sd}2}(t))$ , ${\gamma }_{211}=2{\gamma }_{\mathrm{jd}1}(t){\gamma }_{\mathrm{jd}2}(t)({\gamma }_{\mathrm{sd}1}(t)-{\gamma }_{\mathrm{sd}2}(t)){\sigma }_{\mathrm{n}}^2$ , ${\gamma }_{214}=({\sigma }_{\mathrm{n}}^2{)}^2({\gamma }_{\mathrm{jd}1}(t){\gamma }_{\mathrm{sd}1}(t))-{\gamma }_{\mathrm{jd}2}(t){\gamma }_{\mathrm{sd}2}(t))$ , and ${\gamma }_{215}={\sigma }_{\mathrm{n}}^2{\gamma }_{\mathrm{sd}1}(t){\gamma }_{\mathrm{sd}2}(t)({\gamma }_{\mathrm{jd}1}(t)-{\gamma }_{\mathrm{jd}2}(t))$ ; otherwise, ${\gamma }_{27}={\gamma }_{\mathrm{jd}1}(t){\gamma }_{\mathrm{jd}2}(t)({\gamma }_{\mathrm{jd}1}(t){\gamma }_{\mathrm{sd}2}(t)-{\gamma }_{\mathrm{jd}2}(t){\gamma }_{\mathrm{sd}1}(t))$ , ${\gamma }_{211}=2{\gamma }_{\mathrm{jd}1}(t){\gamma }_{\mathrm{jd}2}(t)({\gamma }_{\mathrm{sd}2}(t)-{\gamma }_{\mathrm{sd}1}(t)){\sigma }_{\mathrm{n}}^2$ , ${\gamma }_{214}=({\gamma }_{\mathrm{jd}2}(t){\gamma }_{\mathrm{sd}2}(t)-{\gamma }_{\mathrm{jd}1}(t){\gamma }_{\mathrm{sd}1}(t))({\sigma }_{\mathrm{n}}^2{)}^2$ , and ${\gamma }_{215}={\sigma }_{\mathrm{n}}^2{\gamma }_{\mathrm{sd}1}(t){\gamma }_{\mathrm{sd}2}(t)({\gamma }_{\mathrm{jd}2}(t)-{\gamma }_{\mathrm{jd}1}(t))$ . Now, we give the detail of the optimal solutions for the seven pairs:

1. $({P_{\mathrm{s}}}^{\mathrm{opt}},{P_{\mathrm{j}}}^{\mathrm{opt}})=(0,0)$ or $(\min (E_{\mathrm{sb}}(t)/\mathrm{\Delta }t,P_{\max },\min (E_{\mathrm{jb}}(t)/\mathrm{\Delta }t,P_{\max }))$ or $(\min (E_{\mathrm{sb}}(t)/\mathrm{\Delta }t,P_{\max },0))$ . The possible optimal solutions are obtained directly.

2. $({P_{\mathrm{s}}}^{\mathrm{opt}},{P_{\mathrm{j}}}^{\mathrm{opt}})=(P_{\mathrm{s}}^{*}(t),0)$ or $(P_{\mathrm{s}}^{*}(t),\min (E_{\mathrm{jb}}(t)/\mathrm{\Delta }t,P_{\max }))$ . Substituting P_j(t) in equation (30) with 0 or $\min (E_{\mathrm{jb}}(t)/\mathrm{\Delta }t,P_{\max })$ , we get a quadratic equation. The roots of the equation are $(-b\pm \sqrt{b^2-4\mathrm{ac}})/2a$ , where $a={\gamma }_{11}{X}_{\mathrm{s}}(t)$ , $b=X_{\mathrm{s}}(t)({\gamma }_{12}+{\gamma }_{13}{P}_{\mathrm{j}}(t))$ , and $c=X_{\mathrm{s}}(t)(({\sigma }_{\mathrm{n}}^2{)}^2+{\gamma }_{14}{P}_{\mathrm{j}}(t)+{\gamma }_{15}{P}_{\mathrm{j}}^2(t))-(\tilde{V}(t)/\ln 2)({\gamma }_{16}+{\gamma }_{17}{P}_{\mathrm{j}}(t))$ . The positive root is the optimal $P_{\mathrm{s}}^{*}(t)$ .

