Optimal Battery Size for a Green Base Station in a Smart Grid with a Renewable Energy Source

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1 Optimal Battery Size for a Green Base Station in a Smart Grid with a Renewable Energy Source Endre Hegland Hjort Kure, Sabita Maharjan, Stein Gjessing, Yan Zhang Abstract Green base stations, powered by renewable energy sources, have mainly been restricted to remote areas. In urban areas, recent technological advancements have strengthened the bidirectional power flow in microgrids (Smart Grids), giving the possibility of green LTE base stations acting as energy traders. To facilitate this trading, batteries are needed to deal with the inherent stochasticity of renewable energy sources. In this paper we provide a new technique for dimensioning batteries for a base station. We show how the solution depends on different boundary conditions set by the microgrid such as energy balancing cost, transaction fee and power price. The technique is based on a Markov decision process and considers uncertainty in traffic, renewable power production and power price in the microgrid. We also show how to quantify the energy balancing cost for the base station, supplementing current research on energy balancing in microgrids. The results show that for certain microgrid conditions, the value of modelling price uncertainty is insignificant. Also, the results show that it is never optimal to invest in more than the minimum required battery size given certain transaction fee and power price regimes. Index Terms LTE, battery, optimization, uncertainty, Markov decision process, Markov process, Smart Grid, renewable energy I. INTRODUCTION LTE base stations (BSs) connected to renewable energy sources are called green BSs and are mainly located in remote areas. BSs in urban areas are powered by the power grid as green BSs are not economically sustainable. However, with the roll-out of microgrids (Smart Grids), small power grids with bidirectional power flow and communication between agents, BSs can act as energy traders. These changes make green BSs more economical, opening up for usage in urban environments. The bidirectional energy trade role for the BSs is a new synergy component, different from the original idea of incorporating information and communications technologies (ICT) for facilitating power systems. When operators install new BSs, they must decide upon the BS s coverage area, the type and the size of renewable power source and if the system should have batteries. While the BS s coverage area and the power source derive from only the environmental conditions, the battery size is also a function of battery management. The microgrid energy trading will effect optimal battery management, making previous research possibly outdated. A battery can also provide the BS with better power robustness, which is an inherent property of local energy storage. The robustness will not be focused upon in this paper. BSs are often deployed in many decentralized locations and altering battery installations after deployment are likely to be costly. If the battery is too small, the system will lack storage Energy flow direction Decision Stochastic process Photovoltaic energy source Microgrid Battery Base station 07 August Fig. 1. The system consists of a battery, a PV power source and a BS connected to a microgrid as shown with black lines. The blue striped arrows denote active decisions for selling or buying energy. The grey filled arrows show stochastic processes, for energy production and demand. capacity, and excess produced energy can not be stored for later use. Likewise, if the battery is too large, parts of the battery will never be used and is therefore an unnecessary investment. As observed by Leithon et al., there is a battery size limit where no further saving, neither in offline or online battery management, can be achieved [1]. In this paper we will investigate a setup consisting of a photovoltaic (PV) power source, an LTE BS and a battery, all connected to a microgrid as given by Figure 1. As shown in Figure 1, there are three main sources of uncertainty; the PV power production e t, the BS power consumption l t and the power price Pbuy zt in the microgrid. The uncertainty in PV power production depends on a range of factors, such as shadowing, temperature, clouds and soiling, that are difficult to predict [2]. An expected relative prediction error is around 40% for hourly day ahead predictions [1]. The uncertainty in a BS s power consumption is related to mobile data traffic. At a global level, it is expected to grow with an annually compound rate of 47% until 2021 [3]. For a single BS, the growth is uncertain as the operators will have to deploy new BSs to cope with global growth. LTE traffic prediction is therefore complex, as it is governed by both stochastic growth and stochastic daily usage. The price uncertainty in power markets, such as Nord Pool, is best understood by the existence of efficient financial markets for financial options 1

