**2. Conceptual model**

#### **2.1. Characterization of SoS interdependencies**

An SoS consists of constituent systems that include individual engineered and actor system (**Figure 1**). An engineered system (ES) represents the performance and capabilities of relevant physical/technological systems. An actor system (AS) represents the development and operation of resources by an actor and, possibly, the management of other actor behavior residing either horizontally at the same SoS level or vertically across level.

The model is described using system model elements that are central to the policy analysis approach [5]. Policies (P) are the set of instruments within the control of the decision makers Dealing with Complexities and Uncertainties in a System-of-Systems: Case Studies on Urban… http://dx.doi.org/10.5772/intechopen.73706 97

**Figure 1.** System-of-systems constituents: (a) engineered system and (b) actor system.

including energy, food, water, transportation, manufacturing, and telecommunications are

From a conceptual point of view, urban infrastructure can be considered as a couple humanengineered system. An engineered system (ES) is *a combination of technological components that* 

Recent structural changes, such as deregulation in the energy sector, have revealed infrastructure's vulnerabilities. Public dissatisfactions began to surface as the lack of capacity, reliability, and vulnerability of such provisions become increasingly prevalent causing politicians to take serious considerations. At the same time, many infrastructures have appeared to be inert to change because these infrastructures have long technical life time and are deeply interwoven in our social, economic, and political structure. Lack of systematic knowledge in addressing uncertainties may contribute to ad hoc political decisions, which may block timely

A system-of-systems (SoS) perspective is an attempt to structure the complexity of humanengineered systems by taking a broader view than just the physical design (i.e., traditional system engineering view) and operational aspect, to include commercial and financial, economic, social, and policy aspects couched within multiple levels. The goal is to improve anal-

A system-of-systems (SoS) consists of multiple, heterogeneous, distributed systems embedded in networks at multiple levels that evolve over time. Complexity in an SoS stems primarily from the heterogeneity of its constituent systems, the distributed nature of these systems. The complexity brought by the system heterogeneity exists both within a domain (e.g., power generation) and across domains (e.g., power generation, energy service economics, and gov-

The chapter first describes a generic SoS conceptual model that characterizes the interdependencies among SoS constituents. Next, a model of uncertainty space evolution over time is presented. Three case studies are then presented to illustrate the framework. In the last sec-

An SoS consists of constituent systems that include individual engineered and actor system (**Figure 1**). An engineered system (ES) represents the performance and capabilities of relevant physical/technological systems. An actor system (AS) represents the development and operation of resources by an actor and, possibly, the management of other actor behavior residing

The model is described using system model elements that are central to the policy analysis approach [5]. Policies (P) are the set of instruments within the control of the decision makers

adaptation and discourage further investment in the infrastructure system [4].

key components in enabling modern society to function.

*work in synergy to collectively perform a useful function* [3].

tion, the added values of SoS framework are discussed.

either horizontally at the same SoS level or vertically across level.

**2.1. Characterization of SoS interdependencies**

ysis for decision-making.

96 System of System Failures

ernmental policy/regulation).

**2. Conceptual model**

that can change the system. External forces (X) refer to factors that are not controllable by the decision maker but may influence the system significantly. The structure of an actor system can be specified as a set of endogenous factors (I) together with relationships (R), which may include functional, behavioral, or causal ones. The results of these interactions, the model outputs, are called outcomes of interest (O). The value systems (W) of decision makers and stakeholders reflect their goals and preferences. Notice that an engineered system is devoid of policy instruments and has no value systems.

An SoS of urban systems can then be constructed via assemblage of some ES and AS systems in a structure resembling the nested system in **Figure 2**. Each element represents an ES, which resides at the lowest level (α level) and an AS, which resides at the β level and above.

Based on the framework, several terms can be defined:


The different types of interdependencies within an SoS are given in **Table 1**. The system model variables in an SoS interact with one another, creating a network of interrelationships. These interdependencies are bidirectional. In one direction, the interdependency can manifest in a form of influence.

#### **2.2. Dealing with uncertainties in a system-of-systems**

To deal with uncertainties in a system-of-systems, a computational approach called exploratory modeling and analysis (EMA) has been proposed [6, 7]. EMA is founded on the idea of exploring multiple hypotheses about the SoS of interest by varying the assumptions underlying the system model. EMA is used to explore the implications of multiple hypotheses about the system by means of computational experiments. A computational experiment is a single computer run of the system model using one set of assumptions.


in the second period, each of which will then lead to multiple possible realizations in the third

Uncertainty spaces at one particular period can be modeled in different ways. For example,

(i.e., the *X, E, R*, and *W* elements) and the same size (i.e., the choice of range of the *X, E, R*, and *W* elements). When a structural change occurs, however, the nature and size of one uncer-

**1.** The degree and scope of uncertainty increases as the time period progresses. This is a default

**2.** The degree and scope of uncertainty actually decreases as time progresses. This occurs, for example, in case the threshold market share of a certain technology is reached, which

**3.** The nature of the uncertainty space in one period is different from that in the subsequent period, which may also depend on the realization preceding it. For example, when a structural change (I, R) occurs, some of the variables become irrelevant and new ones need to be added. Also, the value system of decision makers and stakeholders may change over time. Future decision makers may have different decision criteria or put different weights on the criteria.