3. $({P_{\mathrm{s}}}^{\mathrm{opt}},{P_{\mathrm{j}}}^{\mathrm{opt}})=(P_{\mathrm{s}}^{*}(t),P_{\mathrm{j}}^{*}(t))$ . In this case, we need to solve the equation group which is composed of equations (30) and (31). First, treat P_j(t) as a constant, and solve equation (30) to obtain the expression of P_s(t) satisfying $\partial J(P_{\mathrm{s}}(t),P_{\mathrm{j}}(t))/\partial P_{\mathrm{s}}(t)=0$ . Then, substituting P_s(t) in equation (31) with this expression, we obtain an equation with only one unknown P_j(t). Denote the equation as f(P_j(t)) = 0. This equation is a higher order equation which can only be solved numerically. One way to solve it is given in Appendix 2.

4. $({P_{\mathrm{s}}}^{\mathrm{opt}},{P_{\mathrm{j}}}^{\mathrm{opt}})=(\min (E_{\mathrm{sb}}(t)/\mathrm{\Delta }t,P_{\max }),P_{\mathrm{j}}^{*}(t))$ . Substitute P_s(t) in equation (31) with $\min (E_{\mathrm{sb}}(t)/\mathrm{\Delta }t,P_{\max })$ , we get a quartic equation of P_j(t), which can be solved. See Appendix 3 for the details of the solution.

Case 2

In Case 2, using KKT condition, the optimal power pair $({P_{\mathrm{s}}}^{\mathrm{opt}},{P_{\mathrm{j}}}^{\mathrm{opt}})$ must be one of the following three pairs: $(0,0)$ , $(\min (E_{\mathrm{sb}}(t)/\mathrm{\Delta }t,P_{\max },0)$ , and $(P_{\mathrm{s}}^{*}(t),0)$ . Solve it in the same way as in Case 1.

Case 3

In Case 3, using KKT condition, the optimal power pair $({P_{\mathrm{s}}}^{\mathrm{opt}},{P_{\mathrm{j}}}^{\mathrm{opt}})$ must be one of the following pairs: $(\min (E_{\mathrm{sb}}(t)/\mathrm{\Delta }t,P_{\max }),\min (E_{\mathrm{jb}}(t)/\mathrm{\Delta }t,P_{\max }))$ , $(\min (E_{\mathrm{sb}}(t)/\mathrm{\Delta }t,P_{\max }),B_i)$ , $(P_{\mathrm{s}}^{*}(t),B_i)$ , $(P_{\mathrm{s}}^{*}(t),\min E_{\mathrm{jb}}(t)/\mathrm{\Delta }t,P_{\max })$ , $(P_{\mathrm{s}}^{*}(t),P_{\mathrm{j}}^{*}(t))$ , and $(\min (E_{\mathrm{sb}}(t)/\mathrm{\Delta }t,P_{\max }),P_{\mathrm{j}}^{*}(t))$ , where the subscript of B_i is i = 1 when the confidential information of D₁ is sent; otherwise, i = 2. The optimal power pair that meets the KKT condition can be obtained using the method similar to that in Case 1.

For each TS, all possible solutions of power pair are calculated based on the current system states first, then they are substituted into equation (30) and the corresponding values of $J(P_{\mathrm{s}}(t),P_{\mathrm{j}}(t))$ can be obtained. The power pair with minimum value of $J(P_{\mathrm{s}}(t),P_{\mathrm{j}}(t))$ is the optimal power pair, that is, the solution of P3. When the secrecy transmissions to the two destination nodes are both feasible, the destination node with smaller value of $J(P_{\mathrm{s}}(t),P_{\mathrm{j}}(t))$ is chosen.

Based on the above optimization process, the optimization algorithm is given as follows.

Algorithm 1. Online joint power control algorithm of PLS transmission.

Set the weight V (V ≥ 0), U (U ≥ 0), the upper and lower limits Vmax, Vmin of weight $\tilde{V}(t)$ , the offset δs,δj. In TS t: 1. Observe system states Eja(t), Esa(t), hjd1(t), hjd2(t), hsd1(t), hsd2(t), Xj(t) and Xs(t). 2. According to Lemma 2, determine which destination node to receive confidential data, and get the range of the AN power. 3. Get the pairs of the signal power and the AN power which meet the KKT condition. Calculate values of $J(P_{\mathrm{s}}(t),P_{\mathrm{j}}(t))$ for each pair. The power pair with minimum value of $J(P_{\mathrm{s}}(t),P_{\mathrm{j}}(t))$ is the optimal power pair and the corresponding destination is the legitimate receiver. 4. Calculate the secrecy rate Rs(t) according to equation (8). Update the energy virtual queues Xs(t) and Xj(t) according to equation (16). Calculate ${\overline{R}}_1(t)$ and ${\overline{R}}_2(t)$ , then get the penalty weight of next TS according to equations (25) and (26).

