Energy Efﬁcient Design of Massive MIMO Based on Closely Spaced Antennas: Mutual Coupling Effect

: Massive Multiple Input Multiple Output MIMO technology is a promising candidate for the next generation of communication applications, which essentially group hundreds of transmitting antennas together at the base station and provides the higher energy and spectral efﬁciency. In this article, the transmitting antennas are assumed to be closely spaced at the base station, which in turn results into a mutual coupling effect between the antenna terminals. The optimal system parameters and the energy efﬁciency are computed by considering the mutual coupling effect between the antenna terminals. Mutual coupling effect is deeply investigated on the energy efﬁciency and the other optimal parameters. We propose the domain splitter algorithm for the optimization of energy efﬁciency and the computation of different optimal system parameters in this article. The computational complexity of the proposed domain splitter algorithm is not dependent on the number of transceiver chains, and the detailed comparison is presented between the proposed and the reference algorithms on the basis of the computational complexity, which shows the effectiveness of the proposed domain splitter algorithm.


Introduction
Wireless communication is the most vital and key technology in this modern era and with the ever-increasing trend of advances in the wireless technologies, a large number of users are using these technologies, which in turn require having a network with higher energy and spectral efficiency together with the simple signal processing [1][2][3]. Multiple Input Multiple Output (MIMO) systems have gained a lot of attraction, both in industry and academia, due to their ability to significantly improve the spectral efficiency, but the acquisition of higher spectral efficiency comes at the cost of complex computation and signal processing in the conventional MIMO systems [4]. Massive MIMO can significantly increase the performance of the system by just adopting the linear precoding and decoding schemes for the downlink transmission and the uplink reception due to the asymptotic orthogonality of the propagation channel, as compared to the conventional MIMO systems [5][6][7][8].
In Massive MIMO, hundreds of antennas have to be stationed on the top of a building or tower, serving a comparatively less number of users [9,10]. In the urban environment, the electromagnetic interaction between the antenna elements is inevitable due to the space limitations for the deployment of a large number of antenna elements at the base station, thereby, leads to a phenomena named Mutual Coupling [11]. In [12,13], the authors have investigated the mutual coupling effect on the performance of the conventional MIMO systems by calculating the closed-form expression of the channel capacity, based on closely spaced antenna elements. In [14], the authors have examined and investigated the have designed the energy efficient Massive MIMO by using the correct model of the circuit power consumption and computed the number of optimal user terminals, transmitting antennas and the corresponding consumed as well as the transmitted power. Mutual coupling effect among the antenna elements have been thoroughly investigated on the energy efficiency and other optimal parameters, by varying the length of the transmitting antennas and the spacing between them. The optimization problem of energy efficiency has been formulated in order to calculate the optimal parameters with respect to the different channel conditions. We have proposal a domain splitter algorithm for the optimization of energy efficiency, and calculation of the optimal parameters. A detailed comparison is presented between the reference and the proposed algorithms in terms of the computation complexity. At the end, we have made simulations in order to support and show the effectiveness of the mathematical modelling, where the simulation result shows the optimally achieving energy efficiency and other optimal system parameters, with respect to different channel conditions and the spacing between the antenna elements.
This article is further arranged as follows. First, we have modelled the channel and derived the spectral efficiency of the Massive MIMO, by taking the effect of mutual coupling among the antenna elements, with respect to the different channel situations in Section 2. Next, we have modelled the circuit power consumption of massive MIMO in Section 3. Furthermore, we have formulated the optimization problem of energy efficiency in Section 4. In Section 5, the domain splitter algorithm and the complexity comparison are presented between the proposed and the reference algorithms. The simulation results are provided in Section 6. Section 7 belongs to the conclusion where we summarize and conclude all the discussions.
The notation used in this paper is as follows:(·) −1 , Ci(·), (·) T , Si(·), (·) * and (·) H denotes the inverse, cosine integral, transpose, sine integral, conjugate, and the Hermitian operator respectively, E[·], · 2 denotes the expectation and the norm operation, Z + denotes the positive integers, log 2 (x) and ln(x) denotes the logarithm of x with respect to base 2 and e, (·) , (·) denotes the first order derivative and the second order derivatives, ∼, var(·) denotes the approximation and the variance notation, CN(x, y) denotes the complex normal distribution with mean x and the variance y. Additionally, trace(·) represents the matrix trace operation.

