George A Ropokis
email: georgios.ropokis@centralesupelec.fr

Multi-relay cooperation with self-energy recycling and power consumption considerations

Keywords: Simultaneous Wireless Information and Power Transfer, Amplify and Foward Relaying, Self-energy recycling, Non-linear energy harvesting

We focus on the problem of rate-optimal beamforming for wireless powered cooperative communications systems. We consider the case of systems employing multiple wireless powered Amplify and Forward self-energy recycling relays and relay selection. By employing a generic piecewise linear energy harvesting model, that can adapt to the characteristics of different non linear energy harvesting measurements and models, we present the solution to the rateoptimal beamformer design problem. Our solution is based on solving a set of simple optimization subproblems where the solution of each one of them can be found in a simple closed form. Moreover, for the first time we combine such a realistic energy harvesting model with an equally realistic power consumption model at the self-energy recycling relays that not only accounts for the transmit power but also for other sources of power consumption (e.g. the power consumption of the relay circuits). By means of simulations we quantify the gains that multiple relays can bring validating that significant gains can be delivered for a variety of relay placements.

I. INTRODUCTION

While in conventional Full Duplex (FD) communications the self-interference loop has been recognized as a limiting factor [START_REF] Riihonen | Mitigation of loopback selfinterference in full-duplex MIMO relays[END_REF], within the paradigm of Simultaneous Wireless Information and Power Transfer (SWIPT), we can find ways to exploit the self-interference loop and increase the energy efficiency of communications systems. Focusing particularly on cooperative communications, the capability of FD relays to simultaneously transmit and receive signals can be employed in order to relay data while harvesting energy. This way, also energy radiated from the relays' own transmit antennas can be harvested, therefore enabling the concept of self-energy recycling in wireless communications. Such approaches have been recently studied in [START_REF] Hu | Energy efficiency in energy harvesting cooperative networks with self-energy recycling[END_REF]- [START_REF] Ropokis | Optimal power allocation for energy recycling assisted cooperative communications[END_REF] where system designs for single and multiantenna energy recycling Amplify and Forward (AF) relays were proposed. Moreover, in [START_REF] Ropokis | Cooperative beamforming exploiting energy recycling[END_REF] a new scheme exploiting multiple AF energy recycling relays has been proposed and beamforming solutions for this scheme have been discussed. In addition, more recently communications solutions for energy recycling Detect and Forward relays have also been proposed [START_REF] Demir | Robust optimum and near-optimum beamformers for decode-and-forward full-duplex multi-antenna relay with self-energy recycling[END_REF].

While the majority of the above works considers single relay scenarios, it is true that employing multiple wireless powered relays allows for better exploiting the effects of cooperative diversity as well as the potential for wireless Energy Harvesting (EH) [START_REF] Ropokis | Cooperative beamforming exploiting energy recycling[END_REF]. However, the system design presented in [START_REF] Ropokis | Cooperative beamforming exploiting energy recycling[END_REF] proposes a beamforming scheme based on the simultaneous transmission from the multiple relays, with the beamformer design process itself requiring the use of complicated convex/semidefinite programming methods. As a result, the use of iterative optimization techniques is required such as to solve the resulting beamforming optimization problems and reach near-optimal beamformer designs. More importantly, the algorithmic design presented in [START_REF] Ropokis | Cooperative beamforming exploiting energy recycling[END_REF] is based on an unrealistic linear energy harvesting model, that does not capture the nonlinear characteristics of practical EH circuits, see e.g., [START_REF] Chen | Wireless energy harvesting using signals from multiple fading channels[END_REF], as well as on a simple energy consumption model that ignores the energy consumption of the relay circuits. Deviating from such unrealistic assumptions, in this work we study a multi-relay scheme based on relay selection and present the rate-optimal beamforming solution for this scheme accounting for a realistic power consumption model at the relays. Moreover, we combine this model with a flexible piecewise linear EH model that not only allows for accurately modeling the non-linear EH process but also allows for deriving simple closed form expressions for the optimal beamformer, therefore avoiding the use of advanced, iterative optimization techniques and enabling ease of implementation for practical systems. To the best of our knowledge, our work is the first to approach the problem of optimal beamformer design for such cooperative communications systems while adopting realistic models both for the EH process as well as the energy consumption process at the relays.

In brief, the paper's contribution is summarized as follows: 1) We study a communication scheme exploiting multiple self-energy recycling relays and derive the rate-optimal beamformer for it, accounting not only for the nonlinear characteristics of EH, but also for the power consumption characteristics of the relays. 2) By means of simulations we confirm that this scheme can deliver significant performance gains as compared to single relay schemes, for a wide variety of relay placements. Moreover, by means of simulations we identify the positions close to the source of the system as favorable positions for the relays, since they allow for harvesting sufficient amounts of energy such as to power transmissions.

Concerning the presentation of our work, in Section II we present our system model, while in Section III, we solve the problem of optimizing the beamformer design at the selected relay, such as to maximize the instantaneous communication rate. In Section IV we evaluate the performance of the proposed scheme for different relay placements and relay availability scenarios. Finally in Section V we present briefly our conclusions as well as some future targets of our research.

Notation: Bold lower case letters are used in order to denote vectors and bold upper case letters are used in order to denote matrices. Operator (•)

H stands for the hermitian transpose of a vector/matrix. The K × K identity matrix is expressed as I K×K . Notation w ∼ CN (0, R) is used in order to denote that random vector w follows a Complex Gaussian Distribution with a zero mean, and covariance matrix R.

II. SYSTEM MODEL

We consider a Single Input Single Output (SISO) source S that communicates with a destination D with the help of L relays, denoted as R l , l = 1, . . . , L, each one equipped with M + 1 antennas. As in [START_REF] Ropokis | Cooperative beamforming exploiting energy recycling[END_REF], [START_REF] Zeng | Full-Duplex Wireless-Powered Relay With Self-Energy Recycling[END_REF], each one of the relays is equipped with one rectifier therefore allowing for using one of its antennas for EH purposes. A slotted communication protocol is adopted where for each time slot one relay is selected to assist S -D communication. Furthermore, we adopt the common assumption that the S → D channel is weak enough such that it can be ignored. As a result, during the t-th time slot, communication is achieved purely through the S → R l [t] → D channel, where R l [t] is the selected relay for the t-th time slot, i.e., index l [t] indicates the relay selected during time slot t. The same time, the remaining, non selected, relays, exploit the transmitted signals for EH purposes. The exact communication protocol is an extension of the communication protocol presented in [START_REF] Zeng | Full-Duplex Wireless-Powered Relay With Self-Energy Recycling[END_REF] and [START_REF] Ropokis | Optimal power allocation for energy recycling assisted cooperative communications[END_REF]. According to it, each time slot that is of duration T , is split into two phases, each one of duration T /2. The operation during the two phases is described in what follows, separately for the selected and non selected relays.

A. Operation at the selected relay

In what follows we focus on relay R l [t] . 1) Phase I: Data transmission from S: During phase I, S transmits a data signal x s,1 [t] (where

E |x s,1 | 2 = 1) to relay R l [t] . Relay R l [t]
then uses all M + 1 antennas and applies an MRC combiner v [t] in order to construct the signal:

y r,l ,1 [t] = v H [t] h sr,l [t] P s x s,1 [t] + n r,l ,1 [t] . (1)
In (1), P s stands for the transmit power used by S,

h sr,l [t] is the S → R l [t] , M + 1 × 1 Single Input Multiple Output (SIMO) channel, and n r,l ,1 [t] ∼ CN 0, σ 2 I M +1×M +1 , is the Additive White Gaussian Noise (AWGN) at R l [t] . Since for an MRC combiner it holds that v [t] = h sr,l [t] / h sr,l [t] , by introducing the quantity G sr,l [t] = h sr,l [t]
2 , (1) can also be written as:

y r,l ,1 [t] = G sr,l [t] P s x s,1 [t] + v H [t] n r,l ,1 [t] ñr,l ,1 [t]
.

2) Phase II: Data transmission/energy harvesting from/at R: During phase II, R l [t] amplifies y r,l ,1 [t] and exploiting M of its antennas, say antennas 2, . . . , M + 1, forwards it to D. Concurrently, S transmits another signal which is used by the selected relay for energy harvesting purposes. For this purpose, the relay R l [t] employs antenna 1, by connecting it to the rectifier.

Considering first the data forwarding process, assuming an AF protocol, R l [t] employs a beamformer x [t] of power x [t] 2 = P r [t] at its transmits antennas, and forwards the resulting signal, after normalizing it such as to obtain a signal of power P r [t]. The signal reaching D is then expressed as:

y d [t] = x H [t] h rd,l [t] y r,l ,1 [t] G sr,l [t] P s + σ 2 + n d,2 [t] , (3)
where

h rd,l [t] is the R l [t] → D Multiple
y r,l ,2 [t] = P s h sr,l [t] x s,2 [t] + x H [t] h loop y r,l ,1 [t] G sr,l [t] P s + σ 2 + n r,l ,2 [t] , (4)
where h sr,l [t] is the channel formed between S and the harvesting antenna of R l [t] , and x s,2 [t] is the signal transmitted by S for EH purposes. Moreover, h loop characterizes the MISO channel formed between the transmit antennas of R l [t] and its EH antenna. In what follows, we assume that h loop is deterministic and identical for all relays.

