Sally Ismail
email: sally.ismail@etu.utc.fr

Fabio D'andreagiovanni
email: d.andreagiovanni@hds.utc.fr

Hicham Lakhlef
email: hicham.lakhlef@hds.utc.fr

Youcef Imine
email: youcef.imine@hds.utc.fr

Recent Advances on 5G Resource Allocation Problem using PD-NOMA

Keywords: orthogonal multiple access (NOMA), 5G cellular, power allocation, resource allocation, optimization, energy-efficient, power-efficient, rate-optimal

5G is the fifth generation technology standard for cellular networks, which is already being deployed and offering many advantages for communications, such as the high degree of flexibility, the possibility of management and control of systems and resources, thus allowing to host and execute services in one or more distinct network slices, etc. Non-orthogonal multiple access (NOMA) is being considered as a key enabling technique for 5G cellular systems. The earlier generations are becoming less practical with the emergent requirements such as very high spectral efficiency, very low latency, supporting diverse quality of services (QoS). Researchers are working on different techniques to fulfill these requirements, and NOMA is one of the essentials due to its capacity to increase spectrum efficiency. NOMA schemes exploit channel gain differences to serve multiple users concurrently, by superimposing multiple users' information signals at the transmitter side. In this paper, we review, analyze and classify recent papers on resource allocation problem in 5G networks, and then we discuss the proposed solutions. In particular, the papers are classified into two main categories: power/energy-efficient and rate-optimal. For each paper, the objective, optimization method and main results are discussed.

Finally, some open challenges are highlighted.

I. INTRODUCTION

In existing cellular network systems, orthogonal multiple access (OMA) techniques have been used: frequency division multiple access (FDMA) in first generation (1G), time division multiple access (TDMA) in 2G, code division multiple access (CDMA) in 3G and Orthogonal frequency division multiple access (OFDMA) in 4G. The common between these multiplexing techniques is the orthogonality feature, which means that different users don't interfere when they are accessing the network resources. As for wireless networks such as the 5G and beyond 5G (B5G) are expected to support extremely high data rates and a very large number of users, which is not achievable using OMA. To toggle this problem, non-orthogonal multiple access (NOMA) schemes have been introduced, where multiple users are allowed to share the same resource block (time/frequency) and are alternatively separated in other domains, with a cost of additional complexity at the receivers. NOMA offers several following features: improved spectral efficiency, improved user fairness, low transmission latency, higher cell-edge throughput [START_REF] Shafi | 5G: A tutorial overview of standards, trials, challenges, deployment, and practice[END_REF], [START_REF] Maraqa | A Survey of Rate-optimal Power Domain NOMA Schemes for Enabling Technologies of Future Wireless Networks[END_REF]. Different categories of NOMA has been studied, the main two are power domain NOMA (PD-NOMA) and code domain NOMA (CD-NOMA) [START_REF] Aldababsa | A Tutorial on Nonorthogonal Multiple Access for 5G and Beyond[END_REF]. In power-domain multiplexing, different users are allocated different power coefficients according to their channel conditions. In code-domain multiplexing, different users are allocated different codes and multiplexed over the same timefrequency resources, such as multi-user shared access (MUSA) [START_REF] Yuan | Multi-User Shared Access for 5G[END_REF], sparse code multiple access (SCMA) [START_REF] Nikopour | Sparse code multiple access[END_REF], and low-density spreading (LDS) [START_REF] Hoshyar | Novel low-density signature for synchronous CDMA systems over AWGN channel[END_REF]. On the other hand, resource allocation problem is widely studied in cellular networks. Given the available resources (time, frequency,...), the way you choose to assign these resources to the users will heavily affect the performance of your system. Resource allocation is a key area to achieve the full potential of communication systems with NOMA [START_REF] Zhu | On Optimal Power Allocation for Downlink Non-Orthogonal Multiple Access Systems[END_REF]. With the objective of achieving a certain goal, the resource allocation can vary between: power allocation, channel assignment, transmission scheduling, user selection and joint schemes of these techniques [START_REF] Wei | Power-Efficient Resource Allocation for MC-NOMA with Statistical Channel State Information[END_REF]. In the current work, we will be focusing on recent papers handling the resource allocation problem for PD-NOMA. The rest of the paper is organized as follows: in section II, we provide a brief description of NOMA. Section III surveys the papers on resource allocation. In section IV, a comparative table is provided in addition to some open challenges. Finally, section V concludes the paper.

II. BACKGROUND

PD-NOMA was first introduced to improve the spectral efficiency of wireless networks by allowing multiple users to share both time and frequency domains, while allocating different power levels to each user at the transmitter side, whereas all the receivers (i.e. users) receive their desired signals along with the interference caused by the messages of other users. To obtain the desired signal, each successive interference cancellation (SIC) receiver first decodes all dominant interference (i.e. the ones that are stronger then the receiver's desired signal) and then subtracts them from the superposed signal [START_REF] Md Shipon | Dynamic user clustering and power allocation for uplink and downlink non-orthogonal multiple access (NOMA) systems[END_REF]. To clarify more the idea, let's take the case of two-user down-link (DL) NOMA, i.e. the case where a base station (BS) is sending signals to two users, as shown in figure 1.

In typical PD-NOMA scenario, at the transmitter side, both signals of the two users are superposed upon each other with different power allocation. Usually, the signal of the user with strong channel gain will be allocated with the lower power, and the one of the user with weak channel gain will be allocated with the higher power. Then, at the strong user's receiver, the signal of the weak user, having power higher than his own, will have a high signal-to-noise ratio (SNR), meaning that the strong user can successfully decode and subtract the signal of the weak user before decoding his own signal (i.e.performing SIC). For the weak user side, at his receiver, the signal of the strong user, having less power than his own, will be considered as noise when decoding his signal, so he will be decoding his signal without SIC. Figure 1. An illustration of a two-user down-link PD-NOMA scheme [START_REF] Yuan | Multi-User Shared Access for 5G[END_REF] For the two-user up-link (UL) scenario, where the two users are sending their signal to the BS, the signals are also sent simultaneously with different transmit power. The power share of each user is again based on their channel conditions with respect to the BS. Then, after the transmission, the BS receives the superposed signal containing both users' signal, and starts by decoding the strongest signal, subtracting it from the superposed signal and then decodes the other signal.

