Energy Management in Large-Scale MIMO Systems with Per-Antenna Energy Harvesting

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1 Energy Management in Large-Scale MIMO Systems with Per-Antenna Energy Harvesting Rami Hamdi 1,2, Elmahdi Driouch 1 and Wessam Ajib 1 1 Department of Computer Science, Université du Québec à Montréal (UQÀM), Montréal, QC, Canada 2 École de Technologie Supérieure (ÉTS), Montréal, QC, Canada rami.hamdi.1@ens.etsmtl.ca, {driouch.elmahdi, ajib.wessam}@uqam.ca Abstract This paper investigates the downlink of an energy efficient distributed large-scale MIMO system. The studied system is assumed to be made up of a set of remote radio heads (RRHs), each of which is powered by both an independent energy harvesting source and the grid. The grid energy allows to compensate for the randomness and intermittency of the harvested energy. Hence, the problem of grid power consumption minimization under quality of service (QoS) constraints has to be solved. First, this paper solves the optimal offline version of the problem using linear programming. Next, an iterative link removal algorithm is proposed in order to ensure the feasibility of the problem. Finally, the optimal online energy management algorithm is also proposed to solve the same problem. Simulation results show the performance of the proposed algorithms. The proposed approach in this paper allows efficient use of non-renewable energy to compensate the variability of renewable energy in large-scale MIMO systems. Index Terms-Large-scale MIMO, energy harvesting, power grid, energy management, link removal. I. INTRODUCTION Large-scale multiple-input multiple-output (MIMO) (also known as massive MIMO) systems are advocated as a key technology for next generations of wireless networks [1]. They are based on multiplexing few hundred antennas to serve at the same time-frequency few tens of users. They allow achieving high spectral efficiency using linear transmit and receive techniques [2]. However, systems with co-located antennas may suffer from highly-correlated small-scale fading and identical large-scale fading. Also, the deployment of large number of antennas on the same base station presents many technical and implementation challenges [3]. Alternatively, distributed large-scale MIMO systems can mitigate large-scale fading due to heterogeneous path-loss conditions. They are also shown to be more energy efficient than colocated antenna systems when taking exclusively into account the energy consumption of transmit and reception units [4]. Distributed large-scale MIMO systems can be seen as a set of remote radio heads (RRHs) distributed over a large area. Each RRH contains an antenna and an RF chain and is reliably connected to other RRHs through a central unit. To address their high power consumption, such systems may incorporate energy harvesting that is seen as a promising key technology for greening the future wireless networks since it reduces the network operation cost and carbon footprints [5]. Therefore, each RRH can be powered by both energy harvested from renewable sources such as thermal, wind or solar [6] and grid energy source. The gains offered by large-scale MIMO systems cannot be extracted without adequate resource allocation (e.g., power allocation, antenna selection, etc.) strategies. The studied resource allocation in [7] focuses on antenna selection, power allocation and user scheduling in downlink large-scale MIMO systems considering a non negligible circuit power consumption. In [8], a joint antenna selection and power allocation scheme maximizing the sum-rate in large cloud radio access networks is proposed. In [9], the problem of transmit power minimization and user association is optimally solved for downlink multi-cell large-scale MIMO systems. Resource allocation in cellular systems powered by energy harvesting was also studied in the literature. The authors of [10] investigated a cellular network powered by both grid and renewable energy. The optimal base station on/off policy is obtained using dynamic programming in order to minimize grid power consumption. In [11], the capability of renewable energy to power cellular networks is investigated in terms of coverage. A distributed deployment model of harvesting energy sources is proposed in order to cope with energy spatial random variations. In [12], the optimal offline and on-line energy management settings are derived for