I Introduction
Driven by the rapid development of advanced multimedia applications, nextgeneration wireless networks must support high spectral efficiency and massive connectivity [1]. Due to high data rate demand and massive numbers of users, energy consumption has become a challenging problem in the design of future wireless networks [2]. In consequence, energy efficiency, defined as the ratio of spectral efficiency over power consumption, has emerged as an important performance index for deploying green and sustainable wireless networks [3, 4, 5, 6, 7].
Recently, reconfigurable intelligent surface (RIS)assisted wireless communication has been proposed as a potential solution for enhancing the energy efficiency of wireless networks [8, 9, 10, 11, 12, 13]. An RIS is a metasurface equipped with lowcost and passive elements that can be programmed to turn the wireless channel into a partially deterministic space. In RISassisted wireless communication networks, a base station (BS) sends control signals to an RIS controller so as to optimize the properties of incident waves and improve the communication quality of users. The RIS acts as a reflector and does not perform any digitalization operation. Hence, if properly deployed, an RIS promises much lower energy consumption than traditional amplifyandforward (AF) relays [14, 15, 16]. However, the effective deployment of energyefficient RIS systems faces several challenges ranging from performance characterization to network optimization [17].
A number of existing works such as in [18, 19, 20, 21, 22, 23, 24] has studied the deployment of RISs in wireless networks. In [18]
, the downlink sumrate of an RIS assisted wireless communication system was characterized. An asymptotic analysis of the uplink transmission rate in an RISbased large antennaarray system was presented in
[19]. Then, in [20], the authors investigated the asymptotic optimality of the achievable rate in a downlink RIS system. Considering energy harvesting, an RIS was invoked for enhancing the sumrate performance of a simultaneous wireless information and power transfer aided system [21]. Instead of considering the availability of instantaneous channel state information (CSI), the authors in [22] proposed a twotimescale transmission protocol to maximize the achievable sumrate for an RISassisted multiuser system. Taking the secrecy into consideration, the work in [23] investigated the problem of secrecy rate maximization of an RIS assisted multiantenna system. Further by considering imperfect CSI, the RIS was considered to enhance the physical layer security of a wireless channel in [24]. Beyond the above studies, the use of RISs for enhanced wireless energy efficiency has been studied in [25]. In [25], the authors proposed a new approach to maximize the energy efficiency of a multiuser multipleinput singleoutput (MISO) system by jointly controlling the transmit power of the BS and the phase shifts of the RIS. However, only a single RIS was considered for simplicity in [25]. Deploying a number of lowcost powerefficient RISs in future networks can cooperatively enhance the coverage of the networks. In particular, deploying multiple RISs in wireless networks has several advantages. First, distributed RISs can provide robust datatransmission since different RISs can be deployed geometrically apart from each other. Meanwhile, multiple RISs can provide multiple paths of received signals, which increases the received signal strength. To our best knowledge, this is the first work that optimizes the energy efficiency for a wireless network with multiple RISs.The main contribution of this paper is a novel energy efficient resource allocation scheme for wireless communication networks with distributed RISs. Our key contributions include:

We investigate a downlink wireless communication system with distributed RISs that can be dynamically turned on or off depending on the network requirements. To maximize the energy efficiency of the system, we jointly optimize the phase shifts of all RISs, the transmit beamforming of the transmitter, and the RIS onoff status vector. We formulate an optimization problem with the objective of maximizing the energy efficiency under minimum rate constraint of users, transmit power constraint, and unitmodulus constraint of the RIS phase shifts.

To maximize the energy efficiency for a single user, a suboptimal solution is obtained by using a lowcomplexity algorithm that iteratively solves two, joint subproblems. For the joint phase and power optimization subproblem, a suboptimal phase is obtained by using the successive convex approximation (SCA) method with low complexity, and the optimal power is subsequently obtained in closed form. For the RIS onoff optimization subproblem, the dual method is used to obtain the optimal solution.

To maximize the energy efficiency for multiple users, an iterative algorithm is proposed through solving the phase optimization subproblem, beamforming optimization subproblem, and RIS onoff subproblem, iteratively. For the subproblem of phase optimization or beamforming optimization, we use the SCA method to obtain a suboptimal solution. For the RIS onoff optimization subproblem, we propose a lowcomplexity search method that can determine the onoff status of all RISs.
Simulation results show that our proposed approach can enhance the energy efficiency by up to 33% and 68% gains compared to the conventional RIS scheme and AF relay scheme, respectively.
The rest of this paper is organized as follows. Section II introduces the system model and problem formulation. Section III and Section IV provide the energy efficiency optimization with a single user and multiple users, respectively. Simulation results are provided in Section V and conclusions are given in Section VI.