III. RESOURCE ALLOCATION

Resource allocation is a key area to achieve the full potential of communication systems with NOMA.

Based on what's explained earlier about PD-NOMA, we can see that power allocation and grouping or clustering of users are two important, but not the only, steps for NOMA schemes. It is also worth mentioning that the optimal user clustering usually requires an exhaustive search and may not always be affordable, and for this reason, researchers resort to low complexity solutions to solve the problem through heuristic algorithms which may lead to unpredictable results, or alternatively they use [START_REF] Maraqa | A Survey of Rate-optimal Power Domain NOMA Schemes for Enabling Technologies of Future Wireless Networks[END_REF]:

• The monotonic optimization approach: the monotonic nature of the objective function and constraints allows researchers to limit the exhaustive search to a much smaller region of the feasible set; this accordingly simplifies the problem and can lead to the optimal solution but still with relatively high complexity. • The combinatorial relaxation approach: the selection and association of a particular user to a particular cluster is realized by a binary variable; by relaxing this variable to a continuous one, the original non-convex problem can be transformed to a convex one and solved optimally through classic Lagrangian dual method, but this relaxation results in a performance gap between the original problem and the relaxed one. • The matching theory approach: which is often utilized to deal with problems that have beneficial relationship parties.

• The game theory approaches: which transform the clustering problem into a game. The choice of which resource allocation to use depends on the type of communication system that will be applied. Therefore, it is necessary to choose the optimal strategy according to the specific scenario [START_REF] Wei | Power-Efficient Resource Allocation for MC-NOMA with Statistical Channel State Information[END_REF]. In addition, the performance analysis metrics changes depending on the application, it could be the bit error rate (BER), outage probability, etc... In the current work, the resource allocation will be classified into two main categories:

• Power-efficient or Energy-efficient resource allocation whose goal is to find an optimal scenario in which the energy generated for communication is managed sustainability. Hence, the effect on the environment, economy, living beings, etc is minimized [START_REF] Basnayake | A New Green Prospective of Nonorthogonal Multiple Access (NOMA) for 5G[END_REF]. • Rate-optimal resource allocation whose goal is to optimize the achievable rate and thus the spectral efficiency.

A. Power/Energy-efficient resource allocation

While some papers addressed the power-efficient NOMA and others the energy-efficient NOMA problems, both categories reside under the goal of achieving an environment friendly technology.

1) Power-efficient resource allocation: The authors in [START_REF] Dawadi | Powerefficient resource allocation in NOMA virtualized wireless networks[END_REF] address the power-efficient resource allocation problem in virtualized wireless networks (VWNs). Virtualization of a wireless network is a promising technique, in which the physical wireless infrastructure is shared among multiple service providers, also called slices. Aiming to increase the spectrum and infrastructure efficiency, the main issue in a VWN is to prevent the harmful effects of users of one slice to the users of other slices. The authors wanted to see if NOMA could help improving the network performance of a VWN in terms of power efficiency. So, they compared two transmission modes: NOMA where the whole frequency band of interest is shared among users, and OFDMA where the specific bandwidth is divided into a set of sub-carriers, and each sub-carrier is allocated to a maximum of one user at a time. The objective was to minimize the total transmit power in a VWN while maintaining the minimum required capacity for each slice. The formulated problem, for both cases, was non-convex which is challenging to solve. For NOMA, an efficient algorithm was proposed to solve the problem, where complementary geometric programming and variable relaxation were used to convert the problem into the geometric programming. As for OFDMA, relaxation technique was applied and then Lagrange dual function was used to solve the problem. The simulation results demonstrated that NOMA is more power-efficient than OFDMA in various scenarios, and specifically, the power efficiency was improved by up to 45-54% with NOMA as compared to OFDMA. In [START_REF] Wei | Power-Efficient Resource Allocation for MC-NOMA with Statistical Channel State Information[END_REF], the authors proposed power-efficient resource allocation for multi-carrier (MC) NOMA systems with quality of service (QoS) constraints under statistical channel state information at the transmitter (CSIT). In conventional multi-carrier (MC) systems, a given radio frequency band is divided into multiple sub-carriers (i.e. multicarrier) and each sub-carrier is allocated to at most one user. For NOMA systems, each sub-carrier could be assigned to more than one user. Usually when dealing with resource allocation problems, perfect CSIT is assumed, and this is the case for a vast range of papers on resource allocation, but in practice, perfect CSIT is hard to obtain. So the authors considered an imperfect CSIT scenario, and due to this fact, a SIC policy taking consideration of QoS requirements is proposed, which only allows one user to perform SIC. And based on the considered policy, only two users per sub-carrier were allowed, and the resource allocation was formulated for MC-NOMA systems to minimize the total transmit power. The formulated problem is a mixed combinatorial non-convex problem, and a sub-optimal solution is proposed to solve the power allocation and the user scheduling problems separately.

• For the power allocation problem: the user scheduling policy is supposed given, and the power allocation is performed on each sub-carrier independently. The selection of optimal power allocation in incorporates the SIC decoding policy implicitly to achieve minimum power consumption.

• For the user scheduling problem: an algorithm was proposed based on the agglomerative hierarchical clustering.

As for the results, the proposed user scheduling method provides a significant power saving compared to the random scheduling, and achieves a performance close to the full search scheduling. It achieves a close-to-optimal performance and significantly outperforms a conventional orthogonal multiple access (OMA) scheme.