As explained earlier, the signal y r,l ,2 [t] in ( 4) is exploited at R l [t] for EH purposes. For the EH process we consider a piecewise linear model. According to it, if the EH circuit is fed with a signal x [t] of average power

P in = E |x [t]| 2 ,
the harvested power is related to P in through the expression:

P h (P in ) = η i P in + µ i , P in ∈ [P low,i , P up,i ] , i = 1, . . . , I, (5)
where η i , µ i , are positive parameters related to the EH circuit, and P low,i , P up,i , are the lower and upper points respectively for the i-th segment of the EH model. As we will see in later sections, such a model allows for closely approximating several other composite EH models, such as the accurate models discussed in [START_REF] Chen | Wireless energy harvesting using signals from multiple fading channels[END_REF]. Moreover, it encompasses as a special case the saturated model of [START_REF] Dong | Performance of wireless powered amplify and forward relaying over nakagami-mfading channels with nonlinear energy harvester[END_REF], that is the simplest model capturing the non-linear characteristics of EH circuits. Finally, we also note that such piecewise linear models have also been recently proposed in [START_REF] Alevizos | Sensitive and nonlinear far-field RF energy harvesting in wireless communications[END_REF] for the purposes of the performance analysis of wireless powered communications systems.

As in [START_REF] Zeng | Full-Duplex Wireless-Powered Relay With Self-Energy Recycling[END_REF], we select signal x s,2 as x s,2 [t] = x s,1 [t] e jθ [t] , where θ [t] is the phase of x H [t] h loop . As a result, assuming that the noise power is small enough such that the noise contribution to EH can be ignored, it can be shown that the energy harvested by the relay comes from the transmissions of S and R l [t] itself and is given as:

e W,l [t] = T P h P s g sr,l [t] + √ G sr,l [t]|x H [t]h loop | √ G sr,l [t]Ps+σ 2 2 2 , (6)
where for the ease of presentation we have introduced the notation g sr,l

[t] = |h sr,l [t]| 2 .
As explained in [START_REF] Zeng | Full-Duplex Wireless-Powered Relay With Self-Energy Recycling[END_REF], where a battery architecture for systems that simultaneously transmit and harvest energy is discussed, the amount of energy in (6) harvested during phase II, only becomes available for consumption to R l [t] , at the end of the t-th time slot. Therefore this amount of energy is not directly available for use during timeslot t. However, in what follows we introduce a system design that accounts for knowledge that this amount of energy will become available at the end of the timeslot. In more detail, we consider a system design that operates based on the following two energy consumption constraints, which are similar to the constraints presented in [START_REF] Ropokis | Cooperative beamforming exploiting energy recycling[END_REF].

Energy preservation constraint: We assume that the power level P r [t] is selected such that the energy available at R l [t] at the end of the t-th timeslot, hereby denoted as

E l [t] [t] , t ≥ 1, is at least equal to the amount of energy E l [t] [0]
available to it at the beginning of transmission. By adopting this constraint, it is ensured that none of the relays consumes its own energy resources at any time instance and as a result, after the completion of transmission, the energy budget of the relays has not been impacted by their use for communication.

In order to mathematically express this constraint we start by defining the energy surplus available at relay R l [t] at the beginning of timeslot t as the difference between the energy available to the relay at the beginning of time instance t, and the energy level E l [0], i.e., as:

e s,l [t] = E l [t] [t -1] -E l [t] [0] . (7)
Such an energy surplus may appear in case that some time slots are devoted purely for EH purposes due to the fact that another relay had been selected for transmission. As a further step in order to quantify the energy preservation constraint, we now also introduce our power consumption model at the relay. In more detail we base our model on the energy consumption model presented in [START_REF] Zhou | Greedy relay antenna selection for sum rate maximization in amplify-and-forward MIMO two-way relay channels under a holistic power model[END_REF] that captures both the energy consumption during data reception (i.e. during phase I of the protocol ) and during transmission (i.e. during phase II of the protocol). According to it, the energy consumed by relay R l [t] during a timeslot , is expressed as

E cons = P C,rx T 2 + P r [t] ε + P C,tx T /2. (8)
In the above equation, the term P C,rx corresponds to the power consumption of the receive circuits of the selected relay, which are used for the first T /2 seconds of the timeslot. This term captures the power consumption of the M + 1 receive RF chains associated with the antennas of the selected relay.

On the other hand, focusing on the remaining terms of (8), the term P C,tx corresponds to the power consumption of the transmit circuits, which are used for the T /2 seconds of the second phase of the timeslot. This term captures the power consumption of the M transmit RF chains associated with each one of the transmit antennas of the relay. Finally, we assume that each one of the transmit RF chains is connected to a power amplifier, characterized by a linear input-output power consumption formula. As a result, the actual power consumed by the power amplifiers for a transmission of total transmit power P r [t] is given as P r [t] /ε, where ε ≤ 1 denotes the common (for all amplifiers) efficiency.

Accounting for this power consumption model, the energy preservation constraint is finally expressed as:

P C,rx T 2 + P r [t] ε + P C,tx T /2 ≤ e W,l [t] + e s,l [t] , (9)
or equivalently, after introducing the term P C = P C,tx +P C,rx , as:

P r [t] ε + P C T /2 ≤ e W,l [t] + e s,l [t] . (10)
Using [START_REF] Dong | Performance of wireless powered amplify and forward relaying over nakagami-mfading channels with nonlinear energy harvester[END_REF], it is then straightforward to show that the energy

E l [t] [t] available at R l [t]
at the end of the t-th timeslot is equal to:

E l [t] [t] = E l [t] [0] + e s,l [t] + e W,l [t] - Pr[t] ε + P C T 2 (11)
and is indeed greater than (or equal to)

E l [t] [0].
Energy causality constraint: We further introduce the constraint that the energy consumed during the t-th timeslot is at most equal to the amount of energy available at the battery at the beginning of timeslot t, or equivalently at the end of timeslot t -1. As a result, we have that

1 ε P r [t] + P C T /2 ≤ E l [t] [t -1] = E l [t] [0] + e s,l [t] . (12)
Finally, we note that combining the energy causality and preservation constraints of [START_REF] Dong | Performance of wireless powered amplify and forward relaying over nakagami-mfading channels with nonlinear energy harvester[END_REF] and [START_REF] Zhou | Greedy relay antenna selection for sum rate maximization in amplify-and-forward MIMO two-way relay channels under a holistic power model[END_REF] with the energy harvesting model in [START_REF] Ropokis | Optimal power allocation for energy recycling assisted cooperative communications[END_REF], we can express both constraints in the form of (C1.1) and (C1.2), for each one of the I intervals of the EH model.

B. Operation at the remaining relays

We now describe the mode of operation for each one of the non selected relays, i.e., for relays R l , l = l [t]. For this purpose, we investigate separately the two phases of the communication protocol.

1) Phase I: Harvesting energy from the signal transmitted by S: During phase I, R l uses the antenna connected to the rectifier, i.e., antenna 1, to harvest energy from the signal arriving at it. One can write this signal in the form of the following equation:

x [t] 2 ≤ min    η i ε g sr,l [t] P s + G sr,l [t] P s x H [t] h loop G sr,l [t] P s + σ 2 2 + εµ i + 2εe s,l [t] T , 2εE l [t] [t -1] T    -εP C (C1.1) P low,i ≤ g sr,l [t] P s + G sr,l [t] P s G sr,l [t] P s + σ 2 x H [t] h loop 2 ≤ P up,i , i = 1, . . . , I. (C1.2) y r,l,1 [t] = h sr,l [t] P s x s,1 [t] + n r,l,1 [t] . (13)
In [START_REF] Liu | Power splitting-based SWIPT with decode-and-forward full-duplex relaying[END_REF], h sr,l [t] is the channel between S and the EH antenna of l and n r,l,1 [t] ∼ CN 0, σ 2 is the AWGN at R l . Assuming again that noise power is substantially lower than the power of the transmit signal, based on (5), the amount of energy that can be harvested from signal y r,l,1 [t] is given as:

e w,l,1 [t] = T P h |h sr,l [t]| 2 P s /2. ( 14
)
2) Phase II: Harvesting energy from the signals transmitted by S and R l [t]: During the second phase, relay R l , l = l [t] again harvests the energy of the signal reaching it's EH antenna. This signal can be expressed as:

y r,l,2 [t] = h sr,l [t] P s x s,1 [t] e jθ[t] + x H [t] h l ,l [t] G sr,l [t] P s x s,1 + ñr,l ,1 [t] G sr,l [t] P s + σ 2 + n r,l,2 [t] , (15)
where h l ,l [t] stands for the channel formed between the M transmit antennas of relay R l [t] and the EH antenna of R l . As a result, based again on the assumption that noise power is small enough, the energy that can be harvested can be proven to be equal to [START_REF] Ropokis | Cooperative beamforming exploiting energy recycling[END_REF]:

e w,l,2 [t] = T P h Ps h sr,l [t] e jθ[t] + x H [t]h l ,l [t] √ G sr,l [t] √ G sr,l [t]Ps+σ 2 2 2 . (16)
Hence, the energy available at relay R l at the end of time instance t is equal to:

E l [t] = E l [t -1] + e w,l,1 [t] + e w,l,2 [t] . (17)

III. THE OPTIMAL BEAMFORMING PROBLEM

The focus in this work is on maximizing the instantaneous rate, or equivalently, due to the monotonic relationship between the instantaneous rate and the SNR, on maximizing the instantaneous SNR ρ [t]. Given the form of the received signal at D, the following convenient form can be derived for ρ [t]:

