# Bandwidth Allocation of Cognitive Relay Networks with Energy Harvesting for Smart Grid

Authors: Yupeng Li; Zihao Wang; Ling Luo; Zhiyong Chen; Bin Xia; Hao Luo
Journal: Journal of Computer Networks and Communications
 (2019)

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## Abstract

In this paper, we investigate an energy harvesting scheme in a smart grid based on the cognitive relay protocol, where a primary transmitter scavenges energy from the nature sources and then employs the harvested energy to forward the primary signal. Depending on the intensity of the energy harvesting from nature, a secondary user dynamically acts as a relay node to assist the primary transmission or does not. When the energy is not enough powerful to support the direct transmission between two primary users, the secondary users share the spectrum by assisting the primary transmission. For the relaying scheme, both amplify-and-forward (AF) and decode-and-forward (DF) protocols are investigated. We analytically obtain the exact transmission rates for both primary and secondary networks and derive the exact expressions of the system outage probabilities for both primary and secondary users in the smart grid. Moreover, we develop the analytically optimal bandwidth allocation strategy to maximize the total sum rate of the proposed scheme. Numerical results are presented to demonstrate the performance gain of the proposed scheme over the nonoptimal scheme.


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## Body

## 1. Introduction

Spectral efficiency has attracted much attention in wireless smart grid communications networks due to the increasing demands of wireless services [1]. It has been demonstrated that cognitive radio is a promising way to improve the efficiency of the radio spectrum by allowing spectrum sharing [2–4]. On the contrary, energy efficiency is another aspect worth considering in wireless smart grid systems [5–7]. In order to lower carbon emissions, reducing energy consumption from wireless systems has become more and more important. As a result, combining energy harvesting technology with cognitive radio networks offers a new opportunity to improve both spectral and energy efficiency for the smart grid [8].In smart grid, harvesting energy from nature such as solar, radio frequency, or wind provides a new free source of energy for the wireless nodes. With nature energy harvesting technique, wireless devices can utilize the free renewable energy sources to improve the energy efficiency [9]. In the field of cognitive networks, the energy harvesting has been developed in [10–14]. In [10], the authors investigated the relationship between the optimal sensing threshold and sensing duration in order to maximize the average throughput of the secondary network in a cognitive radio. Park and Hong [11] applied energy harvesting to the cognitive radio network in which the secondary transmitter harvests energy from ambient sources. The theoretical upper bound on the maximum achievable throughput of secondary transmitter was derived, which is a function of the temporal correlation of the primary traffic and the energy arrival rate. In [12], a cognitive radio system was considered. Specifically, an energy harvesting secondary user attempts to access the primary channel randomly. The impact of the multipacket reception capability, the primary queueing delay constraints, and the energy queue arrivals on the maximum secondary throughput was investigated. Park and Hong [13] provided an optimal spectrum sensing policy for an energy harvesting cognitive radio which is in purpose of maximizing the expected total throughput. Besides, an approximate solution to the optimal detection threshold was also presented. Similar researches include two-way channel [15], two-user MIMO interference channels [16], and cognitive radio systems [17–23].In this paper, considering the massive potential of harvesting energy from nature resources, we adopt this technique to cognitive radio network where we assume the primary user lacks of energy and thus harvests energy from the nature resources. Due to the fluctuation of energy harvesting from nature sources, we model and propose an energy harvesting and relaying protocol with dynamic bandwidth allocation in cognitive relay networks. The protocol enables a secondary user dynamically sharing the primary spectrum by assisting the primary transmission if the direction transmission between two primary users cannot be effectively realized. In particular, the energy harvesting primary user first scavenges energy from ambient sources and then decides to cooperate with the secondary network or transmit information directly to its destination based on the intensity of the harvesting energy. If the direct transmission data rate meets the basic threshold requirement, the direct transmission between two primary users occurs and some bandwidth is allocated to the secondary network. If the direct transmission data rate is below the threshold, the secondary user acts as the relay to assist the primary transmission and the secondary network can share some bandwidth for its own transmission in return. We first derive the analytical expressions of the data rates for the primary and secondary networks. Accordingly, the system outage probabilities for PD and SD are then derived for different transmission schemes. In order to maximize the sum data rate, the optimal bandwidth allocation strategies are analyzed and proposed. Furthermore, the correspondingly analytical policies are derived to solve the optimization problems. Finally, numerical results provide us with the valuable insights into the performance under various system parameters, i.e., the energy arrival rate and the location of the relay.An outline of the remainder of the paper is as follows. Section2 describes the system model for the smart grid with the cooperative of cognitive radio technology and discusses the transmission protocol. Section 3 derives the system outage probabilities for PD and SD. Section 4 formulates the optimal bandwidth allocation for maximizing the sum rate and obtains the optimal allocation policy. The numerical results are presented in Section 5, and the conclusions are drawn in Section 6.


## 2. System Model and Protocol Description

In this section, we begin by describing the smart grid system with the cognitive relay model based on energy harvesting.