Approach extended to multi-antenna scenario

The approach proposed in this section can be easily extended to the multi-antenna scenario. The secrecy rate of the system is related to PLS scheme, the channel state, signal power, and AN power. In the multi-antenna scenario, the secrecy rate (equation (8)) in section “System model” needed to be replaced by the formula of the achievable secrecy rate of the PLS scheme adopted in the transmission (such as equation (4) in the work of Cumanan et al. ¹⁹ ). After a derivation similar to that in this section, the optimal solution to signal power and AN power can be obtained.

Performance analysis

Using the Lyapunov optimization framework, the long-term time-averaged secrecy rate maximization problem P1 has been transformed into the per-TS optimization problem P3. If the optimization of the performance in per-TS is realized, the optimization of the time-averaged performance in long-term is also approximately realized. ¹³ The average achievable secrecy rate using the algorithm proposed in this article and the possibly maximum secrecy rate of P1 have the following relationship

$$\underset{T\to \infty }{\lim }sup\frac{1}{T}\sum _{t=0}^{T-1}E[R_{\mathrm{s}}(t)]\ge {{\overline{R}}_{\mathrm{s}}}^{\mathrm{opt}}-\mathcal{O}\left(\frac{1}{\overline{V}}\right)$$

(32)

where ${{\overline{R}}_{\mathrm{s}}}^{\mathrm{opt}}$ represents the maximum secrecy rate that can be achieved by solving optimization problem P1 using other possible algorithms. $\mathcal{O}(x)$ represents the high order infinitesimal of x, and $\overline{V}$ represents the mean value of the penalty weight $\tilde{V}(t)$ throughout the transmission process.

The proof of equation (32) is directly done using the method in Sinha and Chaporkar, ¹³ so we omit it here.

It can be seen from equations (25) and (26) that $\tilde{V}(t)$ is the adjusted version of V, so the average secrecy rate increases as V increase, but the stability of the power level of the batteries will become worse.

Simulation results

This section verifies the performance of the proposed algorithm by simulation. Unless specifically specified, the parameters in the simulation are set as follows. The energy arrival amount per-TS E_sa(t) and E_ja(t) at the source node and the cooperative jammer follow a compound Poisson process with a uniform distribution. The energy arrival rate at the source node is λ_s = 2.5 unit/slot and that at the jammer is λ_j = 0.5 unit/slot. The amount of energy per unit is uniformly distributed in [0, 0.4] J with mean of 0.2 J/unit. Two nodes’ battery capacity is E_max = 5 J. The maximum charging amount per-TS of the batteries is E_c,_max = 1 J, and the maximum discharge rate is P_max = 1 W. All channel coefficients follow the complex Gaussian distribution with mean of 0 and unit variance. The channel coefficients remain unchanged during one TS and randomly change from one TS to the next TS. The variance of channel noise is ${\sigma }_{\mathrm{n}}^2=0.05\mathrm{W}$ . The duration of a TS is Δt = 1 s. We set V = 1, U = 3, δ_s = 4 J, δ_j = 4 J, V_min = 0.1, and V_max = 5. The two step size in the solution of f(P_j(t)) = 0 (in Appendix 2) are d₁ = 0.005 and d₂ = 0.0001. The initial power levels of the two batteries are 5 J.

Performance comparison with two comparison algorithms

To evaluate the performance of the proposed algorithm, we compare it with the following three algorithms:

1. Half-power algorithm (HPA): the system model is the same as that in Figure 1. In each TS, the source node sends confidential information to the destination node which can achieve a higher secrecy rate. The source node uses half of the energy stored in its battery to transmit signal, and the jammer uses half of the energy stored in its battery to transmit AN when AN is helpful for the promotion of secrecy rate. The signal power and AN power are $P_{\mathrm{hs}}(t)=\min (E_{\mathrm{sb}}(t)/2\mathrm{\Delta }t,P_{\max })$ and $P_{\mathrm{hj}}(t)=\min (E_{\mathrm{jb}}(t)/2\mathrm{\Delta }t,P_{\max })$ . We refer to this algorithm as HPA.

