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7/28/2019 Morega
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Constructal Design of Concurrent Power Distribution Networks
Alexandru M. Morega1
and Juan C. Ordonez*,2
1 Department of Electrical Engineering, POLITEHNICA University of Bucharest, Bucharest 060042,
Romania, Tel: +40 21 402 91 53, amm@iem.pub.ro2 Department of Mechanical Engineering and Center for Advanced Power Systems, Florida State
University, Tallahassee, FL 32310, Tel: (850) 644-8405, ordonez@caps.fsu.edu
*Corresponding author
Abstract: This paper reports a strategy to design a
network of higher immunity and capacity of recovery
to faults or other destructive events. The capacity of
survivorship and recovery increases when the system
possesses some redundancy, and this means sources
and lines and switching capability able to take over
areas that loose connectivity to their parent source.
Therefore, a second network, comparable in terms of
size, coverage, and power flow is provided in the
region of interest. The internetworking strategy relies
on the constructal theory, which is concerned with
optimizing the access path from a volume (here, the
territory where the nodes of consumption are located)
to a point (here, the source), under given resources
(the network), and subject to internal and external
constraints (local restrictions, finite size area, ports of
access to the territory). Further to this approach the
structure of the network is an outcome of the
optimization principle.
Keywords: Networks, constructal, power
distribution, tree, numerical simulation, FEM
1. IntroductionThis study’s objective is the identification of a
strategy to design a power network or to modifythe layout of an existing one such as to increase
its immunity and capacity of recovery to faults or
other destructive events. The capacity of
survivorship and recovery increases when the
system possesses some redundancy to overcome
failures, and this means redundant sources and
lines, and an adequate switching capacity able to
take over areas that loose connectivity to their
basic tree source. It is assumed that a second
network functions in the same area, and provides
energy on a regular basis, such that the twonetworks are comparable in terms of size,
coverage, and power flow. The nodes of
consumption are evenly distributed between the
two networks.
The internetworking strategy relies on the
constructal theory [1], [2], which is concerned
with optimizing the access path from a volume
(here, the territory where the nodes of
consumption are located) to a point (here, the
source), under given resources (the network),
and subject to internal and external constraints
(local restrictions, finite size area, ports of access
to the territory). Further to this approach the
structure of the network is an outcome of the
optimization principle.
We rely on the principle that the consumption
nodes and the spanning trees for the twonetworks should be at close distance at all scales
– e.g., as counter and concurrent flow systems
are. To illustrate this process we define a Poisson
boundary value problem for the electric potential
(DC regime). The optimization problem consists
of finding that aspect ratio (shape) for which the
transfer conductance defined through the ratio
between the largest voltage in the region of
interest and the current input is the smallest.
2. Countercurrent flow systems
In counterflow systems two trees are superposedso closely and regularly that each tube in one tree
is in counterflow with a similar tube in the
second tree. Nature offers numerous examples of
systems with streams in counterflow, and the
general opinion is that this structure reduces theconvective current (heat, or mass species) that
flows longitudinally [2]. Figure 1 shows a sketch
of the countercurrent arterial and venous
systems. At all levels, from large vessels to
capillaries the two trees are closely superposed.
Fig. 1 The circulatory tree system [2].
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Weinbaum and Jiji [2] rediscovered the
expression of the longitudinal heat current
Q =
m cP( )2
Up
dT
d x, (1)
as proportional to the temperature gradient (
m is
the mass flow rate, c P is the specific heat at
constant pressure, p is the area of the contact
surface per unit length of the flow current, U is
the total heat transfer coefficient between the
counter currents), in biomedical engineering, andthe model is incorporated in a heat transfer
model of the vascularized tissue to account for
the occurrence of countercurrent pairs of
thermally significant blood vessels. In this
model, the heat longitudinal heat current makes
an additional, convective contribution to the
effect of conduction through the living tissue.
a. First order
ensemble
b. Second order
ensemble.
c. Third order
ensemble.
Fig. 2 Counterflow heat exchanger [2].
Bejan [2] explains by the constructal
formalism the emergent shapes and structures for
a number of countercurrent trees found in both
animated and unanimated systems. Figure 2
shows an engineered countercurrent system that
is constructal optimized for heat transfer.
This system, largely encountered in natural
and engineered systems, is optimized for heat
transfer through fluid flow. The “exchanger”
concept is next adapted and utilized in our proposed solution to the networking problem.
3. Concurrent trees
In constructal theory, the “elemental cell” is the
smallest, simplest system, or “construct”, that
possesses the basic features, properties and
constraints (structure, resources, sources,
physical laws) as related higher order (sizes)
systems. The elemental cell generates – by
growth, in a time-arrow sequence – higher order
ensembles.
