Power and Load Optimization in Interference-Coupled Non-Orthogonal Multiple Access Networks

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Power and Load Optimization in
Interference-Coupled Non-Orthogonal Multiple
Access Networks
Lei Lei1, Lei You2, Yang Yang1, Di Yuan2, Symeon Chatzinotas1, and Bj¨
orn Ottersten1
1Interdisciplinary Centre for Security, Reliability and Trust, Luxembourg University, Luxembourg
2Department of Information Technology, Uppsala University, Sweden
Emails: {lei.lei; yang.yang; symeon.chatzinotas; bjorn.ottersten}@uni.lu; {lei.you; di.yuan}@it.uu.se
Abstract—Towards energy savings in large-scale non-
orthogonal multiple access (NOMA) networks, we investigate
power and load optimization for multi-cell and multi-carrier
NOMA systems in this paper. To capture the coupling relation
of mutual interference among cells, firstly, we extend a load-
coupling model from orthogonal multiple access (OMA) to
NOMA networks. Next, with this analytical tool, we formulate the
considered optimization problem in NOMA-based load-coupled
systems, where optimizing load, power, and determining decoding
order are the key aspects in the optimization. Theoretically,
we prove that the minimum network energy consumption can
be achieved by using all the time-frequency resources in each
cell to deliver users’ demand. To achieve the optimal load
and enable efficient power optimization, we develop a power-
adjustment algorithm. Numerical results demonstrate promising
energy-saving gains of NOMA over OMA in large-scale cellular
networks, in particular for the high-demand and resource-limited
scenarios.
Index Terms—Non-orthogonal multiple access (NOMA), load
coupling, large-scale network, resource allocation, energy opti-
mization
I. INTRODUCTION
In various 5G application scenarios, non-orthogonal multi-
ple access (NOMA) has demonstrated promising performance
improvement over orthogonal multiple access (OMA) [1].
Most NOMA works in the literature focus on single-cell
resource optimization where the inter-cell interference (ICI)
caused by neighboring cells is not present [2]–[5]. For multi-
cell NOMA, resource optimization and performance analysis
are challenging [6], [7]. The difficulty arises not only from
the mutual coupling effect of ICI among cells but also from
the interplay between ICI and the successive interference
cancellation (SIC) process. In SIC, the decoding order of the
co-channel allocated users largely depends on the received
ICI of the channel [6], [7]. Any change of transmit power
and channel usage in a cell may impact the SIC process
and resource optimization in the other cells. The coupling
effect imposes obstacles in problem decomposition, which
poses a challenging optimization task for jointly determining
the decoding order, transmit power, and channel resource
allocation.
In this regard, most multi-cell NOMA works are limited
to some simplified scenarios in order to reduce the analysis
complexity, e.g., two-cell with single-carrier networks [7],
[8], two-user with single-carrier networks [7], [9], and multi-
cell with single-carrier networks [10]–[12]. In some works,
e.g., [10], [12], stochastic geometry has been considered to
model ICI in NOMA. On the one hand, stochastic geometry
is capable of capturing the topological randomness of NOMA
networks [1], and evaluating the average network performance.
On the other hand, the derivation of closed-form expressions
for large-scale network analysis remains challenging [1], [12].
In addition, difficulties remain in studying the specific network
topologies in realistic deployment.
In the context of NOMA resource optimization in practical
multi-cell and multi-carrier systems, there is a lack of ana-
lytical tools to facilitate performance and properties analysis.
In OMA networks, a so called load-coupling model has been
widely used for ICI modeling [13]–[15]. The ICI generated
by a cell is directly proportional to the cell’s load, i.e., the
fractional portion of the used resource units (RUs) in a cell,
ranging from zero to one. When a BS operates at a certain
load level for some time, e.g., operating at load =0.5 for 1
second, the total number of used RUs for data transmission is
fixed, yet, the variation of which RUs are being used (regarding
as RUs allocation) is highly random from time slot to time
slot (in milliseconds). The load-coupling model is proposed to
simplify the ICI modeling without worrying about the short-
time randomness of the RU allocation, and provides average
network performance by considering statistical interference
in ICI modeling. The model is shown to be able to provide
sufficiently accurate approximations for the network-level in-
terference characterization [14], [15]. Recently, the authors in
[16] extended the load-coupling model from OMA to NOMA
to investigate load balancing in arbitrary network topologies.
The energy-saving issues in large-scale multi-cell/multi-
carrier NOMA networks are studied to a limited extent in the
literature. In this work, by adopting the load-coupling model,
we provide analytical results for several key research ques-
tions: to minimize the energy, how to determine the optimal
operating load in each cell; how to optimize transmit power
among BSs to achieve the optimal load; and how to jointly
determine SIC decoding order and optimize power. The main
contributions of this paper are summarized as follows. Firstly,
we formulate the energy minimization problem in a NOMA-
based load-coupling system (NLCS), where optimizing load,
978-1-5386-4727-1/18/$31.00 ©2018 IEEE

power allocation, and determining decoding order are the
key, and intertwined aspects. The problem appears non-linear
and non-convex in nature. By the proposed reformulation
and approximation approaches, we characterize a complete
solution for solving the problem. Secondly, through the derived
analytic expressions, we prove that the energy consumption
can be minimized by operating at full load in each cell. Given
full load or any other load level as a target to be achieved
in network operation, we develop an algorithmic framework
with guaranteed convergence to enable joint decoding order
determination and efficient power adjustment. Thirdly, we use
numerical studies to demonstrate that the network energy-
saving in NOMA can benefit from letting each cell operate
at higher load level and allowing more users to share the
same RU. The results show the effectiveness of the proposed
algorithmic solution for NOMA performance, revealing com-
petitiveness of NOMA over conventional OMA in energy
savings.
II. SYSTEM MOD EL
We consider a downlink NOMA-based network with mul-
tiple BSs. The overall bandwidth in a cell is divided into
multiple subchannels or RUs. We denote an RU as the
minimum time-frequency resource. The main notations are
summarized in Table I. Unlike OMA, each RU in NOMA
can be simultaneously accessed by multiple users. We use the
term “cluster” to represent a co-channel allocated user set, e.g.,
a two-user cluster {k, k′}, k, k′∈ K. A cluster can also be
referred to as a user group/set/pair presented in other works.
In NLCS, we consider uniform transmit power piper RU in
each cell i.
Table I
NOTATIO NS
Nnumber of RUs per cell
Bbandwidth per RU
Inumber of cells, i∈ {1,...,I},I={1,...,I}
Knumber of users, k∈ {1,...,K},K={1,...,K}
Kiset of the associated users in cell i
S∗
iset of the used clusters in cell i
Ui
sset of the users in cluster s∈Siin cell i
pitransmit power per RU in cell i∈ I
pi
ks transmit power for user kin cluster sin cell i
gi
kchannel gain between BS iand user k
ri
ks data rate of user kin cluster sin cell i
Rkuser k’s rate demand
ri
svector consists of rates ri
ks,∀k∈ U i
s, for cell iand cluster s
¯
liload vector [l1,...,li−1, li+1,...,lI]
ppower vector [p1,...,pi,...,pI]
¯pipower vector [p1,...,pi−1, pi+1,...,pI]
liload of cell i,0≤li≤1
li
sload of cluster sin cell i, where 0≤li
s≤1,∑s∈S∗
ili
s=li
ηnoise power
Ckreceived ICI plus noise power for user k
For ICI modeling, we extend the load-coupling model from
OMA to NOMA. Due to the exclusive user-channel allocation
in OMA, i.e., at most one user can access a subchannel/RU at
a time, thus in an OMA based load-coupling system (OLCS), a
cell’s load can be equivalently expressed by the summation of
its associated users’ load [13]–[15]. Unlike OMA, NOMA has
removed this exclusivity. As a result, the load expression for a
cell in OMA is incorrect for NLCS, and the previously derived
conclusions in OLCS may not be applicable to NLCS. In
NOMA, since one RU can accommodate at most one cluster,
we define a cell’s load by the summation of the clusters’ load.
In multi-cell NOMA, co-channel interference consists of
two parts, intra-cell interference and ICI. We treat ICI as noise,
and eliminate part of intra-cell interference by applying SIC
within each cluster. Following the NOMA basis [6] if the ICI
would be known, for cluster sin cell i, the users’ decoding
order in cluster scan be determined by the descending order
of gi
k
Ck, where Ckis the received ICI plus noise power for
user k, defined by j∈I\{i}pjljgj
k+η. We use the product
pjljgj
kin (1) to represent the statistical ICI from cell jto user
k. Intuitively, load ljcan be interpreted as the probability of
a RU in cell ireceiving the interference from cell j. With the
known ICI for cluster s, we use bs(k)to represent the position
of user k∈ Ui
sin the sorted descending sequence in cluster
s. The signal-to-interference-and-noise ratio (SINR) of user k
in cluster s∈ S∗
iis presented below.
SINRi
ks =pi
ksgi
k

h∈Ui
s\{k}:
bs(h)<bs(k)
pi
hsgi
k+
j∈I\{i}
pjljgj
k+η(1)
The entity h∈Ui
s\{k}:bs(h)<bs(k)pi
hsgi
kis the intra-cluster
interference for user kin cluster s. By applying SIC within
each cluster, the intra-cell or intra-cluster interference from
the users h∈ Ui
s\{k}:bs(h)> bs(k)can be decoded and
removed [1], [7]. We use load li
sto represent the proportion of
the RU allocation for cluster s. Since all the users in cluster s
share a common load li
s, thus any user k∈ Ui
sis subject to an
equation system in (2), where power satisfies k∈Ui
spi
ks =pi.
li
s=ri
ks
BN log(1 + pi
ksgi
k
∑
h∈Ui
s\{k}:
bs(h)<bs(k)
pi
hsgi
k+∑
j∈I\{i}
pjljgj
k+η)
,∀k∈ Ui
s
(2)
For illustration, we use a two-user cluster to explain the
rationale and the assumption in (2). Suppose a two-user
cluster swith Ui
s={1,2}in cell iand gi
1
C1≥gi
2
C2, where
C1=j∈I\{i}pjljgj
1+ηand C2=j∈I\{i}pjljgj
2+η.
According to the descending order of gi
k
Ck, user 2 is assumed
to be always able to decode its desired signal x. User 1at
its receiver can decode this signal xonly if it has higher
SINR of signal xat user 1’s receiver than at user 2’s receiver
[6], i.e., pi
2sgi
1
pi
1sgi
1+∑
j∈I\{i}
pjljgj
1+η≥pi
2sgi
2
pi
1sgi
2+∑
j∈I\{i}
pjljgj
2+η. As
a result, user 2 does not perform SIC and the rate for
user 2 is BN log(1 + pi
2sgi
2
pi
1sgi
2+∑
j∈I\{i}
pjljgj
2+η). User 1 can
decode and remove the intra-cluster interference with the
rate BN log(1 + pi
1sgi
1
∑
j∈I\{i}
pjljgj
1+η). Then the load equation

system in cluster sis: li
s=ri
1s
BN log(1+ pi
1sgi
1
∑
j∈I\{i}
pjljgj
1+η)
=
ri
2s
BN log(1+ pi
2sgi
2
pi
1sgi
2+∑
j∈I\{i}
pjljgj
2+η)
, where pi
1s+pi
2s=pi. Without
loss of generality, we normalize BN = 1 for convenience in
the remaining analysis of the paper. For cell i, the cell’s load
is the summation of clusters’ load, i.e., s∈S∗
ili
s=li.
For user clustering, in total there are to the maximum
2|Ki|−1possible clusters in each cell iwith associated |Ki|
users. Traversing all the clusters and obtaining the optimal
clusters can lead to prohibitively high computational com-
plexity. In NOMA, different practical clustering schemes are
adopted according to the constraints in each cell. In this work,
we adopt the practical grouping schemes proposed in [3]. For
example in 2-user clustering, i.e., |Ui
s|= 2, the best-worst
user paring/grouping is applied. That is, we sort the ratio
gi
1
C1, . . . , gi
|Ki|
C|Ki|for all the users in cell i, then the highest ratio
user and the lowest ratio user are paired into a cluster, while
the second highest ratio user and the second lowest ratio user
are grouped into another cluster, and so on. As a result, if two
clusters sand s′are scheduled for cell i, then Ui
s∩ Ui
s′=∅
and Ui
s∪ Ui
s′=Ki. This type of clustering is widely used for
NOMA systems in the literature [1], [3].
III. OPTIMI ZATI ON PROBL EM
With the established load-coupling model, we aim at in-
vestigating the optimal energy-saving strategy to satisfy all
the users’ data demand in large-scale NOMA networks. Be-
fore formulating the optimization problem, we characterize
how the cell’s power p1, . . . , pi, . . . , pIcorrelated with each
other in the load equation system (2). The characterization
will be used to facilitate the problem formulation and the
proofs to be shown later. We introduce a power vector ¯pi=
[p1, . . . , pi−1, pi+1, . . . , pI]collecting power p1, . . . , pIexcept
the ith element pi. Analogously, we define a load vector by
¯
li= [l1, . . . , li−1, li+1 , . . . , lI].
Proposition 1. In (2), power pican be expressed in terms of
¯piby a closed-form expression.
Proof: From the load equations in (2), we can derive pi
ks
for each kin cluster s. Without loss of generality, suppose
Ui
s={1, . . . , ¯
K},¯
K≤K, and the decoding order is
consistent with the user index in cluster s, we have,
pi
1s=(e
ri
1s
li
s−1)(
j∈I\{i}
pjljgj
1+η)/gi
1
. . .
pi¯
Ks =(e
ri
¯
Ks
li
s−1)(
¯
K−1

k′=1
pi
k′sgi
¯
K+
j∈I\{i}
pjljgj
¯
K+η)/gi
¯
K
(3)
From the above, any power expression pi
ks for k≥
2 contains pi
1s, . . . , pi
k−1,s, and thus in pi
2s, . . . , pi¯
Ks , we
can sequentially substitute each pi
ks with the expressions of
pi
1s, . . . , pi
k−1,s. Starting from k= 2, power pi
1sin pi
2scan be
replaced by (e
ri
1s
li
s−1)C1
gi
1
, then pi
2s= (e
ri
2s
li
s−1)((e
ri
1s
li
s−1)C1
gi
1+
C2
gi
2). Next, the substitution is executed for pi
3s, . . . , pi¯
Ks . By
completing the whole substitution process, we can explicitly
express pi=pi
1s+, . . . , +pi¯
Ks by all the other cells’ power in
(4),
pi=
¯
K

k=1
(

j∈I\{i}
pjljgj
k+η
gi
k
−

j∈I\{i}
pjljgj
k−1+η
gi
k−1
)e
∑¯
K
h=kri
hs
li
s
−

j∈I\{i}
pjljgj
¯
K+η
gi
¯
K
=
¯
K

k=1
(Ck
gi
k
−Ck−1
gi
k−1
)e
∑¯
K
h=kri
hs
li
s−C¯
K
gi
¯
K
(4)
where the elements of ¯piare contained in C1, . . . , C ¯
K. Thus,
pihas a close-form expression of ¯pi.
According to (4), we define a function fto express piin
(5). Vector ri
sis the collection of all rate elements ri
ks for
cluster sin cell i,∀k∈ Ui
s.
pi=f(li
s,ri
s,¯
li,¯pi),∀i∈ I,∀s∈ S ∗
i(5)
In the following, we formulate an energy optimization prob-
lem. The objective is to minimize the network energy con-
sumption. Note that the load can reflect the used resource
units either in frequency or time domain. Hence we represent
network energy by the summation of the product of power
and load over all the cells. Constraints (6b) ensure that all the
users’ data demands are satisfied. Constraints (6c) characterize
the load equation system and confine the feasible region for
the load, rate, and power variables. In (6d), the load level of
each cell should be no more than one.
P0: min
pi, li
s, ri
ks
∀i∈I,∀s∈S ∗
i,∀k∈Ki

i∈I
pi
s∈S∗
i
li
s(6a)
s.t. 
s∈S∗
i
ri
ks ≥Rk,∀i∈ I,∀k∈ Ki,(6b)
pi=f(li
s,ri
s,¯
li,¯pi),∀i∈ I,∀s∈ S ∗
i(6c)

s∈S∗
i
li
s≤1,∀i∈ I (6d)
The optimal solution for solving P0 is not straightforward,
due to the non-linearity and non-convexity in (6a) and (6c).
The variables of power, load, and rate in (6c) are intertwined
in a non-linear equation system in each cluster. Moreover, the
product of load and power results in a non-linear objective.
The network energy consumption is determined by transmit
power piper RU and cell’s load li. Specifically in each cell,
in (4) one can observe that power piis exponential with
ri
ks and li
s. Hence, any inappropriate allocation can possibly
result in surge in energy consumption. In addition, we note
that in (6c), the users’ decoding order in a cluster is varying
with the received ICI in optimization, imposing difficulties
in jointly determining optimal decoding order and optimal
transmit power in each cell.

IV. OPT IM AL IT Y CHARACTERIZATIONS
To optimally solve P0, the following questions need to be
addressed. Firstly, what is the optimal load, i.e., the cell’s load
liand the clusters’ load li
s, for energy minimization in NLCS?
Although it has been proved that li= 1,∀i∈ I is optimal for
OMA networks in [15], it is not clear for NOMA networks
whether the same conclusion holds. Secondly, how to optimize
each cell’s transmit power to achieve the target operating load
in network operation? Thirdly, how to jointly determine the
optimal decoding order and the optimal power. In this section,
we provide analyses and solutions for the above questions.
A. Optimal Operating Load in NLCS
We start from dealing with the optimal operating load for
energy minimization in NLCS. The result is formalized in
Theorem 2.
Theorem 2. At the optimum of P0, li= 1,∀i∈ I.
Proof: Suppose at the optimum, there exits a cell iwith
li<1. For simplicity we assume Ui
s={1, . . . , ¯
K}and
gi
1
C1≥, . . . , ≥gi
¯
K
C¯
K. In an arbitrary cluster s, if we increase
cell i’s load by adding li
swith an arbitrary small value β > 0,
we show that the new load li+βwill result in less energy
consumption, which contradicts the optimality of li<1. The
proof is presented below. The product of pili
sis the energy
consumption of cluster sin cell i. Substituting piby (4), pili
s
can be seen as a function of li
s,
f(li
s) = li
s[
¯
K

k=1
(Ck
gi
k
−Ck−1
gi
k−1
)e
∑¯
K
h=kri
hs
li
s−C¯
K
gi
¯
K
](7)
Then the derivative of f(li
s)in li
sis,
f′(li
s) =
¯
K

k=1
[(Ck
gi
k
−Ck−1
gi
k−1
)e
∑¯
K
h=kri
hs
li
s(1 −¯
K
h=kri
hs
li
s
)] −C¯
K
gi
¯
K
(8)
To see the negativity/positivity of f′(li
s), we derive the second
derivative f′′(li
s)which is shown to be non-negative.
f′′(li
s) = ¯
K
k=1[( Ck
gi
k
−Ck−1
gi
k−1
)e
∑¯
K
h=kri
hs
li
s(¯
K
h=kri
hs)2]
(li
s)3≥0,
(9)
Thus f′(li
s)monotonically increases when li
sincreases. One
can observe that lim
li
s→∞ f′(li
s) = 0. If li
sapproaches ∞,f′(li
s)
in (8) becomes C1
gi
1+( C2
gi
2−C1
gi
1)+, . . . , +(C¯
K
gi
¯
K
−C¯
K−1
gi
¯
K−1
)−C¯
K
gi
¯
K
=
0. Therefore we can conclude f′(li
s)≤0for li
s∈(0,1]. When
any cluster’s load li
sin cell ihas been increased, the resulting
transmit power pidecreases, and the product pili
sand the
energy of cell i, i.e., pilidecreases. The minimum energy
is obtained until load achieves one. Hence the conclusion.
From Theorem 2, we establish that the optimality of full
load holds for energy minimization not only for OLCS but
also for NLCS.
B. Updating Power to Achieve the Optimal Load
Being aware of the optimal load solution, i.e., li= 1,∀i∈
Iin P0, a follow-up question is how to find the corresponding
power pifor each cell to achieve the full load or any other
operating load. Next, in order to develop a solution for
computing optimal pi, we first characterize the property of
power by introducing the concept of standard interference
function (SIF). If a function f:Rn
+→Rn
++ satisfies the
following three properties for all input x≥0,fis SIF [17].
•Positivity: f(x)>0;
•Monotonicity: If x≥x′, then f(x)≥f(x′).
•Scalability: αf(x)> f(αx), for all α > 1, .
If f(x)is SIF, starting from any initial point and performing
fixed-point iteration based algorithm, i.e., the iterative algo-
rithm for power (IAP) proposed in [17], the convergence of
the algorithm to the fixed point is guaranteed as long as the
fixed point exists. Next, we prove that the function of piis an
SIF in ¯piin Proposition 3.
Proposition 3. For any li
s,ri
s,¯
liin cell i,pi=f(¯pi;li
s,ri
s,¯
li)
is a standard interference function in ¯pi.
Proof: For the sake of simplicity, again we use Ui
s=
{1, . . . , ¯
K}and gi
1
C1≥, . . . , ≥gi
¯
K
C¯
Kfor cluster sin cell i. From
(3), one can observe the positivity. For monotonicity, if we
increase all the elements in vector ¯piby a positive value β > 0,
i.e., pj+β, ∀j∈ I\{i}, equations in (3) become (10).
pi
1s=(e
ri
1s
li
s−1)(
j∈I\{i}
(pj+β)ljgj
1+η)/gi
1
. . .
pi¯
Ks =(e
ri
¯
Ks
li
s−1)(
¯
K−1

k′=1
pi
k′sgi
¯
K+
j∈I\{i}
(pj+β)ljgj
¯
K+η)/gi
¯
K
(10)
Since all the elements pi
1s, . . . , pi¯
Ks are strictly increased, then
pi=pi
1s+, . . . , +pi¯
Ks increases. Thus f(¯pi+β;li
s,ri
s,¯
li)>
f(¯pi;ls
i,ri
s,¯
li). In terms of scalability, let
αf(¯pi;li
s,ri
s,¯
li) = αpi=αpi
1s+, . . . , +αpi¯
Ks
= (e
ri
1s
li
s−1)(α
j∈I\{i}
pjljgj
1+αη)/gi
1+, . . . ,
+ (e
ri
¯
Ks
li
s−1)(α
¯
K−1

k′=1
pi
k′sgi
¯
K+α
j∈I\{i}
pjljgj
¯
K+αη)/gi
¯
K
(11)
and
f(α¯pi;li
s,ri
s,¯
li) = (e
ri
1s
li
s−1)(α
j∈I\{i}
pjljgj
1+η)/gi
1+, . . . ,
+ (e
ri
¯
Ks
li
s−1)(
¯
K−1

k′=1
pi
k′sgi
¯
K+α
j∈I\{i}
pjljgj
¯
K+η)/gi
¯
K
(12)
We can observe αf (¯pi;li
s,ri
s,¯
li)> f(α¯pi;li
s,ri
s,¯
li), hence
the conclusion.

C. Energy Optimization Framework in NLCS
Algorithm 1 Energy minimization framework for NLCS
Given: target load ¯
li, used clusters s∈ S∗
ifor each cell i
Output:p1, . . . , pi, . . . , pI
1: Initialize: load l1, . . . , lI, power vectors p′and p∗=
[p1, . . . , pI], (||p∗−p′||2> ϵ)
2: repeat
3: p′←p∗
4: for i= 1 : Ido
5: Sort gi
k
Ck,∀k∈ Kiin a descending sequence as the
decoding order
6: repeat
7: Bisection search for pi. For each searched pido
8: for each s∈ S∗
ido
9: Calculate the resulting load li
sby (13)
10: until |s∈S∗
ili
s−¯
li| ≤ ϵ
11: p∗= [p1, . . . , pi, . . . , pI]
12: until ||p∗−p′||2≤ϵ
The derived analysis so far can enable us to outline a
complete solution for optimally solving P0, that is, setting full
load as the target load, then iteratively updating each cell’s
power p1, . . . , pi, . . . , pIto achieve the full load. When a cell
is processed, all the other cells’ power remains unchanged.
The iterations eventually converge to a fixed power vector
which leads to the minimum network energy consumption and
full load in all the cells. If the fixed power point (vector)
exists then it is unique. The convergence rate is linear [18].
Regarding the rate variables, one can observe that transmitting
the minimum rate demand Rkfor all the users is optimal
in energy minimization. In the adopted clustering scheme,
any two clusters sand s′in cell ihave Ui
s∩ Ui
s′=∅and
Ui
s∪Ui
s′=Ki. Then one user’s demand is transmitted by only
one cluster. The rates variables can be therefore equivalent to
Rk,∀k∈ Ui
s. As a result, power piin cell isubmits to the
following expression,
pi=
|Ui
s|

q=1
(C(q)
gi
(q)
−C(q−1)
gi
(q−1)
)e
∑|Ui
s|
h=qR(h)
li
s−C(|Ui
s|)
gi
(|Ui
s|)
,∀s∈ S∗
i
(13)
where index 1, . . . , q, . . . , |Ki|is the descending order after
sorting gi
k
Ck,∀k∈ Ki, and (q)is the user in the qth position
in the order.
Based on Proposition 3, and given the optimality of full
load, the corresponding power solution can be obtained by
means of an alternating power updating approach (or fixed-
point iterations approach) [15], [17]. The steps are summarized
in Algorithm 1. With a target load ¯
liin each cell, from
Line 2 to 12, the power optimization is carried out for cell
i= 1, . . . , I one by one. In each cell i, the SIC decoding order
is determined firstly in Line 5. Then the next optimization
task is to determine piand the optimal load allocation li
s
among the clusters of S∗
i. Since piis uniform over RUs in
a cell, optimizing this single variable can be carried out by
bisection search method. For each searched power value pi,
the corresponding load li
scan be calculated for each cluster
by (13). The bisection search for power piterminates when
the gap between the sum load s∈S∗
ili
sand the target load ¯
li
is less than a predefined tolerance ϵ. We define the algorithm
convergence achieves when the power variation between two
successive iterations is no more than a tolerance ϵ. At the
convergence, the corresponding power solution is organized
in vector p∗.
V. PERFORMANCE EVALUATI ON
In this section, we present numerical studies to verify the
derived theoretical results in previous sections, and evaluate
the energy-saving gains of network NOMA by the proposed
analytical model and algorithm. Table II summarizes the key
parameters. In performance evaluation, all the users in each
cell are randomly and uniformly distributed. We generate two
hundreds instances and consider the average performance.
For performance comparison, we implement the algorithm
proposed in [15] to compute the optimal energy for OMA
networks. In NOMA, we evaluate the performance from 2-
user clustering, i.e., two users are grouped in each RU (or
cluster), to 4-user clustering.
Table II
SIM ULATI ON PARAMETERS.
Parameter Value
Cell radius 200 m
Carrier frequency 2 GHz
Bandwidth per cell 9 MHz
Bandwidth per RU 15 KHz
Number of cells 20
Number of RUs per cell 600
Path loss COST-231-HATA
Shadowing Log-normal, 8 dB standard deviation
Fading Rayleigh flat fading
Noise power spectral density -173 dBm/Hz
Tolerance ϵin Algorithm 1 10−5
Clustering schemes in NOMA [3]
In Fig. 1, we evaluate the energy consumption with re-
spect of user demand. The numerical results contain sev-
eral key information: Firstly, applying NOMA for network
energy savings is more effective in high-demand scenarios
than low-demand cases. The performance gaps (measured
by OM A−NO MA
NO MA ×100%) between OMA and NOMA 4-users
clustering are around 9.7%, 15.7%, 50.1%, 86.8%, 299.3%
for the demands at 2, 3, 4, 4.2, 4.4 Mbps, respectively. For
high-demand instances, all the NOMA schemes demonstrate
superior performance, e.g., 2-3 times of energy decease in
NOMA over OMA in 4.4 Mbps. Secondly, the network energy-
saving can benefit from letting allowing more users share
the same RU. Thirdly, compared to OMA, NOMA is able to
support higher demand in practical power ranges. In Fig. 1, the
energy consumption of OMA increases to an unrealistic value
(>1010J) for 4.4-4.5 Mbps. Although the energy increases
dramatically with the demand for all the schemes, the rate of
increase in the NOMA schemes is much more moderate than
OMA.

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Demand (Mbps)
0
200
400
600
800
1000
Energy (J)
NOMA (4 users per cluster)
NOMA (2 users per cluster)
OMA
2.6 2.7 2.8 2.9 3.0
12
13
14
15
16
17
18
Figure 1. Energy consumption with respect of demand (load =1).
0.5 0.6 0.7 0.8 0.9 1
Load
0
200
400
600
800
1000
1200
Energy (J)
NOMA (4 users per cluster)
NOMA (2 users per cluster)
OMA
0.9 0.92 0.94 0.96 0.98 1
14
16
18
20
22
24
Figure 2. Energy consumption with respect of load (demand =3 Mbps).
In Fig. 2, we examine the energy consumption with respect
to load. Several observations can be noted. Firstly, the results
are in line with Theorem 2. As expected, the minimum energy
is achieved at load =1 in all the NOMA schemes (as well as
in OMA). Secondly, NOMA is able to satisfy users’ demand
but using few resources than OMA. For serving the same
amount of demand, i.e., 3 Mbps in Fig. 2, all the NOMA
schemes consume less than half of RUs (with load 0.3-0.5),
then more RUs can be released for serving the upcoming user
demand, whereas the solution in OMA becomes infeasible
when the load is less than 0.58. Thirdly, the performance
gaps between NOMA and OMA dramatically increase in the
resource-limited scenarios (low-load region).
VI. CONCLUSIONS
We have extended an analytical tool, i.e., load-coupling
model, from OMA to NOMA for studying the performance of
multi-cell and multi-carrier NOMA networks. Towards energy
minimization in NOMA networks, we have concluded that
operating at full load is optimal for energy savings in NOMA
networks, and the less energy can be achieved by applying
more aggressive user clustering scheme. We have developed
an algorithm to jointly determining optimal decoding order and
power-adjustment to achieve the optimal load. The numerical
studies have illustrated the superior performance of NOMA
over OMA in network energy savings, particularly in high-
demand and low-load instances.
VII. ACKN OWLED GM EN TS
This work has been supported by the Luxembourg Na-
tional Research Fund (FNR) CORE project ROSETTA
(C17/IS/11632107).
REF ERENCES
[1] Y. Liu, Z. Qin, M. Elkashlan, Z. Ding, A. Nallanathan, and L. Hanzo,
“Nonorthogonal multiple access for 5G and beyond,” Proceedings of the
IEEE, vol. 105, no. 12, pp. 2347–2381, Dec. 2017.
[2] L. Lei, D. Yuan, and P. Vrbrand, “On power minimization for non-
orthogonal multiple access (NOMA),” IEEE Communications Letters,
vol. 20, no. 12, pp. 2458–2461, Dec. 2016.
[3] M. S. Ali, H. Tabassum, and E. Hossain, “Dynamic user clustering
and power allocation for uplink and downlink non-orthogonal multiple
access (NOMA) systems,” IEEE Access, vol. 4, pp. 6325–6343, Aug.
2016.
[4] J. M. Kang and I. M. Kim, “Optimal user grouping for downlink
NOMA,” IEEE Wireless Communications Letters, pre-print, Mar. 2018.
[5] L. Lei, D. Yuan, C. K. Ho, and S. Sun, “Power and channel allocation
for non-orthogonal multiple access in 5G systems: Tractability and
computation,” IEEE Transactions on Wireless Communications, vol. 15,
no. 12, pp. 8580–8594, Dec. 2016.
[6] Y. Saito, A. Benjebbour, Y. Kishiyama, and T. Nakamura, “System-level
performance evaluation of downlink non-orthogonal multiple access
(NOMA),” in IEEE PIMRC, Sept. 2013, pp. 611–615.
[7] W. Shin, M. Vaezi, B. Lee, D. J. Love, J. Lee, and H. V. Poor, “Non-
orthogonal multiple access in multi-cell networks: Theory, performance,
and practical challenges,” IEEE Communications Magazine, vol. 55,
no. 10, pp. 176–183, Oct. 2017.
[8] ——, “Coordinated beamforming for multi-cell MIMO-NOMA,” IEEE
Communications Letters, vol. 21, no. 1, pp. 84–87, Jan. 2017.
[9] Y. Fu, Y. Chen, and C. W. Sung, “Distributed downlink power control for
the non-orthogonal multiple access system with two interfering cells,” in
IEEE International Conference on Communications (ICC), May 2016,
pp. 1–6.
[10] H. Tabassum, E. Hossain, and J. Hossain, “Modeling and analysis of
uplink non-orthogonal multiple access in large-scale cellular networks
using poisson cluster processes,” IEEE Transactions on Communica-
tions, vol. 65, no. 8, pp. 3555–3570, Aug. 2017.
[11] Z. Yang, C. Pan, W. Xu, Y. Pan, M. Chen, and M. Elkashlan, “Power
control for multi-cell networks with non-orthogonal multiple access,”
IEEE Transactions on Wireless Communications, pre-print, 2017.
[12] Y. Liu, Z. Qin, M. Elkashlan, A. Nallanathan, and J. A. McCann, “Non-
orthogonal multiple access in large-scale heterogeneous networks,” IEEE
Journal on Selected Areas in Communications, pre-print, 2017.
[13] L. Lei, D. Yuan, C. K. Ho, and S. Sun, “Optimal cell clustering and
activation for energy saving in load-coupled wireless networks,” IEEE
Transactions on Wireless Communications, vol. 14, no. 11, pp. 6150–
6163, Nov. 2015.
[14] I. Siomina and D. Yuan, “Analysis of cell load coupling for LTE
network planning and optimization,” IEEE Transactions on Wireless
Communications, vol. 11, no. 6, pp. 2287–2297, June 2012.
[15] C. K. Ho, D. Yuan, L. Lei, and S. Sun, “Power and load coupling
in cellular networks for energy optimization,” IEEE Transactions on
Wireless Communications, vol. 14, no. 1, pp. 509–519, Jan. 2015.
[16] L. You, L. Lei, D. Yuan, S. Sun, S. Chatzinotas, and B. Ottersten,
“Resource optimization with load coupling in multi-cell NOMA,” IEEE
Transactions on Wireless Communications, pre-print, 2018.
[17] R. D. Yates, “A framework for uplink power control in cellular radio
systems,” IEEE Journal on Selected Areas in Communications, vol. 13,
no. 7, pp. 1341–1347, Sept. 1995.
[18] H. R. Feyzmahdavian, M. Johansson, and T. Charalambous, “Contractive
interference functions and rates of convergence of distributed power
control laws,” IEEE Transactions on Wireless Communications, vol. 11,
no. 12, pp. 4494–4502, Dec. 2012.