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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, [email protected] Department of Mechanical Engineering and Center for Advanced Power Systems, Florida State

University, Tallahassee, FL 32310, Tel: (850) 644-8405, [email protected]

*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].

Excerpt from the Proceedings of the COMSOL Users Conference 2006 Boston

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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.


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