2) Energy-efficient resource allocation: Energy-efficient NOMA will be a key ingredient in the upcoming 5G mobile network design to cater for the goal of achieving a low cost, self-sustaining, environment-friendly wireless network. Mathematically, energy efficiency is defined as the amount of information that can be reliably transmitted per Joule of consumed energy; which is denoted using units of bits-per-Joule [START_REF] Basnayake | A New Green Prospective of Nonorthogonal Multiple Access (NOMA) for 5G[END_REF]. In [START_REF] Zhang | Energy-efficient user scheduling and power allocation for NOMA wireless networks[END_REF], the authors proposed an energy-efficient user scheduling or pairing and power allocation for a DL of NOMA wireless network. The problem was formulated with an objective of maximizing the total energy efficiency of the system while respecting the total power constraint. The authors started by finding the optimal user scheduling, and then, based on that, they searched for the optimal power allocation. For the channel pairing, the authors considered the case of two users per sub-channel. They chose to use the channel state sorting-pairing algorithm, which pairs the best channel condition user with the worst channel condition user, with the goal of increasing the system sum rate and decreasing the interference between users of the same sub-channel. After the user pairing is achieved, the BS needs to perform two stages power allocation: power allocation for different subchannels, and power allocation for different users in each subchannel. For the first stage, the author considered having a subchannel with a fixed power allocation and he finds the optimal power distribution between the two users that maximizes the sub-channel transmit rate. As for the second stage, and due to the complexity of the problem (non-convex problem), they transformed it into Dinkelbach representation and proposed an algorithm to find the optimal solution. Finally, simulations of the proposed solution were compared to OFDM scheme and it was proven that the proposed NOMA solution is more energyefficient than OFDM. In [START_REF] Wu | Spectral and energy efficient wireless powered IoT networks: NOMA or TDMA[END_REF], the authors did a comparison between NOMA and TDMA to find out which scheme is more spectral and energy efficient for wireless powered IoT networks. In these kind of networks, multiple energy-limited devices first harvest energy in the DL and then transmit information in the UL (see figure 2).

Figure 2. System model of a wireless powered IoT network [START_REF] Wu | Spectral and energy efficient wireless powered IoT networks: NOMA or TDMA[END_REF] They started by deriving the optimal DL time and power allocation for the spectral efficiency maximization in both cases, and then calculating the corresponding energy consumption and spectral efficiency at these optimal values. For TDMA, the formulated problem was a convex optimization problem satisfying Slater's condition, so the optimal solution was found using Lagrange dual method. The NOMA formulation also had the same structure, so the optimal solution was the same with difference in the constant parts. And then, for these optimal values, energy consumption and spectral efficiency were calculated. In conclusion, it was found that NOMA needs more (or equal) DL time than TDMA, while consuming more (or equal) energy and achieving less spectral efficiency. In [START_REF] Zeb | NOMA Enhanced Backscatter Communication for Green IoT Networks[END_REF], the authors proposed using NOMA in backsacatter communication (BackCom) for green IoT networks. BackCom is a key enabler technique for wireless power communication networks (WPCN), where power is transmitted to IoT devices through radio frequency (RF) signals. A BackCom system is composed of backscatter nodes (BNs) and a reader that transmits a single-tone sinusoidal RF signal. Upon receiving the reader RF signal, the BN reflects it back to the reader by impedance mismatching at the antenna input side. In a BackCom system, the reader may receive reflected signals from multiple BNs during the UL data transmission, which can lead to interference at the receiver. So, to tackle this problem, the authors employed a hybrid TDMA scheme with PD-NOMA for UL communication. A sensor field of K IoT sensors (BNs) and a reader are considered. The reader transmits power to the sensors located in the reader annular coverage zone, and than these sensors can backscatter the incident wave to transmit data back to the reader. After receiving all the backscattered signals from the BNs, the reader transmits the decoded data to a cellular BS. To implement the proposed PD-NOMA, the reader splits the annular coverage zone into N number of virtual subregions as shown in fig. 4, and sensors from different subregions are multiplexed together in a single mini-timeslot of T seconds. It should be mentioned that since the BNs are passive devices, and cannot actively alter the transmitted power as needed in conventional PD-NOMA systems, reflection coefficiens are used instead such as to give the best system performance. To test the proposed scheme, it is compared to an OMAbased TDMA scheme where each sensor is allocated a single dedicated mini-timeslot for transmission. The performance is rated in terms of system outage probability and throughput. The results showed that the proposed PD-NOMA scheme outperforms the conventional TDMA scheme and increases the system throughput.

B. Rate-optimal resource allocation

In [START_REF] Di | Sub-channel assignment, power allocation, and user scheduling for non-orthogonal multiple access networks[END_REF], the authors studied the resource allocation and user scheduling problem for a DL NOMA network. They intended to jointly optimize the sub-channel assignment and power allocation to maximize the weighted total sum-rate while taking into account user fairness. User fairness is considered in terms of capacity, so a weight factor is introduced for each user to adjust its priority when allocating resources. The formulated problem was proven to be NP-hard, so, in order to solve it, they proposed to decouple the power allocation and sub-channel sub-problems and proposed an iterative algorithm in which the power and sub-channel allocations are performed in an iterative way to obtain a joint solution of the problem:

• To solve the power allocation problem, they supposed that the sub-channel allocation is settled, so the problem could be formulated as geometric programming and solved using interior point methods. • For the sub-channel allocation sub-problem, it was formulated as many-to-many matching game with externalities, and to solve it, two matching algorithms were introduced.

With the above stated, the solution of the problem is the following: It starts with an initialization phase were the BS allocates equal power to each user over each sub-channel and the weight factor for each user is set as inversely proportional to the average rate of user in previous time slots. In the resource allocation phase, sub-channel assignment and power allocation are iteratively performed as to obtain a joint solution. The simulations' results proved that the proposed algorithm outperformed traditional OFDMA scheme in terms of both total sum-rate and user fairness. In [START_REF] Sun | Optimal joint power and subcarrier allocation for MC-NOMA systems[END_REF], the authors investigated the resource allocation for MC-NOMA systems. In this paper, the authors aim to treat the MC-NOMA systems, so they tried and managed to find a resource allocation (optimal joint power and sub-carrier allocation policy) algorithm for the weighted system throughput problem of MC-NOMA systems. The formulated problem was a mixed combinatorial non-convex problem, and it was optimally solved using monotonic optimization based resource allocation algorithm. The authors also proposed a low-complexity sub-optimal algorithm that has a polynomial time complexity, to balance between complexity and system performance. Finally, the simulation results proved that the proposed algorithm outperforms conventional MC-OMA in terms of system performance, also, the sub-optimal algorithm achieves similar performance as the optimal one. In [START_REF] Salaun | Weighted sum-rate maximization in multi-carrier NOMA with cellular power constraint[END_REF], the authors treated the problem of weighted sumrate (WSR) maximization in a DL MC-NOMA with cellular power constraint. This problem is important due to the fact that it offers trade-offs between sum-rate performance and user fairness. For these problems, two types of power constraints are considered in the literature: cellular power constraint and individual power constraint. In this paper the authors only considered the cellular power constraint, which represent the total available power at the BS. This is a joint sub-carrier and power allocation optimization problem, and to solve it, the authors divided the problem into two polynomial time solvable sub-problems to take advantage of each one's particular structure, and find an efficient solution, and to finally solve the initial problem by combining the two solutions. The two sub-problems are:

• The single-carrier user selection (SCUS): only one subcarrier is considered with its power budget, and the An efficient heuristic joint subcarrier and power allocation scheme (called JSPA), was designed by combining the solution of SCUS and MCPC, as presented in figure 3. The performance of the algorithm was evaluated by numerical simulations and it was compared to with the near-optimal high complexity benchmark scheme lagrangian duality and dynamic programming (LDDP) introduced in [START_REF] Lei | Power and channel allocation for non-orthogonal multiple access in 5G systems: Tractability and computation[END_REF], and to fractional transmit power control (FTPC) with greedy sub-carrier allocation considered in [START_REF] Lei | Power and channel allocation for non-orthogonal multiple access in 5G systems: Tractability and computation[END_REF]- [START_REF] Parida | Power allocation in OFDM based NOMA systems: A DC programming approach[END_REF]. The results showed that JSPA achieves near-optimal WSR and user fairness.

IV. DISCUSSION AND OPEN CHALLENGES

A. Discussion

After reviewing all these papers, we saw that each one of them exploited the available resources to achieve a certain goal, and the chosen strategy depends on the situation. In most of them, the proposed solution or algorithm outperformed Figure 3. Overview of JSPA [START_REF] Salaun | Weighted sum-rate maximization in multi-carrier NOMA with cellular power constraint[END_REF] the exiting solutions, but there can be situations, such as in [START_REF] Wu | Spectral and energy efficient wireless powered IoT networks: NOMA or TDMA[END_REF], where NOMA was not the preferred candidate, which highlights the importance of the good management of resources depending on the offered service. In order to be able to compare the differences between the papers, we grouped the information in table I, where, for each paper, the type of the link (UL or DL), and the resources that has been treated are specified, the optimization method used to solve the problem is stated along with the main results of the paper.

B. Open challenges

In this paper, we only addressed the resource allocation problem for PD-NOMA, but it should be mentioned that 5G networks are not only based on PD-NOMA. Several technologies are being studied in order to be integrated with NOMA to fulfill the requirements of 5G and beyond 5G networks (B5G), such as MIMO and massive MIMO (mMIMO), millimeter wave (mmWave) communications, coordinated multi-point (CoMP) transmission and reception, cooperative communications, etc [START_REF] Maraqa | A Survey of Rate-optimal Power Domain NOMA Schemes for Enabling Technologies of Future Wireless Networks[END_REF]. The integration of all these different technologies is the topic of research for 5G and B5G networks, and lots of these fields are new and not yet well explored, so there exist a huge number of research direction. In addition, the service-oriented view envisions 5G systems catering to a wide range of services differing in their requirements and types of devices, and going beyond the traditional human-type communications to include various types of machine-type communications. This requires the network to take different forms depending on the service in question, leading naturally to the notion of slicing the network on a per-service basis. Realizing this service-oriented view requires a radical rethink of the mobile network architecture to turn it into a more flexible and programmable fabric, leveraging technologies like software defined networking (SDN), that relies on the separation between the control plane and the data plane to enhances data forwarding efficiency and network programmability, and NFV, that enables each network function, named virtual network function (VNF), to run on generalpurpose hardware to alleviate deployment costs [START_REF] Su | Resource allocation for network slicing in 5G telecommunication networks: A survey of principles and models[END_REF]. Those two last technologies can be used to simultaneously provide a multitude of diverse services over a common underlying physical infrastructure [START_REF] Foukas | Network Slicing in 5G: Survey and Challenges[END_REF].

V. CONCLUSION

There is a major difficulty in designing of efficient resource allocation in 5G networks. In this paper, resource allocation problem using PD-NOMA literature was reviewed, discussed, and classified into two main categories: power/energy-efficient, and rate-optimal. In particular, the paper investigated the objective, optimization methods and the main results of each paper. Furthermore, our paper explores recent solutions on resource allocation problem and provides a comparative table of these solutions. Finally, some open challenges are highlighted, and the mentioned titles could be thoroughly investigated for future research.

978-1-7281-5628-6/20/$31.00 ©2020 IEEE

ACKNOWLEDGEMENT

This work is carried out in the framework of the Labex MS2T, which was funded by the French Government, through the program "Investments for the future managed by the National Agency for Research (Reference ANR-11-IDEX-0004-02)".

Table I COMPARATIVE TABLE OF THE ANALYZED PAPERS

Reference Classification

Link Type Resources Optimization Method Main Finding(s) [START_REF] Dawadi | Powerefficient resource allocation in NOMA virtualized wireless networks[END_REF] Power-efficient DL joint power and sub-carrier allocation

Iterative algorithm based on CGP

The proposed algorithm is more power-efficient than OFDMA [START_REF] Wei | Power-Efficient Resource Allocation for MC-NOMA with Statistical Channel State Information[END_REF] Power-efficient DL Power allocation and user scheduling Sub-optimal solution that treats the two problems separately Close-to-optimal performance and the solution outperforms a conventional orthogonal multiple access (OMA) scheme [START_REF] Zhang | Energy-efficient user scheduling and power allocation for NOMA wireless networks[END_REF] Energyefficient The proposed solution outperformed TDMA scheme and increased the system throughput. [START_REF] Di | Sub-channel assignment, power allocation, and user scheduling for non-orthogonal multiple access networks[END_REF] Rate-optimal DL Joint power and sub-carrier allocation and user scheduling

Interior point method and joint subchannel and power algorithm

The proposed solution outperformed OFDMA in terms of sum-rate and user fairness [START_REF] Sun | Optimal joint power and subcarrier allocation for MC-NOMA systems[END_REF] Rate-optimal DL Joint power and sub-carrier allocation

Algorithm based on monotonic optimization and low complexity sub-optimal algorithm

The proposed algorithm outperforms MC-OMA, and the suboptimal algorithm is close to optimal [START_REF] Salaun | Weighted sum-rate maximization in multi-carrier NOMA with cellular power constraint[END_REF] Rate-optimal DL Joint power and sub-carrier allocation

Heuristic JSPA algorithm The algorithm achieved nearoptimal WSR and user fairness objective is to find the optimal power allocation satisfying the constraints. They proposed an algorithm to solve the sub-problem that is based on dynamic programming (DP). • The multi-carrier power control (MCPC): given a fixed sub-carrier allocation for all sub-carriers, the objective is to maximize the WSR. The problem is formulated as two-stage optimization problem, where in the first stage consists in optimizing the allocated power budget of each sub-carrier, and the second stage performs power allocation on each sub-carrier. To solve the problem, the authors proposed an algorithm based on gradient descent.