### 2.1. System Model

We consider a cognitive network which is composed of four nodes, as shown in Figure1. Let us denote the primary source node as “PS,” the primary destination node as “PD,” the secondary source node as “SS,” and the secondary destination node as “SD.” The PS is capable of harvesting energy from nature resources and has a rechargeable battery which is modeled as a queue. Eu is the amount of Joules per energy packet. We assume that PS has an infinite length buffer to store the arrived energy packets which is a reasonable assumption if the packet length is very small compared with the buffer’s size [21].Figure 1
System model of the energy harvesting and relaying protocol for the cognitive relay networks.We suppose that the secondary user SS has a constant power supply, i.e.,P, while there is no fixed energy provided for the primary node PS, and it thus requires harvesting energy from the nature resources. We adopt a Rayleigh flat fading channel model, and assume that the channel gains remain constant over the duration of a time slot, T. Besides, the channel state information (CSI) is perfect known by the transmitting terminals PS and SS (we can use a short CSI training phase prior to the transmission to obtain the CSI. For example, PD and SD can estimate the channel gains based on the pilot transmitted from PS and SS in a predetermined sequence and then broadcast their corresponding channel gains to PS and SS. This short CSI training phase overhead is reasonably negligible in an assumed quasi-static fading channel). Let h1, h2, h3, and h4 be the channel coefficient between PS and SS, SS and PD, PS and PD, and SS and SD, respectively. We thus have hi∼CN0,Ωi for i=1,2,3,4. The joint instantaneous channel coefficient is denoted by h≜h1,h2,h3,h4. We also assume that σ02 is the additive white Gaussian (AWGN) noise with power per unit frequency. The spectrum licensed to the primary network is denoted as W.The time slotT is divided into small intervals, which has a length of δ. We then have n=T/δ mini slots. Besides, the probability of receiving one energy packet within a mini slot is assumed to be p. Therefore, the number of harvested nature energy packets during one time slot T meets Bernoulli distribution with parameters n and p. If n is large enough and p is small enough, the number of harvested nature energy packets within one time slot T can be approximated as Poisson distribution with a parameter λeT=np, where λe is the energy arrival rate. The probability of harvesting i packets from nature resources within one time slot T is expressed as follows:(1)pIi=λeTiexp−λeTi!,i=0,1,2,…n.Accordingly, the mean transmitted power can be expressed as(2)PS=∑i=0niEupIiT.


### 2.2. Energy Harvesting and Relaying Protocol

Because of the nature fluctuation, PS sometimes harvests high energy, and sometimes, it harvests low energy from the nature sources. Accordingly, we propose the following energy harvesting and relaying protocol:(i)
When the harvesting energy is powerful enough to support the direct transmission between two primary nodes, the primary network operates in the direct transmission mode andw1 spectrum will be allocated to the primary network while the rest to the secondary network. Therefore, the transmission consists of only one phase which lasts for one slot time T.(ii)
If the data rate of the direct transmission between two primary nodes cannot meet the threshold requirement, the secondary node SS acts as the relay to assist the primary transmission. As return, the primary network allocates some bandwidth to the secondary network for its own transmission and the rest bandwidthw1 for the primary transmission. Here, we assume that SS always has data to transmit. In this scenario, the transmission operates in two phases. During the first phase, PS exploits all the harvested energy to transmit the information to SS. During the second phase, SS forwards the received signals along with the information intended for SD based on the AF or DF scheme. In this paper, we assume that each phase is T/2.


#### 2.2.1. AF Scheme

At the beginning of each time slot, PS ascertains the availability of direct transmission between two primary users, and the test function is given as(3)R=Wlog21+PSh32Wσ02,where Wσ02 follows the fact that the unit of power PS is watt, but the unit of σ02 is watt per unit frequency.IfR exceeds the primary data rate threshold Rth, then w1 spectrum is allocated to the primary network and PS sends its information to PD without the assistance of SS. In this scenario, the secondary network has w2 spectrum for its transmission, where w1+w2=W. Therefore, the data rate for primary and secondary network is given by(4)RPDAF=w1log21+PSh32w1σ02,RSDAF=w2log21+Ph42w2σ02.IfR is below Rth, SS acts as the relay to assist the primary transmission. Bandwidth w1 is allocated to the primary network, while rest bandwidth w2 is allocated to the secondary network. During the first phase, the primary node PS sends its information to the relay SS. The received signal of SS in frequency band w1 can thus be formulated as(5)ySS=2PSh1x1+n1,where x1 is the unit-power signal intended for PD. n1∼CN0,σ12 denotes the narrow-band AWGN introduced by the antenna at SS [22]. The coefficient 2 is due to the fact that the time duration of each phase is T/2.During the second phase, SS uses half of its power to broadcast the received primary signals and employs the rest power to transfer signals intended for SD. The information transmitted in frequency bandw1 is given by(6)xSS=β12PySS+nb1,where nb1∼CN0,σb12 denotes AWGN introduced by the signal conversion from passband to baseband at SS, and the coefficient 1/2 is account of the half power allocated to the primary transmission. The power normalization factor β of the relay SS is(7)β=12PSh12+w1σ12+σb12≈12PSh12,where the approximation is tight at high SNR.The corresponding received signal at primary user PD in frequency bandw1 can be expressed as(8)yPD=h2xSS+n2=PPSβh1h2x1+12Pβh2n1+12Pβh2nb1+n2,where n2∼CN0,σ22 denotes the AWGN at PD.Accordingly, we the data rates for PD and SD, respectively, as follows:(9)RPDAF=w12log21+2PPSh12h22Ph22w1σb12+σ12+4PSh12w1σ22,(10)RSDAF=12w2log21+Ph422w2σ02.In fact, the noise introduced by the received antenna is much smaller than that of the baseband noise power. Without loss of generality, we assume the antenna noise power to be zero [23], i.e., σ12=0. Consequently, we have σ22=σb12=σ02. As a result, we rewrite the data rate of PD as follows:(11)RPDAF=12w1log21+2PPSh12h22w1σ02Ph22+4PSh12.


