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Modeling Smart Cities with Hetero-functionalGraph Theory
Wester C.H. SchoonenbergThayer School of Engineering
at Dartmouth CollegeHanover, New Hampshire
Amro M. FaridThayer School of Engineering
at Dartmouth CollegeHanover, New Hampshire
Email: [email protected]
Abstract—In the 21st century, urbanization as a mega-trendwill create many megacities. These highly dense, large populationcenters will have to efficiently deliver essential services including,energy, water, mobility, manufactured goods, and healthcare.While these services may be treated independently, they arein reality interdependent, especially as the need for efficientresource utilization, and consequently integration. This presentsa formidable engineering challenge as the modeling foundationsfor these services have traditionally been discipline specific.Furthermore, efforts to integrate these modeling foundationshave often adopted simplifying constraints which have limitedapplicability to the emerging challenge of smart cities. Thispaper collates an emerging “hetero-functional graph theory”for potential application to integrated smart city infrastructuremodels. It has been recently demonstrated in several applica-tion domains. The paper concludes with the construction of ahetero-functional graph for a smart city model consisting of anintegrated electricity, water, and transportation system. Such agraph has the potential for dynamic modeling, resilience analysis,and integrated decision-making.
Keywords – Hetero-functional Graph Theory, Infrastructure,Smart City, Axiomatic Design, Engineering Systems
I. INTRODUCTION
In the past 50 years, the percentage of global populationliving in cities has increased from 34% to 52% [1]. Thismassive migration has resulted in high density urban areasand increases the stress on the existing infrastructure [2].The existing infrastructure delivers essential services includingenergy, water, mobility, manufactured goods, and healthcare.These infrastructures are not only complex and of a large size,but also interdependent, especially as there is need for efficientresource utilization, and thus integration. The interactionsin this “system of systems” create unpredictable and poorlyunderstood behavior.
Other trends in urban environments, are to include tech-nology and improve efficiency based on the data generatedby the technology. Examples include smart meters, smartthermostats, smart lighting [3], but also smart traffic control[4]. In order to understand the measured behavior of thecomplex systems, a model needs to be developed that includesthe interdependencies amongst the networks. For networkbased systems, such as transportation systems, power grids,water networks, supply chains, and healthcare systems, a graphtheoretic approach has formed the basis of much research [5].
Examples include the maximization of power system resiliency[6]–[8], shortest path calculations [9], hospital operations [10],suppy chain management [11], and even brain function [12].Graph theory offers many advantages, such as proven optimalsolutions for certain types of problems [13]. Disadvantages ofgraph theory include the limited capability to capture morethan one discipline with the existing mathematics.
The network sciences community has made effort to in-troduce the multilayer features to improve the understandingof complex systems [14]. This approach has the potentialto facilitate modeling of “systems of systems” that crossdisciplinary boundaries. However, the paper written by Kivelaalso discusses the constraints that are imposed by existingliterature [14]. These constraints limit the potential applicationdomains of multilayer network theory. For a smart city modelas described in this paper, a different method needs to beintroduced.
Recently, a graph theoretic modeling approach called“Hetero-functional Graph Theory” has developed from roots inthe Axiomatic Design for Large Flexible Engineering Systems.It provides a rigorous platform for modeling systems ofsystems [15]. An essential aspect of this platform is that itdefines mutually exclusive and collectively exhaustive setsof system processes and resources. In so doing, it facilitatesthe translation of systems engineering models (e.g. SysML)to a mathematical engineering systems description. Hetero-functional Graph Theory has been applied in transportation[16], healthcare [17], production systems [18], power grid [19],and resilience studies [15]. Additionally, hetero-functionalgraph theory has proven to be able to support modeling oftwo unlike systems. This paper applies hetero-functional graphtheory to model a smart city test case in a single mathematicalmodel.
The basis for Hetero-functional Graph Theory is introducedin Section II. Section III introduces Hetero-functional GraphTheory for individual infrastructure systems. The combinationof these systems form a Smart City in Section IV. Section Vconcludes the paper and provides an outlook to future work.
II. HETERO-FUNCTIONAL GRAPH THEORY
This section succinctly covers the need for Hetero-functional Graph Theory. It then describes its foundations as
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a modeling approach for multi-layer engineering systems.