small-cell access points. A co-located point-to-point largescale MIMO system powered by single hybrid source was considered in [13]. Energy efficiency is optimized off-line by deriving numerically the optimal number of transmit antennas and the power distribution. Although, resource allocation was extensively investigated for large-scale MIMO and energy harvesting systems, the literature lacks of energy management schemes designed for systems implementing both technologies. Therefore, this paper investigates an energy efficient distributed large-scale MIMO system where the RRHs are assumed to be powered by energy bought from a grid source in addition to energy harvested from renewable sources. We formulate a problem of grid power consumption minimization subject to quality of service (QoS) constraints required by the served users. The optimal off-line and on-line energy management problems are solved. Moreover, the feasibility problem is addressed by proposing a link removal algorithm. Finally, simulation results show the performance of the proposed algorithms. In this paper, (.) H represents the Hermitian of a matrix, /17/$ IEEE

2 Tr{.} denotes the trace of a square matrix, E{.} denotes the mathematical expectation,. is the Euclidean norm of a vector and. F denotes the Frobenius norm of a matrix. The rest of the paper is organized as follows. In Section II, the system model is presented and the grid power consumption minimization problem is formulated. The optimal off-line energy management solution is presented in Section III and the optimal on-line solution is presented in Section IV. Numerical results are presented and discussed in Section V. Finally, we conclude this paper in Section VI. II. SYSTEM MODEL AND PROBLEM FORMULATION A. Channel and Signal Model The downlink of a distributed large-scale MIMO system shown in Fig. 1 is considered. A central unit is connected to N RRHs via error-free links serving K single-antenna users where N K. A given time interval is partitioned into L frames with duration T out. The channel coefficients between the RRHs and the users are represented by a complex matrix G(i) =[g 1 (i), g 2 (i),..., g K (i)] where g k (i) = [g n,k (i)] n=1:n C 1 N is the channel vector of user k at frame i. Since non co-located antennas and non co-located users are assumed, the spatial correlation is neglected for both transmission and reception links. The channel coefficient g n,k (i) is given by g n,k (i) = β n,k h n,k (i), where h n,k (i) is the small-scale fading at frame i and β n,k represents the large-scale fading between user k and RRH n. The smallscale fading h n,k (i) is assumed to be quasi-static Gaussian independent and identically distributed (i.i.d.) slow fading channel. Considering only path loss, the large-scale fading component is expressed as β n,k = ζ d ν n,k, where ν is the path d ν 0 loss exponent, d n,k is the distance between the RRH n and user k, d 0 is the reference distance and ζ is a constant related to the carrier frequency and reference distance. The central unit estimates the channel using the minimum mean square error (MMSE) and thus the estimated channel coefficient satisfies [2]: ĝ n,k (i) = β n,k (ξh n,k (i)+ 1 ξ 2 e), (1) where 0 ξ 1 denotes the reliability of the estimate and e is an error component modeled as Gaussian i.i.d. random variable with zero mean and unit variance. We denotes by w k (i) C N 1 the k th beamforming vector for user k. The low complexity maximum ratio transmitter (MRT) is considered as beamforming technique. The beamforming vector for user k is given by w k (i) = ĝk(i) H η(i), where η(i) = Ĝ(i)H F is the normalization factor. Hence, the received signal-to-interference-plus-noise ratio (SINR) at user k is expressed as: γ k (i) = N n=1 p n,k(i) η 2 (i) g k (i)ĝ k (i) H 2 K N. n=1 pn,m(i) m=1,m k η 2 (i) g k (i)ĝ m (i) H 2 +σ 2 (2) where p n,k (i) is the power allocated to user k on RRH n at frame i and σ 2 is the variance of the noise that is assumed to be additive white Gaussian noise (AWGN) with zero mean. Fig. 1: Distributed large-scale MIMO system with per-rrh energy harvesting. B. Energy Harvesting Model The harvested energy at each RRH is first stored in a battery with maximal capacity B max. It is modeled by a compound Poisson stochastic process. Let E n (i) and X n (i) denote respectively the amount of energy harvested and consumed at RRH n during frame i. The grid energy is required to compensate for