Notations: In this paper, the imaginary unit of a complex number is denoted by . Matrices and vectors are denoted by boldface capital and lowercase letters, respectively. Matrix denotes a diagonal matrix whose diagonal components are . The real and imaginary parts of a complex number are denoted by and , respectively. , , and respectively denote the conjugate, transpose, and conjugate transpose of vector . and denote the th and th elements of the respective vector and matrix . denotes the norm of vector . The distribution of a circularly symmetric complex Gaussian variable with mean and covariance is denoted by .
Ii System Model and Problem Formulation
Iia Transmission Model
Consider an RISassisted MISO downlink channel that consists of one BS, a set of users, and a set of RISs, as shown in Fig. 1. The number of transmit antennas at the BS is , while each user is equipped with one antenna. Such a setting has been used in many practical scenarios such as in InternetofThings networks [18, 19, 20]. Each RIS, , has reflecting elements. The RISs are configured to assist the communication between the BS and users. In particular, the RISs will be installed on the walls of the surrounding highrise buildings.
The transmitted signal at the BS is:
(1) 
where is unitpower information symbol [25] and is the beamforming vector for user .
The power consumption of an RIS depends on both the type and the resolution of the reflecting elements that effectively perform phase shifting on the impinging signal [25, 26, 27]. Considering the power consumption of RISs due to controlling the phase shift values of the reflecting elements [25]
, it is often not energy efficient to turn on all the RISs. We now introduce a binary variable
, where indicates that RIS is on. When , the phase shift matrix of RIS can be optimized through a diagonal matrix with , , and , where captures the effective phase shifts applied by all reflecting elements of RIS . In contrast, when , RIS is off and does not consume any power. Then, with the multiple RISs, the received signal at user can be given by:(2) 
where , , and , respectively, denote the channel responses from the BS to user , from the BS to RIS , and from RIS to user , and is the additive white Gaussian noise.
IiB Power Consumption Model
The total power consumption of the considered RISassisted system includes the transmit power of the BS, the circuit power consumption of both the BS and all users, and the power consumption of all RISs. Consequently, the total power of the system will be given by:
(5) 
where with being the power amplifier efficiency of the BS, is the circuit power consumption of the BS, is the circuit power consumption of user , and is the power consumption of each reflecting element in the RIS. In (5), is the power consumption of RIS .
IiC Problem Formulation
Given the considered system model, our objective is to jointly optimize the reflection coefficients matrix, beamforming vector, and RIS onoff vector so as to maximize the energy efficiency under the minimum rate requirements and total power constraint. Mathematically, the problem for the distributed RISs can be given by:
(6)  
s.t.  (6a)  
(6b)  
(6c)  
(6d) 
where , , , is the minimum data rate requirement of user , and is the maximum transmit power of the BS. The minimum rate constraint for each user is given in (6a) and (6b) represents the total power constraint. The phase shift constraint for each reflecting element is provided in (6c), which can also be seen as the unitmodulus constraint since . The problem in (6) is a mixedinteger nonlinear program (MINLP) even for the singleuser case with . It is generally difficult to obtain the globally optimal solution of the MINLP problem in (6). In the following, we propose two iterative algorithms to obtain suboptimal solutions of problem (6) for the singleuser case and the multiuser case, respectively.
Iii Energy Efficiency Optimization with A Single User
In this section, we consider the singleuser case, i.e., . For , problem (6) becomes:
(7)  
s.t.  (7a)  
(7b)  
(7c)  
(7d) 
Since there is no multiuser interference, it is wellknown that beaming as the the maximum ratio transmission (MRT) at the BS is optimal [28]. That is:
(8) 
where helps satisfy the transmit power constraint at the BS. Substituting the optimal beamforming in (8) into problem (7), it becomes:
(9)  
s.t.  (9a)  
(9b)  
(9c)  
(9d) 
Due to the involvement of integer variable , it is difficult to obtain the globally optimal solution of (9). As such, we propose an iterative algorithm to solve problem (9) suboptimally with low complexity. The proposed iterative algorithm contains two major steps. In the first step, we jointly optimize phase and power with given . Then, in the second step, we update RIS onoff vector with the optimized in the previous step.
Iiia Joint Phase and Power Optimization
For a fixed integer variable , problem (9) becomes:
(10)  
s.t.  (10a)  
(10b)  
(10c) 
From the objective function (10) and the constraint in (10a), we observe that the optimal is the one that maximizes the channel gain, i.e., . With this in mind, the optimal solution of problem (10) can be obtained in two stages, i.e., obtain the value of that maximizes in the first stage and, then, calculate the optimal in the second stage with the obtained in the first stage.
IiiA1 First stage
We first optimize the phase shift vector of problem (10). Before optimizing , we show that , where and . According to problem (10), the optimal can be calculated by solving the following problem:
(11)  
s.t.  (11a) 
Let be the conjugate vector of . The total number of elements for all RISs is denoted by . Denote and . Problem (11) can be rewritten as:
(12)  
s.t.  (12a) 
where .
