Capacitated Multicommodity Flow Problem for Heterogeneous Smart Electricity Metering Communications Using Column Generation
Abstract
:1. Introduction
 We provide a scalable solution that considers a growing population of smart meters, in a way that the network may be flexible regarding the future locations of the nodes selected to be UDAPs;
 We provide a model that takes into account both the wireless links capacities and the capacity demands from the active AMI traffic flows; and
 We demonstrate that it is possible to reduce infrastructure costs in the heterogeneous smart metering network with the proposed CMCFAMI methodology for a neighborhood area network (NAN).
2. Related Work
3. Problem Definition and Proposed Solution
 The sum of flows i through the path l must be equal to ${\phi}_{i}$, which is described by Equation (2).$$\left(i\right)\sum _{i\in F}{X}_{i,l}={\phi}_{i}\phantom{\rule{1.em}{0ex}}\forall i\in F$$
 The sum of all flows routed over a link should not exceed the link’s capacity, which is described by Equation (3).$$\left(ii\right)\sum _{i\in F}\sum _{l\in R}{X}_{i,l}{R}_{i,l}^{e}\le Ca{p}_{e}$$
Algorithm 1 Primary problem: $\phi $. 
Input: $topo$ Output: $obj,x,y,Base,topo$ Data: $E=\parallel topo\parallel ;N=\parallel Coord\_SM\parallel ;A(N+E)=0;ind=1$

Algorithm 2 Secondary problem algorithm: $\mathsf{\Psi}$. 
Input: $y,topo$ Output: $Aggregate,topo$ Data: $N=\parallel topo.N\parallel ;E=\parallel topo.i\parallel ;$ $Aggregate=0;$ $e\_cong=max\left(y\right)N;[topo.i(e\_cong),topo.j(e\_cong\left)\right];$ $costB=topo.costlink;$

Algorithm 3 Matrix generator: $\mathrm{Y}$. 
Input: $topo,conf,radio\_WiFi,radio\_Cellular$ Output: $dist,G,costDist,costlink,Cap$ Data: $n=\parallel topo.N\parallel $;

Algorithm 4 Topology generator: $\mathsf{\Phi}$. 
Input: $Coord\_SM,Coord\_EB,conf,topo$ Output: $topo,conf$ Data: $cost=costlink;cost(cost==inf)=0;\phantom{\rule{0ex}{0ex}}\phantom{\rule{8.11317pt}{0ex}}cost=sparse\left(cost\right);topo.index=index;\phantom{\rule{0ex}{0ex}}\phantom{\rule{8.11317pt}{0ex}}topo.i=i;topo.j=j;p=\left[\phantom{\rule{7.11317pt}{0ex}}\right]$

4. Analysis of Results
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Author, Year  Scalable Deployment  Coverage Scenario  Concentrator’s  Links’ Capacity (Flows, Congestion)  Minimal Costi  Energy Consumption  Others Characteristics 

Guodong Wang et al., 2018 [12]  ✖  ✓  ✖  ✖  ✓  ✓  MaxMin distance SMDAP 
Inga et al., 2017 [6]  ✓  ✓  ✓  ✖  ✓  ✖  Scalable population increase 
Yanxiao Zhao et al., 2018 [13]  ✖  ✓  ✖  ✖  ✓  ✓  Min distance to DAP 
Asif Hassan et al., 2017 [14]  ✖  ✓  ✖  ✖  ✓  ✓  Kmeans Clusterization 
Guodong Wang et al., 2016 [45]  ✖  ✓  ✓  ✖  ✓  ✓  Latency 
Yanxiao Zhao et al., 2017 [15]  ✖  ✓  ✓  ✖  ✓  ✓  Kmeans Clusterization Latency 
Guodong Wang et al., 2018 [45]  ✖  ✓  ✓  ✖  ✓  ✓  MaxMin distance SMDAP 
Sastry Kompella et al., 2007 [20]  ✖  ✖  ✖  ✓  ✓  ✖  Crosslayer Scheduling Matching 
Jihui Zhang et al., 2005 [19]  ✖  ✖  ✖  ✓  ✓  ✖  Routing Scheduling Multiradio 
Proposed work (CMCFAMI)  ✓  ✓  ✓  ✓  ✓  ✓  Capacitated Multicommodity Flow 
Variable  Definition 

e  Link 
i  Flow 
l  Path between nodes 
E  Set of links 
F  Set of flows of data 
R  Set of paths 
${C}_{e}$  Link cost 
${X}_{i}{,}_{l}$  Flow i through the path l 
${R}_{i}^{e}{,}_{l}$  Path of flow i in the link e 
${\phi}_{i}$  The sum of flows i in the path 
$Ca{p}_{e}$  Link capacity 
Variable  Definition 