the randomness and intermittence of the energy harvesting sources to guarantee that all users are served. We consider p n,k (i) =p e n,k (i) +pg n,k (i), where p e n,k (i) and pg n,k (i) denote the power drawn from the energy harvesting source and the power grid respectively. We also define E fix for each RRH as the summation of the required energy to transmit its current battery level and the received pilot signal to the central unit, and the energy consumed by the circuit. The latter includes the power consumed by the digital to analog converters (DACs), mixers and filters. Since the RRHs are powered by both renewable and grid sources, the required energy for the RRHs operation can be written as: E fix = Ef e n(i)+ef g n(i), n =1..N, i =1..L, (3) where Ef e n(i) and Ef g n(i) are the energy drawn from the energy harvesting and the grid source respectively. Hence, the consumed energy per RRH n at frame i can be given by: X n (i) =Ef e n(i)+ef g n(i)+ + p g n,k (i) T out = X e n(i)+ef g n(i)+ p e n,k(i) T out p g n,k (i) T out, (4)

3 where X e n(i) denotes the total energy drawn from the energy harvesting source by RRH n at frame i. Let B n (i) denote the battery level at frame i. The consumed energy from the energy harvesting source cannot exceed the battery level. Hence, the energy causality constraint is given by: X e n(i) B n (i), (5) and the battery level update is expressed as: B n (i +1)=min(B max,b n (i) X e n(i)+e n (i)). (6) We consider that the grid power consumption by RRH n at frame i is weighted by a factor α n,i. Hence, the total grid power consumption is expressed as: Δ tot = where Δ n,i = E fix Ef e n (i) T out L n=1 N α n,i Δ n,i, (7) + K pg n,k (i). C. Frame Structure Each frame consists of 4 phases: 1) During the 1 st phase, users send uplink pilot symbols to the RRHs. 2) During the 2 nd phase, each RRH sends reliably its current battery level and the received pilot signal to the central unit with fixed transmit energy E bs. 3) The central unit performs channel estimation, beamforming, resource management and forwards its decisions towards the RRHs at the 3 rd phase. 4) Data transmission occurs at the 4 th phase. D. Problem Formulation The objective of this work is to minimize the total consumed grid power while making use of the available harvested energy. We assume that each user requires a minimum received SINR to be satisfied. The main problem when channel coefficients, harvested energy arrivals and grid power weights are assumed to be known, can be formulated as (8). Constraints C1 ensure a minimum received SINR, denoted γ th, to each user. Constraints C2 are related to the energy causality, i.e. the consumed harvested energy at RRH n cannot exceed the energy harvested at RRH n. Constraints C3 specify that the harvested energy at the current frame cannot exceed the maximal battery capacity. Constraints C4 specify that the transmit power at each RRH is constrained due to the limited linear domain of the power amplifiers. Finally, constraints C5 and C6 ensure the non-negativity of the allocated amounts of power or energy. The objective function and the constraints of problem (8) are clearly linear. Hence, the optimal energy management is obtained by solving a linear program as discussed in the next section. minimize L {p e n,k (i),pg n,k (i),ef n e(i)} n=1..k,n=1..n,..l N α n,i Δ n,i subject to C1 :γ k (i) γ th, k =1..K, i =1..L, ( ) C2 : Efn(i)+ e p e n,k(i) T out C3 : E n (i), n =1..N, l =1..L, ( ) l 1 E n (i) Efn(i)+ e p e n,k(i) T out B max, n =1..N, l =2..L, C4 : (p e n,k(i)+p g n,k (i)) p max, n =1..N, i =1..L, C5 :p e n,k(i),p g n,k (i) 0, k =1..K, n =1..N, i =1..L, C6 :E fix Efn(i) e 0, n =1..N, i =1..L. (8) III. OFF-LINE ENERGY MANAGEMENT In this section, we distinguish between the scenarios where (1) the harvested energy is high enough and there is no need for consuming the grid power, (2) there is a need for the grid power but the problem is feasible (i.e., all the users can be satisfied) and finally (3) some of the users cannot be satisfied. A. No Need for Grid Power The power drawn from energy harvesting sources can be sufficient to ensure QoS for all users and there is no need for grid power according to the following lemma. Lemma 1: The consumption of grid power tends to zero if the relation below, between the amount of stored energy at the batteries and channel coefficients, is verified: where p k(i) = l E fix + gn,k H (i) 2 η(i) 2 p k(i) T out E n (i), n =1..N, l =1..L, Nσ 2 d + σ2 η(i) 2 1 γ K 1 1 Nσ2 d + σ 2 η(i) 2 th m=1 gm(i)gm(i) H 2 γ th Proof: given in Appendix A. (9) g k (i)g k (i) H 2. (10)