To solve the optimization problem in (12), various methods were proposed by techniques such as semidefinite relaxation (SDR) technique [29] and successive refinement (SR) algorithm [30]. However, the SDR method imposes high complexity to obtain a rankone solution and the SR algorithm requires a large number of iterations due to the need for updating the phase shifts in a onebyone manner. To reduce the computational complexity, we propose the SCA method to solve the phase shift optimization problem (12).
To handle the nonconvexity of objective function (12), we adopt the SCA method and, consequently, objective function (12) can be approximated by:
(13) 
which is the firstorder Taylor series of and the superscript represents the value of the variable at the th iteration.
With the above approximation (13), the nonconvex problem (12) can be approximated by the following convex problem:
(14)  
s.t.  (14a) 
where constraint (12a) is temporarily relaxed as (14a). In the following lemma, we show that (14a) always holds with equality for the optimal solution of problem (14).
Lemma 1
Proof: To maximize in (14), the optimal should be chosen such that is a real number and for any , i.e., the optimal should be given as (15).
From (15) and Lemma 1, we can see that the optimal phase vector should be adjusted such that the signal that goes through all RISs is aligned to be a signal vector with equal phase at each element. We can also see that the optimal phase vector is independent of the amplitude of the channel .
The SCA algorithm for solving problem (12) is summarized in Algorithm 1. The convergence of Algorithm 1 is guaranteed by the following lemma that follows directly from [31, Proposition 3]:
Lemma 2
IiiA2 Second stage
We now obtain the optimal power allocation . With the obtained in Algorithm 1 and defining , problem (10) reduces to:
(16)  
s.t.  (16a) 
where , and . In (16a), is used to guarantee the minimum rate requirement for user 1.
For the energy efficiency optimization problem (16), the Dinkelbach method from [32] can be used. The Dinkelbach method involves solving a series of convex subproblems, which increases the computational complexity. However, the optimal solution of (16) can be derived in closed form using the following theorem.
Theorem 1
The optimal transmit power of the energy efficiency maximization problem in (16) is:
(17) 
where is the LambertW function and .
Proof: The firstorder derivative of the objective function (16) with respect to power is:
(18) 
To show the increasing trend of the objective function (16), we further denote:
(19) 
The firstorder derivative of function is:
(20) 
which indicates that is a monotonically decreasing function. Since and , there must exist a unique such that , where
(21) 
Hence, the objective function (16) first increases in interval and then decreases in interval , which indicates that the optimal solution can be presented as in (17).
From Theorem 1, the optimal power control of problem (16) is obtained in closedform, as shown in (17). According to (17), it is shown that the optimal power control lies in one of three values, i.e., the minimum transmit power, the power with zero firstorder derivative, and the maximum transmit power.
IiiB RIS OnOff Optimization
Substituting the phase and power variables obtained in the previous section, problem (9) becomes:
(22)  
s.t.  (22a)  
(22b) 
We introduce an auxiliary variable and problem (22) is equivalent to:
(23)  
s.t.  (23a)  
(23b)  
(23c) 
where constraint (23b) is used to ensure the minimum rate demand. For the optimal solution of problem (23), constraint (23a) will always hold with equality since the objective function monotonically increases with . Although problem (23) has a simplifier form compared to (22), it is still a nonconvex MINLP. There are two difficulties in solving problem (23). The first difficulty is that objective function (23) has a fractional form, which is difficult to solve. The second difficulty is that constraint (23a) is nonconvex.
To handle the first difficulty, we use the parametric approach in [32] and consider the following problem:
(24) 
where is the feasible set of satisfying constraints (23a)(23c). It was proved in [32] that solving (23) is equivalent to finding the root of the nonlinear function , which can be obtained by using the Dinkelbach method. By introducing parameter , the objective function of problem (23) can be simplified, as shown in (24).
To handle the second difficulty, due to the fact that , we can rewrite the right hand side of constraint (23a) as:
(25) 
where , , and . To solve problem (23), we introduce new variable . Since , constraint is equivalent to:
(26) 
(27) 
for all , . According to (24)(27), problem (23) can be reformulated as:
(28)  
s.t.  (28a)  
(28b)  
(28c)  
(28d)  
(28e) 
where .
Due to constraints (28e), it is difficult to handle problem (28). By relaxing the integer constraints (28e) with , problem (28) becomes a convex problem. For problem (28) with relaxed constraints, the optimal solution can be obtained through the dual method [33]. We show that the dual method obtains the integer solution, which guarantees both optimality and feasibility of the original problem. To obtain the optimal solution of problem (28), we have the following theorem.
Theorem 2
To maximize the objective function in (34), which is a linear combination of and , we must let the positive coefficients corresponding to the and be 1. Therefore, the optimal and are thus given as (29) and (31), respectively.
To optimize from (34), we set the first derivative of objective function to zero, i.e.,
(36) 
which yields
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