Energy-Efficient Resource Allocation in OFDMA Systems with Hybrid Energy Harvesting Base StationThis paper has been presented in part at the IEEE Global Communications Conference (Globecom 2012), Anaheim, California, USA.

# Energy-Efficient Resource Allocation in OFDMA Systems with Hybrid Energy Harvesting Base Station††thanks: This paper has been presented in part at the IEEE Global Communications Conference (Globecom 2012), Anaheim, California, USA.

\authorblockN Derrick Wing Kwan Ng\authorrefmark1, Ernest S. Lo\authorrefmark2, and Robert Schober\authorrefmark1
\authorrefmark1Institute for Digital Communications, Universität Erlangen-Nürnberg, Germany
\authorrefmark2Centre Tecnològic de Telecomunicacions de Catalunya - Hong Kong (CTTC-HK)
Email: kwan@lnt.de, ernest.lo@cttc.hk, schober@lnt.de
###### Abstract

We study resource allocation algorithm design for energy-efficient communication in an orthogonal frequency division multiple access (OFDMA) downlink network with hybrid energy harvesting base station (BS). Specifically, an energy harvester and a constant energy source driven by a non-renewable resource are used for supplying the energy required for system operation. We first consider a deterministic offline system setting. In particular, assuming availability of non-causal knowledge about energy arrivals and channel gains, an offline resource allocation problem is formulated as a non-convex optimization problem over a finite horizon taking into account the circuit energy consumption, a finite energy storage capacity, and a minimum required data rate. We transform this non-convex optimization problem into a convex optimization problem by applying time-sharing and exploiting the properties of non-linear fractional programming which results in an efficient asymptotically optimal offline iterative resource allocation algorithm for a sufficiently large number of subcarriers. In each iteration, the transformed problem is solved by using Lagrange dual decomposition. The obtained resource allocation policy maximizes the weighted energy efficiency of data transmission (weighted bit/Joule delivered to the receiver). Subsequently, we focus on online algorithm design. A conventional stochastic dynamic programming approach is employed to obtain the optimal online resource allocation algorithm which entails a prohibitively high complexity. To strike a balance between system performance and computational complexity, we propose a low complexity suboptimal online iterative algorithm which is motivated by the offline algorithm. Simulation results illustrate that the proposed suboptimal online iterative resource allocation algorithm does not only converge in a small number of iterations, but also achieves a close-to-optimal system energy efficiency by utilizing only causal channel state and energy arrival information.

{keywords}

Energy harvesting, green communication, non-convex optimization, resource allocation.

## I Introduction

The introduction of energy harvesting capabilities for BSs poses many interesting new challenges for resource allocation algorithm design due to the time varying availability of the energy generated from renewable energy sources. In [16] and [17], optimal packet scheduling and power allocation algorithms were proposed for energy harvesting systems for minimization of the transmission completion time, respectively. In [18] and [19], the authors proposed optimal power control time sequences for maximizing the throughput by a deadline with a single energy harvester. However, these works assumed a point-to-point narrowband communication system and the obtained results may not be applicable to the case of wideband multi-user systems. In [20]-[22], different optimal packet scheduling algorithms were proposed for additive white Gaussian noise (AWGN) broadcast channels for a set of preselected users. However, wireless communication channels are not only impaired by AWGN but also degraded by multi-path fading. In addition, dynamic user selection is usually performed to enhance the system performance. On the other hand, although the amount of renewable energy is potentially unlimited, the intermittent nature of energy generated by a natural energy source results in a highly random energy availability at the BS. For example, solar energy and wind energy are varying significantly over time due to weather and climate conditions. In other words, a BS powered solely by an energy harvester may not be able to maintain a stable operation and to guarantee a certain quality of service (QoS). Therefore, a hybrid energy harvesting system design, which uses different energy sources in a complementary manner, is preferable in practice for providing uninterrupted service [23, 24]. However, the results in the literature, e.g. [7]-[22], are only valid for systems with a single energy source and are not applicable to communication networks employing hybrid energy harvesting BSs.