4 B. Linear Programming As discussed before, problem (8) can be solved to optimality using linear programming techniques such as interior point method when it is feasible. We present in the following a new matrix formulation of problem (8). Therefore, we use the following notations. We denote α L N =[α 1,1,..., α 1,L,..., α N,L ] C 1 N L and α K L N = [α 1,1,..., α 1,1,..., α 1,L,..., α 1,L,..., α N,L,..., α N,L ] C 1 N L K. The transmit power vector drawn from energy harvesting sources is denoted p e = [p e 1,1(1),..., p e 1,K (1),..., pe 1,K (L),..., pe N,K (L)] CN L K 1, the transmit power vector drawn from the grid is denoted p g = [p g 1,1 (1),..., pg 1,K (1),..., pg 1,K (L),..., pg N,K (L)] C N L K 1 and the fixed required energy vector is denoted Ef e =[Ef1 e (1),..., Ef1 e (L),..., EfN e (L)] CN L 1. Since the objective function and all the constraints are affine, the problem (8) can be reformulated as: minimize α L N ( E fix Ef e ) + α K L N p g p e 0,p g 0,Ef e 0 T out subject to A 1 p e + B 1 Ef e b 1, A 2 (p e + p g ) b 2, Ef e E fix. (11) Matrices A 1 C N(2L 1) L K N and B 1 C N(2L 1) L N and vector b 1 C N(2L 1) 1 are formed by constraints C2 and C3. Matrix A 2 C L(N+K) L K N and vector b 2 C L(N+K) 1 are formed by constraints C1 and C4. The optimal energy management can be obtained by solving problem (11) using interior-point method implemented in numerical tools such as CVX. C. Link Removal The multi-user interference present in constraints C1 of problem (8) as well as the limitations imposed by constraints C4 on the maximum transmit power at each RRH, involve that, in some cases, it may be impossible to ensure the SINR required by all the users at each frame and hence the problem may be infeasible. Therefore, some users have to be denied service in some frames in order to overcome this infeasibility [14]. More precisely, users with bad channel conditions in a particular frame are the ones who violate the SINR constraint the most, and hence they should not be served in that frame. We propose in this section a link removal algorithm to solve heuristically the infeasibility problem. We mean by link a particular (user, frame) pair. The optimal link removal solution can be obtained only by high complex brute force search algorithm. The proposed low complexity iterative link removal algorithm is given in Algorithm 1. The algorithm is based on the observation that the links with bad channel conditions need large amount of transmit power to attain the required SINR and also they cause high interference to other users. These worst links are iteratively removed one by one based on their received SINRs computed assuming a fixed transmit power allocation. The algorithm terminates once the problem becomes feasible. Accordingly, the removed link at each step is found according to: where (k, l) argmin ψ k,l (12) (k,l) {1,...,K} {1,...,L} [ ] ĝ k (l)ĝ k (l) H 2 ψ k,l = K,i k ĝ. (13) k(l)ĝ i (l) H 2 +σ 2 η 2 (l) Algorithm 1 Iterative Link Removal Algorithm 1: Computation of matrix LR =[ψ k,l ] :K,l=1:L, // initialization (all links are scheduled) 2: Ω {}, // set of removed links 3: feas false, // boolean variable for feasibility test 4: r 0, // number of links already removed 5: while r<l K 1 and Not feas do 6: r r +1, 7: (k, l) argmin LR (k,l) {1,...,K} {1,...,L} 8: Ω Ω {(k, l) }, // remove link (k, l) 9: update matrix LR 10: solve linear program (11) 11: if problem (11) is feasible then 12: feas true 13: end if 14: end while IV. ON-LINE ENERGY MANAGEMENT In this section, we propose to solve problem (11) by an on-line energy management. The central unit is assumed to know the channel coefficients, the harvested energy and the weights of all RRHs but only at the current frame i. Hence, the following problem has to be solved at each frame i: minimize {Efn e(i),pe n,k (i),pg n,k (i)} n=1..k,n=1..n N α n,i Δ n,i subject to γ k (i) γ th, k =1..K, Efn(i)+ e p e n,k(i) T out B n (i), n =1..N, p e n,k(i)+p g n,k (i) p max, n =1..N, p e n,k(i),p g n,k (i) 0, k =1..K, n =1..N, E fix Efn(i) e 0, n =1..N. (14) The optimal on-line energy management algorithm described in Algorithm 2 solves the linear program (14) using interior point method at each frame i. The algorithm uses the maximal available harvested energy at each frame.