Channel Modelling and the Spectral Efficiency
In Massive MIMO, hundreds of antennas have to be stationed on the top of a building or tower, serving comparatively small number of users. Time Division Duplex is the most desirable scheme in the case of Massive MIMO, because of the reciprocity between the uplink and downlink channels during each coherence block, as compared to Frequency Division Duplex (FDD), where pilot overhead in each coherence block is dependent on the number of transmitting antennas at the BS. In FDD, we have to use different frequencies in the uplink and downlink and the pilot overhead of the channel estimation is proportional to the number of transmitting antennas, leading to a very challenging situation at the user terminals to acquire the downlink channel state information and then feedback the acquired channel state information to the base station. The Massive MIMO was basically imagined for TDD protocol, but can be applied also in FDD by using the various techniques. Figure 1 unveils the frame sequence in the case of TDD protocol, the Channel State Information C.S.I at the BS can be estimated with the help of Uplink training signals or pilots for T ul p channel uses and the estimated C.S.I can be exploited during the downlink transmission for T dl p channel uses. The performance of the massive MIMO relies critically on the correctness of the C.S.I during each coherence block U. Consider the M number of the dipole antennas are equipped at the BS, having the length L and separated by a distance s , serving K number of uniformly distributed single antenna users as can be seen in Figure 2. Therefore, the received signal at the base station can be written as:  Consider the M number of the dipole antennas are equipped at the BS, having the length L and separated by a distance s, serving K number of uniformly distributed single antenna users as can be seen in Figure 2. Therefore, the received signal at the base station can be written as: is the matrix of the transmitted signals from the K number of uniformly distributed single antenna users and y is the received signal matrix with the dimension of M × 1 at the base station. p can be deemed as the average transmitted power and n is the Additive White Guassian Noise with the dimension of M × 1 and has the zero mean and unity variance, whereas G is the channel matrix with the dimension of M × K.
Massive MIMO can enhance the performance by exploiting the orthogonality between the channels of the desired and the interference signals, which in turn results into the communication reliability, by using the linear signal processing during the uplink transmission and the downlink reception. Thus, the linear precoding matrix V can be written as:

MMSE
As explained earlier, the transmitting antennas at the base station are closely spaced, leading into the effect of mutual coupling. Thus, the channel matrix by considering the mutual coupling effect among the base station antennas can be expressed as: where C is the mutual coupling matrix with the dimension of M × M and H is the Rayleigh small scale fading matrix. D is a K × K large scale diagonal matrix and can be written as: where β k shows the path loss or the large scale fading between the base station and the kth user terminal. The mutual coupling matrix when the M number of the dipole antennas are equipped at the BS can be written as [13]: where z L and z s represent the load and the self-impedance respectively. z c represents the matrix of mutual impedance, where the diagonal entries showing the self-impedances of the transmitting antennas, and, the non-diagonal entries are showing the mutual impedances. The mutual impedance between two particular transmitting dipole antennas i and j can be written as [35]: where: where s is the distance between two antenna terminals and L is the length of the dipole antenna element as shown in Figure 2. Ci, Si represent the cosine and sine integrals respectively and can be written as: Assume that, the zero forcing linear detection scheme is implemented at the base station, then, the signal received at the base station can be expressed as: Similarly, the signal received from the kth user terminal can be written as: where v k represent the kth column of the detection matrix V and the equivalent noise v H k n can be expressed as: v H k n ∼ CN 0, v H k 2 . The Signal to Noise Ratio (S.N.R) at the corresponding kth link can be expressed as: As we know V 2 = VV H . Thus, the above equation can be written as: The corresponding achievable rate can be illustrated as: Energies 2018, 11, 2029 6 of 25 The corresponding achievable rate can be illustrated as: Proof of the Theorem 1. See Appendix A.
The overall spectral efficiency ( K R ) for the total K number of users can be written as: Put the value of k R from (7) to (8): The overall spectral efficiency by taking the overhead of the training signals into account can be written as: Theorem 1. In Rayleigh fading channel, when M ≥ K + 1 and ZF detector is implemented at the base station, then, the achievable rate for the kth user terminal can be written as: Proof of the Theorem 1. See Appendix A.
The overall spectral efficiency (R K ) for the total K number of users can be written as: Put the value of R k from (7) to (8): The overall spectral efficiency by taking the overhead of the training signals into account can be written as: Whereas, T is the total length of the relative pilot sequence.