In concurrent flows, form the elemental cellup to high order ensembles, the two trees are
conveying current at all levels. As the object of
our work is the concept for a reconfigurable
network, connectivity (electrical current, power
flow paths) is to be outlined first. Therefore, to
illustrate the constructal design we consider, the
DC regime – although less representative for
general networking – provides for a simpler,
satisfactory (connectivity level) analysis.
a. Unconnected networks.
b. Connected networks.
Fig. 3 The elemental cell. Two designs for two
concurrent tree networks. The boundary conditions.
An area of consumption that requires (sinks)
current uniformly and which has a low electrical
conductivity (
"0<<"
p) is fed through two high
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conductivity tree networks, which cover evenly
the region of consumption (Fig. 3). Given are the
amount of high conductivity material, the area of
the cell (i.e., the current demand) and the
structure of the trees. The degree of freedom
used in the constructal optimization problem isthe cell aspect ratio, H / L. The optimization
problem consists of finding the cell aspect ratio
(shape) for which the transfer admittance,
defined through the ratio between the largest
voltage in the cell and the current input to the
cell, is the smallest.
Two closely related designs are shown Fig.
3. The basic difference is that while the two lines
in Fig.3.a are galvanic isolated and may be inter-
connectable on demand only in Fig.3.b they are
seen connected. In this study we focus on the
unconnected network – two concurrent trees.
This design is important when no current pathregularly exists between the two coexisting trees.
However, such paths are possible (generated)
when the trees are to be connected by some
reason, e.g., when some part of one tree losses its
connection to its eigen source by some failure
and has to be powered by the second tree,
presumably active.
As stated, the high conductivity material,
with no current consumption, is given. Although
its partition among fingers (their number and
size) is an optimization problem too, we are
concerned here with only introducing the
network topology through a boundary value
problem. Therefore we investigate only the
constructal morphing problem starting from
elemental cells where the high conductivity
material is divided in four fingers and two bus
bars. The boundary value problem for the
elemental cell is then
In the region that s inks current
"2V
" x2+
"2V
" y2#
$ $ $ w
%0
= 0 , (2)
In the region that conveys the current
"2V
" x2+
"2V
" y2= 0 . (3)
Homogeneous Neumann conditions are assumedon all sides of the cell, except for the inlet ports
(the boundaries that are orthogonal to the
transport lines), where homogeneous Dirichlet
conditions are set. In equations (1), (2) V is the
electrical potential and " " " w is the current sink,
assumed uniform. The current density is related
to the voltage by J p,0 = "# p,0$V within the
transport and sink regions; σ p and σ0 are the
electrical conductivities of the sink and transport
regions, respectively.
Equations (1) and (2) can be written in non-
dimensional form using the cell width, L, as
length scale, (
˜ x = x / L , ˜ y = y / L)
In the region that sinks current
"2 ˜V
"˜ x2+
"2 ˜V
"˜ y2#1= 0, (4)
In the region that conveys the current
"2 ˜V
"˜ x2+
"2 ˜V
"˜ y2= 0 , (5)
where ˜V = V "V
0( ) #V , "V = # # # w L2$0
and V 0
is set to the ground value (0 V).
3. Numerical ImplementationWe used COMSOL [8] multiphysics FEM
software to solve the electric field problem.
Accuracy tests showed that meshes of up to
145000 triangular, Lagrange cubic elements
depending on the ensemble order, provide grid-
independent numerical solutions. Figure 1 shows
a detail of the FEM mesh for a first order ensemble in the unconnected, concurrent current
constructal sequence.
Fig. 4 FEM mesh detail (32514 elements).
4. Results
Figure 5 shows the optimal elemental cell (a)
and the electrical field (b) (absolute value of ˜
V ).The optimal design is that that provides the
minimum voltage (nondimensional).
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a. Optimized cell. b. Voltage distribution.
Fig. 5 The optimal (constructal) elemental cell. Two
unconnected tree networks convey current in a region
of uniform current consumption.
The next step in adding complexity to the
design is to assemble three optimized elemental
cells (Fig.6), which results in a first order ensemble. The elemental cell in this structure is
outlined later, with respect to the corresponding
electrical circuit topology (Fig.8).
a. The grid made of three concurrent tree networks.
b. The voltage. Fig. 6 The first order ensemble. The unconnected
concurrent tree networks convey current in a larger size
region of uniform current consumption independently.
a. The two tree networks are marked in red.
b. The voltage distribution.
Fig.7 Second order ensemble. The disconnected
concurrent tree networks carry current independently in
a larger region of uniform current consumption.