N Lamoureux

P Desgroux

In situ LIF and ex situ CRDS applied to NO measurement in flames: probe perturbation and absolute quantification

Flame-sampling experiments allow describing the species profiles as function of the height above the burner. However, the so-measured species profiles are perturbed by the heat losses due to the probe, and/or species losses inside the probe. In order to attempt to clarify the perturbations induced by a microprobe, NO species profiles were obtained in a stoichiometric low-pressure flame by using in situ and ex situ measurements. An unperturbed NO species profiles was measured locally by LIF. Two shapes of microprobes were used to perform ex situ measurements by CRDS after gas probe sampling. The comparison of the profiles shows that the profiles obtained after probe sampling are shifted downstream, and that the absolute mole fraction of NO may be drastically affected during the sampling.

Introduction

Gas sampling and subsequent analyses are convenient for measuring the chemical composition of gases in combustion devices. The design of the probe is one of the important factors that has to be considered for obtaining accurate measurements. One can distinguish cone probes and microprobes. Cones are characterized by a large area-ratio. Cone sampling is generally associated to molecular-beam mass spectrometry detection. Thanks to the pressure drop and the molecular beam which is generated downstream (supersonic flow), reactions are suddenly cooled down preventing the recombination of the short-lived radicals. However, one of the main drawbacks is the perturbation generated by the cone in the flame and in the probe (flow distortions, heat losses, efficiency of the quenching of chemical reactions). Comparison of experimental species profiles obtained either in situ or after probe sampling highlights spatial distortion of the species profiles, and depending on the cone, divergences of the absolute mole fractions have been observed [1,2]. To restrain the flame perturbations in counterflow diffusion flames, gases can be probed through a fused silica capillary tube. However, its introduction generates disturbance identified by comparing non intrusive measurements with those obtained after gas sampling [3].

Microprobes are routinely used to probe gas samples in flames or in Jet Stirred Reactors (JSR) for further analysis by Gas Chromatography for example. As mentioned by Heitor and Moreira [4], the sampling flow rate and the size of the suction orifice are set as a compromise between the required low pressure to achieve aerodynamic quenching of chemical reactions and the minimum flow rate for the chemical analyzers. Biordi et al. [5] showed that depending on the size of the orifice and the angle of the tip, the species profiles may be affected not only in their shape but also in their absolute mole fraction values. Kramlich and Malte [6] have modeled the pressure and temperature variations inside the microprobe by considering one-dimensional gas flow conditions. They have simulated the NO into  Corresponding author: nathalie.lamoureux@univ-lille.fr Proceedings of the European Combustion Meeting 2019 NO 2 conversion within the tip by taking into account reactions which can occur at the wall and in the gas phase. They found a good agreement with the experimental measurements of NO and NO 2 performed after gas probe sampling in a JSR apparatus. Colket et al. [7] have developed a gas-dynamic model to describe the gas flow and heat transfer inside the probe. They concluded that an aerodynamic quenching is unlikely to occur in microprobes but, chemical quenching rates may be sufficient to freeze reactions. In an atmospheric flame seeded with NO, Zabielski et al. [8] have compared the NO concentration measured in the burned gases either by in situ optical absorption or after probe sampling by chemiluminescent analyzer. No major discrepancies related to processes occurring in probes have been identified. Recently, Li et al. [9] measured NO in flames seeded with NH 3 either by using Laser Saturated Fluorescence calibrated by the NO addition method or by using a chemiluminescent analyzer after gas probe sampling. They observed a difference of NO amounts of 5-7% between the two techniques. They evoked the possible conversion of NO into NO 2 in the probe through recombination reactions in presence of O-atoms as proposed in [10]. Schoenung and Hanson [11] have compared the absolute mole fractions of CO measured either in situ by laser absorption or after gas probe sampling by using a non-dispersive infrared analyzer. Based on these measurements, they have indirectly observed some conversion of CO into CO 2 . They have analyzed the difference of the absolute mole fractions from the chemical equilibrium predictions of CO/CO 2 , and from the aerodynamic model proposed in [7]. They recommended working with reduced pressure downstream the probe (around 50 mbar).

The question of the microprobe disturbance in the flame structure analysis frequently arises. The flame temperature profile is an input data which is required for the chemical flame modeling. In order to match the experimental and the simulated species profiles, it is generally necessary to shift the temperature profiles downstream. On the contrary, such shift is not required for modeling the in situ species profiles [12]. In order to identify the perturbation generated by the microprobe, the present work aims to compare in situ and ex situ measurements of flame species profiles. NO species was chosen because it is very sensitive to possible kinetic reactions that may occur inside the probe, and it is measurable locally and after gas probe sampling.

NO species can be measured by using in situ Laser Induced Fluorescence (LIF). This well-known technique allows measuring in situ species in relative mole fraction with a high sensitivity and spatial resolution. In case of NO species, the relative mole fraction can be converted into absolute one by adding small known amounts (same order than the expected native NO) of NO in the premixed gases [13,14]. The NO standard addition method relies on the main assumption that there is no NO-reburning in the flame. This aspect was verified in particular in lean flame [14].