In this paper, we address the above issues and focus on resource allocation algorithm design for hybrid energy harvesting BSs. In Section II, we introduce the adopted OFDMA channel model and hybrid energy source model. In Section III, we formulate offline resource allocation as an optimization problem by assuming non-causal knowledge of the channel gains and energy arrivals at the BS. The optimization problem is solved via fractional programming and Lagrange dual decomposition which leads to an efficient iterative resource allocation algorithm. The derived offline solution serves as a building block for the design of a practical close-to-optimal online resource allocation algorithm in Section IV which requires only causal knowledge of the channel gains and energy arrivals. In Section V, we show that the proposed suboptimal algorithm does not only have a fast convergence, but also achieves a close-to-optimal performance.

## Ii OFDMA System Model

### Ii-a Notation

A complex Gaussian random variable with mean and variance is denoted by , and means “distributed as”. . . denotes statistical expectation with respect to (w.r.t.) random variable .

### Ii-B OFDMA Channel Model

We consider an OFDMA network which consists of a BS and mobile users. All transceivers are equipped with a single antenna, cf. Figure 1. The total bandwidth of the system is Hertz and there are subcarriers. The transmission time is seconds. We assume that the BS adapts the resource allocation policy (i.e., the power allocation and subcarrier allocation policies) times for a given period . The optimal value of and the time instant of each adaption will be provided in the next section. The downlink symbol received at user from the BS on subcarrier at time instant111In practical systems, the length of the cyclic prefix of an orthogonal frequency-division multiplexing (OFDM) symbol is chosen to be larger than the root mean square delay spread of the channel. Nevertheless, the channel gains from one subcarrier to the next may change considerably. Therefore, we use a discrete model for the frequency domain. On the other hand, the coherence time for a low mobility user is about and an OFDM symbol in Long-Term-Evolution (LTE) systems has a length of 71.3 . Thus, during a transmission time much longer than the coherence time, e.g., 200 , a few thousands of OFDM symbols are transmitted. Therefore, we use a continuous time domain signal model for representing the time variation of the signals. , , is given by

 yi,k(t) = √Pi,k(t)gk(t)Hi,k(t)xi,k(t)+zi,k(t), (1)

where is the symbol transmitted from the BS to user on subcarrier at time . is the transmit power for the link between the BS and user on subcarrier . is the small scale fading coefficient between the BS and user on subcarrier at time . represents the joint effect of path loss and shadowing between the BS and user at time . is the AWGN in subcarrier at user with distribution , where is the noise power spectral density.

### Ii-C Models for Time Varying Fading and Energy Sources

###### Remark 1

We note that the major assumption made in the modelling of the problem is the stationarity and ergodicity of the fading and the energy arrival random processes. In fact, the assumption of particular distributions for the changes in fading gains, time of changes in fading gains, and/or energy arrival times do not change the structure of the algorithms presented in the paper as long as the corresponding distributions are known at the BS. This knowledge can be obtained via long term measurements. The assumption of a Poisson counting process for the energy arrivals is made for illustration of the countability of the incoming energy arrivals.

In the considered model, the transmitter can draw the energy required for signal transmission and signal processing from both the battery444Note that the term “battery” is used interchangeably with the term “energy harvester” in the paper. and the traditional energy source. In particular, the instantaneous total radio frequency (RF) transmit power of the power amplifier (PA) for user in subcarrier at time instant can be modeled as

 Pi,k(t)=PEi,k(t)+PNi,k(t),∀i,k,0≤t≤T, (2)

where and are the portions of the instantaneous transmitted power taken from the energy harvester and the non-renewable energy source for user in subcarrier at time instant , respectively. Furthermore, we model the energy consumption required for signal processing as

 ∫t0(PEC(u)+PNC(u))du=PCt,0≤t≤T, (3)

where and are the portions of the instantaneous power required for signal processing drawn from the energy harvester and the non-renewable energy source, respectively. is the required constant signal processing power at each time instant and includes the power dissipation in the mixer, transmit filters, frequency synthesizer, and digital-to-analog converter (DAC), etc.

Since two energy sources are implemented at the BS, we have to consider the physical constraints imposed by both energy sources, which are described in the following.