2 such as futures and forwards. Since these hedging instruments exist, there has to be uncertainty in the power price. Energy management in a similar setup as ours, with various incorporation of uncertainty has been investigated by several authors [1, 4, 5, 6, 7, 8, 9, 10]. A common strategy is to model the uncertainty either with a Markov processes model or with a statistical distribution. However, none of the published models include uncertainty modelling of all the three sources as our paper does. Some research have been published on optimizing battery size, but are restricted to generic microgrid agents. The main focus of research in this field has been on mitigating the fluctuation in renewable sources [11] or balancing power production and consumption [12, 13, 14]. Our research has three main contributions. Firstly, we investigate a new scenario of using batteries in communication systems in bidirectional microgrids. Secondly, since the scenario has not been focused upon by previous research, we provide a new Markov decision process (MDP) inspired technique to derive a solution. Thirdly, with our new technique we derive the cost for the BS to do its own power balancing, giving new insights into the discussion regarding power balancing in microgrids. The paper is divided into four parts. In Section II we derive the models for the power consumption, the power price and the power production. Then, in Section III our MDP inspired technique is presented. In Section V numerical results are used to derive the BS s power balancing cost and to evaluate the solution s robustness. Finally, in Section VI conclusions are presented. II. SYSTEM MODEL Our model consists of a three step algorithm as shown in Figure 2. We use an MDP to identify how the system should react to uncertainty in order to maximize the operational earnings. We assume that energy is periodically sold in discrete units δ. We also assume Markov property, making our technique independent of the input models used for modelling the uncertainty sources, as long as the input models comply with this property. To simplify the model, we also assume the covariance between the uncertainty sources to be 0. The product of the MDP is a set of decisions called policies. Based on the optimal policy, an irreducible Markov process is created and the corresponding stationary distribution is found. Based on the stationary distribution we find the expected operating cost. The alternative to not having a battery is to buy and sell power directly from the grid. Thus, there is a trade-off between investing in the battery and trading with the grid. By knowing the optimal production strategies and minimizing the investment cost, we derive the optimal battery size. A. Traffic Modelling We use an M/M/C queue to model the uncertainty in the BS s traffic. The model has been used for dimensioning links in Internet [15]. To our knowledge there are no common distributions or models to capture the uncertainty in a single Discretized models for uncertainty in power production, power consumption and price Markov decision process Optimal policy π* Markov process Stationary distribution Optimal battery size Derive total investment cost 07 August Fig. 2. The model consist of three steps. In step 1, an MDP is used to derive optimal response to uncertainty from the system as shown in Figure 1. Based on this response, a statistical distribution of energy in the battery is derived by the associated Markov process as shown in step 2. The statistical distribution is used to derive battery usage as shown in step 3, and the optimal size is derived based on battery size that minimizes the expected total cost. BS s traffic. The connection establishment process for LTE- A follows a Poisson process [16]. A BS with persistent scheduling, allocates Φ resource blocks (RBs) per user per session [17]. With hourly time steps, an average of Φ RBs can be used for all users. The number of available resource blocks RB max is fixed, making the bandwidth consisting of C service stations, each containing Φ RBs. The data traffic follows a self-similar process [18], suggesting that the service rate of a BS can be modelled with the random distribution G. Since data is lacking, there is a problem of choosing the correct distribution. Also, an M/G/C queue can not be obtained in closed