ρ [t] = 1 σ 2 G sr,l [t] P s [t] x [t] H h rd,l [t] 2 G sr,l [t] P s [t] + x [t] H h rd,l [t] 2 + σ 2 . ( 18
)
As a result, exploiting also constraints (C1.1) and (C1.2), the problem of rate-optimal beamforming boils down to solving the subproblems: maximize:

x[t]
ρ [t] subject to: (C1.1), (C1.2), (P1)

for each one of the I intervals of the form [P low,i , P up,i ], and selecting the subproblem solution that results in the maximum SNR. Before continuing with solving (P1), for the purposes of simplifying the presentation, in what follows we drop the dependency of x [t] as well as of all channel realizations on time. We then note that since the SNR is an increasing function of x H h rd,l , solving (P1) such as to maximize the SNR is equivalent to solving subproblem maximize:

x

x H h rd,l subject to: (C1.1), (C1.2), (P2)

for each one of the I intervals of the EH model. In order to solve (P2), and therefore subproblem (P1), the following Lemma will be useful. Lemma 1: A beamformer for solving problem (P2) can be found in the form:

x = γ 1 e jφ1 u 1 + γ 2 e jφ2 u 2 , (21)
where

u 1 = h loop h loop , u 2 = I -u 1 u H 1 h rd,l I -u 1 u H 1 h rd,l , (22)
and

γ i ∈ R + , i = 1, 2 and φ i ∈ [0, 2π) , i = 1, 2.
Proof: Following an approach similar to the one in [START_REF] Demir | Optimum QoS-aware beamformer design for full-duplex relay with self-energy recycling[END_REF], we start by considering an orthonormal basis {u 1 , . . . , u N R } for C M ×1 , with u 1 and u 2 defined as in (22). Using this orthonormal basis, we can write any beamformer in the form:

x = M i=1 γ i e jφi u i . ( 23
)
The objective function in (P2) is then expressed as

x H h rd,l = γ 1 e jφ1 u 1 + γ 2 e jφ2 u 2 H h rd,l , (24)
while (C1.1) and (C1.2) can also be simplified by noticing that

x H h loop = γ 1 u H 1 h loop , and x 2 = N i=1 γ 2 i . (25)
As a result, based on (24), we see that values of coefficients γ i , i ≥ 3 do not influence the utility function value of (P2). On the other hand, based on (25), we see that non zero values of γ i , i ≥ 3 , restrict the feasible values of γ 1 and γ 2 . Therefore, it is straightforward to show that the optimal solution to (P2) is obtained for γ 3 , . . . , γ N = 0. Building on the above Lemma, the following theorem further confines the form of the optimal beamformer. Theorem 1: The optimal solution of (P1) and (P2) is given as:

x = γ 1 e jφ1 u 1 + γ 2 e jφ2 u 2 (26)
where φ 1 , φ 2 ∈ [0, 2π) are selected such that

e -jφ1 u H 1 h rd,l u H 1 h rd,l = e -jφ2 u H 2 h rd,l u H 2 h rd,l (27)
and γ 1 , γ 2 are found to be the solution to problem (P3).

Proof: Using the result of Lemma 1 and restricting ourselves to beamformers of the form (21), it follows that the value obtained for the utility function in (P2) is given as in (24), where this value is maximized if terms e -jφ1 u H 1 h rd,l and e -jφ2 u H 2 h rd,l , are of the same phase, i.e., if property (27) holds. Moreover, based on (25), we can see that for a beamformer of the form (21), the choice of phase parameters φ 1 , φ 2 , does not affect the feasibility of the beamfomer. Hence, the optimal beamformer can be found by selecting phase values φ 1 and φ 2 such as to satisfy property (27). With this choice, we then have that:

x H h rd,l = γ 1 u H 1 h rd,l + γ 2 u H 2 h rd,l , (28)
and the optimal beamformer problem boils down to determining the values of γ 1 and γ 2 such as to maximize the value of the objective function in (28).

Combining equations ( 28) and (25) we then see that solving subproblem (P1) reduces to solving the problem (P3): maximize:

αγ 1 + βγ 2 (P3) subject to: γ 2 1 + γ 2 2 ≤ (A i + B i γ 1 ) 2 + C i , (C3.1) γ 2 1 + γ 2 2 ≤ D, (C3.2) γ 1,low,i ≤ γ 1 ≤ γ 1,up,i , γ 2 ∈ R + (C3.3) where α = u H 1 h rd,l , β = u H 2 h rd,l , (30a)
A i = η i εg sr,l P s , B i = η i εG sr,l P s G sr,l P s + σ 2 u H 1 h loop , (30b)
C i = µ i ε + 2εe s,l [t] [t] T -εP C , D = 2εE l [t] [t -1] T -εP C , (30c)
γ 1,low,i = max    0, G sr,l Ps+σ 2 G sr,l Ps P low,i - √ g sr,l Ps u H 1 h loop    (31) and γ up,i = max    0, G sr,l Ps+σ 2 G sr,l Ps P up,i - √ g sr,l Ps u H 1 h loop    . ( 32
)
Hence, the result of Theorem 1 is proven. In (P3), constraints (C3.1) and (C3.2) provide an equivalent formulation of (C1.1), while (C3.3) corresponds to (C1.2). Moreover, constraint γ 2 ∈ R + can be ignored, since positive values of γ 2 result in higher values for the the utility function in (P3) (with respect to negative values), while the sign of γ 2 does not affect the feasibility of the remaining constraints of (P3). Taking this into account, in the appendix we present our approach for solving (P3). Moreover, we highlight here that the approach followed in the appendix for solving (P3), results in solving a set of simple subproblems where for each one of them, the solution can be expressed in closed form using simple, analytical functions. As a result, the use of advanced optimization techniques, e.g. iterative optimization techniques is avoided, allowing for the ease of implementation of the proposed beamforming scheme.

Finally, the following theorem discusses some additional properties of problems (P1) and (P3), that are exploited in the solution process.

Theorem 2: For the optimal beamformer at least one of constraints (C3.1) and (C3.2) is active.

Proof: Let us denote as γ1 , γ2 , γ2 ≥ 0, an optimal solution for (P3) for which both constraints (C3.1) and (C3.2) are inactive. Since the right hand sides of (C3.1), (C3.2) do not vary with γ 2 , and (C3.3) does not depend on γ 2 , it then holds that another feasible point (γ 1 , γ2 ), with γ2 ≥ γ2 selected such that γ1

2 + γ2 2 = min (A i + B i γi ) 2 + C, D can be found that clearly satisfies one of constraints (C3.1), (C3.2) with equality. Nevertheless, it is obvious that such a point, since γ2 ≥ γ2 , results in a higher value for the utility function of (P3). Hence, no point that results in both constraints (C3.1) and (C3.2) being inactive can be an optimal point. Thus the result of Theorem 2 is proven.

Having presented the solution for the beamforming problem in the following section we present results obtained by using the proposed scheme.

IV. NUMERICAL RESULTS AND DISCUSSION

We first present the considered simulation scenario, starting from the considered energy harvesting model.

A. The considered energy harvesting model

We adopt a piecewise linear approximation of the following highly accurate EH model proposed in [START_REF] Chen | Wireless energy harvesting using signals from multiple fading channels[END_REF]:

P acc h = a -b c P in P in + c , (33)
where P acc h is the harvested power, P in is the input power to the harvesting circuit (measured in mWatts), and parameters a, b, and c are selected to fit real EH measurements. Following [START_REF] Chen | Wireless energy harvesting using signals from multiple fading channels[END_REF] we set a = 2.463, b = 1.635, and c = 0.826.

As a result of the selection of the specific parameter values, function P acc h (P in ) is a concave function of P in . Therefore, for any interval of the form [P low,i , P up,i ], the inequality (34) is satisfied.

P acc h ≥ η i P in + µ i , ∀P in ∈ [P low,i , P up,i ] ,
where η i = P acc h (P up,i ) -P acc h (P low,i ) P up,i -P low,i , and µ i = P acc h (P low,i ) -

η i P low,i . (34)
Based on the EH expression in (33), we model the EH process using [START_REF] Ropokis | Optimal power allocation for energy recycling assisted cooperative communications[END_REF], with η i and µ i defined in (34). Following such an approach, and by selecting P low,i and P up,i by uniformly splitting the interval [0, 1mW atts] in ten intervals, and each one of the intervals [1, 5mW atts] and [5, 20mW atts] in five intervals a piecewise linear model of the form (5) was obtained. We also note that for values of P in larger than 20mW atts, we consider that a saturation effect take place. As a result the harvested power is the same for all values of P in that are higher than (or equal to) 20mW atts. While at a first glance such an approach may lead to loss of accuracy with respect to the exact model in (33), in practice, due to the concave nature of the energy harvesting model in (33), the loss of accuracy caused by this assumption was found to be practically negligible. Indicatively, we note that the accuracy achieved by this piecewise model had a relative approximation error (with respect (33)) that was at most 2.5 % ∀P in ∈ [0, 100mW atts], therefore allowing to accurately model harvesting, with a small number of intervals, i.e., 21 intervals. We finally highlight that in the presence of pathloss and fading, the input power P in to the EH circuit is drastically reduced. As a result, for most practical purposes, only a subset of these 21 intervals, i.e. the 10 intervals corresponding to the range [0, 1mW att], need to be considered. Hence, the complexity is reduced.

B. The considered channel model and simulation scenario

We study a cooperative communications system where the distance d sd between the Source S and the destination D is set to be d sd = 30m. We set S to be at the origin of our coordinate system and we place D at coordinates (d sd , 0). Moreover, we consider a system with two relays R 1 and R 2 positioned symmetrically at coordinates d sd 2 , d sd 4 and d sd 2 , -d sd 4 . To model the wireless channel, we use pathloss and i.i.d. block Rayleigh fading. Hence, for a relay R l positioned at distance d sr,l from S and d rd,l from D, it holds that the pathloss for the S → R l and R l → D channels are expressed as:

ḡsr,l = G t A (d sr,l /d 0 )
, and ḡrd,l =

G t A (d rd,l /d 0 ) , ( 35
)
where G t is the transmit antenna gain that we assume to be common for both S and R. Moreover, A is the pathloss at a nominal distance d 0 . Following the channel model presented in [START_REF] Liu | Power splitting-based SWIPT with decode-and-forward full-duplex relaying[END_REF], we set the pathloss exponent equal to = 2.5, and attenuation A, corresponding to a distance of d 0 = 1m, equal to a value of -30 [dB]. We also set G t to be equal to

with hsr,l [t] ∼ CN (0, I M +1×M +1 ), and similarly hrd,l [t] ∼ CN (0, I M ×M ).