A. Need for Hetero-functional Graph Theory
The emerging challenge of smart cities calls for modelingframeworks that cross disciplinary boundaries. In recent years,the network sciences community has developed significantwork in “multi-layer networks” [14] which has the potential tocross disciplinary boundaries. Nevertheless, a recent review onmulti-layer networks has identified that many of these modelsadopt simplifying constraints. The discussed methods imposeone or more of the following constraints [14]:
1) Alignment of nodes amongst the layers.2) Requirement that the layers need to be disjoint.3) All layers have the same number of nodes.4) The couplings are diagonal.5) The inter-layer couplings consist of layer couplings.6) The inter-layer couplings are categorical.
Such constraints likely limit the number of practical applica-tions, such as modeling smart cities. These integrative modelsshould have the ability to incorporate arbitrary couplingsamongst layers. In that regard, this work collates an emerging“hetero-functional graph theory” for potential application tosmart cities.
B. Foundations of Hetero-functional Graph Theory
This subsection introduces the very elemental basics ofhetero-functional graph theory, a more in depth discussioncan be found in the provided references. The foundationsfor hetero-functional graph theory are ontological. Ontologicalscience encourages the development of models that have theproperties of soundness, completeness, lucidity, and laconicity,which are conceptually understood from Figure 1 [20]. Ulti-mately, these four properties taken together require a 1-to-1mapping of conceptualized abstractions and formal mathemat-ical models [21]. In this work, hetero-functional graph theoryis based upon three abstract concepts which may be foundin the systems engineering and engineering design literature:(1) Allocated Architecture, (2) Hetero-functional AdjacencyMatrix, and (3) Controller Agency Matrix. Each of these isnow addressed in turn.
1) Allocated Architecture: The allocated architecture is astructural description of a system’s capabilities. Mathemati-cally, the allocation of system processes P to resources R isdescribed by a “design equation” [15], [21]–[23]:
P = JS �R (1)
where JS is a binary matrix called a “system knowledgebase”, and � is “matrix boolean multiplication” [15], [21]–[23]. The design equation originates from Axiomatic Designfor Large Flexible Engineering Systems and requires that thesesystem processes and resources be mutually exclusive andcollectively exhaustive [15], [22], [23]. The system resourcesare a combination of transformation, buffering, and transport-ing resources, and the system processes are a combinationof transformation, holding, and transporting processes. The
Figure 1: Graph theoretical representation of mapping betweena models and its abstraction: a soundness, b completeness, clucidity, and d laconicity [20]
knowledge base JS can thus be constructed from the map-pings of these subcategories, where JM is the transformationknowledge base, and JH the refined transportation knowledgebase. Their combinations are provided in Equation 2, and theconstraints matrix is constructed in a similar way (Equation3) [15], [22], [23].
JS =
[JM | 0
JH
](2)
KS =
[KM | 0
KH
](3)
Definition 1. System Knowledge Base [15], [22], [23]:A binary matrix JS of size σ(P ) × σ(R) whose elementJS(w, v) ∈ {0, 1} is equal to one when action ewv ∈ E (inthe SysML sense) exists as a system process pw ∈ P beingexecuted by a resource rv ∈ R. The σ() notation gives thesize of a set.
The system knowledge base itself forms a bipartite graphand has been defined for several different disciplines, such astransportation [16], power [19], mass-customized production[24], water distribution, and health-care delivery systems [17].
The system’s capabilities are quantified as structural degreesof freedom [22], [23]. The existence of these capabilities arerepresented by filled elements in the system knowledge base.Meanwhile, the constraint matrix describes the availability ofthese capabilities in recognition that many systems changetheir capabilities over time.
Definition 2. System Events Constraints Matrix [15], [22],[23]: A binary matrix KS of size σ(P )×σ(R) whose elementKS(w, v) ∈ {0, 1} is equal to one when a constraint eliminatesevent ewv from the event set.