form and will have to be investigated as an embedded Markov chain [19], making it cumbersome to work with. Hence, M/M/C queue is used as an approximation, as it can be obtained in closed form and has already been used for similar problems [15]. The BS s antenna power usage Q ant is a function of number of the RBs used to carry traffic RB used [20]. The antenna only transmits at full power Q max, if RB max are used. RB used is a function of the queue length k and time t. { RBused t Φ k if k C = Φ C otherwise t T (1) Q t ant = RBt used RB max Q max t T (2) The expected utilization of the BS s traffic capacity is given by ρ t and the associated queue states are denoted by k. K is the corresponding set of queue states with probability function P(k) t given time t. The amount of energy required to power the BS Q k depends on queue length k. The linear power model derived by Auer et al. [21] is used, where N T RX is the number of transceiver antennas and Q 0 is the BS s power usage regardless of traffic. 2

3 power is the relationship between total antenna power usage Q k ant and the BS s total power usage Q k. Q k = N T RX Q 0 + power Q k ant k K (3) The probability density function of the P(Q k ) t is therefore a scaled version of P(k) t. The probability density function of BS power consumption P(l t ), is a discrete probability distribution of P(Q k ) t with respect to step size δ. B. Power Price Modelling The power exchange Nord Pool provides hourly spot prices 1 of the system price, representing a calculated average price for the Nordic and Baltic countries. Several distributions, such as double gamma, lognormal, normal, exponential and pareto were fitted based on hourly data for the period and the mean square errors (MSE) were calculated. Between the hours, different types of distributions provided the lowest MSE, where the majority being variations of the normal distribution (such as Johnson s SU). Therefore a normal distribution was fitted for each hour, giving an MSE in the range of The MSE differences between the optimal and the normal distribution were in the range of [0, 69%]. The largest deviations were found in two price peaks where double gamma distribution gave the best fit. However, the double gamma distribution is unsuited for parametrization due to its three parameters. A similar approach using gamma distribution gave a larger range of [0, 100%]. Due to the low MSE, normal distribution was chosen, as it gave a good approximation while being easy to parametrize. The hourly normal distribution has the mean γ and standard deviation ϕ t. To discretize the price deviation d zt onto Z scenarios, the deviation range [±3 ϕ t ] were split into equally sized bins, where the middle value in each bin gave d zt. P(d zt ) was given by the probability of being within the bin s range coverage. With this approach P(d zt ) = 99.7 % and is therefore normalized, but the effect is insignificant. A strong point with this approach is that odd numbered Z will produce scenarios, where one of the scenarios z will be = P buy t mean (both described in Section III), making it suited for comparing cases with and without uncertainty in power price. Pbuy t z C. Renewable Energy Modelling The PV power production can be modelled with Equation (4), where η P V is the efficiency of the PV-system, Λ is the area per m 2 and R(t) is the sun s irradiation measured in watt per m 2. e(t) = η P V Λ R(t) t T (4) The National Renewable Energy Laboratory s (NREL) Physical Solar Model (PSM) 2 contains the Global Horizontal Irradiance (GHI) model used for describing irradiation received at earth s surface both under clear sky and actual conditions, 1 last visited last visited both being good estimates for R(t). Similar approach for the distribution fitting as in Section II-B was used, and a normal distribution gave the best fit for hourly variations of GHI in New York during the period The corresponding MSEs The PV unit has also an upper and a lower power production limit, P V max and P V min, binding the probability density function P(e t ). In the interval between the limits, the probability density function follows a normal distribution with given mean µ P V,t and standard deviation σ P V,t. The distributions are discretized based on δ sized bins. III. MARKOV DECISION PROCESS INSPIRED SOLUTION A. MDP State Space TECHNIQUE Let T be the index set of decision epochs for when power can be traded with t [1..T ] and T <. The entire set of consecutive decision epochs is referred to as an episode. Assume that there is a cyclic pattern in the epochs, such that the consecutive decision epoch to t = T is t = 1. Let N be the index set of episodes with n [1..N] and N <, such that T N describes the full lifetime of the battery. Also, all variables are assumed to be non-negative, unless otherwise stated. With these assumptions the problem can be formulated as a finite horizon MDP with zero termination value. Energy is bought and sold in discrete units of δ amount of energy. Let B be the index set of states in a battery with b [1..B]. The battery energy level at time t is given by w zt. The microgrid s energy price is uncertain, and Z denotes the set of deviation from expected price with z [1..Z]. Each state is denoted s bzt where s [1..S] and S = B Z T. B. Action Space W The charge and discharge rates are bounded by + W and, limiting the movement in the battery s energy levels. w iς w zt + W w zt w iς W i, z Z,ς = (t + 1) mod T, t T The renewable energy production e t [E t..e t ] E t, and energy consumption l t [L t..l t ] L t, are each discretized into δ units with their respectively lower and upper bounds. The discrete probability functions are P(e t ) and P(l t ). In each decision epoch t, the BS can sell u zt, or buy u + zt units of energy, where u zt is a reference to both variables and U(w zt ) is the available action set. C. System Dynamics For decision epoch t, the difference in energy production and consumption e t l t, along with the selected action give the next battery energy level w zt. The battery size is nonnegative and limited by an upper limit W. w zς = w it u it + u+ it + e t l t u it U(w it ), i, z Z, ς = (t + 1) mod T, t T Energy from the grid can be bought for the expected price P buy mean and sold with the expected price P sell mean. The (5) (6) 3

4 difference in the expected prices derive from the transaction fee, (1 α). The battery cost per δ is given by C battery. In order to simplify later comparisons, P buy mean is given as a function of battery cost where γ is the scaling unit. P buy mean = C battery T N γ (7) Deviations in the price d zt, gives the buying price Pbuy zt, and selling price Psell zt, of energy for different price scenarios with associated probability function P(d zt ). d zt can both take negative and positive values as shown in Section II-B Pbuy zt = P buy mean (1 + d zt ) z Z, t T Psell zt = α Pbuy zt z Z, t T The transition probability P(w t+1 w t, u t ) is given by the probability functions P(e t ), P(l t ) and P(d zt ). P(w zς w it, u it ) = e t E t, l t (L t l t=y) P(e t ) P(l t ) P(d zς ) u it U(w it ), i, z Z, ς = (t + 1) mod T, t T where: y = w zς w it u it + u+ it + e t A BS without a battery or a PV power source would draw all its power from the grid. Hence, when w zt = 0 and e t l t < 0 there is no blackout event. Likewise, if the battery is full, the excess net power is sold directly to the grid. Unsolicited power that is pushed or pulled (USPPP) introduces balancing cost for the grid operator [22, 23], and C balancing is the cost the BS has to pay for the grid s flexibility. With given setup, the reward function in Equation (10) becomes stochastic and the expected reward is used in the MDP. ( ) E r(w zt, u zt ) =Psell zt u zt Pbuy zt u + zt + where: D. Objective Function e t E t, l t L t u zt U(w zt ), z Z, t T (8) (9) (r 1 + r 2 ) r 1 =P(e t ) P(l t ) max ( w zt + u + zt + e t l t u zt W, 0 ) (P ) zt sell C balancing r 2 =P(e t ) P(l t ) min ( w zt + u + zt + e t l t u zt, 0 ) (P ) zt buy + C balancing (10) The BS aims at maximizing the reward of operations for each state v(s). The problem is formulated with a Bellman equation with termination value v N and solved with dynamic programming. β is the discounting value for an episode. v n (s) = max {r(s, u) + β } P(j s, u) v n+1 (j) u U(s) s S where: v N (s) = 0 s S subject to: (5), (6), (9) and (10) j S (11) Let policy π n be the set of policies that maximize decisions in the episode n. π n will change for each iteration n until it is stabilized with policy π when π n+1 = π n. IV. MARKOV PROCESS Let matrix A be the right stochastic matrix of the irreducible Markov process deducted based on the optimal policy π. If T > 1 the process is also cyclic. The stationary probability x t per epoch t of matrix A is derived by using the scheme method [24, p ] and eigenvalue diagonalization [25, p. 274]. The stationary distribution vector x t (b) gives the probability of how much time the process spends in battery level b where w zt is the energy level of the battery. x t (b) = ( ) P w zt b δ w zt < (b + 1) δ z Z (12) t T The total duration of time spent in a battery state b P(s = b), is a function of the stationary probability of state b in each of the epochs, as shown in Equation (13). A. Optimal Battery Size P(s = b) = t T x t (b) (13) The optimal battery size is derived by comparing the total expected