We follow a similar block Rayleigh fading approach for modeling the channel formed between the transmit antennas of relay l 1 , l 1 = 1 . . . , L, and the energy harvesting antenna of relay l 2 , l 2 = 1, . . . , L, with l 1 = l 2 . In more detail, we consider that the R l1 → R l2 channel is of the form:

h l1,l2 [t] = ḡl1,l2 hl1,l2 [t] , with ḡl1,l2 = G t A (d l1,l2 /d 0 ) , (37)
where d l1,l2 is the distance between relay R l1 and relay R l2 , and hl1,l2 [t] ∼ CN (0, I M ×M ). Concerning the selfenergy recycling loop channel h loop , we set it equal to h loop = √ δ1 M ×1 where δ = -15[dB], as in [START_REF] Zeng | Full-Duplex Wireless-Powered Relay With Self-Energy Recycling[END_REF]. Moreover, we set the number of antennas at the relays to be equal to M + 1 = 5 and the noise power to σ 2 = -90[dBm] [START_REF] Liu | Power splitting-based SWIPT with decode-and-forward full-duplex relaying[END_REF]. Finally, the time slot duration T was set to T = 1 second.

Based on the above simulation parameters, in Figure 1 we present the performance of the proposed beamforming scheme, measured in terms of the average (over 10 4 channel realizations) achievable rate, as a function of transmit power P s . Concerning the additional power consumption P C of the relay, we base our selection of value on recent results related to the design of low power transceivers for EH applications (e.g. [START_REF] Rajavi | An energy harvested ultra-low power transceiver for internet of medical things[END_REF], [START_REF] Kim | An ultra-low-power RF energy-harvesting transceiver for multiple-node sensor application[END_REF]) that propose transceiver designs characterized by power consumption of the order of tens of µW atts. With this in mind, we set P C = 50µW atts. Concerning relay selection, we assume that for each time instance, the relay resulting in the maximum instantaneous SNR is selected. Finally, we mention that in Fig. 1, for the purposes of comparison, we also include results corresponding to the case that only relay R 1 is available and again the proposed beamforming scheme is used, as well as results corresponding to the case that an additional relay R 3 is also available. We place this additional relay in close proximity to R 2 (i.e., at coordinates d sd 2 , -d sd 4 + 1m ) such that we can consider that the effects of pathloss are nearly identical for all three relays. We finally note, that in case of an infeasibility in the optimal beamforming problem for all available relays, we exploit the corresponding time slot for the purposes of energy harvesting, at all available relays. As we can see in Figure 1, the addition of the second and third relays results in significant performance improvement. Indicatively, for a value of P s equal to 30dBm, the addition of R 2 results in a performance improvement of nearly 100%, doubling the transmit rate (with respect to the single relay case) while the addition of R 3 results in a performance improvement of nearly 200% practically tripling the rate (with respect to the single relay case).

As a second test, we now investigate the effect of relay placement on the benefits that relay selection can deliver.

To this end, we consider a system with two relays namely R 1 and R 2 and as in the previous case, we place relay R 1 at coordinates (d sd /2, d sd /4). On the other hand, for relay R 2 we consider placements of the form (d sd /2, -y) and set different values for y. Finally, concerning the transmit power P s we investigate the case that P s is selected to be equal to 30[dBm] as well as the case that P s = 25[dBm] 1 . For the two above values of P s , in Fig. 2 we present the relative performance gain that relay R 2 delivers (with respect to the case that only R 1 is present) as a function of the normalized distance y/d sd . We measure this performance gain as:

Performance Gain = (C 1,2 -C 1 ) /C 1 × 100%, (38)
where C 1 stands for the average rate in case that only R 1 is available, and C 1,2 for the average rate in case that both R 1 and R 2 are available. We observe that for both values of P s , a very similar trend for the performance gain as a function of y appears. Moreover, for values of y less than d sd /2 a performance gain of at least nearly 50% is achieved, indicating that significant gains can be delivered for several relay placements. On the other hand, for y > d sd the performance gains delivered by the additional relay nearly vanish. Finally, we further investigate the effect of relay positioning by considering the case of two available relays and placements of the form (x, d sd /4) for relay R 1 and (x, -d sd /4) for relay R 2 . For such placements, after setting P s = 30[dBm], in Fig. 3 we plot the achievable rate as a function of the normalized distance x/d sd . The remaining system parameters were once more set to the values specified earlier. Based on the results presented in Fig. 3, we observe that placements of the two relays closer to S result in higher achievable rate. This implies that the achievable rate is mostly influenced by the ability of the relays to harvest sufficient energy for their transmissions.

V. CONCLUSIONS AND FUTURE WORK

We have presented a multi-relay scheme employing selfenergy recycling relays and we have developed a new 1 The remaining simulation parameters are set to the values specified earlier. beamformer that is based on realistic EH and energy consumption models at the relays. Our performance analysis results have illustrated that the proposed multi-relay scheme can deliver significant benefits as compared to single relay systems for a wide variety of relay placements. Finally, we set as a future objective the design of optimal relay selection policies for maximizing the average rate of our scheme.

Fig. 1 .

 1 Fig. 1. The transmit rate as a function of Ps for different numbers of relays.

Fig. 2 .AverageFig. 3 .

 23 Fig. 2. Performance gain delivered by the second relay as a function of y.

  Input Single Output (MISO) channel formed between the M transmit antennas of R l [t] , (i.e., antennas 2, . . . M + 1) and the receive antenna of D, and n d,2 [t] ∼ CN 0, σ 2 is the AWGN at D. Focusing now on the EH process at R l [t] , the signal that reaches R l [t] during phase II, is expressed as:

  10[dB]. The actual communication channels between S → R l and R

l → D are then expressed as: h sr,l [t] = ḡsr,l hsr,l [t] , and h rd,l [t] = ḡrd,l hrd,l [t] ,

APPENDIX A SOLVING OPTIMIZATION PROBLEM (P3)

After dropping constraint γ 2 ∈ R + , we can solve problem (P3) by separately considering the following two cases.

A. Case I: D ≤ A 2

i + C i In this case (C3.1) is redundant. Moreover, from Theorem 2 we have that the optimal value is obtained when γ 2 2 = D -γ 2 1 . (P3) is therefore equivalently expressed as: maximize:

and the optimal γ 1 is easily obtained as:

provided that γ 1 respects (C4.1). Otherwise the optimal solution is obtained as the value in the set γ 1,low,i , min γ 1,upi , √ D that maximizes the utility function in (P3). The optimal γ 2 is then γ 2 = D -γ 2 1 . Having investigated this first case, we now move to the investigation of the second one.

B. Case II: D ≥ A 2

i + C i In this case both constraints (C3.1) and (C3.2) have to be taken into account. Employing then the result of Theorem 2 we discriminate the following two cases.

1) Solving problem (P3) with constraint (C3.2) being active: In this case, by setting γ 2 2 = D -γ 2 1 , after some straightforward manipulations we can rewrite problem (P3) in the equivalent form: maximize:

Clearly, the solution to (P5) is the value γ 1 in (40), provided that it is feasible. Otherwise, the solution to (P5) is selected among the values of the set max γ 1,low,i ,

, min γ 1,up,i , √ D as the feasible one that results in the highest utility value. The value of γ 2 is then selected based on the fact that (C3.2) is active.

2) Solving problem (P3) with constraint (C3.1) being active: In this case we can set

The combination of this constraint with constraint γ 1 ∈ [γ 1,low,i , γ 1,up,i ] then leads to the following problem: maximize:

subject to:

To solve this problem, we start by finding the roots of the derivatives of its utility function, by solving the equation:

Candidate roots for the above equation can be found by squaring both sides. We then obtain that roots of equation (43) need also be roots of the polynomial equation

where

Note that due to the fact that the polynomial in (44) is of second degree, its solutions can be easily expressed in closed form. The solution to (P6) is then obtained by comparing the values for the utility function of problem (P6) achieved by the feasible solutions of (44), with the values achieved by the endpoints of constraint (C6.1) and selecting among these points the feasible solution that achieves the best value. The value of γ 2 is then determined by the the fact that (C3.1) is active. Having obtained the solutions for the case that constraints (C3.1) and (C3.2) are respectively active, we select the solution that results in maximizing the objective of (P3).


Jean Philippe Praene

Vanessa Rakotoson
email: vanessa.rakotoson@univ-reunion.fr

Energy-related carbon emission from electricity sector: past trends and futures for Madagascar

Keywords: Madagascar, Scenario, CO2 emission, LCA, LMDI, KAYA, Clustering

To meet the increase in energy demand, many developing countries, such as Madagascar, have turned to fossil fuels for electricity generation. This dependence has a significant environmental impact. Moreover, in Madagascar, the current electricity mix does not allow for stable and serene economic development. Thus, in order to discuss an energy planning strategy of the territory, this paper intends to analyze the past trends of CO2 emissions from electricity generation in Madagascar over the 1990-2015 period. In this study, the logarithmic mean Divisia index (LMDI) method is used to first quantify the driving forces of changes by examining past trends. Then, the combination of the LMDI with Kaya's identity through the concept of IPAT, allows us to link CO2 emissions to anthropogenic activity. This makes it possible to define Madagascar's future energy needs and the distribution of associated consumption by 2030. Starting from the classic business-as-usual trend scenario, two other scenarios have been implemented. The two cases selected come from Madagascar's new energy planning visions (NPE and IEM). The impact of political instability is discussed from the perspective of energy consumption and the economy.