2. Greedy algorithm (GA): the system model is the same as that in Figure 1. The algorithm is basically the same as HPA, except that the signal power and AN power are the maximum values supported by the energy stored in the batteries of the source node and the jammer, that is, $P_{\mathrm{gs}}(t)=\min (E_{\mathrm{sb}}(t)/\mathrm{\Delta }t,P_{\max })$ and $P_{\mathrm{gj}}(t)=\min (E_{\mathrm{jb}}(t)/\mathrm{\Delta }t,P_{\max })$ . We refer to this algorithm as GA.

3. The algorithm proposed by Lei and Wang. ²⁰ The system model considered in this algorithm is similar to that in this article, but there is no cooperative jammer. The Lyapunov optimization method is used to control the transmission power of the source node for the maximization of the secrecy rate. In each TS, the source node transmits the secrecy information to the destination node with better channel quality and optimizes the transmission power using Lyapunov optimization framework. We refer to this algorithm as without-jammer algorithm (WJA).

In the simulation of the two comparison algorithms, the energy arrival, channel characteristics, and the features of the batteries are the same as those in the simulation of the proposed algorithm.

Figure 2 shows the simulation results of time-averaged achievable secrecy rate versus TS t. The time-averaged secrecy rate of each TS is the average value of the secrecy rate from the beginning of simulation to the current TS. It can be clearly seen that the proposed algorithm has a significant advantage over other three algorithms. HPA and GA do not do any optimization according to the channel states and the power levels of the batteries. All energy stored in the batteries is consumed in GA, and its performance is the worst among the three algorithms for the system with a jammer. HPA uses half of the energy stored in the batteries in the current TS, and the remained energy is reserved for the future, so its performance is superior to that of GA. Compared with WJA, an extra jammer sends AN in the system of the proposed algorithm. The simulation results show that the achievable secrecy rate can be significant improved by elaborately controlling the power of AN.

Figure 2.

Time-averaged secrecy rate versus TS.

Figure 3 shows the time trajectory of the power levels of the batteries of the four algorithms. At the beginning of simulation, all batteries are fully charged. Figure 3 shows that although the energy can be supplemented in the transmission period, the energy stored in the two batteries in HPA and GA drops to a very low level (the level is lower in GA) in a short time and then remains at this level, while those in other two algorithms can be maintained around a middle level (there is no jammer in WJA).

Figure 3.

Time trajectory of the power levels of the batteries: (a) source node’s battery and (b) jammer’s battery.

Effects of the system parameters

In this section, we evaluate the effects of the system parameters on the performance of the proposed algorithm. The simulation period is T = 10⁵ TS.

Figure 4 shows the effects of energy arrival rates λ_s and λ_j at the long-term time-averaged secrecy rate. It can be found that as the energy arrival rate λ_s at the source node increases, the secrecy rate increases. This is because the more energy arriving per-TS, the more energy is available to transmit data, and the higher transmission rate can be achieved. With the increase in the arrival rate λ_j at the jammer, the secrecy rate first increases and then remains. It’s because when the energy arrival rate is small, the average power of AN increases with the increase in the arrival rate λ_j, which is beneficial for the promotion of the secrecy rate. As the analysis in section “Solution of the optimization problem” shows, there is no need to transmit AN or the optimal power of AN is small in many cases. So, when the energy arrival rate λ_j is large enough, the secrecy rate no longer increases with the increase in the arrival rate λ_j.

Figure 4.

Long-term time-averaged secrecy rate versus energy arrival rates.

Figure 5 shows the effects of the offsets of energy virtual queues δ_s and δ_j on the system performance. The purpose of setting the offsets is to ensure that the power level of the battery fluctuates around a certain level to accommodate random variations in EH and channel quality. By setting the offsets, in each TS, enough energy is stored in the battery, which can support the transmission of information or AN, while there is enough free space to store the harvested energy. It can be seen from Figure 5(a) that as the offset of the energy virtual queues at the source node δ_s increases, the average secrecy rate increases first and then decreases slightly. When δ_s is small, the average amount of power reserved in the source node battery is small too. According the water-filling theory, the better the channel quality is, the higher the transmission power should be. Because the energy stored in the battery is not enough, the high transmission power cannot be supported, and the capacity of the channel is not fully utilized. So, the average secrecy rate is low. As δ_s increases, the energy stored in the battery increases too, which results in the increase in the secrecy rate. Since the available energy is also limited by the amount of the harvested energy, the average secrecy rate does not always increase with the increase in the δ_s. However, the increase in the δ_s leads to the decrease in the remaining capacity of the battery, and the possibility increases that the amount of the harvested energy exceeds the remaining capacity of the battery (i.e. energy overflow). Some harvested energy is discarded once the energy overflow occurs, and it is harmful to the system performance. So, the average secrecy rate decreases slightly with the increase in the δ_s when it is larger than 4.6. The secrecy rate only increases slightly as δ_j increases. The reason is that the energy consumption of AN is lower than that of the signal, and the change of the energy stored in the jammer’s battery has a small impact on the system performance. It can be seen from Figure 5(b) and (c) that the time-averaged battery energy saved by the source node and jammer is near the offset.