Computation of the Spectral Efficiency under the Scenario of Imperfect Channel Situation
The performance of the Massive MIMO is significantly dependent on the acquisition of the C.S.I. However, having a perfect (C.S.I) is almost unfeasible in terms of real-world scenario, which in turn leads to interference among the user terminals. During the uplink transmission, the C.S.I is computed by using the pilot signals, and, let the uplink power of the pilot signal for the kth user terminal be Energies 2018, 11, 2029 7 of 25 σ 2 p β k and the total length of the orthogonal pilot signals be T ul p K as shown in Figure 1. The estimated channel matrixĜ can be expressed as: where ε is the channel estimation error during the acquisition of the frequency response of propagation channel. Therefore, the signal received at the BS can be written as: Theorem 2. In Rayleigh fading channel, when M ≥ K + 1 and ZF detector is implemented at the BS, then, the achievable rate for the kth user terminal by computing the estimated channel with the help of the MMSE estimator for the imperfect channel condition can be expressed as: Similarly, the overall spectral efficiency by taking the overhead of the training signals into account for the imperfect channel conditions can be expressed as:

Power Consumptions
Total power consumption can be expressed as an additive summation of the overall circuit power consumption, and the transmitted or power amplifier power. We have modeled the total circuit power consumption as well as the transmitted power in this section. The total transmitted power P t can be expressed as: where η is the efficiency of the PA's and Bα 2 is the overall noise power. Assume that the users are distributed uniformly in between the minimum d min and the maximum distance d max , then, the total transmitted power by using the uniform distribution can be expressed as: where α represent the path loss factor and the circuit power consumption can be written as the power consumed in the transceiver chains (P TRC ) (filters, convertors and the mixers powers), power consumption during the site cooling and maintenance P f x , coding and the decoding power P cod/doc , linear processing power P L.P , power consumed during the estimation of the channel P es and the oscillator power P o . Thus, the total power consumption can be expressed as: P c = P f x + P TRC + P o + P cod/doc + P L.P + P es (14) Energies 2018, 11, 2029 8 of 25 The consumed power in the transceiver chains P TRC can be written as: where P T , P R represent the consumption of power at the transmitter and the user end. The power consumed during the decoding and coding of the signals is dependent on the number of bits, so, the power consumed during this process can be written as: where P doc , P cod is the power consumption during the process of decoding and coding of the received and the transmitted signals. As explained earlier, massive MIMO relies significantly on the accurate acquisition of the frequency response, so, the timely acquisition of the frequency response is of timely important, and the consumed power during the channel estimation P es can be expressed as [34]: where B U represent the number of coherence blocks per second, η Bs , η ue is the computational efficiencies at the base station and the user end during the estimation of the channel. The power consumption during the linear processing P L.P can be interpreted as [36]: Thus, the overall power consumption P tot by using the (13)-(18) can be written as: Let: Thus, the total power consumption (19) by using the above given substitutions can be expressed as:

Problem Definition
In this section, we have formulated the problem and derived the mathematical expressions of the energy efficiency for the perfect and imperfect channel situation. Energy efficiency can be expressed as the ratio of total spectral efficiency and the total power consumption i.e., Thus, the Energy efficiency of Massive MIMO EE P (M, K, p) by using the (9) and (20) under the perfect channel condition can be expressed as: Energy efficiency needs to be optimized, so, the optimization problem of Energy efficiency can be formulated as: Max Similarly, the energy efficiency of Massive MIMO EE Im,P (M, K, p) by using the (11) and (20) along with the optimization problem for the Imperfect channel condition can be expressed as: We need to calculate the optimal number of BS antennas, users, and their corresponding transmitted power and energy efficiency. The M, K and p are closely associated and related to each other as it can be seen in the (21) and (23). Take the following substitutions and their corresponding interpretations (Table 1) into account for the simplification of EE P (M, K, p) and EE Im,P (M, K, p). Table 1. Substitutions and physical interpretations.

