IEEE Power & Energy Magazine - September/October 2019 - 79

systems. On the one hand, they derive concepts from MBSE
to address the heterogeneity of interdependent infrastructure
systems; on the other, they draw their quantitative nature
from the formal mathematics of graph theory.
Figure 7(a) shows a small network depicting a multiinfrastructure system. The four nodes represent a water
treatment facility, a solar photovoltaic (PV) array, a house,
and a parking lot at a work location. These nodes are connected by power lines, a water pipeline, and a road. The
nodes of the graph presented in Figure 7(a) are not differentiated mathematically on the basis of the functions of
the four resources in the system. In contrast, heterofunctional graphs [Figure 7(b)] differentiate themselves from
conventional graphs, as they introduce a new ontology.
More specifically, the nodes and edges of the graph are
no longer physical resources. Instead, the nodes (denoted
by }1 -}11) of the heterofunctional graph are system capabilities, and the directed edges represent their sequential
couplings (i.e., that the input of one capability follows the
output of another).
Here, the system capabilities are defined as the mapping
of a system function onto system form. This creates a new
framework. For example, instead of being portrayed as a single node, the house is now represented by three capabilities
(shown as capabilities }3, }4, and }5). Similarly, the edges

of the traditional network, in Figure 7(a) (e.g., the power line
from the solar array to the house) are also expressed as capabilities. The physical resource power line transports power
from the solar array to the house. Node }9, therefore, represents the capability "power line transports power from the
solar array to the house."
The system's capabilities are then coupled sequentially
to describe how operands can move through the system. All
of the feasible ordered pairs of capabilities (e.g., }1, }7) can
be organized into a heterofunctional graph or network. This
sequential coupling adheres to physical continuity laws and
to the reference architecture of the system. For example, two
capabilities precede the capability }3, "house consumes
water." First, water needs to be transported from the water
treatment facility to the house with capability }7 . Second,
electric power needs to be transported from the solar array
to the house with capability }9 .
In traditional graphs, edges may be either directed one
way or undirected two way. This difference is accounted for
in heterofunctional graphs with one or two transportation
capabilities, respectively. The undirected graph edge performs two functions: transport from house to work location
and transport from work location to house.
The simple example serves to demonstrate the ability of
heterofunctional graphs to explicitly model and differentiate

ψ1

ψ3

ψ7
ψ4

n4
(a)

Legend
Nodes:
n1: Water Treatment Facility
n2: Solar PV
n3: House
n4: Work Location
Edges:
e1: Water Pipeline
e2: Power Line 1
e3: Power Line 2
e4: Road

ψ8
ψ9
ψ2

ψ10

ψ5

ψ11

ψ6
(b)

Degrees of Freedom:
ψ1: Treat Water at Water Treatment Facility
ψ2: Generate Electricity at Solar PV
ψ3: Consume Water at House
ψ4: Charge EV at House
ψ5: Park EV at House
ψ6: Park EV at Work Location
ψ7: Transport Water From Water Treatment Facility to House With Water Pipeline
ψ8: Transport Power From Solar PV to Water Treatment Facility With Power Line 1
ψ9: Transport Power From Solar PV to House With Power Line 2
ψ10: Discharge EV From House to Work Location With Road
ψ11: Discharge EV From Work Location to House With Road

figure 7. A traditional versus heterofunctional graph representation of a small network: (a) the original network topography and (b) the heterofunctional adjacency matrix. EV: electric vehicle; PV: photovoltaic.
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IEEE Power & Energy Magazine - September/October 2019

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