#### Ii-C1 Energy Harvesting Source

There are two inherent constraints on the energy harvester:

 C1:nF∑i=1K∑k=1∫tEb−δ0εsi,k(u)PEi,k(u)duEnergy from % energy harvester used in PA +∫tEb−δ0PEC(u)du ≤ b−1∑j=0Ej,∀b∈{1,2,…}, (4)
 C2:d(t)∑j=0Ej−nF∑i=1K∑k=1∫t0εsi,k(u)PEi,k(u)du−∫t0PEC(u)du ≤ Emax,0≤t≤T, (5)

where is an infinitesimal positive constant for modeling purpose555The integral in equation (4) is defined over a half-open interval in . As a result, the variable is used to account for an infinitesimal gap between the upper limit of integration and the boundary . , , is the binary subcarrier allocation indicator at time , and is constant which accounts for the inefficiency of the PA. For example, when , 100 Watts of power are consumed in the PA for every 10 Watts of power radiated in the RF. In other words, the power efficiency is . Constraint C1 implies that in every time instant, if the BS draws energy from the energy harvester to cover the energies required at the PA and for signal processing, it is constrained to use at most the amount of stored energy currently available (causality), even though more energy may possibly arrive in the future. Constraint C2 states that the energy level in the battery never exceeds in order to prevent energy overflow in(/overcharging to) the battery. In practice, energy overflow may occur if the BS is equipped with a small capacity battery.

#### Ii-C2 Non-renewable Energy Source

In each time instant, a maximum power of Watts can be provided by the non-renewable energy source to the BS. In other words, a maximum of Joules of energy can be drawn from the non-renewable source from time zero up to time . As a result, we have the following constraint on drawing power/energy from the non-renewable energy source at any time instant:

 C3: nF∑i=1K∑k=1εsi,k(t)PNi,k(t)Power from non-renewable source used in PA +PNC(t)≤PN,0≤t≤T. (6)

## Iii Offline Resource Allocation and Scheduling Design

In this section, we design an offline resource allocation algorithm by assuming the availability of non-causal knowledge of energy arrivals and channel gains.

### Iii-a Channel Capacity and Energy Efficiency

In this subsection, we define the adopted system performance measure. At the BS, the data buffers for the users are assumed to be always full and there are no empty scheduling slots due to an insufficient number of data packets at the buffers. Given perfect channel state information (CSI) at the receiver, the channel capacity666In general, if the future CSI is not available at the BS, the randomness of the multipath fading causes resource allocation mismatches at the BS which decreases the system capacity. For instance, if only causal knowledge of multipath coefficients is available at the BS, the BS may transmit exceedingly large amounts of power at a given time instant and exhaust all the energy of the energy harvester, even though there may be much better channel conditions in the next fading block which deserve more transmission energy for improving the system energy efficiency/capacity. between the BS and user on subcarrier over a transmission period of second(s) with subcarrier bandwidth is given by

 Ci,k=∫T0si,k(t)Wlog2(1+Pi,k(t)Γi,k(t))dt where Γi,k(t)=gk(t)|Hi,k(t)|2N0W. (7)

The weighted total system capacity is defined as the weighted sum of the total number of bits successfully delivered to the mobile users over a duration of seconds and is given by

 U(P,S)=K∑k=1αknF∑i=1Ci,k, (8)

where and are the power and subcarrier allocation policies, respectively. is a positive constant provided by upper layers, which allows the BS to give different priorities to different users and to enforce certain notions of fairness. On the other hand, we take into account the total energy consumption of the system by including it in the optimization objective function. For this purpose, we model the weighted energy dissipation in the system as the sum of two dynamic terms

 UTP(P,S)=∫T0(ϕPEC(t)+PNC(t))dt+K∑k=1nF∑i=1∫T0si,k(t)ε(ϕPEi,k(t)+PNi,k(t))dt, (9)

where is a positive constant imposed on the use of the harvested energy. The value of can reflect either a normalized physical cost (e.g., relative cost for maintenance/operation of both sources of energy) or a normalized virtual cost (e.g., energy usage preferences), w.r.t. the usage of the non-renewable energy source [28]. In practice, we set to encourage the BS to consume energy from the energy harvesting source. The first term and second term in (9) denote the total weighted energy consumptions in the signal processing unit and the PA, respectively. Hence, the weighted energy efficiency of the considered system over a time period of seconds is defined as the total average number of weighted bit/Joule

 Ueff(P,S) = U(P,S)UTP(P,S). (10)