Substitution
Interpretation r 1 = K Number of optimal user terminals. r 2 = M/K Number of optimal active antennas per user terminal. r 3 = pK Total transmitted power.
Following the above mentioned substitutions, the EE P (M, K, p) and EE Im,P (M, K, p) can be modified as: EE Im,P (r 1 , r 2 , r 3 ) = Energies 2018, 11, 2029 10 of 25 Thus, the corresponding mathematical optimization problems for the perfect and imperfect channel situation can be simplified as: And: Max EE Im,P (r 1 , r 2 , r 3 ) Subject to r 1 ∈ + , r 2 ∈ + r 3 > 0

Computation of the Optimal Parameters and the Complexity Comparison
In this section, we have proposed and discussed the domain splitter algorithm to calculate the optimal parameters and the comparison is presented between the proposed and the reference algorithms, in terms of the computation complexity. We have named the proposed algorithm as the domain splitter algorithm because the proposed algorithm splits the domains of EE P (r 1 , r 2 , r 3 ) and EE Im,P (r 1 , r 2 , r 3 ), and then find the solution in each respective domain as shown in Figure 3. The EE P (r) and EE Im,P (r) shows the response of the quasi concave and depicts the one and the only one zero crossing at the points EE P (r) = 0 and EE Im,P (r) = 0 in each respective domain r = (r 1 , r 2 , r 3 ), as per the Appendixs C and D. In the Appendixs C and D, the solution of the energy efficiency is derived in each respective domain r = (r 1 , r 2 , r 3 ) and the optimal values are computed at the points EE P (r) = 0 and EE Im,P (r) = 0 by using these solutions. The Figure 3 shows the domain splitter algorithm and the simulation methodology where the optimal values of r = (r 1 , r 2 , r 3 ) are computed until the convergence. Finally, the acquired optimal values are used to compute the number of optimal user terminal, optimal transmitters along with the transmitted power, energy and the spectral efficiency. Table 2 shows the computation complexity between the proposed and the reference schemes and W(n) is the lambert omega operation. It can be seen from Table 2 that the computational complexity of the reference algorithms is dependent on the multiplication of the number of transmitting antennas at the BS with the number of user terminals (M × K). As Massive MIMO is dependent on the theory of large numbers of transmitting antennas at the BS, so, the computational complexity behind the reference algorithms is so high. The proposed algorithm (domain splitter algorithm) significantly improves the computation complexity where the energy efficiency gets saturated at the third iteration as shown in Figure 12.

Simulations
We have performed simulations in order to test and demonstrate the effectiveness of the domain splitter algorithm along with computation of different optimal system parameters. Table 3 unveils the basic simulation parameters used for the simulations. We have used the acronyms of PCSI and ICSI for the perfect and imperfect channel state information respectively in the simulation figures. Figure 4 unveils the input impedances by varying the length of the antenna, where the real part of the antenna impedance is showing the power that is either absorbed or radiated away and the imaginary part is showing the stored power. Input impedance depends significantly on the length of the antenna terminal, as it can be seen from Figure 4. Figure 5 unveils the mutual impedance between the antenna terminals with respect to the spacing among the antennas, calculated by using the (3), at different lengths of the antenna terminal. 3 n ln n 