cost of batteries with given sizes. This is essentially a problem of how much storage is needed to minimize the expected cost of hedging. If a battery is in state b, it also stores energy in state b-1, as this was modelled in the MDP. Hence, the amount of time energy is stored in a given level b Π b, is therefore a function of the total duration of time in state b P(s = b), and all states above as shown in Equation (14). Π b = B P(s = i) (14) i=b For a battery of size b, the total investment cost, I b, will consist of the cost related to the PV power source, the battery and the BS s operations. Power stored in the battery comes from the PV power source and has the cost price P P V, as P P V < Pbuy zt. When the energy production is larger than size b, the excess power is sold for the price Psell mean. The sold power is bought back at a later time for the price P buy mean, if Π b+1 > 0. Hence, the expected loss C b+1 alt of not having a battery state b + 1 is given by the time the system exists in 4

5 state b+1 and the states above. If USPPP is allowed, the total expected investment cost of a battery with size b is given by Equation (15). I b = P P V δ b i=1 where: C b+1 alt = (Π i i) + C battery δ b + C b+1 alt B i=( b+1) T N Π i δ γ (1 α) Cbattery T N (15) Equation (15) can be simplified, as shown in Equation 16, by setting C battery = 1 and assuming P P V = 0. The optimal battery size is found when the investment cost over the entire lifespan of the battery is minimized. As all components of I b are linear, the simplex algorithm can be used to find optimal b. {( min b + γ (1 α) b B subject to: (14) B i=(b+1) ) } Π i δ (16) Markets that do not allow USPPP, can be modelled by setting C balance to an extremely high value. This forces the MDP to search for other strategies than the ones that might lead to unscheduled power movement. Since USPPP is not acceptable, Equation (16) is bounded and must be adapted. In energy consumption, the worst case scenario is if the system buys L t energy, but uses L t energy. Hence, a battery must have at least Lt = L t L t stored energy to account for fluctuation in traffic. Likewise, in energy production, the worst case scenario is that the system sells E t energy but produces E t energy. Therefore, a battery must have at least the capacity to store Et = E t E t energy. The smallest allowed battery size must therefore facilitate both worst case scenarios. Equation (16) is therefore modified to Equation (17) which can also be solved with the simplex algorithm. { ( B )} min b + γ (1 α) Π i δ b B Y i=(b+1 E ) where: Y : b max{( Lt + Et ) t T } Lt = L t δ subject to: (14) and Et = E t δ (17) 1) Investment boundary: Based on Equations (16) and (17), it is never optimal to invest in more that the minimum required battery size given that γ 1 1 α. If γ = 1 1 α then Equation (16) is reduced to: { ( B )} min b + Π i b B i=(b+1) (18) The expression s value grow with b since Π i 1 and the b index increase with units of 1. Therefore it will never be preferable to invest in any battery size above the required minimum. This relationship is also valid for when P P V > 0. 2) Trading boundaries: The trading potential, to sell power when prices are high and to buy power when prices are low, is only present in the model when the price uncertainty is included. The transaction fee and the relative battery cost affects the trading potential in opposite directions. If the transaction fee (1-α) is high, the relative selling price decreases; if it is low, the relative selling price increases. This effect is counteracted, as the relative cost of not having a battery decreases with the fee. Hence, for a certain price uncertainty standard deviation (std) ϕ, given buying price to selling price ratio α, and buying price to battery cost ratio γ, the trading potential will be insignificant, and the difference in battery size between including and excluding price uncertainty will revert back to similar values. V. NUMERICAL RESULTS The system holds a large degree of complexity in traffic and renewable power as both have a mean and a std that change during the day by following a cyclic pattern. The problem is simplified by setting T = 1 hour and N = 87600, i.e., a 10 year battery life. The BS has most flexibility during day, while night time can be viewed as a day case with extra restrictions on energy production (no production). Therefore the cyclic pattern is bounded by the day case. Also, as the price mean is fixed, there are no arbitrage opportunities for intra day power trading. However, the price will fluctuate, due to uncertainty, and the system can trade on this fluctuation, but the associated risks are then accounted