#### 2.2.2. DF Scheme

IfR meets need of the primary rate threshold Rth, the data rate for the primary and secondary network are given by(12)RPDDF=w1log21+PSh32w1σ02,RSDDF=w2log21+Ph42w2σ02,respectively.IfR is below the primary rate threshold, the transmission mechanism is the same with the AF scheme. The data rate achieved at PD and SD can be given by(13)RPDDF=minR1,R2,(14)RSDDF=12w2log21+Ph422w2σ02,where R1=1/2w1log21+2PSh12/w1σ02 denotes the rate from PS to SS in the first phase and R2=1/2w1log21+Ph22/2w1σ02 denotes the rate from SS to PD in the second phase.


## 2.1. System Model

We consider a cognitive network which is composed of four nodes, as shown in Figure1. Let us denote the primary source node as “PS,” the primary destination node as “PD,” the secondary source node as “SS,” and the secondary destination node as “SD.” The PS is capable of harvesting energy from nature resources and has a rechargeable battery which is modeled as a queue. Eu is the amount of Joules per energy packet. We assume that PS has an infinite length buffer to store the arrived energy packets which is a reasonable assumption if the packet length is very small compared with the buffer’s size [21].Figure 1
System model of the energy harvesting and relaying protocol for the cognitive relay networks.We suppose that the secondary user SS has a constant power supply, i.e.,P, while there is no fixed energy provided for the primary node PS, and it thus requires harvesting energy from the nature resources. We adopt a Rayleigh flat fading channel model, and assume that the channel gains remain constant over the duration of a time slot, T. Besides, the channel state information (CSI) is perfect known by the transmitting terminals PS and SS (we can use a short CSI training phase prior to the transmission to obtain the CSI. For example, PD and SD can estimate the channel gains based on the pilot transmitted from PS and SS in a predetermined sequence and then broadcast their corresponding channel gains to PS and SS. This short CSI training phase overhead is reasonably negligible in an assumed quasi-static fading channel). Let h1, h2, h3, and h4 be the channel coefficient between PS and SS, SS and PD, PS and PD, and SS and SD, respectively. We thus have hi∼CN0,Ωi for i=1,2,3,4. The joint instantaneous channel coefficient is denoted by h≜h1,h2,h3,h4. We also assume that σ02 is the additive white Gaussian (AWGN) noise with power per unit frequency. The spectrum licensed to the primary network is denoted as W.The time slotT is divided into small intervals, which has a length of δ. We then have n=T/δ mini slots. Besides, the probability of receiving one energy packet within a mini slot is assumed to be p. Therefore, the number of harvested nature energy packets during one time slot T meets Bernoulli distribution with parameters n and p. If n is large enough and p is small enough, the number of harvested nature energy packets within one time slot T can be approximated as Poisson distribution with a parameter λeT=np, where λe is the energy arrival rate. The probability of harvesting i packets from nature resources within one time slot T is expressed as follows:(1)pIi=λeTiexp−λeTi!,i=0,1,2,…n.Accordingly, the mean transmitted power can be expressed as(2)PS=∑i=0niEupIiT.


## 2.2. Energy Harvesting and Relaying Protocol

Because of the nature fluctuation, PS sometimes harvests high energy, and sometimes, it harvests low energy from the nature sources. Accordingly, we propose the following energy harvesting and relaying protocol:(i)
When the harvesting energy is powerful enough to support the direct transmission between two primary nodes, the primary network operates in the direct transmission mode andw1 spectrum will be allocated to the primary network while the rest to the secondary network. Therefore, the transmission consists of only one phase which lasts for one slot time T.(ii)
If the data rate of the direct transmission between two primary nodes cannot meet the threshold requirement, the secondary node SS acts as the relay to assist the primary transmission. As return, the primary network allocates some bandwidth to the secondary network for its own transmission and the rest bandwidthw1 for the primary transmission. Here, we assume that SS always has data to transmit. In this scenario, the transmission operates in two phases. During the first phase, PS exploits all the harvested energy to transmit the information to SS. During the second phase, SS forwards the received signals along with the information intended for SD based on the AF or DF scheme. In this paper, we assume that each phase is T/2.