5 Algorithm 2 Optimal On-line Energy Management B n (1) E n (1), n =1..N, // initialization Δ tot 0, // initial grid power consumption for i =1:L do solve linear program (14) Δ tot Δ tot + N n=1 α n,i Δ n,i, for n =1:N do X e n(i) Ef e n(i)+ K pe n,k (i) T out, n=1..n, // Consumed harvested energies computation B n (i + 1) min(b max,b n (i) X e n(i) + E n (i)), n =1..N, // batteries level update end for end for V. NUMERICAL RESULTS In this section, monte carlo simulations are used to evaluate the performance of the proposed algorithms. The simulation parameters are summarized in Table I. We consider that the distributed large-scale array system adopts a circular topology. The RRHs are uniformly deployed along a circle of radius r a and the users are uniformly distributed within the circular cell of radius r c with r a <r c. The grid power consumption weights α n,i are randomly generated according to a standard uniform distribution. TABLE I: Simulation Parameters. Symbol Description Value p c circuit power per RF chain 30 dbm [15] ν path loss exponent 3.7 r a antenna array radius 40 m r c cell radius 500 m p max maximal transmit power per-rrh 1W B max maximal battery capacity 50 J N number of RRHs 80 K number of users 8 noise PSD -174 dbm/hz In Fig. 2, we compare the achieved grid power consumption with optimal off-line and optimal on-line energy management. The achieved grid power consumption approaches zero for SINR target lower than 6dB, the available harvested energy is sufficient to satisfy the QoS constraints. Obviously, the performance gap between the off-line and online solutions increases when the SINR target or the number of frames increases. In Fig. 3, we investigate the performance of the iterative link removal algorithm by showing the percentage of removed links versus the minimum received SINR. The optimal values are shown for only limited number of users K =4 and of frames L =2due to the extremely high exponential complexity of the exhaustive search optimal algorithm. Achieved grid power consumption (W) Optimal off-line energy managment Optimal on-line energy managment L=5 L=10 Minimum received SINR γ th (db) Fig. 2: Achieved grid power consumption with optimal offline and on-line energy management. The performance gap between the proposed algorithm and optimal link removal is tight and does not change too much when the SINR target increases. As expected, the percentage of removed links is higher when the QoS constraints are more stringent. Increasing per-rrh maximal transmit power p max allows to increase the total transmit power and thus to admit more links. Hence, the percentage of removed links decreases particularly at low SINR target. However, this performance gain vanishes at high SINR target due to the increase of multi-user interference. Iterative removal algorithm p max =1W Optimal removal algorithm p max =1W Iterative removal algorithm p max =3W Optimal removal algorithm p max =3W Minimum received SINR γ th (db) Fig. 3: Percentage of removed links under iterative and optimal link removal algorithms. VI. CONCLUSION This paper investigated energy efficient distributed largescale MIMO systems where the RRHs are powered by energy bought from a grid source in addition to energy harvested from renewable sources. A minimization problem of grid power consumption subject to quality of service constraint