INTRODUCTION

Since the COP21, many countries around the world have set ambitious new targets for reducing greenhouse gas (GHG) emissions. For developing countries, these objectives are all the more ambitious as they must both achieve the energy demand to develop the territory and mitigate the impact of economic and anthropogenic activity. Reliable power generation is one of the main drivers of economic growth. Moreover, the electricity sector is among the most GHG emitting sectors. In 2016, electricity and heat generation accounting for 42% of the total global emissions, which corresponds to 13.41 Gt of CO2 [1]. As a significant contributor to CO2 emissions, the power sector is particularly in energy planning scenario to achieve low carbon development [START_REF] Cansino | Driving forces of Spain׳s CO2 emissions: A LMDI decomposition approach[END_REF][3] [START_REF] De Oliveira-De | Effect of generation capacity factors on carbon emission intensity of electricity of Latin America & the Caribbean, a temporal IDA-LMDI analysis[END_REF]. The benefits of higher renewable energy sources (RES) penetration in electricity mix are commonly acknowledged. The fact that Africa contributes less than 4% of global GHG emissions while being the continent most vulnerable to climate change contrasts the need for an energy transition with regional policymakers' decisions. The challenges and opportunities are not the same whether these African states are developed, emerging or developing. Madagascar is currently in the third category and aims to become an emerging country through new political ambitions. Poor access to electricity remains a significant barrier to most businesses and economic growth in sub-Saharan Africa's regions [START_REF] Chakamera | Electricity crisis and the effect of CO2 emissions on infrastructure-growth nexus in Sub Saharan Africa[END_REF] [START_REF] Cdc | What are the links between power , economic growth and job creation ? What ' s in this report ?[END_REF]. Under this context, understanding the driver's forces in the power sector of Madagascar the sector appears to be essential in order to establish a diagnosis of the current situation of the country in order to be able to draw the outlines of possible future opportunities. Among the existing decomposition methods applied to energy, the LMDI approach is the one most used by the authors [START_REF] Ang | Decomposition analysis for policymaking in energy: Which is the preferred method?[END_REF] [START_REF] Goh | Quantifying drivers of CO2 emissions from electricity generation -Current practices and future extensions[END_REF]. This paper aimed to first provide a general picture of Madagascar's energy situation in the 1990-2015 period. It investigates the relationship between electricity consumption, economic growth, population, and CO2 emissions. The remainder of this paper is arranged as follows. Section 2 summarizes an overview of Madagascar socio-economic and energy situation. Section 3 presents the methods used for the decomposition of CO2 emissions and those used for the implementation of the scenarios. Section 4 describes the results of LMDI, factorial analysis, and the different scenarios and discusses the meaning of these results. Section 5 concludes this article and provide some policy recommendations for Madagascar.

MADAGASCAR AT A GLANCE

Located between the Mozambique channel and Indian Ocean, Madagascar is the fourth largest island in the world with total area of 586,295 km². Subjected to subtropical climate conditions, Madagascar spans 14 degrees of latitude. The territory thus offers a wide diversity of micro-climates between the highland areas and the 5600 km of coastline. From a global point of view, the climatic zoning of the island reveals three main areas: dry and arid (west), wet and rainy (east) and highlands characterized by low temperature and precipitation. Madagascar is heavily populated with an estimated population of 25.57 million people in 2017 [START_REF] Bank | Madagascar profile[END_REF]. In 2012, the poverty rate reached 70.7%. In rural areas where two-thirds of the population lives, this rate increases to 82%. The main consequence is a massive exodus of people to the six main urban areas of the territory. From an economic point of view, recurrent political instability has weakened the territory's growth and development over the past 30 years, see Figure 1. In order to put the economy back on a healthy and sustainable growth path at a level at least equal to 5%, the IMF1 decided in 2016 to grant an extended credit facility of 305 million USD. This fund was backed by a three-year reform programme (2016-2019), whose energy is one of the cornerstones. The reliability of electricity supply is a strategic issue for the development of Madagascar's urban areas. Indeed, the chronic load shedding problems in large cities make it difficult for the country to emerge economically. The energy sector is characterized by the predominance of fuel wood and charcoal in final energy consumption (83% in 2017), which is the origin of the degradation of the natural forests (~ 36,000 ha/ year). The reliability of electricity supply is a strategic issue for the development of Madagascar's urban areas. Indeed, the recurrent load shedding problems in large cities make it difficult for the country to emerge economically. Adaptation and strengthening of the regulatory and institutional framework and business environment to achieve the NPE vision by ensuring effective coordination between the various entities involved, including ministries involved in boosting the energy sector, as well as public and private partners [START_REF]Ministère de l'Energie et des Hydrocarbures[END_REF]. The Emergence for Madagascar initiative of the very recent Malagasy President is an ambitious vision of the territory by 2023. We focused on the aspirations on the energy issue. This new policy aims to commit the country by 2023 to a resilient territory. Emerging countries such as China inspire IEM's development model. This plan is first and foremost a new one aimed at contributing to development and prosperity for all in a generation. This vision of emergence is based on the in-depth economic reforms needed to move from a counter-economy to an economy that exports finished products with high added value. This re-foundation is the subject, in all African countries, and all Asian countries in the emergence phase or having successfully emerged, of skills and performance in terms of reforms.

Regarding the question of the energy situation, no precise planning has yet been established.

The only quantified targets are on the one hand to increase the electrification rate from 16% to 50% by 2023. In addition, at the beginning of 2019, several hydroelectric power plant construction projects are aimed at contributing to the penetration of renewable energies into the electricity mix. In addition to the development of RES, the project also intends to control the cost of electricity production, which is one of the highest in the Indian Ocean region. The last point is the production guarantee in order to limit the risks of power outages in Madagascar's major cities.

MATERIALS AND METHOD

Environmental assessment LCA Methodology. Although Madagascar is a non-issuing country with a low contribution of 0.2% of world emissions in 2000, Madagascar's Intended Nationally Determined Contribution (INDC) was established as the outcome of the COP21 in Paris in 2015. The INDC aims to reduce GHG emissions by 14% compared to the Business-as-usual (BAU) scenario, projections based on the 2000 emissions inventory [START_REF]Republic of Madagascar[END_REF].

In the context of the recent energy policies, the first part of this study involves in defining the level of emissions from electricity sector. This approach is to determine the basis of the energy scenarios. The Life cycle assessment (LCA) method used in this study is a standard approach which allows to identify environmental burdens of a product through its life cycle [START_REF]Environmental management -Life cycle assessment -Requirements and guidelines[END_REF]. It comprises four steps: (1) goal and scope definition phase, (2) inventory analysis phase, (3) impact assessment phase and (4) interpretation phase. The goal is to determine the environmental impacts of electricity sector at the territorial level. The functional unit (FU) is defined by the amount of annual electricity produced by this sector. The impacts per 1 kWh are also calculated to measure the change in the electricity mix. The study's boundaries are from cradle-to-gate, including raw material acquisition, transport, infrastructure construction and electricity production.

A number of methods and tools are available to assess environmental impacts and can assist in decision-making. Software databases of current LCA tools, as seen in [START_REF] Jursová | Greenhouse Gas Emission Assessment from Electricity Production in the Czech Republic[END_REF], [START_REF] Brizmohun | Life cycle assessment of electricity generation in Mauritius[END_REF] where Simapro is used or in [17] [18] with GaBi, do not generally integrate systems used in insular territories or developing countries. An evaluation tool that can be adapted to each territory, was developed, [START_REF] Rakotoson | Integration de l'analyse de cycle de vie dans l'étude de la production d'électricité en milieux insulaires[END_REF], to assess electricity production, based on the LCA methodology and the adaptation of the GEMIS database [START_REF]Öko-Institut, Global Emissions Model of Integrated Systems (GEMIS)[END_REF]. At first, the methodology consists in describing the production facilities with the specific characteristics related to the study area: the installed capacity for each technology, the supply distance of raw materials for petroleum products, the output generated by each plant. Then, the collected data are treated on the basis of the corresponding emission factors, at each life cycle stage, according to Equation (1):

(1)

where C represents the emission value of each technology i, k the life cycle stage, EF the corresponding emission factor for 1 FU of the life-cycle stage and qp the quantity of the product expressed in the FU.

The obtained results will allow to determine the environmental indicators of both total electricity production and ratio values. This methodology was applied to non-interconnected French territories, demonstrating the assessed environmental impacts of the system and the comparison of territories on a global scale [START_REF] Rakotoson | A Life cycle assessment approach to the electricity generation of French overseas region[END_REF].

KAYA-LMDI methods

Madagascar's electricity-related carbon dioxide (CO2) emissions are analysed using an approach which results from the combination of KAYA's identity and the LMDI [START_REF] Ang | LMDI decomposition approach: A guide for implementation[END_REF] The objective is to be able to identify and quantify the links between CO2 emissions and human activities.