### Iii-B Optimization Problem Formulation

The optimal power allocation policy, , and subcarrier allocation policy, , can be obtained by solving

 maxP,SUeff(P,S) (11) s.t. C1, C2, C3 C4:∫t0(PEC(u)+PNC(u))du=PCt,0≤t≤T,C5:K∑k=1nF∑i=1Ci,k≥Rmin, C6:nF∑i=1K∑k=1Pi,k(t)si,k(t)≤Pmax,0≤t≤T,C7:si,k(t)={0,1},∀i,k,0≤t≤T, C8:K∑k=1si,k(t)≤1,∀i,0≤t≤T,C9: PNi,k(t),PEi,k(t),PNC(t),PEC(t)≥0,∀i,k,0≤t≤T,

where C4 ensures that the energy required for signal processing is always available777 In the considered system, the BS always has sufficient energy for CSI estimation despite the intermittent nature of energy generated by the energy harvester. Indeed, the BS is able to extract energy from both the traditional power supply (from the power generator) and the energy harvester. In the worst case, if the energy harvester is unable to harvest enough energy from the environment, the BS can always extract power from the traditional power supply for supporting the energy consumption of signal processing in the BS. . C5 specifies the minimum system data rate requirement which acts as a QoS constraint for the system. Note that although variable in C5 is not an optimization variable in this paper, a balance between energy efficiency and aggregate system capacity can be struck by varying . C6 is a constraint on the maximum transmit power of the BS. The value of in C6 puts a limit on the transmit spectrum mask to control the amount of out-of-cell interference in the downlink at every time instant. Constraints C7 and C8 are imposed to guarantee that each subcarrier will be used to serve at most one user at any time instant. C9 is the non-negative constraint on the power allocation variables.

###### Remark 2

We note that an individual data rate requirement for each user can be incorporated into the current problem formulation by imposing the individual data rate requirements as additional constraints in the problem formulation [29, 30], i.e., , where is a set of delay sensitive users and is a constant which specifies the minimal required data rate of user . The resulting problem can be solved via a similar approach as used for solving the current problem formulation.

### Iii-C Transformation of the Objective Function

The optimization problem in (11) is non-convex due to the fractional form of the objective function and the combinatorial constraint C7 on the subcarrier allocation variable. We note that there is no standard approach for solving non-convex optimization problems. In order to derive an efficient power allocation algorithm for the considered problem, we introduce a transformation to handle the objective function via nonlinear fractional programming [31]. Without loss of generality, we define the maximum weighted energy efficiency of the considered system as

 q∗=U(P∗,S∗)UTP(P∗,S∗)=maxP,SU(P,S)UTP(P,S). (12)

We are now ready to introduce the following Theorem.

###### Theorem 1

The maximum weighted energy efficiency is achieved if and only if

 maxP,S U(P,S)−q∗UTP(P,S)=U(P∗,S∗)−q∗UTP(P∗,S∗)=0, (13)

for and .

Proof: The proof of Theorem 1 is similar to the proof in [32, Appendix A].

Theorem 1 reveals that for any objective function in fractional form, there exists an equivalent objective function in subtractive form, e.g. in the considered case, which shares the same optimal resource allocation policy. As a result, we can focus on the equivalent objective function for finding the optimal offline resource allocation policy in the rest of the paper.

### Iii-D Iterative Algorithm for Energy Efficiency Maximization

In this section, an iterative algorithm (known as the Dinkelbach method [31]) is proposed for solving (11) by exploiting objective function . The proposed algorithm is summarized in Table I. Its convergence to the optimal energy efficiency is guaranteed if we are able to solve the inner problem (III-D) in each iteration.

Proof: Please refer to [32, Appendix B] for a proof of convergence.