for. A micro BS is parametrized as given by Auer et al. [21] with N T RX = 2, Q 0 = 56, power = 2.6 and RB max = 75. Based on arguments in Section II-A Φ = 5 is used. For the PV power source P V max = 400W h and P V min = 0 are used. Tesla s Power Wall s 3 charging and discharging are used with adaption to W = 2000W h, giving charge = 500W h and discharge = 500W h. An annual discount rate of 5% is assumed, giving β = To comply with computational limits of available tools δ = 5 and Z = 3. Our method was implemented in Python 3.5 with the MDPtoolbox and the SciPy libraries [26, 27] Figure 3 shows each of the components of Equation (16). For battery size less than optimum size, the total cost is driven by the expected cost of selling and buying back energy, while for larger values, battery installation cost is the main driver. In Figure 4, the sensitivity analyses are based on the difference between Equation (16) and Equation (17). The differences display the cost increase in battery size for the BS if the microgrid does not allow USPPP. Therefore, they also represent the power balancing cost of the BS and give the robustness of the solution. As previous published research have not focused on price uncertainty, each analysis were performed twice to quantify the effect of including the price uncertainty. Based on the sensitivity analyses in Figure 4, we made four main observations. The first observation is when the price uncertainty is not included, there is a larger variation in battery 3 last visited

6 07 August Total cost Expected operational cost Battery investment Cost per δ Battery size in Wh Optimal size: 410 Wh Fig. 3. The grey dotted line denotes the expected operational costs, while the striped green line is the investment cost of the battery. The solid orange line is the total cost and is a function of the two other lines. In this example ρ = 30%, µ P V = 130W h, σ P V = 30W h, α = 85% and γ = 10. Associated π is found with n 200, suggesting that π represents the system s operation of the battery through out the battery s lifetime. size between when USPPP is allowed and not. This is due to the flexibility in price variation that can be capitalized, reducing the effect of USPPP. Further, we observe that the PV mean and the PV std have the largest effect on difference in needed battery size. This is coherent with the minimum required limit given by possible energy movement. We also observe the trading and investment boundaries in cases 3 and 4. The investment boundary can be observed in both cases where the difference is the bare minimum as required by Equation (17). The trading boundary can be observed as the lack of the price uncertainty s effect on the solution. In case 3 the differences are equal between having and not having uncertainty in power price. In case 4 the differences do not change with ϕ. Finally, we observe that our approach gives a tangible value for the cost of hedging uncertainty and can be used to quantify the energy balancing cost for a BS. In Figure 4 this cost is given as Wh of a battery. VI. CONCLUSIONS In this work we have shown a new technique to calculate the optimal battery size given uncertainty in the power price, the power production and the BS s traffic. Our technique consists of reuse of well defined methods connected in new ways. We show how the model should be parametrized, and provide two data driven models for modelling the uncertainty in renewable power production and power price. The parametrized model is used numerically to find the power balancing cost for the BS, giving new insights to energy balancing in microgrids. The work is extended with sensitivity analyses that identify parameters that have the highest impact on optimal battery size such as renewable power mean and std. Further, we provide an analytical expression for market conditions, such as transaction Fig. 4. Comparison between sensitivity analyses for cases with and without USPPP, Equation (17) - Equation (16). Values as given in Figure 3 and ϕ = 10% are used unless otherwise stated. The heat maps on the left side in cases 1-3 do not include uncertainty in price, while the others do. The two plots in case 1 show sensitivity analyses based on µ P V and σ P V. The preceding two heat maps in case 2 show sensitivity based on ρ and σ P V. The two plots in case 3 show sensitive analysis based on γ and α, while the heat map for case 4 shows sensitivity based on α and ϕ. All plots adhere to the same colouring scheme, given above the heat maps. Light green denotes a small difference, while dark green denotes a large difference. The associated investment boundary and trading boundary as discussed in Sections IV-A1 and IV-A2 are also given. fee and relative power price, that make it undesirable to invest in battery, regardless of the power production or the BS s traffic. We also show numerically that the effect of modelling price uncertainty is insignificant given certain market conditions, and it could be excluded from the model with minor effect on the results. 6