$Coord\_SM,Coord\_EB$  Georeferenced coordinates SM, BE 
$radio\_WiFi,radio\_Cellular$  Coverage radius WiFi, cellular 
$WiFi,Cellular$  Indices 
$super$  Vector of SMs with dual technology 
B  Initial feasible base in the simplex method 
C  Vector of first solution that is basic and feasible 
$conf.\left\{capacity\phantom{\rule{4pt}{0ex}}cost\phantom{\rule{4pt}{0ex}}costdist\phantom{\rule{4pt}{0ex}}factor\right\}$  Vectors: constraints 
$obj$  Value of the objective function in each iteration 
$dist$  Haversine distance 
$topo$  Wireless topology 
$ind$  Index referred to SM 
$link\_disp$  Set of SMs considered for cellular connectivity 
$temp$  Temporary variable 
M  Total number of BEs 
N  Number of SMs 
G  Connectivity matrix 
$\phi $  Primary problem: vobj, x, y, base, topology (topo) 
$\mathsf{\Psi}$  Secondary problem, new topo 
$\mathsf{{\rm Y}}$  Matrix: dist, G, CostDist, CostLink, Cap 
$\mathsf{\Phi}$  Topology: topo, conf 
$x,y$  Primal solution, dual of the simplex method 
$Base$  Set of feasible elements 
$CostDist$  Cost in relation to distance 
$Costlink$  Cost of each link 
$Aggregate$  Binary variable: 1, added node; 0, no added node 
$Cap$  Matrix of capacities per node 
$vobj$  Vector objective function 
K  Increment factor related to distance 
$pred$  Path predecessor tree function, Dijkstra 
E  Link set 
A  Generation of column matrix 
$e\_cong$  Congested link 
$path,nodo$  Set of SM in a path, index of SM 
$CostB$  Matrix of costs of the topology 
$SM\_Cellular$  Index of SMs with cellular technology 
$optimo\_SM\_Cellular$  Optimal index of SMs with cellular technology 
$totalcost$  Total cost of the links at the end of the program 
p  Link to be discarded due to congestion 
e $cost$  Vector: cost factor 
Scenario  Population  Cellular  WiFi  

#  # of SMs  LTE  UMTS  GPRS  802.11 
1  32  15  12  6  29 
2  64  20  16  8  60 
3  96  35  28  14  89 
4  128  50  40  20  118 
5  160  65  52  26  147 
6  192  80  64  32  176 
Scenario  Population  Chargeability WiFi  Wireless Technology  

#  # of SMs  Average  Cost: SMCellular  Cost: SMWiFi 
1  32  3.08  3  29 
2  64  3.21  4  60 
3  96  3  7  89 
4  128  3.02  10  118 
5  160  2.92  13  147 
6  192  2.84  16  176 
Scenario  Population  Link Use  Occup Link 

#  # of SMs  Mean  Mean % 
1  32  1.97  10.7 
2  64  2.5  14.1 
3  96  2.56  14.7 
4  128  2.93  16.8 
5  160  3.22  18.9 
6  192  3.46  20.3 
Scenario  Population  Iterations  Column Size  

#  # of SMs  1 Stage  2 Stage  Total  $\mathit{x}$  $\mathit{y}$  Base 
1  32  56  179  235  534  316  316 
2  64  56  208  264  999  712  316 
3  96  56  329  385  2080  1148  316 
4  128  56  169  225  2282  1574  316 
5  160  56  133  189  3476  2002  316 
6  192  56  62  118  3972  2494  316 
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Inga, E.; Hincapié, R.; Céspedes, S. Capacitated Multicommodity Flow Problem for Heterogeneous Smart Electricity Metering Communications Using Column Generation. Energies 2020, 13, 97. https://doi.org/10.3390/en13010097
Inga E, Hincapié R, Céspedes S. Capacitated Multicommodity Flow Problem for Heterogeneous Smart Electricity Metering Communications Using Column Generation. Energies. 2020; 13(1):97. https://doi.org/10.3390/en13010097
Chicago/Turabian StyleInga, Esteban, Roberto Hincapié, and Sandra Céspedes. 2020. "Capacitated Multicommodity Flow Problem for Heterogeneous Smart Electricity Metering Communications Using Column Generation" Energies 13, no. 1: 97. https://doi.org/10.3390/en13010097