As shown in Table I, in each iteration of the main loop, we solve the following optimization problem for a given parameter :

 maxP,SU(P,S)−qUTP(P,S) s.t.C1, C2, C3, C4, C5, C6, C7, C8, C9. (14)

#### Solution of the Main Loop Problem

Although the objective function is now transformed into a subtractive form which is easier to handle, there are still two obstacles in solving the above problem. First, the equivalent problem in each iteration is a mixed combinatorial and convex optimization problem. The combinatorial nature comes from the binary constraint C7 for subcarrier allocation. To obtain an optimal solution, an exhaustive search is needed in every time instant which entails a complexity of and is computationally infeasible for . Second, the optimal resource allocation policy is expected to be time varying in the considered duration of seconds. However, it is unclear how often the BS should update the resource allocation policy which is a hurdle for designing a practical resource allocation algorithm, even for the case of offline resource allocation. In order to strike a balance between solution tractability and computational complexity, we handle the above issues in two steps. First, we follow the approach in [33] and relax in constraint C7 to be a real value between zero and one instead of a Boolean, i.e., . Then, can be interpreted as a time-sharing factor for the users to utilize subcarrier . For facilitating the time sharing on each subcarrier, we introduce three new variables and define them as , , and . These variables represent the actual transmitted powers in the RF of the BS on subcarrier for user under the time-sharing assumption. Although the relaxation of the subcarrier allocation constraint will generally result in a suboptimal solution, the authors in [10, 34] show that the duality gap (sub-optimality) becomes zero when the number of subcarriers is sufficiently large for any multicarrier system that satisfies time-sharing888The proposed offline solution is asymptotically optimal when the number of subcarriers is large [10, 34]. In fact, it has been shown in [2] via simulation that the duality gap is virtually zero for only 8 subcarriers in an OFDMA system. Besides, the number of subcarriers employed in practical systems such as LTE is in the order of hundreds. In other words, the solution obtained under the relaxed time-sharing problem formulation is asymptotically optimal with respect to the original problem formulation. On the other hand, we note that although time-sharing relaxation is assumed, the solution in (24) indicates that the subcarrier allocation is still a Boolean which satisfies the binary constraint on the subcarrier allocation of the original problem. . Second, we introduce the following lemma which provides valuable insight about the time varying dynamic of the optimal resource allocation policy.

###### Lemma 1

The optimal offline resource allocation policy999Here, “optimality” refers to the optimality for the problem formulation under the time-sharing assumption. maximizing the system weighted energy efficiency does not change within an epoch.

Proof: Please refer to the Appendix for a proof of Lemma 1.

As revealed by Lemma 1, the optimal resource allocation policy maximizing the weighted system energy efficiency is a constant in each epoch. Therefore, we can discretize the integrals and continuous variables in (III-D). In other words, the number of constraints in (III-D) reduce to countable quantities. Without loss of generality, we assume that the channel states change times and energy arrives times in the duration of . Hence, we have epoch(s) for the considered duration of seconds. Time instant is treated as an extra fading epoch with zero channel gains for all users to terminate the process. We define the length of an epoch as where epoch is defined as the time interval , cf. Figure 2. Note that is defined as . For the sake of notational simplicity and clarity, we replace all continuous-time variables with corresponding discrete time variables, i.e., , , , , , , , , and . Then, the weighted total system capacity and the weighted total energy consumption can be re-written as

 U(P,S) = K∑k=1αknF∑i=1L∑j=1ljCi,k[j] and (15) UTP(P,S) = L∑j=1lj(ϕPEC[j]+PNC[j])+K∑k=1nF∑i=1L∑j=1ljε(ϕ~PEi,k[j]+~PNi,k[j]), (16)

respectively, where is the channel capacity between the BS and user on subcarrier in epoch . As a result, the optimization problem in (III-D) is transformed into to the following convex optimization problem:

 maxP,SU(P,S)−qUTP(P,S) (17) C1:nF∑i=1K∑k=1e∑j=1ljε~PEi,k[j]+e∑j=1PEC[j]lj≤e∑j=1Ein[j],∀e∈{1,2,…,M+N}, C2:r∑j=1Ein[j]−nF∑i=1K∑k=1r−1∑j=1εlj~PEi,k[j]−r−1∑j=1ljPEC[j]≤Emax,∀r∈{2,…,M+N+1}, C5:K∑k=1nF∑i=1L∑j=1ljCi,k[j]≥Rmin,C6:nF∑i=1K∑k=1le~Pi,k[e]≤lePmax,∀e, C7:0≤si,k[e]≤1,∀e,i,k,C8:K∑k=1si,k[e]≤1,∀e,i,C9:PNi,k[e],PEi,k[e],PNC[e],PEC[e]≥0,∀i,k,e,

where in C1 is defined as the energy which arrives in epoch . Hence, for some if event is an energy arrival and if event is a channel gain change, cf. Figure 2. The transformed problem in (17) is jointly concave w.r.t. all optimization variables101010We can follow a similar approach as in Appendix A to prove the concavity of the above problem for the considered discrete time model., and under some mild conditions [35], solving the dual problem is equivalent to solving the primal problem.