### 2.2.1. AF Scheme

At the beginning of each time slot, PS ascertains the availability of direct transmission between two primary users, and the test function is given as(3)R=Wlog21+PSh32Wσ02,where Wσ02 follows the fact that the unit of power PS is watt, but the unit of σ02 is watt per unit frequency.IfR exceeds the primary data rate threshold Rth, then w1 spectrum is allocated to the primary network and PS sends its information to PD without the assistance of SS. In this scenario, the secondary network has w2 spectrum for its transmission, where w1+w2=W. Therefore, the data rate for primary and secondary network is given by(4)RPDAF=w1log21+PSh32w1σ02,RSDAF=w2log21+Ph42w2σ02.IfR is below Rth, SS acts as the relay to assist the primary transmission. Bandwidth w1 is allocated to the primary network, while rest bandwidth w2 is allocated to the secondary network. During the first phase, the primary node PS sends its information to the relay SS. The received signal of SS in frequency band w1 can thus be formulated as(5)ySS=2PSh1x1+n1,where x1 is the unit-power signal intended for PD. n1∼CN0,σ12 denotes the narrow-band AWGN introduced by the antenna at SS [22]. The coefficient 2 is due to the fact that the time duration of each phase is T/2.During the second phase, SS uses half of its power to broadcast the received primary signals and employs the rest power to transfer signals intended for SD. The information transmitted in frequency bandw1 is given by(6)xSS=β12PySS+nb1,where nb1∼CN0,σb12 denotes AWGN introduced by the signal conversion from passband to baseband at SS, and the coefficient 1/2 is account of the half power allocated to the primary transmission. The power normalization factor β of the relay SS is(7)β=12PSh12+w1σ12+σb12≈12PSh12,where the approximation is tight at high SNR.The corresponding received signal at primary user PD in frequency bandw1 can be expressed as(8)yPD=h2xSS+n2=PPSβh1h2x1+12Pβh2n1+12Pβh2nb1+n2,where n2∼CN0,σ22 denotes the AWGN at PD.Accordingly, we the data rates for PD and SD, respectively, as follows:(9)RPDAF=w12log21+2PPSh12h22Ph22w1σb12+σ12+4PSh12w1σ22,(10)RSDAF=12w2log21+Ph422w2σ02.In fact, the noise introduced by the received antenna is much smaller than that of the baseband noise power. Without loss of generality, we assume the antenna noise power to be zero [23], i.e., σ12=0. Consequently, we have σ22=σb12=σ02. As a result, we rewrite the data rate of PD as follows:(11)RPDAF=12w1log21+2PPSh12h22w1σ02Ph22+4PSh12.


### 2.2.2. DF Scheme

IfR meets need of the primary rate threshold Rth, the data rate for the primary and secondary network are given by(12)RPDDF=w1log21+PSh32w1σ02,RSDDF=w2log21+Ph42w2σ02,respectively.IfR is below the primary rate threshold, the transmission mechanism is the same with the AF scheme. The data rate achieved at PD and SD can be given by(13)RPDDF=minR1,R2,(14)RSDDF=12w2log21+Ph422w2σ02,where R1=1/2w1log21+2PSh12/w1σ02 denotes the rate from PS to SS in the first phase and R2=1/2w1log21+Ph22/2w1σ02 denotes the rate from SS to PD in the second phase.


## 2.2.1. AF Scheme

At the beginning of each time slot, PS ascertains the availability of direct transmission between two primary users, and the test function is given as(3)R=Wlog21+PSh32Wσ02,where Wσ02 follows the fact that the unit of power PS is watt, but the unit of σ02 is watt per unit frequency.IfR exceeds the primary data rate threshold Rth, then w1 spectrum is allocated to the primary network and PS sends its information to PD without the assistance of SS. In this scenario, the secondary network has w2 spectrum for its transmission, where w1+w2=W. Therefore, the data rate for primary and secondary network is given by(4)RPDAF=w1log21+PSh32w1σ02,RSDAF=w2log21+Ph42w2σ02.IfR is below Rth, SS acts as the relay to assist the primary transmission. Bandwidth w1 is allocated to the primary network, while rest bandwidth w2 is allocated to the secondary network. During the first phase, the primary node PS sends its information to the relay SS. The received signal of SS in frequency band w1 can thus be formulated as(5)ySS=2PSh1x1+n1,where x1 is the unit-power signal intended for PD. n1∼CN0,σ12 denotes the narrow-band AWGN introduced by the antenna at SS [22]. The coefficient 2 is due to the fact that the time duration of each phase is T/2.During the second phase, SS uses half of its power to broadcast the received primary signals and employs the rest power to transfer signals intended for SD. The information transmitted in frequency bandw1 is given by(6)xSS=β12PySS+nb1,where nb1∼CN0,σb12 denotes AWGN introduced by the signal conversion from passband to baseband at SS, and the coefficient 1/2 is account of the half power allocated to the primary transmission. The power normalization factor β of the relay SS is(7)β=12PSh12+w1σ12+σb12≈12PSh12,where the approximation is tight at high SNR.The corresponding received signal at primary user PD in frequency bandw1 can be expressed as(8)yPD=h2xSS+n2=PPSβh1h2x1+12Pβh2n1+12Pβh2nb1+n2,where n2∼CN0,σ22 denotes the AWGN at PD.Accordingly, we the data rates for PD and SD, respectively, as follows:(9)RPDAF=w12log21+2PPSh12h22Ph22w1σb12+σ12+4PSh12w1σ22,(10)RSDAF=12w2log21+Ph422w2σ02.In fact, the noise introduced by the received antenna is much smaller than that of the baseband noise power. Without loss of generality, we assume the antenna noise power to be zero [23], i.e., σ12=0. Consequently, we have σ22=σb12=σ02. As a result, we rewrite the data rate of PD as follows:(11)RPDAF=12w1log21+2PPSh12h22w1σ02Ph22+4PSh12.