Definition 3. Structural Degrees of Freedom [15], [22],[23]: The set of independent actions ψi ∈ ES that completely
2
Table I: Types of Sequence-Dependent Production Degree of Freedom Measures [15], [25], [26]
Type Measures Processes Resources Knowledge Base ConstraintMatrix
PerpetualConstraint
Measure Function
I DOFMMρ PµPµ M,M JMMρ =[JM · KM
]V [JM · KM
]V TKMMρ K1 = K2 〈JMMρ, KMMρ〉F
II DOFMHρ PµPη M,R JMHρ =[JM · KM
]V [JH · KH
]V TKMHρ k1 − 1 =
(u1 − 1)/σ(BS)〈JMHρ, KMHρ〉F
III DOFHMρ PηPµ R,M JHMρ =[JH · KH
]V [JM · KM
]V TKHMρ k1 − 1 =
(u1 − 1)&σ(BS)〈JHMρ, KHMρ〉F
IV DOFHHρ PηPη R,R JHHρ =[JH · KH
]V [JH · KH
]V TKHHρ (u1 − 1)%σ(BS)
= (u2−1)/σ(BS)〈JHHρ, KHHρ〉F
ALL DOFρ PP R,R Jρ =[JS · KS
]V [JS · KS
]V TKρ All of the Above 〈JSρ, KSρ〉F
defines the available processes in a large flexible engineeringsystem. Their number is given by:
DOFS = σ(ES) =
σ(P )∑w
σ(R)∑v
[JS KS ] (w, v) (4)
=
σ(P )∑w
σ(R)∑v
AS(w, v) (5)
2) Hetero-functional Adjacency Matrix: The Hetero-functional Adjacency Matrix (Aρ) describes how the systemcapabilities are connected and quantifies these interconnec-tions as sequence dependent degrees of freedom (DOFρ).
Aρ = Jρ Kρ (6)
where the potential for interconnected capabilities is describedby the system sequence knowledge base (Jρ) and the availabil-ity of these connections is described by the system sequenceconstraints matrix (Kρ).
Definition 4. Sequence-Dependent Degrees of Freedom[15], [25], [26]: The set of independent pairs of actionszψ1ψ2
= ew1v1ew2v2 ∈ Z of length 2 that completely describethe system language. The number is given by:
DOFρ = σ(Z) =
σ(ES)∑ψ1
σ(ES)∑ψ2
[Jρ Kρ](ψ1, ψ2) (7)
=
σ(ES)∑ψ1
σ(ES)∑ψ2
[Aρ](ψ1, ψ2) (8)
where Jρ and Kρ as defined in Definition 5 and 6.
Definition 5. System Sequence Knowledge Base [15], [25],[26]: A square binary matrix Jρ of size σ(P )σ(R)×σ(P )σ(R)whose element Jρ(ψ1, ψ2) ∈ {0, 1} is equal to one when stringzψ1,ψ2
exists. It may be calculated directly as
Jρ =[JS · KS
]V [JS · KS
]V T(9)
where ()V is shorthand for vectorization (i.e. vec()).
Definition 6. System Sequence Constraints Matrix [15],[25], [26]: a square binary constraints matrix Kρ of sizeσ(P )σ(R) × σ(P )σ(R) whose elements K(ψ1, ψ2) ∈ {0, 1}are equal to one when string zψ1ψ2 = ew1v1ew2v2 ∈ Z iseliminated.
It is important to recognize that the System Sequence Knowl-edge Base includes at a minimum the physical continuityconstraints described in Table I [15]. These constraints aretranslated in the System Sequence Constraints Matrix, asdefined in Definition 6. The construction of this matrix re-quires tracking all four constraints for each of the combina-tions of the capabilities in a system. A straightforward wayof calculating this matrix is a scalar implementation usingFOR loops while adhering to the following relationships ofindices ψ = σ(P )(v − 1) + w. v = k ∀k = [1 . . . σ(M)].w = [σ(Pη)(g − 1) + u] + j [15], [25], [26].
C. ConclusionHetero-functional graph theory differentiates itself in that
it centers itself on the allocated architecture. This implicitlyrequires mutually exclusive and collectively exhaustive setsof system processes and resources. It also quantifies a sys-tem’s capabilities in terms of structure degrees of freedom.The hetero-functional adjacency matrix then couples thesecapabilities into a combined network. Because there is anexplicit differentiation of system processes, hetero-functionalgraph theory provides a basis upon which networks with unlikefunction can be combined into a single mathematical modelof system structure. Prior work has also used this hetero-functional graph theory to quantify resilience measures [15],[19].
III. APPLICATION OF HFGT TO INFRASTRUCTURESYSTEMS
Before the construction of an integrative model of threenetworks of engineering systems, this section discusses eachof the systems individually. The three networks presented inthis paper are of special interest in a smart cities context [2].For each of the three networks, the allocated architecture isdefined, and a knowledge base is constructed.
A. Electric Power SystemsPrevious work has modeled power systems with the use
of Hetero-functional Graph Theory [19]. The power systemresources are defined as: RE = ME ∪ BE ∪ HE , where thetransformation resources ME are for example generators, theIndependent Buffers BE are the buses, and the TransportersHE are power lines. The power system processes are definedas: PE = PµE ∪ PηE ∪ PγE , where the transformation pro-cesses PµE are for example “generate power”, the transporta-tion processes PηE are “transport power from Bus 1 to Bus 4”,
3
and the holding processes PγE are “carry power at 132 kV”.The mapping of processes and resources is achieved with theknowledge base JSE , following Definition 1. This knowledgebase is the result of a combination of the transformationknowledge base JME and the refined transportation knowledgebase JHE , as demonstrated in general in Equation 2. Thecorresponding constraints matrix KSE is similarly constructedin general in Equation 3.B. Transportation Systems
Previous work has modeled transportation systems with the useof Hetero-functional Graph Theory [26]. The transportationsystem resources are defined as: RT = MT ∪BT ∪HT , whereMT are the transformation resources (e.g. stations that enterpeople in and out of the system), BT are the independentbuffers (e.g. parking lots), and HT the transporters (e.g.roads). The transportation system processes are defined as:PT = PµT ∪ PηT ∪ PγT , where PµT are the transformationprocesses (e.g. enter the transportation system), PηT are thetransportation processes (e.g. transport a person form node 1to node 3), and PγT are the holding/charging processes (e.g.charge the vehicle). The transportation system knowledge baseand the constraints matrix can be constructed using Equations2 and 3.
861 91314 11 12 199 15793122 119 118
10 85
9
87
88
67
66
65
69
76
58
72 73
71
77
79
68
84
92
63
70
64
17
90 62
4
59
60
61
57
195
40173148139748075
142
89
81
82
193
78
137
154
134194
176
197
201200 123
126
127
124
56
20
49
45
41
36
31
4448302833
99
164
171
179
150151
188
15
172
178
7
2
65
167
187
13
175
16
9694
115105 110
97112
188
192
168
169
177
143
156
132149 141
152
159
128
155
130
147 135
131
129
140
136
138
133
153
113181
114
116
22
95
189
111
198
158
160
184163
190
146
170 162
183
186
174166
196
180
185
191
182
161
165
102
107108
522119
35
117
125
38
29
25
54
27
101
121
120
98
32
100
103
104
39
145
50
53
23
46144
26
51
37 34
24
42
47
55
43
83
109
106
Legend:ChargingStation
ChargingStation & Water
Consumption Bus
Electrified RoadPower Bus
Electrified WaterConsumption Bus
Electrified Road Power and WaterConsumption Bus
Conventional PowerSystem Bus
Figure 2: Power Grid for Symmetrica [27]
C. Water Distribution Systems
Previous work has modeled water distribution systems with theuse of Hetero-functional Graph Theory [15]. The water systemresources are defined as RW = MW ∪ BW ∪ HW , where
MW are the transformation resources (e.g. a water treatmentfacility), BW are the independent buffers (e.g. pipe junctions),and HW are the transporters (e.g. water pipes). The watersystem processes are defined as PW = PµW ∪ PηW ∪ PγW ,where PµW are the transformation processes (e.g. generatewater), PηW are the transportation processes (e.g. transportwater from junction 2 to junction 13), and PγW are the holdingprocesses (e.g. pressurize water). Similar to the previoussystems, the water distribution system knowledge base andconstraints matrix are constructed in Equations 2 and 3.
IV. HFGT INFRASTRUCTURE MODEL FOR SMART CITYINFRASTRUCTURES
A. Case study: power, transportation, and water system
The smart city case study is based on the Symmetrica casestudy, which has been used in previous work in transportation-electrification [16]. The Symmetrica study combines a 201-buspower system (Figure 2), a symmetric 13×13 node transporta-tion system (Figure 3), and a 125 node potable water system(Figure 4). The data for the test case is openly available [28].
1) Power System: The Symmetrica power grid lay-out (Fig-ure 2) is derived from the 201-bus distribution system test case[29], [30]. The transformation resources ME are a generator atNode 201, and consuming resources on all other nodes. Someof these consumption nodes are coupled to the transportationand water distribution systems. The transportation resourcesare the power lines as indicated in the figure.
Legend: ChargingStation
ElectrifiedRoad
Figure 3: Transportation Network for Symmetrica [27]
2) Transportation System: The Symmetrica transportationsystem lay-out (Figure 3) was defined in previous work as atest case for transportation studies, comparable to the test casesused in power systems [27]. The transport network consists
4
of 169 nodes; all of which are independent buffers (BT ).The holding processes differentiate the buffers to allow forcharging at 53 charging nodes. The transportation resourcesare the roads between the nodes, but notice that each road inthe figure represents two transportation resources, as the roadsare bidirectional and allow a two way flow of traffic. Someroads are enabled to wirelessly charge the driving vehicles,this is again achieved by including a holding process at the104 electrified roads.
3) Water Network: The water network is based on the Any-town Water Distribution Network lay-out [31]. The networkis adjusted and now consists of 5 water treatment facilities,10 water storage tanks, and 110 water consumption nodes(Figure 4). The treatment facilities and the consumption nodesare the transformation resources MW , and the storage tanksare independent buffers BW . The pipes are the transportingresources HW . Note that each of the nodes in the system isconnected to the electric grid, as they all include electric waterpumps.
Figure 4: Water Network for Symmetrica [28]
4) Integration of Multiple Infrastructure Systems: Basedupon the figures of the individual infrastructures, and theirassociated data [28], the knowledge base for the multipleinfrastructure systems can be constructed.
JSC =
JME | 0JMW | 0
JHEJHTJHW
(10)
Collectively, the nodes and edges in each infrastructureconstitute the system resources. The legend in Figure 2 con-tains the couplings between the power grid and the other twonetworks. Care must be taken to avoid the double countingof resources which exist in multiple networks. For example,node 71 in the power grid, node 1 in the transportation networkand node 20 in the water network effectively represent one cityresource (e.g. building). Consequently, in the knowledge basefor the entire city, resource 71 contains capabilities for powerconsumption, vehicle charging, and water consumption. Alsonote that all nodes in the water network consume power, inthis situation there is a one-to-one mapping from water nodesonto power nodes. The transportation knowledge bases arerefined individually and aggregated afterwards to maintain thedefinition of holding processes. The number of capabilities inthe system as defined in Equation 10 is 2,248. The capabilitiesare projected as nodes in a three dimensional space in Figure5. The top layer is the water system, the middle layer is thepower grid, and the bottom layer is the transportation system.
Figure 5: Representation of the Smart City Degrees of Free-dom, with layer specific sequence dependent degrees of free-dom
B. Network Visualization
After constructing the knowledge base for the full city, theallocated architecture is now used to form the basis for thecalculation of the hetero-functional adjacency matrix and thecorresponding sequence dependent degrees of freedom. Forclarity, they are presented in Figure 5 as the connectionsbetween the nodes for each layer. Figure 6 also contains thesequence dependent degrees of freedom in between the layers.Their total number is 21,145.
V. CONCLUSION AND FUTURE WORK
This paper introduces hetero-functional graph theory asa method to model multi-disciplinary engineering systems.Hetero-functional graph theory allows for the integration oflayers of networks without the constraints imposed by previousmethods. The paper applies hetero-functional graph theory
5
Figure 6: Representation of the full set of sequence dependentdegrees of freedom
to create a three layer smart city infrastructure model. Inthe model, function and form are represented unambiguously.Finally, the paper shows a visual representation of all systemcapabilities, and includes the allowed sequences of thesecapabilities.
Future work includes the expansion of the model to includemore infrastructures. Additionally, these hetero-functionalgraphs may form the basis for measuring multi-layer infras-tructure resilience.
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