KAYA identity: The concept of the IPAT, which is becoming increasingly popular, consists in dividing environmental impacts (I) into three factors: population (P), affluence (A) and technology (T). By establishing links between the environmental and socio-economic dimensions [START_REF] Mahony | Decomposition of Ireland ' s carbon emissions from 1990 to 2010 : An extended Kaya identity[END_REF], it allows to measure the environmental impacts of human activities [START_REF] Mahony | Decomposition of Ireland ' s carbon emissions from 1990 to 2010 : An extended Kaya identity[END_REF]- [START_REF] Lima | A cross-country assessment of energy-related CO2 emissions: An extended Kaya Index Decomposition Approach[END_REF], as summarized in the following equation:

(2)

Based on the IPAT equation, Kaya's identity applies this principle to GHG emissions, promoting the assessment of energy-related CO2 emission factors [START_REF] Rosa | Human drivers of national greenhouse-gas emissions[END_REF], [START_REF] Ma | Carbon abatement in China's commercial building sector: A bottom-up measurement model based on Kaya-LMDI methods[END_REF], according to :

(3)

From Equation (3), the effects of the different variables and their evolution over time can be studied regardless of the geographical scale chosen (city, region, country, world). In addition, many researchers have extended or modified the basic Kaya equation to include other explanatory variables depending on the nature and scope of their research [START_REF] Ma | Carbon abatement in China's commercial building sector: A bottom-up measurement model based on Kaya-LMDI methods[END_REF] [27][28] [START_REF] Remuzgo | Environmental Science & Policy International inequality in CO 2 emissions : A new factorial decomposition based on Kaya factors[END_REF].

In the current case, regarding electricity consumption, Madagascar is a country characterized by a low electricity rate and a dependence on fossil fuels. As a result, the vision of the country's energy policy is mainly focused on increasing access to electricity and promoting renewable energy [START_REF]Ministère de l'Energie et des Hydrocarbures[END_REF]. Thus, in order to integrate the specificities of the Malagasy electricity sector into the analysis, the Kaya equation has been modified as follows:

(4)

where Ci represents the carbon emission for the energy source type i, FFi the electricity generated by the energy source type i, FF the total electricity production, E the total electricity consumption, P the number of inhabitants and Pelec the number of inhabitants with access to electricity.

The "Technology" group refers to the technology of the electricity production system: the CO2 emission factor by energy source (Ci/FFi), the electricity mix (FFi/FF) and the energy losses during distribution (FF/E). The "Affluence" group expresses the country's richness in relation to its electricity consumption through electricity consumption per GDP (E/GDP) and the ratio of GDP per capita to the electrification rate (GDP/Pelec). The "Population" group denotes the characteristics related to the social dimension: the electrification rate (Pelec/P) and the number of inhabitants (P).

The modified Kaya equation then passes through a decomposition method that quantifies the effect of the parameters considered on the evolution of the CO2 emission. The most commonly applied method for this is the LMDI [30] [31].

LMDI:

The LMDI (Logarithmic Mean Divisia Index) was preferred to other decomposition methods due to its advantages [START_REF] Ang | LMDI decomposition approach: A guide for implementation[END_REF], which range from perfect decomposition (no residual terms) to the ability to deal with zero values (replacement with small positive values). In addition, as it is easy to formulate, LMDI I was the most recommended methodology [START_REF] Ang | Decomposition analysis for policymaking in energy: Which is the preferred method?[END_REF], [START_REF] Ang | LMDI decomposition approach: A guide for implementation[END_REF]. Therefore, in accordance with Equation 3, the decomposition by the LMDI I is performed through seven effects that are categorized into three groups: technology, affluence, population. Each effect is defined in Table 1. The decomposition formulas, presented in Table 2, will be performed for each pair of years during the period considered. Thus, as can be seen in Table 2, the analysis will be carried out using both additive and multiplicative decomposition to obtain both quantities and change ratios.

PCA-based Clustering

Principal component analysis (PCA): PCA and HCA (Hierarchical Clustering Analysis) are usually used concomitantly in many studies in the field of biology. PCA is part of the group of multidimensional descriptive methods called factorial methods widely used in the literature for the analysis of quantitative data [START_REF] Jolliffe | Principal Component Analysis. Second Edition[END_REF]. This approach makes it possible to better identify the data on which we are working, and to detect possible outliers, [START_REF] Beh | Exploratory multivariate analysis by example using R[END_REF]. The PCA makes it possible to define new factors (the principal components) by synthesizing a certain percentage of the total variance contained in the initial set of data. The criterion of an eigenvalue λ>1 was used to select the number of relevant components on which to conduct the analysis, [START_REF] Kaiser | The Application of Electronic Computers to Factor Analysis[END_REF]. This first step allows us to define the variables considered significant to our analysis but also to define the projection plan of the individuals. In our case this corresponds to the countries. Thus, the first plan consisting of two first components will be retained for the next step.

Hierarchical Clustering on Principal Components (HCPC):

The second step of the consists in the implementation of clustering based on the K-means partitioning method, which is one of the most commonly used clustering algorithms. Clustering is an unsupervised learning process, often used for data analytics. The method is intended to identify homogeneous groups called clusters from a set of parameters. In other words, data clustering aims to minimize the total within-cluster variation. The objective of this clustering is to study the socio-economic characteristics of different developing or emerging countries. Understanding the current situation of Madagascar is useful to implement policy scenario proposed the government and thus see which trajectory the country could take in the coming years. All statistical analyses investigated in this study are performed using the FactoMineR package developed by [START_REF] Husson | Exploratory Multivariate Analysis by Example Using R[END_REF].

Data collection Impact assessment and decomposition analysis:

To establish a diagnostic of the electricity sector, the data over a period of 25 years from [START_REF] Jirama | Evolution annuelle des productions, ventes et nombre d'abonnes 1987-2015[END_REF], [START_REF] Ore | Synoptique Parc de Production[END_REF], shown in Figure 2, have been used. These data describe the evolution of the Malagasy electricity mix, in Figure 2.a. The installed capacity for each technology is illustrated in Figure 2.b. They are used to determine the emission value for each type of technology. To note that only JIRAMA's data have been used in this study, not including off-grid rural electrification. All these data concerning the power generation system are injected into the equations of the decomposition method. Thus, as required by the combined Kaya-LMDI method, they are enhanced by socio-economic data from 1990 to 2015: GDP, electrification rate and number of inhabitants obtained from [START_REF]Total population[END_REF]- [START_REF]The World Bank Open Data[END_REF] and the electrified population calculated on the basis of the electrification rate and the number of inhabitants. [START_REF] Jirama | Evolution annuelle des productions, ventes et nombre d'abonnes 1987-2015[END_REF], [START_REF] Ore | Synoptique Parc de Production[END_REF]).

RESULTS AND DISCUSSION

CO2 emission drivers

Environmental burdens: The main environmental indicators obtained from this evaluation are related to the Global Warming Potential (GWP), the acidification potential, the tropospheric ozone precursor potential, the eutrophication potential, the cumulative energy used and the material expenditures. In fact, these are the most recurrent indicators in LCA studies [START_REF] Pehnt | Dynamic life cycle assessment (LCA) of renewable energy technologies[END_REF], [START_REF] Asdrubali | Life cycle assessment of electricity production from renewable energies: Review and results harmonization[END_REF]. However, this study will focus on results based on climate change, expressed by GWP in CO2 equivalent (CO2-eq).

As can be seen on Figure 3, total emissions show an increasing trend, which is explained by the increase in production, but also by the increase in the use of fossil fuels in the Malagasy electricity mix. Indeed, the environmental loads of renewable energies are mainly attributed to the infrastructure construction. To note that construction phase represents only 7% of the total emissions. It can be observed that the emission rate varies over the last 25 years in quasi-sine wave manner. This is due to the continue slight increase of hydroelectric production that contributes at an average of 67% of the electric mix, and also due to the variation of fossil fuel production, even the installed capacity has increased. The electrical emission rate value in 2015 equals 602 g CO2-eq/kWh, which represents a 20% increase over 25 years. This assessment categorizes Madagascar, in 2015, at the lower bound of the countries with a high GHG emission rate, as classified by [START_REF] Herbert | A typology for world electricity mix: Application for inventories in Consequential LCA (CLCA)[END_REF]. Regarding the total emission, according to the production growth, it can be observed that value in 2015 is equal to nearly 4 times of value in 1990 due to the high installed capacity in gasoil infrastructure. However, other parameters related to CO2 emission are identified by Kaya's identity coupled with the LMDI presented in the following paragraph.

Decomposition analysis: Decomposition was performed using both additive and multiplicative decomposition. However, the analysis focuses mainly on the interpretation of variations in CO2 emission in terms of quantities. For this reason, only the results of additive decomposition are presented.

As outlined in the materials and method section, the decomposition analysis highlights the relation between electricity production, CO2 emissions and the socio-economic situation in Madagascar. The results obtained are summarized in Figure 4.

Whether for annual or aggregate variations, the evolution of CO2 emission follows the trend of variations generated by the technology of the electricity generation system. Indeed, regarding the LCA, by its formula (Equation 1) the CO2 emission of the electricity sector is linked to the electricity mix.

However, the analysis by the combined Kaya-LMDI method shows the importance of exogenous factors on the environmental impact of the electricity sector. On one hand, the results reveal that the increase in CO2 emissions is mainly due to changes in the country's demographic situation, represented by the Population group. As shown in Figure 4, the trend of the population group effect and the total effect are of the same slope. On the other hand, Madagascar's economic situation in relation to its electrification rate, expressed by the Affluence group, contributes to the reduction of CO2 emitted by the electricity sector.

From 1990 to 2015, CO2 emissions increased by +688 kt with an impact of +649 kt for the Population, -222 kt for the Affluence and +261 kt for the Technology.

In addition, as illustrated in Figure 4, the GWP variation from 1990 to 2015 can be subdivided into five periods of alternating increases and decreases (1990-1992; 1992-1997; 1997-2006; 2006-2010; 2010-2015).

The first period considered, 1990-1992, shows an increase in CO2 emissions. This is mainly due to the Technology group, which generated an increase of 48,486 tonnes, essentially caused by an energy-mix effect of 40,059 tonnes. In addition, the positive population growth effect (+14,890 tonnes) is followed by a negative electrification rate effect (-11,743 tonnes).