Three first order ensembles may be merged
then to produce then a second order ensemble(Fig.7), and so on.
This simple replication of the elemental cell
is not proven to produce an optimized first order ensemble. Therefore, to find the optimum design
(highest admittance) we performed a sequence of
simulations where the shape factor, H / L,was
varied.
Fig. 8 Optimization sequence for unconnected trees
(non-dimensional).
Figure 8 summarizes this optimization
sequence. At each level, from the elemental cellto the highest order ensemble (here, 2), different
aspect ratios of the computational domains were
used while keeping constant the “amount” of material (the areas of low and high conductivity).
In each case, the minimum voltage indicates the
optimal, highest admittance design. For a better
comparison, the voltages are divided by the
minimum values recorded for each sequence.
In this study, topology is of interest, i.e., the
branches and branching nodes rather then the
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nature of the branches or their real geometry).
Figure 8 depicts, in a topological manner, the
elemental cell, the first and second order
ensembles for the equivalent electrical network.
The consumption nodes convey the current to
equal size, evenly distributed patches of territory.The nodes used for switching purposes only, to
enable supplementary path between the two
independent (unconnected) trees, are not shown.
The nodes that deliver power to the
elemental patches are assumed equal in terms of
load, I 0, P 0, and areas of dependence, 2l 0×2l 0.
The networks are traced in different colors (black
and blue, respectively).
a. Elemental cell. b. First order ensemble.
c. Second order ensemble.
Fig. 8 Unconnected concurrent electrical networks. The
elemental cell is evidenced in the first and second order
ensembles
5. Conclusions
The DC model provides a fairly accurate
approximation of the steady state behavior of a
power grid. It is vastly popular in the electric
power literature, being the model of choice anytime that many power flow computations are
needed, or in the network design phase when
topology and connectivity are evaluated.
The constructal theory, used as networking
strategy, unifies the extremely wide class of
engineered and natural flows that connect an
infinite number of points (volume, area) to one
or more discrete points (sources, sinks). For this
type of problems, optimization of access under
global constraints indicates that a tree-like
structure minimizes global resistance.
The optimization problem we solved is
related to a field model, where the sink iscontinuously distributed throughout the domain.
Although different from the electrical network
problem, where the sink is discrete (a set of
nodes of consumption), directly connected to the
grid, this field problem provides an important
message and a conceptual solution to the grid
problem: the electrical network should bear the
same major features – two concurrent current
network trees, either connected or not, the
constructal design where the elemental cell
generates higher order ensembles that posses at
each scale the same topologic properties.
The tools developed as part of this effort areuseful also in the evaluation and selection of
planning routes or specific “access path” to
selected nodes or group of nodes in the network
– one way of conceiving the network
reconfiguration.
8. References
1 Poirier H., “Une theorie explique l'intelligence de
la nature”, Science & Vie, 1034 (2003) 44-63.
2 Bejan, A. Shape and Structure, from Engineering
to Nature, Cambridge Univ. Press, 2000.
3 Bejan, A., Errera, M.R., “Convective trees of
fluid channels for volumetric cooling”, Int. J. Heat Mass Transfer , 43, (2000) 3105-3118.
4 Vargas, J.V.C., Ordonez, J.C., Bejan, A.
“Constructal Flow Structure for a PEM Fuel
Cell ,” International Journal of Heat and Mass
Transfer , 47 (2004) 4177–4193.
5 Ordonez, J.C., Bejan, A., Cherry, R.S.,
“Designed Porous Media: Optimally Nonuniform
Flow Structures Connecting One Point With One
Or More Points,” International Journal of
Thermal Sciences, 42 (2003) 857-870.
6 Morega, A.M., Bejan, A., “A constructal
approach to the optimal design of photovoltaic
cells,” Int. J. Green Energy, 2, 3 (2005) 233-242.
7 Arion, V., Cojocari, A. and Bejan, A.,
“Constructal tree shaped networks for the
distribution of electrical power,” Energ.
Conversion Management , 44 (2003) 867-891.
8 Arion, V., Cojocari, A. and Bejan, A., “Integral
measures of electric power distribution networks:
load-length curves and line-network multipliers,”
Energ. Conversion Management , 44 (2003)
1039-1051.
9 Comsol A.B., v.3.2b, Sweden, 2006.
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9. Acknowledgements
The authors acknowledge with gratitude the
support from the U.S. Department of Energy,
Office of Electricity Delivery and EnergyReliability, Award number DE-
FG0205CH11292, the Office of Naval Research
(ONR), and the CEEX grant 05-D11-25/2.10.05.
Excerpt from the Proceedings of the COMSOL Users Conference 2006 Boston
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