###### Remark 3

Mathematically, on both sides of the (in)equalities in C3, C4, and C6 in (17) can be cancelled. Nevertheless, we do think that it is desirable to keep in these constraints since they preserve the physical meaning of C6; the energy consumption constraints in the system in time duration .

### Iii-E Dual Problem Formulation

In this subsection, we solve transformed optimization problem (17). For this purpose, we first need the Lagrangian function of the primal problem. Upon rearranging terms, the Lagrangian can be written as

 L(\boldmathγ,\boldmathβ,ρ,\boldmathμ,\boldmathν,\boldmathψ,\boldmathη,P,S)=L∑j=1K∑k=1αknF∑i=1lj(wk+ρ)Ci,k[j]−ρRmin+L∑j=1nF∑i=1K∑k=1ηi,j −L∑j=1γj(nF∑i=1K∑k=1j∑m=1εlm~PEi,k[m]+j∑m=1lmPEC[m]−j∑m=1Ein[m])−L∑j=1νjlj(PEC[j]+PNC[j])
 −q(L∑j=1lj(ϕPEC[j]+PNC[j])+K∑k=1nF∑i=1L∑j=1ljε(ϕ~PEi,k[j]+~PNi,k[j]))+L∑j=1νjljPC −L+1∑j=2βj(j∑m=1Ein[m]−nF∑i=1K∑k=1j−1∑m=1εlm~PEi,k[m]−j−1∑m=1lmPEC[m]−Emax)−L∑j=1nF∑i=1K∑k=1ηi,jsi,k[j] −L∑j=1μj(nF∑i=1K∑k=1εlj~PNi,k[j]+ljPNC[j]−ljPN)−L∑j=1ψj(nF∑i=1K∑k=1lj~Pi,k[j]−ljPmax), (18)

where is the Lagrange multiplier vector associated with causality constraint C1 on consuming energy from the energy harvester and has elements , . is the Lagrange multiplier vector corresponding to the maximum energy level constraint C2 in the battery of the energy harvester with elements where . is the Lagrange multiplier corresponding to the minimum data rate requirement in C5. , , and have elements , , and are the Lagrange multiplier vectors for constraints C3, C4, and C6, respectively. is the Lagrange multiplier vector accounting for subcarrier usage constraint C8 with elements . Note that the boundary constraints C7 and C9 are absorbed into the Karush-Kuhn-Tucker (KKT) conditions when deriving the optimal solution in Section III-F.

Thus, the dual problem is given by

 min\boldmathγ,\boldmathβ,ρ,\boldmathμ,\boldmathψ,\boldmathη≥0 maxP,SL(\boldmath%$γ$,\boldmathβ,ρ,\boldmathμ,\boldmathν,\boldmathψ,\boldmathη,P,S). (19)

Note that is not an optimization variable in (19) since C4 in (17) is an equality constraint.

### Iii-F Dual Decomposition and Subproblem Solution

By Lagrange dual decomposition, the dual problem is decomposed into two parts (nested loops): the first part (inner loop) consists of subproblems where subproblems have identical structure; the second part (outer loop) is the master dual problem. The dual problem can be solved iteratively where in each iteration the BS solves subproblems (inner loop) in parallel and solves the master problem (outer loop) with the gradient method.