The population is increasing but the electrification rate is decreasing: the growth of the electrified population is therefore slight. For the impact of the Affluence group, the increase in electricity consumption followed by a decrease in GDP led to a positive energy-intensity effect. Thus, as GDP and electrification rates decline, the economic-related electrification rate effect is also declining. Nevertheless, this energy-intensity effect (ΔCe) passes through two remarkable phases between 1990-1992: a drop of -22,684 t CO2-eq from 1990 to 1991 followed by an increase of 25,979 t CO2-eq between 1991 and 1992. In addition to its impacts on the country's social situation, the political crisis of 1991 strongly affected Madagascar's economic development [START_REF] Hugon | La stagnation de l'économie malgache : le rôle des crises et des facteurs sociopolitiques en longue période[END_REF]: the ΔCe involved is -53,751 t CO2-eq; at the end of the crisis, an economic recovery is observed, the ΔCe rises to 49,584 t CO2-eq.

From 1992 to 1997, the environmental impact of the electricity sector was reduced of -71,862 t CO2-eq. Although the Population group makes a positive contribution of 88,753 t CO2-eq, Affluence and Technology respectively had a negative effect of -48,568 t CO2-eq and -112,047 t CO2-eq. During this period, the population and electrification rate increased, so the number of electrified people increased, leading to a rise in the demand for electricity. However, to meet this demand, it was decided to increase hydropower generation rather than thermal generation.

For the 1997-2006, the growing trend of CO2 emissions from the electricity sector is resuming again, reaching a total change of 407,424 t CO2-eq with a Technology effect of 210,303 t CO2-eq, an Affluence group effect of -21,010 t CO2-eq and a Population group effect of 407,424 t CO2-eq. As the population is constantly changing, the variation in CO2 emissions follows the instability of the electricity mix and the economic situation. The same pattern applies for the 2006-2010 and 2010-2015 periods. A total decrease of -157,134 t CO2-eq can be observed between 2006-2010 which is due to a Technology effect of -198,279 t CO2-eq, an Affluence group effect of -67,052 t CO2-eq and a Population group effect of 108,198 t CO2eq. This is followed by an upward trend for 2010-2015 with an observable peak of increase for 2011. The total variation in CO2 emissions for 2010-2015 amounts to 454,800 t CO2-eq with a contribution of 312,629 t CO2-eq for the Technology, -88,354 t CO2-eq for the Affluence and 230,525 t CO2-eq for the Population.

In conclusion, fossil fuels are playing an increasingly important role in Madagascar's electricity mix from 1990 to 2015. As a result, CO2 emissions increase. However, the country's socio-economic situation prevents CO2 emissions from rising further: Madagascar is a developing country with a low GDP growth rate. As electricity consumption is correlated to GDP [START_REF] Cream | Etude sur l'énergie à Madagascar[END_REF], this situation limits the country's electricity consumption and production and therefore the environmental impact of the electricity sector. Furthermore, since the electrification rate is low for the 25 years studied, population growth is not accompanied by an extension of the electricity distribution network. The interpretation of the results also revealed that the evolution of GDP in relation to the electrified population is mainly correlated with the evolution of GDP per capita. The evolution of the electrification rate did not have a significant influence on the evolution of Madagascar's level of development during the period studied.

Energy Scenarios

Hypothesis: Based on these results and taking into account the current energy policies in Madagascar, prospective scenarios can be modelled on the basis of the qualitative and quantitative targets of energy policies. Three scenarios are therefore depicted: the BAU scenario, the NPE scenario and the IEM scenario.

The BAU scenario, also known as the reference scenario, is based on a projection of the different effects obtained by the decomposition method according to their past trend. In other words, it is a scenario without any particular intervention. The forecast is performed by extrapolating the average trend of each effect that characterizes the evolution of the CO2 emission.

The NPE Scenario is defined by the quantitative and qualitative objectives of the new energy policy established in 2015, for the 2015-2030 period [START_REF]Ministère de l'Energie et des Hydrocarbures[END_REF]. The targets set are focused on the following parameters: increasing the electricity generation, with an ambition to produce 7900 GWh by 2030, raising the country's electrification rate by up to 70% in 2030, increasing the contribution of renewable energies in 2030 (75% hydropower, 5% solar energy, 5% wind power, 15% of fossil fuels). At a middle level, in 2023, additional criteria are considered, through the performance contract presented by the Ministry of Energy and Hydrocarbons: a twofold production of electricity compared to the reference year, which we have taken as 2015 and an electrification rate of 50% [START_REF]Ministère de l'Energie de l'Eau et des Hydrocarbures (MEEH)[END_REF]. The remaining parameters are based on the BAU scenario.

The IEM scenario allowed us to translate the ambitions of the Malagasy government into a scenario. The initial difficulty is that this program is currently being developed in the form of a national vision. No quantified targets have so far been proposed. Our prospective exercise focused on translating this ambition into quantitative constraints or objectives. However, it is important to note that the President Andry Rajoelina's initial objective have been adjusted from five to ten years. This choice is a direct result of the exploratory analysis of data from developing and emerging countries. This hypothesis will be discussed in the results section. We have thus selected some parameters that make it possible to characterize the current socioeconomic situation of a country. We have then selected a list of developing and emerging countries. Our methodology, therefore, consists in classifying these countries first, identifying the group to which Madagascar belongs, then seeing to which cluster the country would be most likely to migrate over time. Thus, it is the characteristics of the paragon of the cluster that will be attributed to Madagascar by 2030.

The technique proposed in this paper is the one initially used by Lebart et al. [START_REF] Lebart | Statistique exploratoire multidimensionnelle[END_REF]. This exploratory data analysis approach will combine on the one hand the principal component analysis that allows us to synthesize the information contained in a data set. Then, besides, an ascending hierarchical classification is performed, the partition of which is optimized by the K-means algorithm. The assumptions of the IEM scenario take into account the increase in three main elements: population, electrification rate, and energy intensity. Regarding the electrification rate, the evolution is similar to that of the NPE from 2019 to 2023. However, this rate will not exceed 60% by 2030. The analysis of clusters defines the evolution of energy intensity.

Scenarios discussion:

The driving factors in the three scenarios, including electricity consumption, are calculated based on the previous assumptions. As seen in Figure 5, the BAU scenario estimates electricity consumption at 1372 GWh by 2030, taking into account the trend observed over the period 1990-2015. These results confirm that the consideration of a BAU scenario over the long term cannot reflect the evolution of electricity demand. Indeed, according to the data observed in 2017 by JIRAMA, electricity consumption already reaches 1.146 TWh [START_REF] Jirama | Statistiques de Ventes , Abonnés et Productions JIRAMA -2017 provisoire[END_REF] compared to 1.027 TWh for the BAU projection. So, noteworthy, the importance of the BAU scenario is that it serves as a basis for comparing and interpreting scenarios established in terms of political ambitions.

The NPE scenario projects an electricity consumption of 6193 GWh by 2030. The NPE electricity consumption projection focuses on the importance given to the exponential increase in the number of subscribers due to the electrification improvement objectives. As population and economic growth are at the same level as those of the BAU, the increasing electrification rate to 70% raises electricity demand to 4.5 times higher than that of the BAU scenario.