6 per user was formulated for downlink transmissions. The optimal off-line energy management was solved by linear programming. The formulated problem could be infeasible due to the users requirement and per-antenna power constraints. Hence, an iterative link removal algorithm was proposed in order to overcome the feasibility problem. An optimal on-line energy management algorithm was also proposed for solving the problem in each frame independently. Furthermore, the proposed approach allows efficient use of non-renewable energy in hybrid energy large-scale MIMO systems. Future work will focus on proposing on-line solutions that consider the dependence of the energy arrival and the weights over time. APPENDIX A PROOF OF LEMMA 1 Considering the case where energy harvesting is sufficient to ensure the SINR requirements for each user at each frame, the allocated power drawn from energy harvesting sources for user k at frame i verifies: p k (i) g k(i)g k (i) H 2 K m=1,m k p m(i) g k (i)g m (i) H 2 +σ 2 η 2 (i) = γ th. (15) The interference term in large-scale MIMO systems can be asymptotically approximated when K and N are large but finite, [16] as: m=1,m k p m(i) g k (i)g m(i) H 2 p out(i)e{ g k (i)g m(i) H 2 } Np out(i)σd, 2 (16) where p out (i) is the total transmitted power at frame i and σd 2 is the variance of the channel. Hence, we obtain: [3] E. G. Larsson, O. Edfors, F. Tufvesson and T. L. Marzetta, Massive MIMO for Next Generation Wireless Systems, IEEE Commun. Mag., vol. 52, no. 2, pp , Feb [4] C. He, B. Sheng, P. Zhu, and X. You, Energy efficiency comparison between distributed and co-located MIMO systems, Int. J. Commun. Syst., vol. 27, no 1, p , [5] S. Ulukus, A. Yener, E. Erkip, O. Simeone, M. Zorzi, P. Grover and K. Huang, Energy harvesting wireless communications: A review of recent advances, IEEE J. Sel. Areas Commun., vol. 33, no. 3, March [6] K. N. R. Prasad, E. Hossain and V. K. Bhargava, Energy Efficiency in Massive MIMO-Based 5G Networks: Opportunities and Challenges, IEEE Wireless Commun. Mag., Jan [7] R. Hamdi, E. Driouch and W. Ajib, Resource Allocation in Downlink Large-Scale MIMO Systems, IEEE Access, vol. 4, no. 1, pp , Dec [8] A. Liu and V. K. N. Lau, Joint Power and Antenna Selection Optimization in Large Cloud Radio Access Networks, IEEE Trans. Signal Process., vol. 62, no. 5, March [9] T. V. Chien, E. Bjornson and E. G. Larsson, Joint Power Allocation and User Association Optimization for Massive MIMO Systems, IEEE Trans. Wireless Commun., vol. 15, no. 9, Sep [10] Y. Che, L. Duan and R. Zhang, Dynamic Base Station Operation in Large-Scale Green Cellular Networks, IEEE J. Sel. Areas Commun., Aug [11] S. Hu, Y. Zhang, X. Wang and G. B. Giannakis, Weighted Sum-Rate Maximization for MIMO Downlink Systems Powered by Renewables, IEEE Trans. Wireless Commun., vol. 15, no. 8, Aug [12] A. Yadav, T. M. Nguyen and W. Ajib, Optimal Energy Management in Hybrid Energy Small Cell Access Points, IEEE Trans. Commun., [13] Z. Zhou, S. Zhou, J. Gong and Z. Niu, Energy-efficient antenna selection and power allocation for large-scale multiple antenna systems with hybrid energy supply, in proc. of IEEE Global Commun. Conf. (GLOBECOM), Dec [14] L. B. Le and E. Hossain, Resource Allocation for Spectrum Underlay in Cognitive Radio Networks, IEEE Trans. Wireless Commun., vol. 7, no. 12, Dec [15] R. Kumar and J. Gurugubelli, How Green the LTE Technology can be?, in proc. of Int. Conf. on Wireless Commun., Veh. Technol., Inform. Theory and Aerosp. Electron. Syst. Techn., [16] L. Zhao, Hu. Zhao, F. Hu, K. Zheng and J. Zhang, Energy Efficient Power Allocation Algorithm for Downlink Massive MIMO with MRT Precoding, in proc. of IEEE Veh. Technol. Conf. (VTC), pp. 1-5, Sept p γ th k(i) = g k (i)g k (i) H 2 (Np out(i)σd 2 + σ 2 η 2 (i)) (17) Using p out (i) = K p k (i) and (17), we obtain the expression of p out (i) as: p out (i) = Nσ 2 d + σ2 η 2 (i) 1 γ K 1 1. (18) th g k (i)g k (i) H 2 By replacing the expression of p out (i) in (17), we obtain p k (i) that ensure the SINR requirements for each user. Then, we replace p k (i) in the energy causality constraints C2 and we obtain the relation (9). This completes the proof of Lemma 1. REFERENCES [1] J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong and J. C. Zhang, What Will 5G Be?, IEEE J. Sel. Areas Commun., vol. 32, no. 6, June [2] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta, O. Edfors and F. Tufvesson, Scaling up MIMO: Opportunities and Challenges with Very Large Arrays, IEEE Signal Process. Mag., vol. 30, no. 1, Jan