 [(s2 )1

Dealing with Complexities and Uncertainties in a System-of-Systems: Case Studies on Urban…

] can have exactly the same nature

http://dx.doi.org/10.5772/intechopen.73706

99

] and the *uncertainty space2*

Three main characteristics of SoS uncertainty space are described as follows:

characteristics since the longer the time span the greater the uncertainty.

reduces the range of the share of other technologies (i.e., *lock-in effect*).

period, and so on until the last period is reached.

tainty space may be totally different with the other.

 [(s1 )1

**Figure 3.** The evolution of SoS uncertainty space.

the *uncertainty space1*

**Table 1.** Interdependencies among SoS elements.

#### *2.2.1. Characterization of the evolution of SoS uncertainty space over time*

**Figure 3** shows an 'uncertainty space funnel' that models how the future realizations of the model variables will unfold in a time horizon of *n* periods. In a future of multiple discrete time periods, one unique realization in the first period leads to multiple possible realizations Dealing with Complexities and Uncertainties in a System-of-Systems: Case Studies on Urban… http://dx.doi.org/10.5772/intechopen.73706 99

**Figure 3.** The evolution of SoS uncertainty space.

*2.2.1. Characterization of the evolution of SoS uncertainty space over time*

**Table 1.** Interdependencies among SoS elements.

98 System of System Failures

**Figure 3** shows an 'uncertainty space funnel' that models how the future realizations of the model variables will unfold in a time horizon of *n* periods. In a future of multiple discrete time periods, one unique realization in the first period leads to multiple possible realizations

(**p**ASj;i≠j)

(**o**AS j;i≠j)

(**o**AS j;i≠j)

**Linkage of interdependencies Formalism** An outcome of an ES is an external factor for another ES **o**ES <sup>i</sup> = **x**ES j;i≠<sup>j</sup> An outcome of an ES is an external factor for an AS **o**ES <sup>i</sup> = **x**AS <sup>i</sup> An outcome of an ES influences actor decision **p**AS <sup>i</sup> = *f*(**o**ES i) An actor's decision is an external factor for an ES **p**AS <sup>i</sup> = **x**ES <sup>i</sup> An actor's decision is an external factor for another actor **p**AS <sup>i</sup> = **x**AS j;i≠<sup>j</sup> An actor's decision influences another actor's decision **p**AS <sup>i</sup> = *f*

**Figure 2.** A system-of-systems consisting of engineered (ES) and actor systems (AS).

The realization of an outcome of interest of an actor affects the decision of another actor **p**AS <sup>i</sup> = *f*

A value system of one actor becomes an exogenous factor of another **x**AS <sup>i</sup> = **w**AS j;i≠<sup>j</sup> The outcomes of one actor system affect the value system of another actor **w**AS <sup>i</sup> = *f*

in the second period, each of which will then lead to multiple possible realizations in the third period, and so on until the last period is reached.

Uncertainty spaces at one particular period can be modeled in different ways. For example, the *uncertainty space1* [(s1 )1 ] and the *uncertainty space2* [(s2 )1 ] can have exactly the same nature (i.e., the *X, E, R*, and *W* elements) and the same size (i.e., the choice of range of the *X, E, R*, and *W* elements). When a structural change occurs, however, the nature and size of one uncertainty space may be totally different with the other.

Three main characteristics of SoS uncertainty space are described as follows:


This way of characterizing SoS uncertainty space highlights its path dependency nature. Suppose that *(s1 ) 0* is the realization of the uncertainty space in period<sup>0</sup> (i.e., the initial condition). In period1 , the uncertainty space that is path dependent on the initial condition (s1 ) 0 is the uncertainty space1 [*(s1 ) 0* ]. The evolution of uncertainty space progresses over time until the end of analysis' time horizon, *Periodn* . In *Periodn* , there will be a set of uncertainty spaces, each originated from the realizations of all the preceding periods (i.e., uncertainty space<sup>1</sup> *([(si ) n-1],…, [(si ) 2 ], [(si )1 ], [(si )0 ]*). One future path (illustrated by the dashed arrows), for example, can be represented by *(s1 ) 0 (s1 ) 1 (s1 ) 2 … (s1 ) n* .