7 REFERENCES [1] J. Leithon, T. J. Lim, and S. Sun, Renewable Energy Management in Cellular Networks : An Online Strategy based on ARIMA Forecasting and a Markov Chain Model, no. IEEE Wireless Conference and Networking Conference, [2] D. Thevenard and S. Pelland, Estimating the uncertainty in long-term photovoltaic yield predictions, Sol. Energy, vol. 91, pp , [3] Cisco VNI Mobile, Cisco Visual Networking Index (VNI) Update Global Mobile Data Traffic Forecast , Cisco Public Information, Tech. Rep., [4] J. Leithon, T. J. Lim, and S. Sun, Energy Management Strategies for Base Stations Powered by the Smart Grid, 2013 IEEE Int. Conf. Smart Grid Commun. SmartGrid- Comm 2013, pp , [5], Energy exchange among base stations in a Cellular Network through the Smart Grid, in 2014 IEEE Int. Conf. Commun., 2014, pp [6] A. Kwasinski and A. Kwasinski, Role of energy storage in a microgrid for increased use of photovoltaic systems in wireless communication networks, 2014 IEEE 36th Int. Telecommun. Energy Conf., no. 1, pp. 1 8, [7] J. Peng, P. Hong, and K. Xue, Optimal power management under delay constraint in cellular networks with hybrid energy sources, Comput. Networks, vol. 78, pp , [8] Y. K. Chia, S. Sun, and R. Zhang, Energy cooperation in cellular networks with renewable powered base stations, IEEE Trans. Wirel. Commun., vol. 13, no. 12, pp , [9] M. J. Farooq, H. Ghazzai, A. Kadri, and H. Elsawy, Energy Sharing Framework for Microgrid- Powered Cellular Base Stations, in IEEE Glob. Commun. Conf. (Globecom 2016), Washington, DC USA, no. December, [10] X. Liu, T. Han, and N. Ansari, Intelligent Battery Management for Cellular Networks with Hybrid Energy Supplies, in IEEE Wirel. Commun. Netw. Conf. WCNC, [11] S. Teleke, M. E. Baran, a. Q. Huang, S. Bhattacharya, and L. Anderson, Control Strategies for Battery Energy Storage for Wind Farm Dispatching, IEEE Trans. Energy Convers., vol. 24, no. 3, pp , [12] S. X. Chen, H. B. Gooi, and M. Q. Wang, Sizing of energy storage for microgrids, IEEE Trans. Smart Grid, vol. 3, no. 1, pp , [13] R. Atia and N. Yamada, Sizing and Analysis of Renewable Energy and Battery Systems in Residential Microgrids, IEEE Trans. Smart Grid, vol. 7, no. 3, pp , [14] J. Walrand, N. Edalat, and M. Motani, Sizing and control of residential solar panel and battery, 2015 IEEE Int. Conf. Smart Grid Commun., pp , [15] V. B. Iversen, The Internet Erlang Formula, ACM SIGCOMM Comput. Commun. Rev., vol LNCS, no. 1, pp , [16] R. R. Tyagi, F. Aurzada, K. D. Lee, and M. Reisslein, Connection Establishment in LTE-A Networks: Justification of Poisson Process Modeling, IEEE Syst. J., pp. 1 12, [17] F. Capozzi, G. Piro, L. A. Grieco, G. Boggia, and P. Camarda, Downlink packet scheduling in LTE cellular networks: Key design issues and a survey, IEEE Commun. Surv. Tutorials, vol. 15, no. 2, pp , [18] V. Paxson and S. Floyd, Wide Area Traffic : The Failure of Poisson Modeling, {IEEE/ACM} Trans. Netw., vol. 3, no. 3, pp , [19] H. Khazaei, J. Mišić, and V. B. Mišić, Modelling of cloud computing centers using M/G/m queues, Proc. - Int. Conf. Distrib. Comput. Syst., pp , [20] J. Gong, J. S. Thompson, S. Zhou, and Z. Niu, Base Station Sleeping and Resource Allocation in Renewable Energy Powered Cellular Networks, IEEE Trans. Commun., vol. 62, no. 11, pp , [21] G. Auer, V. Giannini, C. Desset, I. Gódor, P. Skillermark, M. Olsson, M. A. Imran, D. Sabella, M. J. Gonzalez, O. Blume, and A. Fehske, How much energy is needed to run a wireless network? IEEE Wirel. Commun., vol. 18, no. 5, pp , [22] H. K. Nguyen, A. Khodaei, and Z. Han, Distributed Algorithms for Peak Ramp Minimization Problem in Smart Grid, in IEEE Int. Conf. Smart Grid Commun., [23] N. Gensollen, V. Gauthier, M. Marot, and M. Becker, Submodular Optimization for Control of Prosumer Networks, in IEEE Int. Conf. Smart Grid Commun., [24] W. J. Stewart, Introduction to the Numerical Solution of Markov Chains. New Jersey: Princeton University Press, [25] C. H. J. Edwards and D. E. Penny, Elementary Linear Algebra. Harlow: Pearson Education Limited, [26] Chades I., G. Chapron, M. Cros, F. Garcia, and R. Sabbadin, MDPtoolbox: a multi-platform toolbox to solve stochastic dynamic programming problems. Ecography (Cop.)., vol. 37, no. 9, pp , [27] Jones E, E. Oliphant, and P. Peterson, SciPy: Open Source Scientific Tools for Python, [Online; accessed ]. 7