Each one of the subproblems with identical structure is designed for one subcarrier and can be expressed as

 maxP,S Li(\boldmathγ,\boldmathβ,ρ,\boldmathμ,\boldmathν,\boldmathψ,\boldmathη,P,S) (20)

for a fixed set of Lagrange multipliers where

 +L+1∑j=2βjK∑k=1j−1∑m=1εlm~PEi,k[m]−q(K∑k=1L∑j=1ljε(ϕ~PEi,k[j]+~PNi,k[j])+L∑j=1lj(ϕPEC[j]+PNC[j]))
 −L∑j=1ψj(K∑k=1lj~Pi,k[j])−L∑j=1ηi,j(K∑k=1si,k[j])−L∑j=1μj(K∑k=1εlj~PNi,k[j]+ljPNC[j]) −L∑j=1ljνj(PEC[j]+PNC[j])−L∑j=1γj(K∑k=1j∑m=1εlm~PEi,k[m]+j∑m=1lmPEC[m]). (21)

Let , and denote the solution of subproblem (20) for event . Using standard optimization techniques and the KKT conditions, the power allocation for signal transmission for user on subcarrier for event is given by

 ~PE∗i,k[j] = si,k[j]PE∗i,k[j]=si,k[j][W(αk+ρ)ln(2)(∑Le=jγeε−∑Le=jβe+1ε+qϕε+ψj)−1Γi,k[j]]+and ~PN∗i,k[j] = si,k[j]PN∗i,k[j]=si,k[j][W(αk+ρ)ln(2)(qε+μjε+ψj)−1Γi,k[j]−~PE∗i,k[j]]+, (22)

for . The power allocation solution in (III-F) can be interpreted as a multi-level water-filling scheme as the water levels of different users can be different. Interestingly, the value of depends on . As can be seen in (22), decreases the water-level for calculation of the value of . In other words, reduces the amount of energy drawn from the non-renewable source for maximization of energy efficiency. Besides, it can be observed from (III-F) that the BS does not always consume all available renewable energy in each epoch for maximization of the weighted energy efficiency and the value of determines at what point the water-level is clipped. On the other hand, in order to obtain the subcarrier allocation, we take the derivative of the subproblem w.r.t. , which yields , where is the marginal benefit [36] for allocating subcarrier to user for event and is given by

 W(αk+ρ)(log2(1+Γi,k[j](PE∗i,k[j]+PN∗i,k[j]))−Γi,k[j](PE∗i,k[j]+PN∗i,k[j])ln(2)(1+Γi,k[j](PE∗i,k[j]+PN∗i,k[j]))). (23)

Thus, the subcarrier selection on subcarrier in event is given by

 s∗i,k[j]={1if k=argmaxc Qi,c[j] 0 otherwise. (24)

It can be observed from (23) that only the user who can provide the largest marginal benefit on subcarrier in epoch is selected by the resource allocator, for transmission on that subcarrier. This is because the channel gains of different users are generally different due to uncorrelated fading across different users. We note that a larger marginal benefit is not necessarily equivalent to a larger system throughput since the marginal benefit includes a notion of fairness.

After solving the subproblems with identical structure, we calculate the amount of power used for signal processing in each of the two energy sources. We substitute into (20) which yields the following KKT condition for :

 ∂Li(…)∂PE∗C[j]=−ljL∑e=jγe+ljL∑e=jβe+1−qljϕ+qlj+μlj{≥0,PE∗C[j]≥0<0, otherwise. (25)

It can be observed from (25) that the Lagrangian function is an affine function in . In other words, the value of must be one of the two vertexes of a feasible solution set created by the associated constraints. As a result, the powers used for signal processing drawn from the energy harvester and the non-renewable source are given by

 PE∗C[j] = [∑ja=1Ein[a]−∑nFi=1∑Kk=1∑ja=1laε~PE∗i,k[a]−∑j−1m=1PEC[m]lmlj]PC0and (26) PN∗C[j] = PC−PE∗C[j], (27)

respectively. The numerator of variable in (26) represents the residual energy level in the battery, i.e., the vertexes (feasible set) created by the associated constraints on . Equations (26) and (27) indicate that if the amount of energy in the energy harvester is not sufficient to fully supply the required energy , i.e, , then the BS will also draw energy from the non-renewable energy source such that .

### Iii-G Solution of the Master Dual Problem

For solving the master minimization problem in (19), i.e, to find , , , , and for given and , the gradient method can be used since the dual function is differentiable. The gradient update equations are given by:

 γj(ς+1) = [γj(ς)−ξ1(ς)×(j∑m=1Ein[m]−j∑m=1PEC[m]lmnF∑i=1K∑k=1j∑m=1lmε~PEi,k[m])]+,∀j, (28)