Joint Resource Optimization for Multicell Networks with Wireless Energy Harvesting Relays

Joint Resource Optimization for Multicell Networks
with Wireless Energy Harvesting Relays

Ali A. Nasir, Duy T. Ngo, Xiangyun Zhou, Rodney A. Kennedy, and Salman Durrani Ali A. Nasir, Xiangyun Zhou, Rodney A. Kennedy and Salman Durrani are with the Research School of Engineering, the Australian National University, Canberra, ACT 2601, Australia (Email: {ali.nasir, xiangyun.zhou, rodney.kennedy, salman.durrani}@anu.edu.au). Duy T. Ngo is with the School of Electrical Engineering and Computer Science, the University of Newcastle, Callaghan, NSW 2308, Australia (Email: duy.ngo@newcastle.edu.au).
Abstract

This paper first considers a multicell network deployment where the base station (BS) of each cell communicates with its cell-edge user with the assistance of an amplify-and-forward (AF) relay node. Equipped with a power splitter and a wireless energy harvester, the self-sustaining relay scavenges radio frequency (RF) energy from the received signals to process and forward the information. Our aim is to develop a resource allocation scheme that jointly optimizes (i) BS transmit powers, (ii) received power splitting factors for energy harvesting and information processing at the relays, and (iii) relay transmit powers. In the face of strong intercell interference and limited radio resources, we formulate three highly-nonconvex problems with the objectives of sum-rate maximization, max-min throughput fairness and sum-power minimization. To solve such challenging problems, we propose to apply the successive convex approximation (SCA) approach and devise iterative algorithms based on geometric programming and difference-of-convex-functions programming. The proposed algorithms transform the nonconvex problems into a sequence of convex problems, each of which is solved very efficiently by the interior-point method. We prove that our algorithms converge to the locally optimal solutions that satisfy the Karush-Kuhn-Tucker conditions of the original nonconvex problems. We then extend our results to the case of decode-and-forward (DF) relaying with variable timeslot durations. We show that our resource allocation solutions in this case offer better throughput than that of the AF counterpart with equal timeslot durations, albeit at a higher computational complexity. Numerical results confirm that the proposed joint optimization solutions substantially improve the network performance, compared with cases where the radio resource parameters are individually optimized.

{keywords}

Convex optimization, multicell interference, resource allocation, successive convex approximation, wireless energy harvesting

I Introduction

Multicell networks with universal frequency reuse play an important role in meeting the ever increasing demand of ubiquitous wireless coverage and high data throughput in the near future [1, 2, 3]. One of the challenges in such networks is to maintain the quality of service requirements for cell-edge users due to the interference from the neighboring cells [2, 1]. The deployment of relays is regarded as a viable solution in eliminating coverage holes in areas that are otherwise difficult for BSs’ signals to penetrate [4, 5]. In addition, the performance of multicell networks can be further enhanced by utilizing coordinated multipoint transmission and reception (CoMP) techniques [6, 7], in which BSs and relays cooperate with one another to best serve the cell-edge users.

Due to random positions and mobility of users, relays need to be opportunistically deployed where most needed. This can be achieved if relays do not require a wired power connection and are powered using alternative ‘green’ energy resources. Recently, radio frequency (RF) or wireless energy harvesting has emerged as an attractive solution to power wireless nodes [8]. While energy harvesting from ambient sources may not be sufficient to power relay nodes, carefully designed wireless power transfer links can be used to power relay nodes [8, 9, 10]. In this regard, it is crucial to ensure that the very different information decoding and power transfer power sensitivity requirements are met at the receiver (e.g., dBm for information receivers and dBm to dBm for energy receivers [8]).