## 2.2.2. DF Scheme

IfR meets need of the primary rate threshold Rth, the data rate for the primary and secondary network are given by(12)RPDDF=w1log21+PSh32w1σ02,RSDDF=w2log21+Ph42w2σ02,respectively.IfR is below the primary rate threshold, the transmission mechanism is the same with the AF scheme. The data rate achieved at PD and SD can be given by(13)RPDDF=minR1,R2,(14)RSDDF=12w2log21+Ph422w2σ02,where R1=1/2w1log21+2PSh12/w1σ02 denotes the rate from PS to SS in the first phase and R2=1/2w1log21+Ph22/2w1σ02 denotes the rate from SS to PD in the second phase.


## 3. System Outage Probability Analysis

In this section, we investigate the system outage performance in Rayleigh fading environments. If the transmission rates of PD or SD is lower than the given target rate, the system occurs an outage event. Therefore, we have the following proposition for the primary user PD.Proposition 1.
Leta1=w1σ02/2PS, b1=2w1σ02/P, and c1=1/22Rth/w1−1. Given a target rate Rth, the system outage probability for the primary user PD with the AF scheme can be expressed as
(15)
P
out
PD,AF
=
1
−
exp
−
2
R
th
/
W
−
1
W
σ
0
2
P
S
Ω
3
1
−
∫
0
c
1
/
b
1
1
y
2
Ω
2
exp
−
1
y
Ω
2
exp
−
a
1
Ω
1
c
1
−
b
1
y
d
y
+
exp
−
2
R
th
/
W
−
1
W
σ
0
2
P
S
Ω
3
1
−
exp
−
2
R
th
/
w
1
−
1
w
1
σ
0
2
P
S
Ω
3
.
And for the DF scheme, we have
(16)
P
out
PD,DF
=
exp
−
2
R
th
/
W
−
1
W
σ
0
2
P
S
Ω
3
1
−
exp
−
2
R
th
/
w
1
−
1
w
1
σ
0
2
P
S
Ω
3
+
1
−
exp
−
2
R
th
/
W
−
1
W
σ
0
2
P
S
Ω
3
1
−
exp
−
a
1
c
1
Ω
1
exp
−
b
1
c
1
Ω
2
.Proof.
Accordingly, we have the outage probability
(17)
P
out
PD,AF
=
Pr
R
PD
AF
<
R
th
=
Pr
R
PD
AF
<
R
th
∣
R
≤
R
th
+
Pr
R
PD
AF
<
R
th
∣
R
>
R
th
,
(18)
P
out
PD,AF
=
Pr
W
log
2
1
+
P
S
h
3
2
W
σ
0
2
≤
R
th
Pr
1
2
w
1
log
2
1
+
2
P
P
S
h
1
2
h
2
2
w
1
σ
0
2
P
h
2
2
+
4
P
S
h
1
2
<
R
th
+
Pr
W
log
2
1
+
P
S
h
3
2
W
σ
0
2
>
R
th
Pr
w
1
log
2
1
+
P
S
h
3
2
w
1
σ
0
2
<
R
th
.
LetX=1/h12 and Y=1/h22, where the corresponding probability density function (PDF) can be presented as(19)PrXx=1Ω1x2exp−1Ω1x,PrYy=1Ω2y2exp−1Ω2y.
Therefore, we have(20)Pr12w1log21+2PPSh12h22w1σ02Ph22+4PSh12<Rth=Pra1X+b1Y>c1=1−∫0c1/b1∫0c1−b1y/a1PrXxPrYydxdy=1−∫0c1/b11y2Ω2exp−1yΩ2exp−a1Ω1c1−b1ydy.Based on (17) and (20), we can easily to obtain the proposition.Similarly, the system outage probability for the secondary user SD is given by the following proposition.Proposition 2.
For both the AF and DF scheme, the system outage probability for the secondary user SD is(21)PoutSD=exp−2Rth/W−1Wσ02PSΩ31−exp−2Rth/w2−1w2σ02PΩ4+1−exp−2Rth/W−1Wσ02PSΩ31−exp−22Rth/w2−12w2σ02PΩ4.Proof.
Similar to the proof of Proposition 1, we can have this proposition.


## 4. Optimization Problem: Sum Data Rate

In this section, we attempt to maximize the sum rate of the whole system for both the AF and DF scheme. However, this depends on the value ofR. If R exceeds the data threshold, then the exact value is given by (4) and (12). If R is below the threshold, the primary network will allocate some bandwidth to the secondary network for its own transmission. The data rate in (10), (11), (13), and (14) depends on bandwidth distribution.