The line-of-sight pulsed-CRDS (Cavity Ring-Down Spectroscopy) technique has already been applied to measure NO species. In situ measurements can be perturbed by the fact that NO being a stable species present in the burned gases may persist in the cold gases surrounding the flame as shown in low-pressure flames [15]. In atmospheric flames, Sepman et al. [16] have performed NO measurements by using pulsed-CRDS. They were suffering from a strong background signal due to CO 2 and H 2 O absorption around 226 nm. Evertsen et al. [17] have quantified NO in absolute mole fraction after sampling gases at the exhaust of a diesel engine. These studies [15,17] were undertaken by probing weak absorption transitions to avoid saturation effects [18,19]. These effects will be discussed later in the paper.

The present work aims to identify the microprobe perturbation effects in the flame. Species profiles of NO in a stoichiometric low pressure flame of CH 4 /O 2 /N 2 obtained either in situ by LIF or ex situ by CRDS were compared one to each other to highlight the probe perturbation in the flame. The absolute mole fractions determined in the burned gases by two techniques are also discussed. NO-LIF was calibrated in situ in the burned gas of a stoichiometric low pressure flame by using the NO standard addition method. Results were compared to NO mole fraction measured after gas probe sampling by using pulsed-CRDS technique applied in an external absorption cell.

Experimental setup

Low pressure burner and gas supply Experiments were undertaken in a stoichiometric laminar CH 4 /O 2 /N 2 premixed flame stabilized at 5.33 kPa (40 Torr) on a 6-cm diameter bronze water-cooled McKenna burner. Volumetric flow rates (in standard (273.15K, 1.013×10 5 Pa) L/min, slpm) of the gas mixture are 0.48/0.96/3.35 for CH 4 /O 2 /N 2 respectively. An additional line with a low range Mass Flow Controller (MFC, 0.1 slpm) is connected to the premixed gases inlet of the burner to seed the mixture with known amounts of NO from a commercial preparation of 2001 ppmv NO diluted in N 2 (standard 1 with ±2% precision). The temperature profile of the flame has previously been obtained by NO-LIF thermometry [20]. Details concerning the burner and gas supply were given in [21].

The burner enclosure is equipped with different optical accesses allowing in situ laser diagnostics (LIF). For the in situ measurements, the height above the burner (HAB) is defined by the distance between the laser axis and the burner surface. For the probe sampling measurements, it is defined by the distance between the microprobe tip and the burner surface.

Probe sampling and absorption cell Gas sampling was performed through quartz microprobe in order to perform ex situ measurements of NO molecule by using pulsed-CRDS technique in an external absorption cell. Two shapes of microprobe were examined. They are designed with a tip angle extended to a cylindrical tube over a total length of 50 mm. This first section is connected to a second tube with an inner diameter of 9.8 mm. Both are opened with a 200 µm diameter orifice. The first microprobe is very sharp with an inner tip angle about 10°, and an area ratio of 255 (microprobe 1). The second one has an inner tip angle about 20°, and an area ratio of 750 (microprobe 2). The absorption cell of 39.5 cm long is ended by two mirrors. Gases are injected through a Teflon tube either from the probe gas sampling or from a commercial preparation of 10.6 ppmv NO diluted in N 2 (standard 2 with ±5% precision) as shown in Fig. 1. In order to prevent the adsorption/desorption phenomena at the wall, gases in the cell are in continuous flowing. The pressure was stabilized at the desired value by using an automatic regulating valve (Leybold, CMOVE1250). The gas temperature was controlled by a thermocouple (type K) at the inner wall of the cell.

Laser techniques

Two laser techniques were implemented in this work; in situ LIF, and ex situ pulsed-CRDS. The pulsed laser system consists of a seeded frequency-doubled Nd:YAG laser (Q-smart) pumping a dye laser (Quantel TDL90+). Wavelengths around 226 nm were provided by doubling the fundamental dye radiation (around 572 nm) and by mixing it with YAG residual radiation at 1064 nm. The spectral laser bandwidth at 226 nm was estimated to be around 0.12 cm -1 .

LIF and CRDS signals were collected through a photomultiplier tube (PMT) (Photonis, XP2020Q) and recorded on a digital oscilloscope (Lecroy HDO4000, 12bit vertical resolution, 1GHz bandwidth, 1.25 GS/s). During spectrum acquisition, laser wavelength (fundamental) was scanned at a rate equal to 0.25 pm s -1 and signals were averaged over 15 laser shots. For the species profiles measurements, signals were averaged over 300 laser pulses. Signals from the scope and the step motor of the dye laser were transferred in real time on a PC, and treated with Labview programs.

In situ LIF The laser beam was introduced unfocused parallel to the burner surface and shaped using a horizontal slit of 300 µm height in front of the burner enclosure and parallel to the burner surface. The LIF signal was collected by a two-lenses system (f=200 and f=350 mm) and focused on the entrance slit of a 0.5 m monochromator (SpectraPro 2500i). The entrance slit (550 µm width, 10 mm height) was parallel to the laser axis. The magnified image of the entrance slit in the vertical axis matches the size of the laser height in the flame, corresponding to the spatial resolution (300 µm). The output slit was adjusted to obtain a collection bandwidth of 9 nm, centered at 246 nm in order to collect the complete (0-2) vibrational band of NO.

The relative NO mole fraction profiles were obtained by prompt-LIF in the linear regime of fluorescence. The prompt-LIF signal is measured at the peak value of the temporal LIF signal. As the LIF signal is proportional to the rotational population, conversion into relative mole fraction has been completed by dividing the signal by the Boltzmann fraction f B (J",T), easily calculated using the experimental temperature data [20]. Since NO is a stable species, the relative mole fraction can be converted into absolute mole fractions by using the standard addition method [13,14]. In order to limit the impact of the temperature uncertainties on the Boltzmann factors calculation, NO was probed along the line Q 2 (26.5) (225.58 nm) for in situ LIF measurements. The spectroscopic parameters of NO proposed by Reisel et al. [22] were adopted in the present work.