A multicell network with energy harvesting relays poses interesting design challenges, such as: (i) How to effectively manage intercell interference, (ii) How to allocated limited power at the base stations (BSs), (iii) How to design wireless power transfer links for amplify-and-forward (AF) and decode-and-forward (DF) relays, and (iv) How the harvested RF energy is utilized at the relays. Existing research in the literature has partially addressed these important issues. The design of wireless energy harvesting relays in point-to-point single-cell systems is considered in [11, 12, 13, 14, 15, 16, 17]. Assuming simultaneous wireless information and power transfer in a single-cell network, the power control problem for multiuser broadband wireless systems without relays is studied in [18]. In [19], a similar problem is examined, albeit in the context of multiuser multi-input-multi-output (MIMO) systems. Considering relays in a single-cell network, resource allocation schemes for the remote radio heads are specifically developed in [20]. In the downlink of a multicell multiuser interference network, coordinated scheduling and power control algorithms for the macrocell BSs only are proposed in [21, 22]. Recently, in [23], an optimal power splitting rule is devised for energy harvesting and information processing at the self-sustaining relays of multiuser interference networks. However, [23] does not consider the important issue of allocating the transmit powers at the BSs and the relays.

In this paper, we consider a multicell network in which the BS of each cell communicates with its cell-edge user via a wireless energy harvesting relay node. The relay is equipped with an energy harvesting receiver and information transceiver. We assume that the energy harvesting receiver implements a power-splitting (PS) based receiver architecture [24], i.e., the relay uses a portion of the received signal power for energy harvesting and the remaining signal energy as input to the information transceiver. Using the harvested energy, the information transceiver employs either AF or DF relaying to forward the received signal to its corresponding user. The BSs in the multicell network adopt CoMP, i.e., they share the channel quality measurements and schedule the transmissions, allowing for more efficient radio resource utilization.

First, we formulate three new resource optimization problems for multicell networks with EH-enabled AF relays, namely, sum-rate maximization, minimum-throughput maximization, and sum-power minimization111 A preliminary version of this work, which considers the sum-rate maximization problem for AF relaying only, has been accepted for presentation at the 2015 IEEE International Conference on Communications (ICC), London, U.K. [25].. The objective is to jointly optimize the transmit powers at the BSs and the relays and also find the optimal power splitting rule at the relays. Our formulations directly target the critical issue of multicell interference, at the same time as meeting the stringent constraints on the available transmit powers at the BSs and the relays. Since the optimization variables are strongly coupled with many nonlinear cross-multiplying terms, the formulated problems are highly nonconvex. To the best of our knowledge, there exists no practical method that guarantees to offer the true global optimality to these challenging problems.

Then, we exploit the problem structure and adopt the successive convex approximation (SCA) method to transform the highly nonconvex problems into a series of convex subproblems. Here, we specifically tailor the generic SCA framework via the applications of geometric programming (GP) and difference-of-convex-functions (DC) programming. At each step of our proposed iterative algorithms, we efficiently solve the resulting convex problem by the interior-point method. We analytically prove that our developed algorithms generate a sequence of improved feasible solutions, which eventually converge to a locally optimal solution satisfying the Karush-Kuhn-Tucker (KKT) conditions of the original problems. Note that the general convergence analysis of SCA method is established in [26] and SCA-based solutions have been empirically shown to often achieve the global optimality in many practical applications, e.g., in wireline DSL networks [27], wireless interference networks [28, 29], and small-cell heterogeneous networks [30].

Finally, we show that the proposed SCA-based approach can be extended to the more general case of variable timeslot durations with DF relaying. Numerical examples with realistic network parameters confirm that our joint optimization solutions significantly outperform those where the radio resource parameters are individually optimized.

The rest of this paper is organized as follows: Sec. II presents the system model and states the key assumptions used throughout this work. Sec. III presents the signal model for AF relaying and equal timeslot durations. Sec. IV formulates the nonconvex resource allocation problems and introduces the generic SCA framework. Secs. V and VI propose the GP-based and DC-based SCA solutions for AF relaying, respectively. Sec. VII extends our results to the case of variable timeslot durations with DF relaying. Sec. VIII presents numerical results to confirm the advantages of our proposed algorithms. And Sec. IX concludes the paper.