### 4.1. AF Scheme

The Optimization Problem 1 (OP1) for the AF scheme can be formulated as(22)OP1:maxw1h,λe,w2h,λeRPDAF+RSDAF,s.t.1w1h,λe+w2h,λe=W,20≤w1h,λe,w2h,λe≤W,3RPDAF≥Rth,where we use w1h,λe and w2h,λe to replace w1 and w2 since w1 and w2 are in dynamic adjustment in correspondence with the instantaneous channel coefficients and energy arrival rate.In order to simplify the expression ofRPDAF and RSDAF, we rewrite RPDAF+RSDAF as follows:(23)w1h,λe2log21+φ1w1h,λe+w2h,λe2log21+ν1w2h,λe,R≤Rth,w1h,λelog21+φ3w1h,λe+w2h,λelog21+2ν1w2h,λe,R>Rth.where φ1, φ3, and ν1 are given in Table 1.(1)
R≤Rth: first, we do not consider the restricted condition of RPDAF≥Rth. Then, we use Giwih,λe to represent the data rate of each network user, where Giwih,λe≜1/2wih,λelog21+mi/wih,λe is a concave and increasing function of wih,λe and mi is a constant, i.e., m1=φ1andm2=ν1, i=1,2. The first part of (23) resembles the classical water-filling power distribution problem [24]. As a result, we obtain the requirement for the optimal solution of the problem, denoted by wiAF†bh,λe:(24)∂G1w1h,λe∂w1h,λew1h,λe=w1AF†bh,λe=∂G2w2h,λe∂w2h,λew2h,λe=w2AF†bh,λe,and we have(25)∂Giwih,λe∂wih,λewih,λe=wiAF†bh,λe=12log21+miwiAF†bh,λe−miln2wiAF†bh,λe+mi=YmiwiAF†bh,λe,where the function Yx≜1/2log21+x−x/2ln21+x is monotonically increasing. From (24), we obtain that(26)m1w1AF†bh,λe=m2w2AF†bh,λe,as w1AF†bh,λe+w2AF†bh,λe=W. Furthermore, from (26), we have the optimal solution for bandwidth allocation:(27)w1AF†bh,λe=m1Wm1+m2=4PSWh12h22Ph22+4PSh12h42+4PSh12h22,where, we take the restricted condition of RPDAF≥Rth into consideration. G1w1h,λe must exceed the threshold Rth, so we suppose that w1AF‡bh,λe is the solution to G1w1h,λe=Rth. Since G1w1h,λe is increasing in w1h,λe, the optimal w1AF∗bh,λe must exceed w1AF‡bh,λe. As a result, we obtain the optimal bandwidth for the primary network:(28)w1AF∗bh,λe=maxw1AF†bh,λe,w1AF‡bh,λe.(2)
R>Rth: similar to the situation of R≤Rth, we have the optimal bandwidth for the primary network:(29)w1AF∗oh,λe=maxw1AF†oh,λe,w1AF‡oh,λe,where(30)w1AF†oh,λe=φ3Wφ3+2ν1=PSWh32PSh32+Ph4,w1AF‡oh,λe is the solution to w1h,λelog21+φ3/w1h,λe=Rth.Table 1
Symbol notations.
Symbol
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### 4.2. DF Scheme

The Optimization Problem 2 (OP2) for the DF scheme can be formulated as(31)OP2:maxw1h,λe,w2h,λeRPDDF+RSDDF,s.t.1w1h,λe+w2h,λe=W,20≤w1h,λe,w2h,λe≤W,3RPDDF≥Rth.For the sake of simplicity, we rewriteRPDDF+RSDDF as follows:(32)minR1,R2+12w2h,λelog21+ν1w2h,λe,R≤Rth,w1h,λelog21+φ3w1h,λe+w2h,λelog21+2ν1w2h,λe,R>Rth,where(33)R1=12w1h,λelog21+ν2w1h,λe,R2=12w1h,λelog21+φ2w1h,λe,where φ2 and ν2 are given in Table 1.(1)
R≤Rth: similar to the AF scheme, we obtain the optimal bandwidth allocation policy for the primary network user:(34)w1DF∗bh,λe=maxw1DF†bh,λe,w1DF‡bh,λe,where(35)w1DF†bh,λe=minw1DF†1bh,λe,w1DF†2bh,λe,w1DF†1bh,λe=ν2ν2+ν1=4PSh124PSh12+ph42,w1DF†2bh,λe=φ2φ2+ν1=h22h22+h42,w1DF‡bh,λe is the solution to minR1,R2=Rth.(2)
R>Rth: the optimal bandwidth allocation solution is the same as the AF scheme:(36)w1DF∗oh,λe=maxw1DF†oh,λe,w1DF‡oh,λe,where(37)w1DF†oh,λe=PSWh32PSh32+Ph42,w1DF‡oh,λe is the solution to w1h,λelog21+φ3/w1h,λe=Rth.


## 4.1. AF Scheme

The Optimization Problem 1 (OP1) for the AF scheme can be formulated as(22)OP1:maxw1h,λe,w2h,λeRPDAF+RSDAF,s.t.1w1h,λe+w2h,λe=W,20≤w1h,λe,w2h,λe≤W,3RPDAF≥Rth,where we use w1h,λe and w2h,λe to replace w1 and w2 since w1 and w2 are in dynamic adjustment in correspondence with the instantaneous channel coefficients and energy arrival rate.In order to simplify the expression ofRPDAF and RSDAF, we rewrite RPDAF+RSDAF as follows:(23)w1h,λe2log21+φ1w1h,λe+w2h,λe2log21+ν1w2h,λe,R≤Rth,w1h,λelog21+φ3w1h,λe+w2h,λelog21+2ν1w2h,λe,R>Rth.where φ1, φ3, and ν1 are given in Table 1.(1)
R≤Rth: first, we do not consider the restricted condition of RPDAF≥Rth. Then, we use Giwih,λe to represent the data rate of each network user, where Giwih,λe≜1/2wih,λelog21+mi/wih,λe is a concave and increasing function of wih,λe and mi is a constant, i.e., m1=φ1andm2=ν1, i=1,2. The first part of (23) resembles the classical water-filling power distribution problem [24]. As a result, we obtain the requirement for the optimal solution of the problem, denoted by wiAF†bh,λe:(24)∂G1w1h,λe∂w1h,λew1h,λe=w1AF†bh,λe=∂G2w2h,λe∂w2h,λew2h,λe=w2AF†bh,λe,and we have(25)∂Giwih,λe∂wih,λewih,λe=wiAF†bh,λe=12log21+miwiAF†bh,λe−miln2wiAF†bh,λe+mi=YmiwiAF†bh,λe,where the function Yx≜1/2log21+x−x/2ln21+x is monotonically increasing. From (24), we obtain that(26)m1w1AF†bh,λe=m2w2AF†bh,λe,as w1AF†bh,λe+w2AF†bh,λe=W. Furthermore, from (26), we have the optimal solution for bandwidth allocation:(27)w1AF†bh,λe=m1Wm1+m2=4PSWh12h22Ph22+4PSh12h42+4PSh12h22,where, we take the restricted condition of RPDAF≥Rth into consideration. G1w1h,λe must exceed the threshold Rth, so we suppose that w1AF‡bh,λe is the solution to G1w1h,λe=Rth. Since G1w1h,λe is increasing in w1h,λe, the optimal w1AF∗bh,λe must exceed w1AF‡bh,λe. As a result, we obtain the optimal bandwidth for the primary network:(28)w1AF∗bh,λe=maxw1AF†bh,λe,w1AF‡bh,λe.(2)
R>Rth: similar to the situation of R≤Rth, we have the optimal bandwidth for the primary network:(29)w1AF∗oh,λe=maxw1AF†oh,λe,w1AF‡oh,λe,where(30)w1AF†oh,λe=φ3Wφ3+2ν1=PSWh32PSh32+Ph4,w1AF‡oh,λe is the solution to w1h,λelog21+φ3/w1h,λe=Rth.Table 1
Symbol notations.
Symbol
Notation
φ
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2
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## 4.2. DF Scheme

The Optimization Problem 2 (OP2) for the DF scheme can be formulated as(31)OP2:maxw1h,λe,w2h,λeRPDDF+RSDDF,s.t.1w1h,λe+w2h,λe=W,20≤w1h,λe,w2h,λe≤W,3RPDDF≥Rth.For the sake of simplicity, we rewriteRPDDF+RSDDF as follows:(32)minR1,R2+12w2h,λelog21+ν1w2h,λe,R≤Rth,w1h,λelog21+φ3w1h,λe+w2h,λelog21+2ν1w2h,λe,R>Rth,where(33)R1=12w1h,λelog21+ν2w1h,λe,R2=12w1h,λelog21+φ2w1h,λe,where φ2 and ν2 are given in Table 1.(1)
R≤Rth: similar to the AF scheme, we obtain the optimal bandwidth allocation policy for the primary network user:(34)w1DF∗bh,λe=maxw1DF†bh,λe,w1DF‡bh,λe,where(35)w1DF†bh,λe=minw1DF†1bh,λe,w1DF†2bh,λe,w1DF†1bh,λe=ν2ν2+ν1=4PSh124PSh12+ph42,w1DF†2bh,λe=φ2φ2+ν1=h22h22+h42,w1DF‡bh,λe is the solution to minR1,R2=Rth.(2)
R>Rth: the optimal bandwidth allocation solution is the same as the AF scheme:(36)w1DF∗oh,λe=maxw1DF†oh,λe,w1DF‡oh,λe,where(37)w1DF†oh,λe=PSWh32PSh32+Ph42,w1DF‡oh,λe is the solution to w1h,λelog21+φ3/w1h,λe=Rth.


## 5. Numerical Results

In this section, we present some numerical results to verify the effectiveness of the proposed protocol and the analytical optimization solutions. Besides, the derived analytical results are employed to provide insights into the different parameter’s impact on the whole system. In the simulation, the distance between PS and PD is normalized. Letd1, d2, d3, and d4 denote the distance between PS and SS, SS and PD, PS and PD, and SS and SD, respectively. Considering the large-scale fluctuation, we have Ω1=1/d13, Ω2=1/d23, Ω3=1/d33, and Ω4=1/d43 for h1, h2, h3, and h4, respectively. The parameters used to generate the figure are T=1ms, Eu=0.006J, W=20kHz, n=20, and σ02=10−6W/Hz.In Figure2, the analytical results for system outage probability are evaluated. For either PD and SD, the analytical expressions are given in (15), (16), and (21). We observe that the system outage probability of PD with the AF scheme is very close to that of the DF scheme. Also, we find that SD can achieve a better outage probability than PD.Figure 2
System outage probability vs.PS/Wσ02 (d4=1/2, Rth=10000bps, and P=0.0632W).In Figure3, the analytically results for optimized sum rates of the AF and DF scheme are examined and verified through simulation under different distance d1. For either AF or DF scheme, the analytical optimization solutions are given in (28), (29), (34), and (36). To test the optimized results, we present one pair of AF and DF results with fixed bandwidth allocation, i.e., w1/W=1/2. We notice from Figure 3 that the optimized sum data rates achieve evident improvement compared with fixed bandwidth allocation. For the AF scheme, the sum rate first increases with increase of d1 and then decreases with increase of d1, and it achieves maximum when d1=0.85. While for the DF scheme, the situation is inverse. This can be explained as following: for the AF scheme, the harvested energy of PS is impotent compared with SS’s transmitted power. As a result, the SS’s power become dominating which generates the situation where the distance between SS and PD has a vital influence on the sum rate. When d1 increases, the distance between SS and PD decreases, which leads to less path loss and more sum rate. When d1 is too large, PS and SS cannot communicate effectively which leads to reduction of the sum data rate. For the DF sheme, the PS’s power is dominating and the distance between PS and SS has a major influence on the sum rate. Therefore, with increase of d1, the pass loss between PS and SS increases which results in less sum rate. Moreover, when d1 is below approximate 0.5, DF scheme outperforms the AF scheme. The situation is opposite when d1 exceeds 0.5.Figure 3
Sum data rate vs. distanced1 (d4=1/2, λe=2packets/s, Rth/W=1/2bps/Hz, and P=1W).In Figure4, we present numerical examples of sum data rates under different energy arrival rates. Compared with fixed w1/W, the optimized results are extremely improved. Besides, Figure 4 shows that the sum data rates of the AF and DF scheme increase as the energy arrival rate increases. This is due to the fact that large energy arrival rate leads to more energy allocated to data transmission which increases the sum data rates. Furthermore, compared with direct transmission, we find that the proposed protocol can achieve a better transmission data rate of the primary user. Figure 5 depicts the impact of P on the performance of sum data rates. We firstly observe that the optimized results are conspicuously enhanced. Besides, Figure 5 illustrates that, with increase of P, the sum rate for either case increases, which can be interpreted by its physical meaning. Larger P means that more power is employed to assist the primary and secondary transmission which boosts the sum data rate. Moreover, the DF scheme outperforms the AF scheme which is on account of specific d1.Figure 4
Sum data rate vs. energy arrival rateλe (d1=1/3, d4=1/2, Rth/W=1/2bps/Hz, and P=1W).Figure 5
Sum data rate vs. the relay’s powerP (d1=1/3, d4=1/2, λe=2packets/s, and Rth/W=1/2bps/Hz).In Figure6, we evaluate the impact of λe on the optimal w1. It is interesting to note that w1/W decreases when λe increases. This is because when λe is small, more bandwidth is allocated to the primary network in order to meet the threshold requirement, thus resulting in large w1/W. We also note that when λe increases, the difference of w1/W between AF and DF scheme shrinks. This can be explained as follows: when λe is large, more energy is harvested to assist the primary transmission, which tends to do direct transmission. As result, the gap between AF and DF decreases. Besides, when λe is large enough, the curve of the DF scheme changes smoothly. This is due to the fact that, for large λe, the primary data rate is more likely determined by the rate between SS and PD, which is a constant value. Therefore, w1/W varies slowly when λe increases in the large range of values. Moreover, compared with the DF scheme, w1/W is larger under the AF scheme. This is because for the AF scheme, the secondary signals interfere with the primary transmission which leads more bandwidth distributed to the primary network so as to meet the threshold requirement. Furthermore, for each scheme, w1/W under Rth/W=1bps/Hz is larger than the situation under Rth/W=1/2bps/Hz. This is on account of the reason that larger Rth means more threshold requirement for the primary transmission which leads to more bandwidth allocated to the primary network.Figure 6
Bandwidth allocation vs. energy arrival rateλe (d1=1/3, d4=1/2, and P=1W).


## 6. Conclusion

In this paper, we have investigated an energy harvesting and relaying protocol in the smart grid with cognitive relay networks. The proposed protocol enables the secondary network opportunistically share the spectrum licensed to the primary network by assisting the primary transmission if the direct transmission between two primary users cannot meet the threshold requirement. The exact expressions of data rate for both primary and secondary networks are derived under different relaying schemes, namely, AF and DF schemes. We then derive the system outage probability of PD and SD for both two transmission schemes. Furthermore, in order to improve the performance of whole system, we formulate optimization problems which are in purpose of achieving the maximum sum data rates. For each scheme, we derive the correspondingly analytical solution. Subsequently, the numerical results verify the accuracy of theoretical derivations. Besides, simulation results demonstrate that the proposed spectrum sharing with energy harvesting in this protocol can achieve better transmission rate compared with the direct transmission. Finally, the numerical results provide us with valuable insights into the influence of various system parameters on the system performance.



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*Source: 5038963-2019-04-28.xml*