Simulations
We have performed simulations in order to test and demonstrate the effectiveness of the domain splitter algorithm along with computation of different optimal system parameters. Table 3 unveils the basic simulation parameters used for the simulations. We have used the acronyms of PCSI and ICSI for the perfect and imperfect channel state information respectively in the simulation figures. Figure 4 unveils the input impedances by varying the length of the antenna, where the real part of the antenna impedance is showing the power that is either absorbed or radiated away and the imaginary part is showing the stored power. Input impedance depends significantly on the length of the antenna terminal, as it can be seen from Figure 4. Figure 5 unveils the mutual impedance between the antenna terminals with respect to the spacing among the antennas, calculated by using the (3), at different lengths of the antenna terminal.  Figure 4. Input Impedance. Figure 4. Input Impedance. Figure 6 unveils the total number of optimal BS antennas with respect to the maximum distance between the base station and the users, at different circuit power consumption and at different spacing among the BS antennas. When the BS antennas are closely spaced, then it leads to mutual coupling and a higher number of the transmitting antennas are required. The effect of the mutual coupling and different antenna spacing on the optimal number of transmitters under the different channel situations can be seen in Figure 6.        When the maximum distance between the base station and the end users gets increased, then, the number of optimal BS antennas required for the system will be increased, in order to cover more distance as shown in Figure 6. Similarly, Figure 7 shows the optimal number of user terminals with respect to the maximum distances (100-500 m) between the BS and the users, at different circuit power consumption and at different spacing among the BS antennas. More transmitted power is required when the BS antennas are closely spaced to overcome the mutual coupling effect along with the imperfect channel conditions as shown in Figure 8. Furthermore, high circuit power consumption leads to have more transmitted power to accommodate the higher number of BS antennas and the end users, and it is significantly dependent on the maximum distance between the BS and the users as shown in Figures 8 and 9. Figure 10 unveils the effect of mutual coupling on the achievable spectral efficiency of the system, and when the BS antennas are more closely spaced, then, the overall spectral efficiency gets reduced. As the system requires to have more transmitted power when the maximum distances between the BS and users gets increased, thereby, leads to have the reduction in the overall spectral efficiency of the system as shown in Figure 10. When the maximum distance between the base station and the end users gets increased, then, the number of optimal BS antennas required for the system will be increased, in order to cover more distance as shown in Figure 6. Similarly, Figure 7 shows the optimal number of user terminals with respect to the maximum distances (100-500 m) between the BS and the users, at different circuit power consumption and at different spacing among the BS antennas. More transmitted power is required when the BS antennas are closely spaced to overcome the mutual coupling effect along with the imperfect channel conditions as shown in Figure 8. Furthermore, high circuit power consumption leads to have more transmitted power to accommodate the higher number of BS antennas and the end users, and it is significantly dependent on the maximum distance between the BS and the users as shown in Figures 8 and 9. Figure 10 unveils the effect of mutual coupling on the achievable spectral efficiency of the system, and when the BS antennas are more closely spaced, then, the overall spectral efficiency gets reduced. As the system requires to have more transmitted power when the maximum distances between the BS and users gets increased, thereby, leads to have the reduction in the overall spectral efficiency of the system as shown in Figure 10. Similarly, Figure 11 unveils the mutual coupling effect on the overall energy efficiency of the system with respect to the different maximum distances between the BS and the end users, and imperfect channel situation leads to have a reduction in the overall achievable energy efficiency, due to the higher consumed as well as the transmitted power. Figure 12 unveils the computational complexity and the convergence of the proposed domain splitter algorithm, where the convergence of the energy efficiency can be examined with respect to the number of iterations. The energy efficiency goes into the saturation at the 3rd iteration, so, the computational complexity of the domain splitter algorithm in terms of Landau's big O notation can be written as: detailed comparison between the proposed and the reference algorithms in terms of computation complexity has been presented and discussed in Table 3. Similarly, Figure 11 unveils the mutual coupling effect on the overall energy efficiency of the system with respect to the different maximum distances between the BS and the end users, and imperfect channel situation leads to have a reduction in the overall achievable energy efficiency, due to the higher consumed as well as the transmitted power. Figure 12 unveils the computational complexity and the convergence of the proposed domain splitter algorithm, where the convergence of the energy efficiency can be examined with respect to the number of iterations. The energy efficiency goes into the saturation at the 3rd iteration, so, the computational complexity of the domain splitter algorithm in terms of Landau's big O notation can be written as: 3 n 3 ln n 3 . A detailed comparison between the proposed and the reference algorithms in terms of computation complexity has been presented and discussed in Table 3.

Conclusions
This article mainly concentrates on the energy efficient designing along with the selection of optimal parameters, with respect to different channel conditions and circuit power consumption levels, by considering the scenario of the urban environment for the Massive MIMO. The energy efficiency and the other optimal system parameters are calculated by using the realistic modelling of the circuit power consumption. The effect of mutual coupling has been deeply investigated on the energy efficiency and other optimal parameters of the system, by varying the length and the interdistance spacing of the antenna terminals. Closed form expressions of the energy and spectral efficiency are formulated and derived by taking the mutual coupling effect into account. The simulation result shows the mutual coupling effect on the overall optimum energy efficiency and other system parameters. According to the simulation results, we can conclude that it is useful to have the large cell coverage area together with the less power consumption of the circuit. As per the simulation results, the proposed algorithm (domain splitter algorithm) works efficiently without much computational effort. The detailed comparison (Table 2) is presented between the proposed and the reference algorithms. Contrary to the existing researchers, the computation complexity of the proposed domain splitter algorithm is not dependent on the number of the transceiver chains. The mathematical modelling and the simulation results can be so much more beneficial for the future trend of research in order to have the energy efficient communication technologies by controlling the power consumption of the communication devices.