For the IEM Scenario, the initial dataset of 75 countries and 9 variables (one of which is qualitative) was reduced to 72. Indeed, the particular cases of India, China and Russia were removed because their situation and projection did not allow a good observation of the rest of the data sample. Details of the variables used in this analysis are provided in Table 3. Three categories are also added as a qualitative parameter: UM, LM, L respectively for Upper medium, Lower medium, low country. As shown, in Figure 5, the classification made on individuals reveals 3 clusters. Each country is represented on the graph by its ISO alpha-3 code. Madagascar belongs according to the 2016 indicators to Cluster 1 in green (MDG at the bottom left of the figure), given the position of the barycenters of each cluster, a migration to Cluster 3 seems to be the most likely. The cluster 1 is made of individuals such as ETH, HTI, MMR, NER and YEM. This group is mainly characterized by low value for all the freedom index. Low values characterize the cluster for a large number of variables. Electricity consumption per capita is around 250 kWh in the category compared to an overall mean of all countries of around 1517 kWh. The observation is the same for the electrification rate, which does not exceed 52% compared to 81% of the overall mean. Also, it can be noted that Madagascar is still far from the mean trend of Cluster 1: the country barely reaches 42 kWh of electricity per capita for an electrification rate not exceeding 16%. This clearly indicates that Madagascar's fundamental transformations in this scenario, even if ambitious, will not change its cluster in the first place. Instead, it will be a matter of getting closer to the average trends of Cluster 1. The cluster 3 made of individuals such as BGR, CRI, HRV, MUS, MNE and ROM. This group is characterized by: • high values for the variables H.FI, Eco.FI, HDI, Perso.FI, ELEC.Sh, GDP.CAP and ELEC.CONS (variables are sorted from the strongest). • low values for the variables CO2.ELEC and POP (variables are sorted from the weakest). The paragon of cluster 3 is Dominica. That means that if Madagascar's evolution tends towards group 3, the country's common characteristics would be closer to those of Dominica. Thus, the characteristics of the latter in terms of electrification and energy intensity have been taken as an objective for 2030. The cluster 2 is not considered for the scenario as it is mainly defined by high values for energy and economic development. And it has low value for all the freedom index. The choice to limit the increase in the electrification rate to 10% over the period 2023-2030 is enlightened by the analysis of the data from the cluster 3 countries. Indeed, most of the countries in this group have experienced an improvement in the electrification of their territory over the past twenty years. However, this rate never exceeds 10% per 10-year period. As shown in Figure 6, the ambitions of the IEM scenario are reflected in an increase in the logarithmic trajectory of electricity demand that differs significantly from the other two scenarios (NPE and BAU). Indeed, we can see that the consumption in this case if it amounts to nearly 54.7 TWh which is almost thirty-three times higher than the consumption obtained in the NPE scenario. Although the needs obtained seem exorbitant given current values, the territory has a hydroelectric power generation capacity of around 180,000 GWh according to recent prospecting studies. The question is, therefore, about proposing a financial engineering plan which aims at establishing a closer link-up between public and private financing to encourage the development of renewables power plant on the territory. According to the country's observations over the last twenty years, this issue of financing and securing electricity production is the major challenge for Madagascar's economic development for the next years. The evaluation of the CO2 emission from the Malagasy electricity sector for the period 2015-2030 is then carried out by the combined Kaya-LMDI method (Table 2). Figure 7 shows the CO2 emissions of electricity production in Madagascar under the BAU and NPE scenarios. In the BAU scenario, the aggregated CO2 emissions predicted to be 899,243 t CO2-eq, based at the year 1990, representing a 54% increase over 2014 and a 31% over 2015. The electrical emission rate value in 2030 equals 440 g CO2-eq/kWh, representing a 27% decrease compared to 2015. This reduction is a consequence of considering the average trend of the last 25 years: the year 2015 shows a peak in CO2 emission which, according to the BAU scenario, reaches the average trend from 2016 onwards, which leads to a reduction in CO2 emission. The Population group and the Technology group effects increase by 834,842 t CO2eq and 257,584 t CO2-eq respectively, whereas, the Affluence group effect decreases CO2 emissions by -193,174 t CO2-eq. This is an extension of the past situation: population growth and the electricity mix favour an increase in CO2 emissions while the effect of the electrification rate and the level of development reduce it. The CO2 emissions in the NPE scenario are considerably higher than that in the BAU scenario. The total aggregated emissions value reaches 1,677 kt CO2-eq, based at the year 1990. This is mainly due to a positive Population group effect of 2,900 kt CO2-eq and a negative Technology effect of -1,134 kt CO2-eq. However, the NPE scenario estimates the electrical emission rate value at 242 g CO2-eq/kWh by 2030. These results support those resulting from the breakdown of the NPE scenario for 2015-2030: the Malagasy government's energy policy, which promotes the use of renewable energies, has reduced the CO2 emission factor to 45% less than the BAU scenario. The NPE is therefore achieving the emission factor reduction targets set at COP21 [START_REF]Republic of Madagascar[END_REF]. As stated before, energy consumption and GDP nexus in Madagascar has been established in literature [START_REF] Cream | Etude sur l'énergie à Madagascar[END_REF]. However, several researches have reported the relevance of political stability on economic growth. [START_REF] Tang | The impacts of tourism, energy consumption and political instability on economic growth in the MENA countries[END_REF] studied a panel of Middle East and North African region, in order to analyze the parameters that influence economic growth. It has been shown that tourism and energy consumption have positive influence in economic growth, whereas political instability impedes the economic and development growth. Authors in [START_REF] Tabassam | Nexus between Political Instability and Economic Growth in Pakistan[END_REF] have used ARCH and GARCH model to examine the political uncertainty on the economic growth with parameters as election, regime and strikes. They conclude that the short-term change in the political targets of a territory demonstrates political instability. Authors in [START_REF] Gurgul | Political instability and economic growth: Evidence from two decades of transition in CEE[END_REF] studied the transition period of 10 CEE countries, and have stated that the causal dependence between economic growth and political instability is unidirectional, from political instability to growth rate. Effects of political instability has been examined by [START_REF] Aisen | How does political instability affect economic growth?[END_REF], in order to identify the key drivers from political instability to economic growth. The variables included amongst others trade, the economic freedom index, GDP per capita, and polity scale. As known, Madagascar has also witnessed some political instability in 1992, 2002 and 2009. The outcomes in this study provide the basis for studying the impacts of political instability on economic growth. However, it can already be observed that the periods of strikes mentioned above in the country have an impact on electricity consumption, which is marked by shifts in consumption Figure 1.b. Over the past 30 years, Madagascar has been the scene of three "coups d'état" in 1992, 2002 and 2009. These instabilities have a substantial impact on the economy. As shown in Figure 1-b, for the three years concerned, the impact on GDP growth is well illustrated. This effect is also sharply observed in electricity consumption. The decrease in consumption is very pronounced. This change is particularly significant for small and medium-sized enterprises.

CONCLUSION

The economic growth of developing countries must meet a dual challenge of transforming and implementing change while securing the transition. These changes concern an increase in electricity demand and, at the same time, the control of CO2 emissions. Embarking on the path of the energy transition is today an essential step in the emergence of Madagascar's territory. The present work used a combined Kaya -LMDI method to decompose Madagascar's electricity consumption from 1990 to 2015. It is well known that electricity generation is a significant component of GHG emissions. At the end of this first stage, this enabled us to define three energy scenarios, two of which are directly inspired by national strategies.

The LCA approach allows identifying the CO2 emissions level through the studied period. This study focused on GWP but over indicators can be decomposed. The decomposition of electricity consumption highlights that the increase in CO2 emissions is mainly due to the increase of the share of fossil fuels in the electricity mix and population growth. Madagascar's socio-economic situation, characterized by a low electrification rate and a low level of development, prevents CO2 emissions from increasing further. Between 1990 and 2015, GDP did not increase at a constant rate. This period is systematically marked by declines in growth due to political crises. The decomposition shows the impact of the percentage of the population electrified on these emissions. This result is particularly crucial given national ambitions to increase access to electricity from 16 to 50% by 2030. The constancy of the population's energy intensity means that the increase in CO2 emissions can be explained more by an increasingly carbon-intensive electricity mix. In 2017, for the first time in its history, electricity production depended more on fossil fuels than on renewable resources. The prospective approach allowed us to define three scenarios.

The NPE scenario estimates a CO2 emissions variation of 1,677 kt CO2-eq by 2030, which is about twice the BAU scenario. Increasing the electrification rate to 70% has considerably increased the total CO2 emission. However, the NPE scenario estimates the electrical emission rate value at 242 g CO2-eq/kWh by 2030, 45% less than the BAU scenario, which is the result of an electricity mix dominated by renewable energies. The results of the IEM scenario, which corresponds to the most ambitious vision, highlight an energy requirement that takes on very high values until it reaches more than 54 TWh. This highlights the difficulty of achieving the target over a period as long as 2019-2023. To sum up, the Malagasy economy must initiate its transition to a better energy-efficiency status and reduce its import of fossil fuels for electricity generation. At the same time, increasing access to electricity requires that we consider controlling household energy intensity in order to mitigate the increase in total electricity production.

Figure 1 .

 1 Figure 1. (a) -Electricity mix in 2017, (b) -Economic and electricity consumption evolution, (c) -Electricity generation

Figure 2 .

 2 Figure 2. (a) Electricity production by fuel source (b) Installed capacity for each technology from 1990 to 2015 (Data from[START_REF] Jirama | Evolution annuelle des productions, ventes et nombre d'abonnes 1987-2015[END_REF],[START_REF] Ore | Synoptique Parc de Production[END_REF]).

Figure 3 .

 3 Figure 3. Evolution of total CO2-eq emissions and emission rate per kilowatt-hour due to electricity sector between 1990-2015

Figure 4 .

 4 Figure 4. Additive decomposition results of the electricity-related CO2 emissions of Madagascar, based on the reference year 1990

Figure 5 .

 5 Figure 5. Hierarchical clustering projected on first two components.

Figure 6 .

 6 Figure 6. Electricity consumption results from the BAU, NPE and IEM scenarios

Figure 7 .

 7 Figure 7. Decomposition of BAU and NPE scenarios

Table 1 .

 1 Definition of LMDI effects

	Group	Effect		Notation Variable Significance
	Technology Emission-factor effect	Cef		Effect of the CO2 emission
						factor for each type of energy
						source
		Energy-mix effect		Cem		Effect of the share of each
						energy source in electricity
						production
		Energy-loss effect		Cel		Effect of energy losses in
						electricity distribution
	Affluence	Energy-intensity effect Cei	I1=	Effect	of	electricity
						consumption per GDP
		Economic-related		Ce	I2=	Effect of economic growth in
		electrification	rate			relation to the electrification
		effect				rate
	Population Electrification	rate	Cer	I3=	Effect of changes in the
		effect				electrification rate
		Population effect		Cpop	P	Effect due to the population
						growth

Table 2 .

 2

	LMDI decomposition formulas
	Additive decomposition	Multiplicative decomposition
	Change scheme

Table 3 .

 3 Summary of variables used for PCA based clustering

	Variables	Unity	Designation
	Country		72 countries designed by iso-code alpha 3
	HDI	[-]	Human Development Index
	POP	[-]	Population, in the base year of 2016
	GDP.CAP	[$/capita]	Gross domestic product per capita, 2016
	ELEC.Sh	[%]	Electrification rate, 2016
	ELEC.CONS [kWh/capita] Electricity consumption per capita, 2014
	CO2.ELEC	[kt CO2]	CO2 emissions due to electricity production, 2014
	H.FI	[-]	Human Freedom Index, 2016
	Eco.FI	[-]	Economic Freedom Index, 2016
	Perso.FI	[-]	Personal Freedom Index, 2016
	Category	[-]	Classification of countries by income , 2016:

L : Low income/ LM : Lower middle income/ UM : Upper middle income

International Monetary Fund

Jiro sy rano malagasy

ACKNOWLEDGMENT

The authors would like to thank JIRAMA for providing us with access to all multi-year power generation data for Madagascar. The finalization of this work was made possible thanks to the mobility of researchers between Reunion Island and Madagascar as part of an ERASMUS+ project.

NOMENCLATURE