Ii System Model and Assumptions

Fig. 1: A multicell network consists of cells and a central processing (CP) unit. Each cell has a base station, a relay and a cell-edge user. For clarity, we only show the interfering scenarios in Cell 1, i.e., at relay and user . In general, interference happens at all relays and users.

Consider the downlink transmissions in an -cell network with universal frequency reuse, i.e., the same radio frequencies are used in all cells. Adopting CoMP, we assume that the base stations (BSs) are connected to a central processing (CP) unit which coordinates the multicellular transmissions and radio resource management. The network under consideration is illustrated in Fig. 1. Note that although square-cells are shown in Fig. 1, the analysis and proposed solutions in this paper are valid for any cellular network geometry.

Let denote the set of all cells. In each cell , the BS attempts to establish communication with its cell-edge users. We assume that these users are located in the ‘signal dead zones’, where no direct signal from their serving BS can reach. A relay node is deployed in each cell to assist in forwarding the signal from the BS, extending the network coverage to the distant users. We assume that orthogonal channels are assigned to users in each cell (e.g., by means of TDMA, FDMA or OFDMA); hence, the intracell interference is eliminated. Therefore, we only focus on the resource allocation in one channel, which corresponds to only one user in a cell. By BS , relay and user , we mean the BS, the relay and the single user of cell , respectively.

We assume that the relays are energy-constrained nodes and they harvests energy from the RF signals of all BSs, using the power-splitting based receiver architecture. While each BS has a maximum power limit available for transmission, it must transmit with a minimum transmit power to ensure that the energy harvesting circuit at the relay is activated. The harvested energy is used by a relay transceiver to process and forward the BS signal to its intended user. We further assume that the relays are mounted on the building rooftops to have a line-of-sight link from the serving BSs.

Let be the channel coefficient from the BS to relay and be the channel coefficient from the relay to user . We assume that all the BSs send the available channel state information (CSI) to the CP unit via a dedicated control channel. In this paper, we assume perfect knowledge of CSI at the BSs, allowing for a benchmark performance to be determined.

Iii Signal Model with AF relaying

We first consider the case of AF relaying where we divide the total transmission block time into two equal timeslots. The first timeslot includes BS-to-relay transmissions and energy harvesting at the relays. During the first timeslot, the relays do not transmit. The second timeslot includes signal processing at the relays and relay-to-user transmissions. In this second timeslot, the BSs do not transmit. The operations in each timeslot are illustrated in Fig. 2, which will be further discussed in the following.

Fig. 2: BS-to-user communication assisted by a wireless energy harvesting AF relay.

Iii-a BS-to-Relay Transmissions and Wireless Energy Harvesting at Relay Receivers

In the first timeslot , let be the normalized information signal to be sent by BS , i.e., , where denotes the expectation operator and the absolute value operator. Let denote the transmit power of BS , the distance between BS and relay , and the path-loss exponent. Assuming that is the zero-mean additive white Gaussian noise (AWGN) with variance at the receiving antenna of relay , the received signal at relay can be expressed as:

(1)

We assume that each relay is equipped with a power splitter that determines how much received signal energy should be dedicated to the energy harvester and the signal processing receiver [24, 23, 11, 12]. As shown in Fig. 2, the power splitter at relay divides the power of into two parts in the proportion of . Here, is termed as the power splitting factor. The first part is processed by the energy harvester and stored as energy (e.g., by charging a battery at relay ) for the use in the second timeslot. The amount of energy harvested at relay is given by:

(2)

where is the efficiency of energy conversion and , is the effective channel gain from BS to relay (including the effects of both small-scale fading and large-scale path loss).

The second part of the received signal is passed to an information transceiver. In Fig. 2, denotes the AWGN with zero mean and variance introduced by the baseband processing circuitry. Since antenna noise power is very small compared to the circuit noise power in practice [31], has a negligible impact on both the energy harvester and the information transceiver of relay . Thus, for simplicity, we will ignore the effect of in the following analysis by setting . The signal at the input of the information transceiver of relay can be written as:

(3)

where the first term in (3) is the desired signal from BS , and the second term is the total interference from all other BSs.