Ex situ pulsed-CRDS Measurements of NO were performed in an external absorption cell. The quasi-plane cavity (39.5 cm long) is ended by two highly reflective mirrors (Layertec GmbH, 25 mm diameter, -6 m radius of curvature, reflectivity>0.993). Typically, the empty ringdown decaytime was equal to 200 ns. As shown in Fig. 1, the laser beam was shaped with a system consisting of a 100 µm pinhole aperture and two lenses (f=100 and f=175 mm) in order to match approximately the TEM 00 transverse mode of the optical cavity. At the exit of the cell, a beamsplitter allows reflecting a fraction of the light towards an ICCD camera for controlling the perfect alignment of the cavity [23]. The transmitted light was collected by a PMT in front of which a 226-Corion filter was placed, to measure the temporal CRD signal. Ringdown decaytime was determined from an exponential fitting procedure over 800 ns [20], which correspond to 4 times the off-resonance decay. By using a 12-bit vertical resolution system, the on-resonance decay as low as 130 ns (fitting period/6) can be reasonably measured [24].

The absolute species concentration of NO was calculated from the integrated spectral absorptivity determination A() by using CRDS technique according to Eq. ( 1) [START_REF] Mercier | Cavity Ring-Spectrosc. Tech. Appl[END_REF].

( ) = ∫ ( ) = ( ⁄ ) " ( ", )
Eq. ( 1) where L() are the net losses per pass along the absorption length l s (in cm), ⁄ is equal to 1.13×10 12 cm -1 , f J"J' is the rotational oscillator line strength for absorption, and N tot is the total population of NO (in molecule cm -3 ). Due to the relatively important ratio between the laser bandwidth and the Doppler linewidth of NO at 300K (close to 1.2), the monoexponential behaviour of the ringdown decay can be easily affected with the increase of the absorbance [18,19]. To prevent this, NO was probed along the line S R 21 (7.5) (at 225.87 nm) which is weak enough to properly measure the absorbance of NO at room temperature. By considering the radiative lifetime of 201 ns [22], f J"J' was calculated equal to 8.226×10 -5 .

Experimental results

Validation of the pulsed CRDS setup The procedure of quantification by CRDS was validated by introducing the commercial preparations of NO diluted in nitrogen in continuous flowing in the cell, under controlled temperature and pressure. First, the gas standard 1 (2001 ppm NO) was diluted with nitrogen (from 5 to 30 ppm in N 2 ). The dilution was undertaken through 2 MFC implemented in the inlet gas lines of the burner set-up. The cold diluted preparation was injected in the burner enclosure from which it was probed through the microprobe 1. This arrangement allows controlling the mixing efficiency of the gases in the burner inlet line. The pressure in the cell was maintained at 1.3 kPa, allowing varying the NO concentration from 2 to 8.10 12 molecule.cm -3 . Second, using the gas standard 2 (10.6 ppm NO), the concentration of NO was varied from 1.5 to 10.10 12 molecule.cm -3 , by changing the pressure in the cell from 0.5 to 4 kPa, independently of the burner setup. Figure 2 shows the determined concentration as function of the known injected concentration of NO in the absorption cell. Regardless of the procedure, the evolution of the measured NO concentrations with the injected ones is identical. This indicates the absence of NO losses in the sampling line or inside the probe tip using a non reactive mixture of NO/N 2 at room temperature.

For NO concentration lower than 6×10 12 molecule cm -3 (corresponding to net losses per pass equal to 2×10 - 3 at the peak transition), the measured concentration was found in very good agreement with the injected one (linear fit with a slope equal to 0.97). At higher concentration of NO, the measured value is slightly lower than the injected one. When the absorption coefficient increases with the increase of the NO concentration, the ringdown decay is no more mono-exponential, yielding to an apparent decrease of the measured NO concentration, a drawback well described in [18,19]. In order to ensure that we were working in the appropriate range of concentrations, the pressure of the probed gas in the cell was maintained at 1.333 kPa (10 Torr with a reading accuracy of 0.01 Torr), and the maximum mole fraction of NO should not exceed 18.5 ppm. The uncertainties for measuring the NO concentration in the CRD cell have been estimated to be ±14% due to the pressure gauge (0.5% f.s.), the spectroscopic parameters (2%), and the statistical error for CRD measurements (6%), additionally of those due to the mixture dilution and preparation (5%).

NO calibration in the burned gases

The absolute calibration of NO was performed in the flame at HAB=20 mm with different approaches either by LIF or by CRDS. The first one was undertaken in situ from the standard addition method by LIF. The others were undertaken after gas sampling using CRDS.

In situ LIF The prompt-LIF signal of NO by exciting the Q 2 (26.5) transition was converted into absolute mole fraction by using the standard addition method detailed in [14,21]. For that, NO was substituted to N 2 in the premixed gases within a small range of mole fractions (0 to 40 ppm) by using the gas standard 1 and the same arrangement of the RDM as previously. Figure 3 shows the variation of the prompt-LIF signal as function of the known mole fractions of NO added in the premixed gases. By extrapolating the linear fit to y=0, the native NO mole fraction in the flame was found equal to 10.6±1.1 ppm. The overall uncertainty was estimated to be ±10%, due to the pressure gauge and MFC accuracies (0.5% f.s. for each apparatus), the standard gas preparation (2%), linear regression (1%), and statistical error for LIF measurements (5%). Ex situ after gas probe sampling by CRDS Gases were continuously sampled from the flame at HAB=20 mm. The 2 microprobes were tested. The determined concentration of NO by using CRDS was converted into mole fraction knowing the temperature (room T) and pressure (10 Torr) in the cell. NO mole fraction was quantified from the measurement of the integrated spectral absorptivity of transition S R 21 (7.5), according to Eq. ( 1). With the microprobe 1, the measured mole fraction of NO after gas sampling was found equal to 6.7±0.7 ppm, much lower than the value obtained by LIF. With the microprobe 2, the NO mole fraction was determined equal to 11.4±1.1 ppm, in very good agreement with the value obtained by LIF calibrated by NO addition.

With the microprobe 1, further investigations were undertaken to identify the source of the discrepancies between the absorptivity measurements performed with both microprobes. First, the standard addition method (with the gas standard 1 in the premixed gases) was applied similarly to what was performed during the LIF calibration. Figure 3 shows the variation of the losses per pass as function of the NO added in the premixed gases. Here, the NO addition was limited to 10 ppm to ensure that the CRD measurements were satisfyingly performed (in the linear regime of absorption). The mole fraction of NO was found equal to 7.2±1.0 ppm, in good agreement with the value previously obtained from the integrated spectral absorptivity (6.7 ppm). Second, small amounts of NO were added to the sampling line just after the microprobe. No evidence of NO losses in the sampling line could be seen. The overall uncertainty was estimated to be ±15% (for measurements performed with NO addition) and ±10% (without NO addition).