Figure 5.

Effect of energy virtual queue offsets δ_s and δ_j on system performance: (a) long-term time-averaged secrecy rate, (b) long-term time-averaged power level of the source node’s battery, and (c) long-term time-averaged power level of the jammer’s battery.

Figure 6 shows the effects of weights U and V on the system performance. Figure 6(a) gives the effect on long-term time-averaged secrecy rate. Figure 6(b) and (c) gives the effects on the standard deviations of the power level of the two batteries, that is, $\overline{\mathrm{\Delta }{E}_i}=\sqrt{(1/T)\sum _{t=0}^{T-1}{(E_i(t)-{\overline{E}}_i)}^2}(i=\mathrm{sb},\mathrm{jb})$ , which reflects the fluctuation of the power level. Figure 6(d) gives the effect on the normalized root mean square (RMS) of the difference between the secrecy rates of the two destination nodes, that is, $\overline{\mathrm{\Delta }{R}_{\mathrm{s}}}=\sqrt{(1/T)\sum _{t=0}^{T-1}{({\overline{R}}_1(t)-{\overline{R}}_2(t))}^2}/{\overline{R}}_{\mathrm{s}}$ . The results show that the time-averaged secrecy rate increases as V increases, while the fluctuation of the battery power increases. The increases in the V makes the optimization focus more on the optimization of the secrecy rate, while that of the stability of battery power declines. The increases in the U leads to a slightly decrease in the average secrecy rate. When U increases, more consideration is given to the fairness between the two destination nodes. It can be clearly seen from Figure 6(d) that the average transmission rate difference between the two destination nodes decreases with the increase in the U, which confirms that the transmission fairness can be improved by adjusting the penalty weight.

Figure 6.

Effect of weights U and V on system performance: (a) long-term time-averaged secrecy rate, (b) standard deviations of the power level of the source node, (c) standard deviations of the power level of the jammer, and (d) normalized RMS of the difference between the secrecy rates of the two destination nodes.

Figure 7 shows the effects of upper and lower limits of weight V_max and V_min on the system performance. Figure 7(a) gives the effect on long-term time-averaged secrecy rate, and Figure 7(b) gives the effect on $\overline{\mathrm{\Delta }{R}_{\mathrm{s}}}$ . The increase in the V_max or the decrease in the V_min expands the adjustment range of weight V, which is beneficial to the reduction of the difference between the secrecy rates of the two destination nodes, but the long-term time-averaged secrecy rate slightly declines.

Figure 7.

Effect of V_max and V_min on system performance: (a) long-term time-averaged secrecy rate and (b) normalized RMS of the difference between the secrecy rates of the two destination nodes.

Conclusion

This article studies the power control problem in an EH wireless communication system with a cooperative jamming node. The system model includes an EH source node, an EH jamming node, and two destination nodes. The information sent to one destination node is kept secret from the other destination node. Per-TS, based on channel states and battery states, the source node selects the destination node as the legitimate receiver which can achieve a lower drift-plus-penalty function and sends the confidential information to it. In order to utilize the harvested energy efficiently, the signal power and AN power are jointly controlled to minimize the drift-plus-penalty function. The original optimization problem is transformed into a per-TS optimization problem using the Lyapunov optimization framework, and the decision of power control is only based on the current system states. The constraint on the battery operation is converted into the stability constraint of the energy virtual queues. In addition, the fairness between the transmissions of the two destination nodes’ information is also considered in the optimization process. The optimal power pair of the signal and AN in different cases is analyzed, and the solutions are given. The simulation results show that the proposed algorithm can efficiently utilize the harvested energy and a higher averaged secrecy rate can be achieved compared with two comparison algorithms.