Iii-B Signal Processing at Relays and Relay-to-User Transmissions

In the second timeslot , the information transceiver amplifies the signal prior to forwarding it to user . Denote the transmit power of relay transceiver as . With the harvested energy in (55), the maximum power available for transmission at relay is given by , which means that:

(4)

The transmitted signal from relay to user can then be written as:

(5)

where the denominator of (5) represents an amplifying factor that ensures power constraint (4) be met.

Now, the received signal at user is:

(6)

where denotes the distance between relay and user , and the AWGN with zero mean and variance at the receiver of user . Substituting in (5) into (6) yields:

(7)

With defined in (3), we can then write (7) explicitly as:

(8)

The first term in (III-B) represents the desired signal from BS to its serviced user , whereas other terms represent the intercell interference and the noise.

Without loss of generality, let us assume . The signal-to-interference-plus-noise ratio (SINR) at the receiver of user can be derived from (III-B) as:

(9)

where we define

(10)

where . For notational convenience, let us also define , and . From (9), the achieved throughput in bps/Hz (bits per second per Hz) of cell is given by

(11)

An important observation from (9) and (11) is that by dedicating more received power at relay for energy harvesting (i.e. increasing ), one might actually decrease the end-to-end throughput in cell . This can be verified upon dividing both the numerator and the denominator of in (9) by . However if one opts to decrease , the transmit power available at the information transceiver of relay will be further limited [see (4)], thus potentially reducing the corresponding data rate . Similarly, increasing the BS transmit power or the relay transmit power does not necessarily increase the throughput of cell . The reason is that and appear in the positive terms in both the numerator and the denominator of . This suggests the importance of the resource allocation problem in this context, which will be addressed in the next section.

Iv Joint Resource Optimization Problems for AF Relaying

In this paper, we aim to devise an optimal tradeoff of all three parameters, transmit power at BSs, , transmit power at relays, , and power splitting factor at relays, , to maximize the performance of the multicell network under consideration. Specifically, we will study the following problems which jointly optimize for three different design objectives.

Iv-a Problem (P1): Sum-Rate Maximization

We assume that is the maximum power available for transmission at each BS. Also, is the minimum transmit power required at each BS to ensure the activation of energy harvesting circuitry at the relay. The problem of sum throughput maximization is formulated as follows.

(12a)
s.t. (12b)
(12c)
(12d)

In this formulation, (12a) is the total network throughput whereas (12b) are the constraints for the power splitting factors for all relays. Also, (12c) and (12d) ensure that the transmit powers at the BSs and relays do not exceed the maximum allowable.

Iv-B Problem (P2): Max-Min Throughput Fairness

In Problem (P1), the network sum-rate is maximized without any consideration given to the throughput actually achieved by the individual users. It might happen that users with more favorable links conditions are allocated with most of the radio resources, leaving nothing for others to fulfill their bare minimum QoS requirements. The latter includes cell-edge users who are the victims of strong intercell interference. In the following, we formulate a max-min fairness problem where the throughput of the most disadvantaged user is maximized.

(13a)
s.t.

From the network design perspective, (13) can be regarded as the problem of maximizing a common throughput:

(14a)
s.t. (14b)

where is an auxiliary variable that denotes the common throughput.

Iv-C Problem (P3): Sum-Power Minimization

Different from Problems (P1) and (P2), our objective here is to minimize the total transmit power consumption subject to guaranteeing some minimum data throughput for each user:

(15a)
s.t. (15b)

This problem is of particular interest for “green” communications, where one wishes to reduce the environmental impacts of the large-scale deployment of wireless communication networks. At the same time, the performance of all cell-edge users is protected with constraint (15b).