Conclusions
This article mainly concentrates on the energy efficient designing along with the selection of optimal parameters, with respect to different channel conditions and circuit power consumption levels, by considering the scenario of the urban environment for the Massive MIMO. The energy efficiency and the other optimal system parameters are calculated by using the realistic modelling of the circuit power consumption. The effect of mutual coupling has been deeply investigated on the energy efficiency and other optimal parameters of the system, by varying the length and the inter-distance spacing of the antenna terminals. Closed form expressions of the energy and spectral efficiency are formulated and derived by taking the mutual coupling effect into account. The simulation result shows the mutual coupling effect on the overall optimum energy efficiency and other system parameters. According to the simulation results, we can conclude that it is useful to have the large cell coverage area together with the less power consumption of the circuit. As per the simulation results, the proposed algorithm (domain splitter algorithm) works efficiently without much computational effort. The detailed comparison (Table 2) is presented between the proposed and the reference algorithms. Contrary to the existing researchers, the computation complexity of the proposed domain splitter algorithm is not dependent on the number of the transceiver chains. The mathematical modelling and the simulation results can be so much more beneficial for the future trend of research in order to have the energy efficient communication technologies by controlling the power consumption of the communication devices. Acknowledgments: This research was supported and facilitated by Suranaree University of Technology along with the National Research University Project of Thailand and the Higher Education Research Promotion, Office of the Higher Education Commission. We are grateful to the editor and all the reviews for their crucial and significant comments on this article.

Conflicts of Interest:
We declare that there is no conflict of interest associated with this article.

Appendix A
Proof of the Lemma 1: Equation (6) can be written as: For a m × m Wishart central complex matrix (W ∼ W m (n, I n )) with n degrees of freedom and provided that n > m, we get: By using the identity (A2), (A1) can be simplified as:

Appendix B
Proof of the Lemma 2: The mean and variance of the estimated channelĜ k for the kth user terminal, with the help of MMSE estimator can be expressed as [38]: KpT ul   Therefore, the variance of the channel estimation error ε can be written as: Thus, the achievable rates of the kth user terminal can be written as: By using the results of (A1) and (A2), (A3) can be written as: Calculation of the optimal parameter (r 1 ) for EE P (r 1 ) in terms of r 1 while the remaining dimensions (r 2 , r 3 ) in the interval [0, ν] are constant and fixed. The energy efficiency (EE P (r 1 )) in terms of r 1 can be written as: With the following substitutions: Differentiate EE P (r 1 ) with respect to r 1 in order to calculate the optimal parameter: d dr 1 (EE P (r 1 )) = d dr 1 Equate the above equation to zero in order to get the optimal parameter: At r 1 = 0; EE P (0) = νd 3 At r 1 = ν; EE P (ν) = −ν d 3 + d 4 r 1 + d 5 r 2 1 + d 6 r 3 Thus, EE P (r 1 ) follows the quasi concave response because EE P (r 1 ) is first positive and then negative with the optimal value must be existed at EE P (r 1 ) = 0, and the EE P (r 1 ) must be less than zero: Calculation of the optimal parameter (r 2 ) for EE P (r 2 ) in terms of r 2 while the remaining dimensions (r 1 , r 3 ) in the interval [1, ∞) are constant and fixed. Thus, the energy efficiency (EE P (r 2 )) in terms of r 2 can be written as: With the following substitutions: ln (2) , Differentiate EE P (r 2 ) with respect to r 2 to calculate the optimal value: d dr 1 (EE P (r 2 )) = d dr 2 Equate the above equation to zero to get the optimal value: At r 2 = ∞; EE P (∞) < 0 Thus, EE P (r 2 ) follows the quasi concave response because EE P (r 2 ) is first positive and then negative with the optimal value must be existed at EE P (r 2 ) = 0, and the EE P (r 2 ) must be less than zero: EE P (r 2 ) < 0 • Calculation of the optimal parameter (r 3 ) for EE P (r 3 ) in terms of r 3 while the remaining dimensions (r 1 , r 2 ) in the interval [1, ∞) are constant and fixed. Thus, the energy efficiency (EE P (r 3 )) in terms of r 3 can be written as: With the following substitutions: ln (2) , Equate the above equation to zero in order to get the optimal value: At r 3 = ∞; EE P (∞) < 0 Thus, EE P (r 3 ) follows the quasi concave response because EE P (r 3 ) is first positive and then negative with the optimal value must be existed at EE P (r 3 ) = 0, and the EE P (r 3 ) must be less than zero: