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Linear State Estimation via 5G C-RAN Cellular Networks using Gaussian Belief Propagation

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Machine-type communications and large-scale information processing architectures are among key (r)evolutionary enhancements of emerging fifth-generation (5G) mobile cellular networks. Massive data acquisition and processing will make 5G network an ideal platform for large-scale system monitoring and control with applications in future smart transportation, connected industry, power grids, etc. In this work, we investigate a capability of such a 5G network architecture to provide the state estimate of an underlying linear system from the input obtained via large-scale deployment of measurement devices. Assuming that the measurements are communicated via densely deployed cloud radio access network (C-RAN), we formulate and solve the problem of estimating the system state from the set of signals collected at C-RAN base stations. Our solution, based on the Gaussian Belief-Propagation (GBP) framework, allows for large-scale and distributed deployment within the emerging 5G information processing architectures. The presented numerical study demonstrates the accuracy, convergence behavior and scalability of the proposed GBP-based solution to the large-scale state estimation problem.
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arXiv:1710.08671v1 [cs.IT] 24 Oct 2017
Linear State Estimation via 5G C-RAN Cellular
Networks using Gaussian Belief Propagation
Mirsad Cosovic, Dejan Vukobratovic, Vladimir Stankovic
Abstract—Machine-type communications and large-scale
information processing architectures are among key
(r)evolutionary enhancements of emerging fifth-generation (5G)
mobile cellular networks. Massive data acquisition and
processing will make 5G network an ideal platform for
large-scale system monitoring and control with applications in
future smart transportation, connected industry, power grids,
etc. In this work, we investigate a capability of such a 5G
network architecture to provide the state estimate of an
underlying linear system from the input obtained via
large-scale deployment of measurement devices. Assuming that
the measurements are communicated via densely deployed
cloud radio access network (C-RAN), we formulate and solve
the problem of estimating the system state from the set of
signals collected at C-RAN base stations. Our solution, based
on the Gaussian Belief-Propagation (GBP) framework, allows
for large-scale and distributed deployment within the emerging
5G information processing architectures. The presented
numerical study demonstrates the accuracy, convergence
behavior and scalability of the proposed GBP-based solution to
the large-scale state estimation problem.
I. INTRO DUC TI ON
With transition towards fifth generation (5G), mobile
cellular networks are evolving into ubiquitous systems for
data acquisition and information processing suitable for
monitoring and control of large-scale systems. At the
forefront of this evolution is the transformation of radio
access network (RAN) to support massive-scale
machine-type communications (MTC) [1] and transformation
of core network (CN) to support large-scale centralized or
distributed information processing through Cloud-RAN
(C-RAN) and Fog-RAN (F-RAN) architecture [2], [3]. MTC
services in 5G will offer both massive-scale data acquisition
from various machine-type devices through massive MTC
(mMTC) service, but also, provide ultra reliable and
low-latency communication (URLLC) service for
mission-critical applications [4]. Complemented with
ultra-dense RAN deployment and flexible and virtualized
signal processing architecture, novel 5G network services
that are particularly suitable for large-scale system
monitoring and control of various smart infrastructures are
emerging [5].
In this work, we focus on a generic state estimation
problem placed in the context of a future 5G-inspired
M. Cosovic is with Schneider Electric DMS NS, Novi Sad, Serbia
(e-mail: mirsad.cosovic@schneider-electric-dms.com). D. Vukobratovic
is with Department of Power, Electronic and Communications
Engineering, University of Novi Sad, Novi Sad, Serbia (e-mail:
dejanv@uns.ac.rs). V. Stankovic is with Department of Electronic and
Electrical Engineering, University of Strathclyde, Glasgow, UK (e-
mail:vladimir.stankovic@eee.strath.ac.uk).
C-RAN-based cellular network. We consider an underlying
large-scale physical system characterized by the state vector
sthat contains values of Nsystem state variables. The state
variables are observed through the set of Mmeasurements x
of physical quantities collected at the measurement devices
spread across the system. This paper considers linear system
model in which measured quantities are linear functions of
the (sub)set of state variables. Further, we assume
measurements are wirelessly communicated across
C-RAN-based cellular network. In C-RAN, large number of
spatially distributed remote radio heads (RRH) constitutes an
ultra-dense RAN infrastructure that receives signals from
densely populated MTC devices (e.g., the measurement
devices under consideration) [2]. The signal vector y
collected at RRHs is forwarded via backhaul links to a
central C-RAN location where it is fed into a collection of
base-band units (BBU) for signal detection and recovery. In
the standard C-RAN signal detection problem, the goal is to
recover the signal xtransmitted by the set of MTC devices
from the signal yreceived at RRHs and gathered centrally at
BBUs [6] [7]. However, in this paper, focusing on widely
applicable linear system state estimation, we extend this goal
and investigate the problem of recovering the system state s
directly from the signal ycollected across the C-RAN.
The problem we observe represents a concatenation of the
two well-studied problems: the linear system state estimation
problem (see, e.g., [8], for the case of power system state
estimation) and the problem of uplink signal detection in
C-RAN [6]. For the joint problem, it is straightforward to
derive (and implement at a central location) the standard
minimum mean-square error (MMSE) estimator, however,
such a solution comes with prohibitive O(N3)-complexity
that hinders its application for large-scale systems. By
exploiting inherent sparsity within both of the component
problems, an approximate MMSE solution for each problem
can be obtained using the tools from probabilistic graphical
models, as recently investigated for both (power system)
state estimation [9] and uplink signal detection in C-RANs
[7]. In particular, an instance of the Belief-Propagation (BP)
algorithm, called Gaussian BP (GBP) [10], can be applied to
produce an exact MMSE estimate with O(N)-complexity,
thus scaling the MMSE solution to large-scale system
scenarios.
In this paper, we motivate, formulate and solve the linear
system state estimation problem considered jointly with the
signal detection problem in C-RAN-based cellular networks.
We cast the problem of estimating the system state sfrom
the received vector yinto an equivalent maximum
a-posteriori (MAP) problem, and place it into the framework
of a popular class of probabilistic graphical model called
factor graphs. The state estimate ˆ
sis then derived as a
solution of the GBP algorithm applied over a specific
bi-layer structure of the factor graph. Throughout the paper,
we use state estimation in power systems with the
measurements collected via 5G-inspired C-RAN network as
a running example. Our initial numerical results demonstrate
the viability of the proposed approach, both in terms of
accuracy and convergence.
The paper is organized as follows. In Section II, we present
the joint state estimation and C-RAN uplink communication
system model. In Section III, this model is mapped into a
corresponding factor graph, and the state estimate is obtained
via GBP. Section IV provides numerical results of the proposed
GBP state estimator. The paper is concluded in Section V.
II. SY S TE M MO DE L
We consider a generic state estimation problem where a
set of state variables sis estimated from a set of observed
noisy linear measurements x. However, unlike the traditional
setup where the measurements in xare assumed available at
a central node, here we assume they are transmitted via radio
access network (RAN) of a mobile cellular system based on
a cloud-RAN (C-RAN) architecture. The received signal yis
collected at a large-number of densely deployed remote radio
heads (RRHs) and jointly processed at the C-RAN base-band
units (BBUs). The problem we consider is that of estimating
the system state sfrom the received signal y.
Linear system measurements model: We consider a
system described via the set of Nstate variables
s= (s1, s2,...,sN)TCN×1. The system is observed via a
set of Mmeasurements x= (x1, x2,...,xM)TCM×1.
Each measurement is a linear function of the state variables
additionally corrupted by the additive noise, i.e.,
xi=ai·s+ni,1iM, where
ai= (ai,1, ai,2,...,ai,N )C1×Nis a vector of coefficients,
while niCis a complex random variable. Overall, the
system is represented via noisy linear observation model
x=A·s+n, where the matrix ACM×Ncontains
vectors ai,1iM, as rows, while
n= (n1, n2,...,nM)TCM×1is a vector of additive noise
samples. We assume noise samples niare independent
identically distributed (i.i.d.) Gaussian random variables with
variance σ2
ni. For simplicity, we assume the measurement
noise variances are equal, i.e., σ2
ni=σ2
n,1iM. In
other words, nrepresents a complex Gaussian random vector
with zero means µn=0and the covariance matrix
Σn=σ2
nICM×M(Iis an identity matrix).
C-RAN uplink communication model: In the standard
state estimation models, measurements are either assumed
available, or they are communicated to the central node,
where the state estimation problem is solved.
Communication models typically involve point-to-point
communication links between the measurement devices and
the central node, affected by communication impairments
such as delays, packet losses, limited bit rates, etc. In the
cellular networks context, this assumes reservation of uplink
resources and subsequent non-orthogonal transmission,
which typically incurs significant communication delays.
Inspired by the recent evolution of massive MTC and
ultra-dense C-RAN architectures in upcoming 5G mobile
cellular networks, in this work, we consider different
grant-less and non-orthogonal communication model, as we
detail next. Note that the following C-RAN cellular network
model could provide ultra-low latency for the state
estimation application under consideration. In other words,
such an architecture could produce the system state estimate
at the central network node with very low delay after the
measurements are acquired, which is crucial for emerging
mission-critical 5G MTC use cases [11].
In the mobile cellular system under consideration,
measurements are collected by the measurement devices that
we refer to as MTC user equipment (MTC-UE). We consider
uplink transmission of Msingle-antenna MTC UEs towards
Lsingle-antenna RRHs. We assume both MTC UEs and
RRHs are randomly and uniformly distributed across a given
geographic area (note that this placement model is somewhat
refined in the numerical results section). The signal
xCM×1, representing the set of collected noisy
measurements, is transmitted1by MMTC UEs, while the
signal y= (y1, y2,...,yL)CL×1is received at LRRHs,
where y=H·x+m. The matrix HCL×Mrepresents
the channel matrix, where hi,j represents a complex channel
coefficient between the j-th MTC UE and the i-th RRH,
while m= (m1, m2,...,mL)CLis a vector of additive
noise samples. As for the measurement process, for the
communication process we also assume noise samples mi
are i.i.d. zero-mean Gaussian random variables with variance
σ2
m, i.e., the mean value and the covariance matrix of mis
given as µm=0and Σm=σ2
mICL×L, respectively.
Linear system state estimation problem: From the
received signal ycollected across RRHs, we are interested in
finding an estimate ˆs of the state vector s. In this paper, we
focus on the centralized C-RAN architecture, where all the
BBUs are collocated at the central C-RAN node. Thus, due
to availability of yat the central location, we consider
centralized algorithms for the state estimation problem.
However, the solution we propose in this paper is based on
GBP framework, thus it is easily adaptable to a distributed
scenario that we refer to as fog-RAN (F-RAN), where BBUs
are distributed across different geographic locations closer to
the MTC UEs. We refer the interested reader to our recent
overview of distributed algorithms for solving the state
estimation problem in the context of upcoming 5G cellular
networks [5].
A common centralized approach to solve the above state
estimation problem is to provide the corresponding minimum
mean-square error (MMSE) estimate. One can easily obtain
the MMSE estimate ˆs for the underlying linear model in the
form:
ˆs = (HA)HΣ·(HA)Σ(HA)Hy,(1)
1Note that, at this point, one can insert specific linear modulation scheme
xm=fm(x). For simplicity, we assume fm(x) = x.
C-RAN BBUs
C-RAN RRHs
Power System
Fig. 1. System model example: State estimation in Smart Grid via C-RAN-based mobile cellular network.
where (·)His the conjugate-transpose matrix operation, and
Σ=nHH+Σm. However, solving (1) scales as O(N3)
which makes linear MMSE state estimation inapplicable in
large-scale systems which are of interest in this paper.
In the following, we cast the MMSE estimation into an
equivalent MAP state estimator as follows:
ˆs = arg max
sCNP(s|y),(2)
where P(s|y)is the posterior probability of the state safter
the signal yis observed at the RRHs. As we will demonstrate
in the sequel, if certain sparsity arguments are applicable in the
system model under consideration, the solution of the MAP
problem can be efficiently calculated using the framework of
factor graphs and belief-propagation (BP) algorithms.
System model example (Smart Grid): Before
continuing, it is useful to consider an example of the above
state estimation setup. We consider the state estimation
problem in an electric power system, where the goal is to
estimate the state of the power system s, containing complex
voltages of Nsystem buses, via the set of measurements x
obtained using measurement devices. Measurement devices
are geographically distributed across the power system and
we assume they are equipped with wireless cellular
interfaces, i.e., they represent MTC UEs connected to the
C-RAN based cellular network. The signal yreceived from
the set of RRHs densely deployed across the cellular
network coverage area is processed centrally within the
C-RAN system architecture. Fig. 1 illustrates the smart grid
example that we will further refine in Section IV and use as
a running example throughout this paper.
III. STATE EST IM ATI ON V IA GAUS SIA N BELIEF
PROPAG ATION
In this section, we provide a solution to the combined state
estimation and uplink signal detection problem defined in (2),
by applying factor graphs and GBP framework. In fact, for
both constituent scenarios: the conventional state estimation
(in case of power systems) and the uplink signal detection in
C-RAN, the GBP has already been proposed and analyzed (see
details in [9] and [7]). Thus in this work, we propose using
GBP in a joint and combined setup of extracting the system
state directly from the observed C-RAN signals.
We note that the various properties of the proposed GBP
approach (e.g., complexity, convergence, etc.) will strongly
depend on the structure of the underlying factor graph. For
example, in terms of complexity (we will come back to
convergence in the next section), for GBP to scale well to
large-scale systems, it is fundamental that both matrices A
and Hdefining the two linear problems are sparse, i.e., that
for both Aand H, the number of non-zero entries scales as
O(N). In many real-world scenarios, the sparsity typically
arises from geographic constraints and reflect locality that is
typically present in both the measurement and the
communication part of the system model. More detailed
account on the sparsity of matrices Aand Hclearly depends
on the specific scenario under consideration, and we relegate
these details to Section IV where we will explicitly deal with
the smart grid example introduced earlier.
Factor Graph System Representation: The MAP problem
under consideration can be rewritten as follows:
ˆs = arg max
sCNP(s|y)arg max
sCNP(s,y) = (3)
= arg max
sCN,xCMP(s,x,y)(4)
Assuming that the system state shas a Gaussian prior, and
given that the measurement and communication noise is
assumed Gaussian, the distribution P(s,x,y)is jointly
Gaussian. In addition, due to the problem structure where s
and yare conditionally independent given x, we obtain:
ˆs = arg max
sCN,xCMP(y|s,x)P(s,x) = (5)
= arg max
sCN,xCMP(y|x)P(x|s)P(s).(6)
As noted before, in many real-world systems of interest, a
measurement xjis a linear function of a small subset of local
state variables sN(xj), where N(xj)is the index set of the
state variables that affect xj, and sN(xj)={si|i N (xj)}.
In other words, the row-vector ajhas non-zero components
only on a small number of positions indexed by the set N(xj),
thus making the matrix Asparse2. Using this fact and the fact
that the measurements xjare mutually independent, we obtain:
P(x|s) =
M
Y
j=1
P(xj|sN(xj)).(7)
On the other hand, in the C-RAN communication part,
although in theory the received signal yidepends on all the
transmitted symbols in x, the channel coefficients between a
RRH and a geographically distant MTC UE can be
considered negligible, thus leading to matrix Hsparsification
[7]. Upon distance-based sparsification proposed in [7], the
received symbol yidepends only on a small number of
symbols xN(yi), where N(yi)is the index set of symbols
transmitted by the set of MTC UEs in geographic proximity
of the i-th RRH. Taking the channel sparsification into
account, we obtain:
P(y|x) =
L
Y
i=1
P(yi|xN(yi)).(8)
Finally, assuming that the state vector is apriori given as
a set of i.i.d Gaussian random variables, we obtain the final
factorized form of the initial MAP problem:
ˆs = arg max
sCN,xCM
L
Y
i=1
P(yi|xN(yi))·
M
Y
j=1
P(xj|sN(xj))·
N
Y
k=1
P(sk).(9)
2More precisely, the number of state variables that affect certain
measurement is limited by a constant, independently of the size Nof the
system.
The factor graph representation of the MAP problem
follows the factorization presented in (9) and is illustrated in
Fig. 2. Factor graph G=G(V ∪ F,E)is a bipartite graph
consisting of the set of variable nodes V, the set of factor
nodes F, and the set of edges E. In our setup, the set Vcan
be further divided as V=S ∪ X ∪ Y ,where
S={s1, s2,...,sN}is the set of state nodes,
X={x1, x2,...,sM}is the set of measurement nodes,
while Y={y1, y2,...,yL}is the set of received symbol
nodes. The set of factor nodes can be divided as
F=FH∪ FA∪ Fy∪ Fx∪ Fs,where
FH={fh1, fh2,...,fhL}and FA={fa1, fa2,...,faM}
represent factor nodes that capture linear relationships
between variable nodes described by the rows of matrices H
and A, respectively. In addition, Fy={fy1, fy2,...,fyL}
and Fs={fs1, fs2,...,fsN}represent the factor nodes that
provide inputs due to observations of yand the prior
knowledge about x, respectively, while
Fx={fx1, fx2,...,fxM}serve as virtual inputs needed for
initialization of measurement nodes. Similarly, the set of
edges Ecan be divided as E=EH∪ EA∪ Ey∪ Ex∪ Es,
where EH∪ Ey∪ Exand EA∪ Es∪ Excan be considered as
the set of edges of two bipartite subgraphs
GH= (Y ∪ X ∪ FH∪ Fy∪ Fx,EH∪ Ey∪ Ex)and
GA= (X ∪ S ∪ FA∪ Fs∪ Fx,EA∪ Es∪ Ex), obtained as
the subgraphs of Ginduced from the set of factor nodes
FH∪ Fy∪ Fxand FA∪ Fs∪ Fx, respectively3.
.
.
.
y
F
Y
x
F
A
F
H
F
S
X
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Fig. 2. Factor graph representation of the system model.
As noted earlier, the state estimation problem using GBP
over factor graph GA, and the uplink C-RAN signal detection
problem using GBP over factor graph GH, have been recently
investigated in detail in [9] and [7], respectively.
Gaussian Belief-Propagation and GBP Messages: To
estimate the state variables s, we apply message-passing
GBP algorithm [10]. GBP operates on the factor graph Gby
exchanging messages between factor nodes and variable
3As noted in footnote 1, if the signal xis modulated prior to transmission,
one can easily add an additional “layer” to the factor graph in Fig. 2 containing
a set of Mmodulated signal variable nodes Xmconnected via modulation
factor nodes Fmwith the corresponding measurement nodes X.
nodes in both directions. As a general rule, at any variable or
factor node, an outgoing message on any edge is obtained as
a function of incoming messages from all other edges, using
the message calculation rules presented below. In general, the
underlying factor graph describing joint state estimation and
signal detection problem (Fig. 2) will contain cycles, thus the
resulting GBP will be iterative, which means that all nodes
will iteratively repeat message updates on all of the outgoing
edges according to a given message-passing schedule. We
provide details on message-passing schedule, correctness and
convergence of GBP on loopy graphs later in this section.
Let us consider a variable node vi∈ V incident to a factor
node fj∈ F. Let N(vi)denote the index set of factor nodes
incident to vi, and N(fj)denote the index set of variable
nodes incident to fj. We denote messages from vito fjand
from fjto vias µvifj(vi) = (mvifj, σ2
vifj)and
µfjvi(vi) = (mfjvi, σ2
fjvi), respectively. Note that, in
the GBP scenario, all messages exchanged across the factor
graph represent Gaussian distributions defined by the
corresponding mean-variance pairs (m, σ2). Thus to describe
processing rules in a variable and a factor node, it is
sufficient to provide equations that map input (m, σ2)-pairs
into the output(m, σ2)-pair, as detailed below.
Message from a variable node to a factor node: the
equations below are used to calculate
µvifj(vi) = (mvifj, σ2
vifj):
mvifj= X
k∈N (vi)\j
mfkvi
σ2
fkvi!σ2
vifj(10a)
1
σ2
vifj
=X
k∈N (vi)\j
1
σ2
fkvi
.(10b)
Message from a factor node to a variable node: In the
setup under consideration, factor nodes represent linear
relations between variable nodes. Thus, e.g., for a factor
node fj, we can write the corresponding linear relationship
as:
fj(vN(fj)) = Civi+X
k∈N (fj)\i
Ckvk.(11)
With this general notation, the equations below provide
µfjvi(vi) = (mfjvi, σ2
fjvi):
mfjvi=1
Ci X
k∈N (fj)\i
Ckmvkfj!(12a)
σ2
fjvi=1
C2
i X
k∈N (fj)\i
C2
kσ2
vkfj!.(12b)
Calculation of marginals: Applying the above rules in
variable and factor nodes of the factor graph results in the
sequence of updates of messages exchanged across the edges
of the graph. To complete description of loopy GBP, we
need to define message initialization at the start, and
message scheduling during the course of each iteration,
which is done next. After sufficient number of GBP
iterations, the final marginal distributions of the random
variables corresponding to variable nodes is obtained as:
ˆmvi= X
k∈N (vi)
mfkvi
σ2
fkvi!σ2
vi(13a)
1
ˆσ2
vi
=X
k∈N (vi)
1
σ2
fkvi
.(13b)
GBP Message-Passing Schedule, Correctness and
Convergence: We adopt standard synchronous GBP
schedule in which variable node processing is done in the
first half-iteration, followed by the factor node processing in
the second half-iteration. The iterations are initialized by
input messages from Fygenerated from the received signal
y, and initial messages from Fxand Fsthat follow certain
prior knowledge (as detailed in the next section).
GBP performance on linear models defined by loopy
factor graphs is fairly well understood. For example, if the
GBP converges, it is known that the GBP solution will
match the solution of the MMSE estimator. The convergence
criteria can also be derived in a straightforward manner, by
deriving recursive fixed point linear transformations that
govern mean value and variance updates through the
iterations and investigating spectral radius of such
transformations. Due to space restrictions, we leave the
details of the convergence analysis in our scenario for the
future work.
IV. NUM ERI CA L CASE ST UDY: SMA RT GRID STATE
EST IM ATI ON IN 5G C- RAN
In this section, we specialize our state estimation setup for a
case study in which we perform power system state estimation
by collecting measurements via 5G-inspired C-RAN.
Power system state estimation - DC model: For the sake
of simplicity, in the following, we consider the linear DC
model of a power system. The DC model is an approximate
model obtained as a linear approximation of the non-linear
AC model that precisely follows the electrical physical laws
of the power system. In the DC model, the power system
containing Nbuses is described by Nstate variables
s= (s1, s2,...,sN)T, where each state variable si=θi
represents the voltage angle θi(in the DC model, the
magnitudes of all voltage phasors are assumed to have unit
values). In the DC model, the measurements include only
active power flow Prk at the branch (r, k)between the bus r
and the bus k, active power injection Printo the bus r, and
the voltage angle θr. Collecting Mof such arbitrary
measurements across the power system, we obtain the
measurement vector x= (x1, x2,...,xM)T, where each
measurement xi∈ {Prk , Pr, θr}is a linear function4of the
(sub)set of state variables s, additionally corrupted by
additive Gaussian noise of fixed (normalized) noise standard
deviation of σnper unit (p.u.). The noisy measurements x
are then transmitted via C-RAN network as described below.
4More precisely, we have that Prk =brk (θrθk)and Pr=
Pk∈Nrbrk (θrθk), where Nris the set of adjacent buses of the bus
rand brk is susceptance of the branch (r, k).
We illustrate the methodology using the IEEE test bus case
with 30 buses shown in Fig. 3 (N= 29, since one of the bus
voltage angles is set to the reference value zero) that we use
in the simulations. The example set of Mmeasurements is
selected in such a way that the system is observable with the
redundancy M/N. For each simulation scenario, we generate
1000 random (observable) measurement configurations.
10 21
22
24
25
26
27
28
29
30
6
20
17
9
11
15 18 19
23
13 12
14
16
8
1 3 4
5
7
2
Fig. 3. The IEEE 30 bus test case divided into disjoint sub-rectangles.
C-RAN cellular network model: The set of M
MTC-UEs simultaneously transmit their measurements to the
set of LRRHs during a given allocated time-frequency slot
shared by all MTC-UEs. We assume MTC-UEs and RRHs
are placed uniformly at random following independent
Poisson Point Process (PPP) in a unit-square area, however,
with slight refinement of the PPP placement strategy.
Namely, to account for neighboring relations within logical
topology of IEEE 30 bus test case, we first divide a
unit-square into w×qdisjoint sub-rectangles as shown in
Fig. 3, and then we assign MMTC-UEs to one of w·q
sub-rectangles. We also balance the number of RRHs per
sub-rectangle, thus allocating L/(w·q)RRHs per
sub-rectangle. Finally, all RRHs and MTC-UEs allocated to
a given sub-rectangle are placed using the PPP within a
given sub-rectangle.
After the placement, we assume MMTC-UEs transmit
their signals x, where each measured signal is normalized to
its expected normalization value5. For the channel
coefficients between the MTC UEs and RRHs, we assume
the following model:
hi,j =γi,j dα
i,j (14)
where γi,j is the i.i.d. Rayleigh fading coefficient with zero
mean and unit variance, di,j is distance between i-th
MTC-UE and j-th RRH and αis the path loss exponent. We
5We assume normalization constants are known in advance at MTC-UEs
and C-RAN nodes, either as a prior knowledge or by long-term averaging.
use channel sparsification approach proposed in [7], with
threshold distance set to d0=pw2+q2(i.e., equal to the
diagonal length of each sub-rectangle). The received signal
y= (y1, y2,...,yL)collected at LRRHs is additionally
corrupted by additive Gaussian noise, whose standard
deviation is selected so as to provide fixed and pre-defined
signal-to-noise ratio (SNR) value. Finally, noisy received
signal yis forwarded via high-throughput backhaul links to
C-RAN BBUs.
GBP-based State Estimation: Using the approach
presented in Section III, we apply GBP across the factor
graph illustrated in Fig. 2 to recover the state estimate x
from the received signal y. More precisely, for each random
measurement configuration, we generate the part of the
factor graph GAand, similarly, from known MTC-UE and
RRH random positions, we derive6the part of the factor
graph GH. Upon reception of y, the GBP runs until it
converges. We adopt a synchronous scheduling of GBP
messages where messages are synchronously flooded from
the factor nodes to variable nodes and back within a single
GBP iteration. For a linear model, it is well known that if
the GBP converges, it will converge to the minimum
mean-square error (MMSE) estimate of the state x.
Simulation Results: In the first set of experiments, we fix
the relative RRH density L/M = 1,SNR = 10, and
redundancy M/N = 3. We investigate the accuracy of the
GBP solution of state estimate as a function of different
values of measurements noise σn={101,102,103,
104}.
101102103104
0.00
2.00
4.00
6.00
8.00
Measurement noise σn(p.u.)
RMSE
Fig. 4. The root mean square error of estimate vector of power system state
variables obtained by C-RAN and without C-RAN model.
Fig. 4 shows the root mean square error
RMSE = (1/N)||ˆscˆs¯c ||2, where ˆscand ˆs¯c are estimate
vectors of power system state variables obtained with and
without C-RAN model discussed in this paper, respectively,
for different values of measurement noise σn. For the case
without C-RAN model, we assume measurements xare
available at BBUs as they are, i.e., without additional noise
or errors. In practice, this could be obtained via standard
grant-based uplink procedures where each MTC UE is
6We note that, in case the small-scale fading is included in the model, one
can assume that the channel state information is available at the C-RAN.
allocated separate orthogonal resources. However, such a
strategy incurs significant delay as the underlying system
scales, due to message exchange delay, resource allocation
delay, as well as ARQ-based error-correction strategies. Note
that the C-RAN model described in this paper admits very
low latency as all MTC UEs transmit their signals
immediately and concurrently. According to the box plot in
Fig. 4, the C-RAN approach is able to reach nearly identical
solution as the approach without C-RAN (e.g., RMSE 0),
if the value of measurements noise is sufficiently low. Note
that the typical value (standard deviation) of the
measurement noise for devices located across a power
system are in the range between 102p.u.and 103p.u., for
legacy measurement devices, and between 104p.u.and
105p.u., for phasor measurement units. Consequently, the
presented approach is suitable for the state estimation in
power systems.
In the next simulation experiment, we investigated the
system observability as a function of the number of RRHs L
deployed in the system, for different values of redundancy
M/N. We start with L/M = 0.2and increase the RRH
density in order to evaluate its effect on the system
observability.
0.2 0.4 0.6 0.8 1 1.2
0
50
100
Relative RRH density L/M
Unobservable Topologies (%)
M/N
1.5
2.0
2.5
3.0
3.4
Fig. 5. The fraction of unobservable system topologies for different values
of measurement redundancies M/N versus relative RRH density L/M.
Fig. 5 shows the fraction of instances GBP was not able
to converge due to insufficient rank of the underlying system
as a function of the number of base stations L. Note that the
fundamental condition for the system to have full rank is
that LN. By slightly expanding this condition, we get
(L/M)·(M/N)1. For all the points in Fig. 5 for which
this condition is not satisfied, the system is unobservable. If
the condition is satisfied, then in each simulation run, a
random measurement configuration is verified to provide an
observable system, thus the rank insufficiency may only
appear as a consequence of the C-RAN topology and the
channel matrix sparsification. According to Fig. 5, for the
parameters used in our simulations, we can see that GBP
generally performs well, however, in the region where
(L/M)·(M/N)is slightly above 1, rank insufficiency may
deteriorate the performance.
Overall, for systems with large number of MTC UEs M,
the state can be estimated with relatively small number of
RRHs L. In contrast, for the scenario where a number of
MTC UEs is small, the number of RRHs must be increased
for successful reconstruction. In addition, simulation results
point to capability of the proposed scheme to provide
successful reconstruction if the underlying system is
observable (i.e., of full rank), while the accuracy of
reconstruction (i.e., the accuracy of the state estimator) will
depend on the parameters such as channel sparsification,
SNR, measurement standard deviations and number of MTC
UEs and RRHs. We leave detailed study of these
inter-dependencies for our future work.
V. CO NC LUS IO N S
Motivated by the development of 5G massive MTC and
large-scale distributed 5G C-RAN architecture, in this paper,
we proposed a scalable and efficient linear state estimation
framework. The proposed framework is based on the GBP
algorithm and jointly combines linear state estimation with
signal detection in 5G C-RANs. The advantage of GBP
solution is accuracy that matches the MMSE estimation, low
complexity due to lack of scheduling MTC-UE
transmissions, low latency due to simultaneous data transfer,
scalability to large-scale systems (due to the fact that the
underlying factor graph is usually sparse), and ease of
parallelization and distributed implementation in future
distributed F-RAN architectures. For the future work, we aim
to provide rigorous convergence analysis of GBP in the
presented framework, motivated by similar analysis in [7]
and [9], and provide extensive numerical simulation study.
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