All three problems (P1), (P2) and (P3) are highly nonconvex in because the throughput in (11) is highly nonconvex in those variables. Even if we fix and and try to optimize the BS transmit power alone, would still be highly nonconvex in the remaining variable due to the cross-cell interference terms. Simultaneously optimizing and will be much more challenging due to the nonlinearity introduced by the cross-multiplying terms, e.g., in (9) and in (12d).

To efficiently solve Problems (P1), (P2) and (P3), we propose to adopt the successive convex approximation (SCA) approach [32, 27, 28, 29, 30, 26] to transform the original nonconvex problems into a sequence of relaxed convex subproblems. The key steps of the generic SCA approach are summarized in Algorithm 1 for our formulated optimization problems. However, in applying the SCA approach, there remain two key questions: (i) How to perform the approximation in Step 2 in generic Algorithm 1? (ii) Given that the approximation is known, how to prove that the iterative algorithm is convergent to an optimal solution? We will provide the answers for those questions in the following sections. Specifically, we will exploit the structure of the formulated problems to propose two types of approximations, one based on GP programming and the other DC programming. We will demonstrate that with the given objective functions and constraints, it is possible to apply both approximations to solve the formulated nonconvex problems under the same SCA framework.

1:Initialize with a feasible solution .
2:At the -th iteration, form a convex subproblem by approximating the nonconcave objective function and constraints of (P1), (P2) and (P3) with some concave function around the previous point .
3:Solve the resulting convex subproblem to obtain an optimal solution at the -th iteration.
4:Update the approximation parameters in Step 2 for the next iteration.
5:Go back to Step 2 and repeat until converges.
Algorithm 1 Generic Successive Convex Approximation Algorithm

V Solutions for AF Relaying: SCA Method Using GP

To implement Step 2 in Algorithm 1, in this section we will make use of the single condensation approximation method [28] to form a relaxed geometric program (GP), instead of directly solving the nonconvex Problems (P1), (P2) and (P3). A GP is expressed in the standard form as [33, p. 161]:

(16a)
s.t. (16b)
(16c)

where are posynomials and are monomials222A monomial is defined as , where , , and . A posynomial is a nonnegative sum of monomials. [33]. A GP in standard form is a nonlinear and nonconvex optimization problem because posynomials are not convex functions. However, with a logarithmic change of the variables and multiplicative constants, one can easily turn it into an equivalent nonlinear and convex optimization problem (using the property that the log-sum-exp function is convex) [33, 28].

V-a GP-based Approximated Solution for Problem (P1)

First, we express the objective function in (12a) as:

(17a)
(17b)

where (17b) follows from (17a) since is monotonically increasing function. Upon substituting in (9) to (17b) and replacing by an auxiliary variable , it is shown that Problem (P1) in (12) is equivalent to:

(18a)
s.t. (18b)
(18c)
(18d)

where .

It can be seen that (18) is not yet in the form of (16) because (18a) and (18d) are not posynomials. For notational convenience, let us define:

(19)
(20)

where . The objective function in (18a) can then be expressed as:

(21)

Since and are both posynomials, is not necessarily a posynomial, confirming that (18a) is not a posynomial.

To transform Problem (P1) into a GP of the form in (16), we would like the objective function (21) to be a posynomial. To this end, we propose to apply the single condensation method [28] and approximate with a monomial as follows. Given the value of at the -th iteration, we apply the arithmetic-geometric mean inequality to lower bound at the -th iteration by a monomial as [28, Lem. 1]:

(22)

It is straightforward to verify that . In fact, is the best local monomial approximation to near in the sense of the first-order Taylor approximation. With (V-A), the objective function in (18a) is approximated by . The latter is a posynomial because is a monomial and the ratio of a posynomial to a monomial is a posynomial. The upper bound of (21) is also a posynomial because the product of posynomials is a posynomial.