In the burned gases, (HAB=20 mm), the absolute mole fraction of NO obtained by CRDS after gas sampling with the microprobe 2 is close to the one derived from the addition NO in LIF, respectively equal to 11.4±1.1 and 10.6±1.1 ppm. This consistency suggests that the NO-reburning which could affect the standard addition method is negligible, and that the NO losses are very weak inside the microprobe 2 and in the sampling line On the contrary, the values of the NO mole fractions determined by CRDS (with different methodologies) with the microprobe 1 are 40% lower than with the one measured with the microprobe 2. This suggests that the NO losses occur inside the tip of this microprobe during the sampling of the burned gases at flame temperature.

Calibrated NO profiles and discussions

In situ and ex situ NO profiles were measured as function of HAB after exciting the molecule at the peak of the selected transition (Q 2 (26.5) in the flame or S R 21 (7.5) in the CRD cell). Figure 4 presents the three NO profiles independently obtained in the studied flame as previously explained. Each profile is calibrated as follow. The in situ relative LIF profile was corrected for the Boltzmann factor and temperature variations along the flame axis. It was converted into absolute mole fraction at HAB =20 mm, from the calibrated value obtained by the standard addition method. As a matter of comparison, the NO profile previously obtained in [20] is also reported. One can see that the NO profiles obtained by LIF after several years were found in good agreement one to each other. The ex situ profile obtained with a given microprobe was calibrated by using the integrated absorptivity measurements at 20 mm with this microprobe. The shape of the NO profiles obtained by using three experimental approaches differ one from each other. Clearly, NO profiles obtained after probe sampling are shifted downstream. The shape of the NO profile obtained by CRDS is less disturbed by using the microprobe 1 than the microprobe 2 (according to the unperturbed LIF profile). The gradient of the NO profile obtained by CRDS with the microprobe 2 is about 1.8 times sharper than the ones observed with the microprobe 1 or by LIF. This indicates that spatially the flame is less perturbed by the sharp tip. However, both microprobes induce a shift of the reaction zone. Indeed, the inflection points of the NO profiles obtained by CRDS are shifted downstream by 2.2 and 2.6 mm by using the microprobes 1 and 2, respectively. The present observations are in agreement with previous ones [5], indicating that the gradients of the species profiles steepen and are further shifted downstream with the increase of the tip angle.

Here, the microprobes differ by their tip angle and their area ratio, which can affect the efficiency of the quenching of chemical reactions [START_REF] Smith | [END_REF], the rate of surface reaction [6], the residence time of the gases in the tip. The flow rate of the sampling gas was found identical with the two microprobes (6.54 sccm) at HAB=20 mm. This corresponds to a mass flow rate equal to around 1.6 µg s - 1 , in agreement with the recommendations given by Fristrom and Westenberg [START_REF] Fristrom | Flame Structure[END_REF]. The surface/volume ratio of the microprobe 1 is almost twice higher than the one of the microprobe 2. Reactions at the surface are certainly favored in the microprobe 1. Allen [10] described that NO quickly react with O-atoms at the wall, yielding NO 2 . Meantime, if the aerodynamic quenching in the probe is not efficient, remaining O-atoms will react with NO 2 yielding back NO. Kramlich and Malte [6] have computed the species variation in the probe along the axis by considering wall reactions for the atoms recombination and gas-phase reactions. Rate of the surface reactions are increasing with decreasing the area ratio (increasing the surface/volume ratio), yielding to a faster recombination of O and H atoms into O 2 , H 2 and HO 2 . The conversion of NO into NO 2 is mainly explained by the reaction between NO and HO 2 .

Conclusions

In the present work, NO profiles in a stoichiometric flame were measured by using either in situ LIF or ex situ CRDS techniques. In situ NO-LIF profile was converted into absolute mole fractions from the standard NO addition method. Ex situ measurements were undertaken after probing gases from the flame through a quartz microprobe. Two shapes of microprobes were examined. Gases were sampled in the same conditions of flow rate and pressure inside the probe, and analyzed in an external absorption cell. Absolute LIF calibration was performed in a stoichiometric flame with standard addition method up to 40 ppm.

The unperturbed LIF profile was compared to the ex situ CRDS profiles obtained independently with the two microprobes. In the burned gases, the NO mole fraction value obtained by using CRDS after probing the gases through the sharper microprobe was found 40% lower than the values measured either in situ by LIF or ex situ but with a larger microprobe. This seems to indicate that the quenching (aerodynamically and/or chemically) inside the narrower tip is not as efficient as in the wider one. Reactions at the inner surface of the sharper microprobe are assumed to be more important, promoting the conversion of NO into NO 2 . NO profiles obtained after probe sampling are shifted downstream. The absolute mole fraction of NO in the burned gases is unaffected when working with a large angle of the tip.

The shape of the species profile obtained with the sharp angle probe is close the in situ one.

This work shows that the design of the microprobe should be specified in the literature, especially for NO species measurements. Further investigations on the chemical reactivity occurring during the sampling in the gas phase and at the wall would be helpful to fully understand the difference between the two microprobes. Extension of this work to atmospheric flames is intended.

Fig. 1 .

 1 Fig. 1. CRDS setup with the standard gas and the probe sampling introduction to the cell.

Fig. 2 .

 2 Fig. 2. Calibration of NO in the CRD cell at room temperature (300K). The slope of the fitted linear line is equal to 0.97.

Fig. 3 .

 3 Fig.3. NO calibration in the flame from NO addition in the premixed gases by using in situ LIF or ex situ CRDS.

Fig. 4 .

 4 Fig. 4. NO mole fraction profiles in flame obtained either by LIF in situ or by CRDS after gas probe sampling.

Acknowledgments

This work is a contribution to the CPER research project CLIMIBIO. The authors thank the French Ministère de l'Enseignement Supérieur et de la Recherche, the Hauts de France Region and the European Funds for Regional Economical Development for their financial support to this project.