Next, we will approximate constraint (12d) by a posynomial to fit into the GP framework (16). Again, we lower bound posynomial by a monomial as [28, Lem. 1]:

(23)

It is clear that the ratio is now a posynomial. Upon substituting (V-A) and (23) into (18), we can formulate an approximated subproblem at the -th iteration for Problem (P1) as follows:

(24a)
s.t. (24b)

Comparing with (16), we see that (24) belongs to the class of a geometric program, i.e., a convex optimization problem. In (24a), since [see (V-A)], we are actually minimizing the upper bound of the original objective function in (18a). With (23), constraint (24b) is stricter than (12d) as:

(25)

V-B GP-based Approximated Solution for Problem (P2)

By substituting in (11) and carrying out simple algebraic manipulations, constraint (14b) of Problem (P2) can be rewritten as:

(26)

where denotes the natural logarithm. By introducing the auxiliary variable and with and defined in (19)-(20), it is shown that Problem (P2) is equivalent to:

(27a)
s.t. (27b)
(27c)

As seen, (27) is not yet in the form of the standard GP (16) because constraints (27b) and (12d) are not posynomials. Using the similar approach in Sec. V-A, we can transform (27b) and (12d) into posynomials by the approximations in (V-A) and (23). The resulting subproblem at the -th iteration of Problem (P2) can be expressed in the standard GP form as:

(28a)
s.t. (28b)
(28c)

where (28b) follows directly from (27b) by replacing with [see in (V-A)], and (24b) is used in lieu of (12d).

V-C GP-based Approximated Solution for Problem (P3)

By introducing an auxiliary variable and applying monomial approximation [in (V-A)] for [in (20)], we can transform the nonconvex constraint (15b) in Problem (P3) into a posynomial form as:

(29)

Again, we use (24b) instead of (12d) and arrive at the following GP, which is an approximated problem for Problem (P3) at the -th iteration:

(30a)
s.t. (30b)

V-D Proposed GP-based SCA Algorithm for Joint Resource Allocation

It should be noted that GP problems (24), (28) and (30) are the convex approximations of the original Problems (P1), (P2) and (P3), respectively. In Algorithm 2, we propose an SCA algorithm in which a (convex) GP is optimally solved at each iteration.

1:Initialize .
2:Choose a feasible point .
3:Compute the value of according to (20).
4:repeat
5:     Using , form the approximate monomial according to (V-A).
6:     Using the interior-point method, solve one GP, i.e., (24) or (28) or (30) to find the -th iteration approximated solution for Problem (P1) or (P2) or (P3), respectively.
7:     Compute the value of according to (20).
8:     Set .
9:until Convergence of or no further improvement in the objective value (24a) or (28a) or (30a)
Algorithm 2 Proposed GP-based SCA Algorithm
Proposition 1

Algorithm 2 generates a sequence of improved feasible solutions that converge to a point satisfying the KKT conditions of the original problems (i.e., Problems (P1), (P2) and (P3)).

Proof:

We will prove that Proposition 1 holds for the case of GP (24) and its corresponding Problem (P1). The proofs for GP (28) (hence Problem (P2)) and GP (30) (hence Problem (P3)) are similar and will be omitted. From (23), we have that . This means that the optimal solution of the approximated problem (24) always belongs to the feasible set of the original Problem (P1).

Next, since , it follows that:

(31)

where the last equality holds because . As the actual objective value of Problem (P1) is non-increasing after every iteration, Algorithm 2 will eventually converge to a point .

Finally, it can be verified that

(32)
(33)

where denotes the gradient operator. The results in (32)-(33) imply that the KKT conditions of the original Problem (P1) will be satisfied after the series of approximations involving GP (24) converges to the point . This completes the proof. \qed

Vi Solutions for AF Relaying: SCA Method Using DC Programming

Vi-a DC-based Approximated Solution for Problem (P1)

In the GP-based approach proposed in Sec. V, we have eliminated the logarithm function in the objective function to form a posynomial [see (17)] and solve the resulting (convex) GP. In the current approach, we propose to keep the logarithm function and rewrite the throughput expression as:

(34)

where we define and with and given in (19) and (20), respectively. We also recall that , and .

Using the following logarithmic change of variables:

(35)

for all , we can further write