An Agent-Based Optimization Framework for
Engineered Complex Adaptive Systems with
Application to Demand Response in Electricity Markets
by
Moeed Haghnevis
A Dissertation Presented in Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
Approved April 2013 by the
Graduate Supervisory Committee:
Ronald Askin, Co-Chair
Dieter Armbruster, Co-Chair
Pitu Mirchandani
Tong Wu
Kory Hedman
ARIZONA STATE UNIVERSITY
August 2013
ABSTRACT
The main objective of this research is to develop an integrated method to study
emergent behavior and consequences of evolution and adaptation in engineered complex
adaptive systems (ECASs). A multi-layer conceptual framework and modeling approach
including behavioral and structural aspects is provided to describe the structure of a class
of engineered complex systems and predict their future adaptive patterns. The approach
allows the examination of complexity in the structure and the behavior of components as
a result of their connections and in relation to their environment. This research describes
and uses the major differences of natural complex adaptive systems (CASs) with artificial/engineered CASs to build a framework and platform for ECAS. While this framework
focuses on the critical factors of an engineered system, it also enables one to synthetically
employ engineering and mathematical models to analyze and measure complexity in such
systems. In this way concepts of complex systems science are adapted to management
science and system of systems engineering. In particular an integrated consumer-based optimization and agent-based modeling (ABM) platform is presented that enables managers
to predict and partially control patterns of behaviors in ECASs. Demonstrated on the U.S.
electricity markets, ABM is integrated with normative and subjective decision behavior
recommended by the U.S. Department of Energy (DOE) and Federal Energy Regulatory
Commission (FERC). The approach integrates social networks, social science, complexity theory, and diffusion theory. Furthermore, it has unique and significant contribution
in exploring and representing concrete managerial insights for ECASs and offering new
optimized actions and modeling paradigms in agent-based simulation.
i
TABLE OF CONTENTS
Page
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
CHAPTER
1 INTRODUCTION AND LITERATURE REVIEW . . . . . . . . . . . . . . . .
1
1.1
Motivations to study the US power system as an ECAS . . . . . . . . . . .
4
1.2
Engineered Complex Systems and Complex Adaptive Systems . . . . . . .
9
1.3
Agent-based Modeling of Engineered Complex Adaptive Systems . . . . . 11
2 A FRAMEWORK FOR ENGINEERED COMPLEX ADAPTIVE SYSTEMS . . 13
2.1
Dissection of Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Exponential Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Logistic Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Dis-uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2
Entropy vs. Dis-uniformity, Source of Self-organization . . . . . . . . . . . 22
2.3
Emergence as the Effect of Interoperability . . . . . . . . . . . . . . . . . . 35
2.4
Evolution Because of Updates in the Traits . . . . . . . . . . . . . . . . . . 40
3 AN AGENT-BASED OPTIMIZATION PLATFORM . . . . . . . . . . . . . . . 43
3.1
Agent-based Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Decision Makers/Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Environment/Patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2
Agent-based Optimization Engine . . . . . . . . . . . . . . . . . . . . . . 52
Process Overview and Scheduling . . . . . . . . . . . . . . . . . . . . . . 52
Decision Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Termination Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
ii
CHAPTER
Page
4 SIMULATION OF ELECTRICITY MARKETS AND ANALYZING BEHAVIORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1
Specifying the Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2
Analyzing Behaviors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
The Effect of The Awareness-Threshold . . . . . . . . . . . . . . . . . . . 66
Establishing a BaseLine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Improving Effect of the Network . . . . . . . . . . . . . . . . . . . . . . . 69
Externalities and Irrationality . . . . . . . . . . . . . . . . . . . . . . . . . 70
Effects of Saturated Interrelationships . . . . . . . . . . . . . . . . . . . . 73
Significant Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3
Expansion to other ECASs . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
iii
LIST OF TABLES
Table
Page
2.1
Summary of emergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2
Prior and Posterior joint probabilities . . . . . . . . . . . . . . . . . . . . . . . 39
3.1
Interoperability between consumer agents . . . . . . . . . . . . . . . . . . . . 56
4.1
State Variables for Running the Simulation . . . . . . . . . . . . . . . . . . . . 64
4.2
Measuring the Interoperabilities . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3
Estimated effects and coefficients of ∑ Xi , Kr , and U . . . . . . . . . . . . . . . 75
4.4
Estimated effects and coefficients of topology and max link generation . . . . . 76
iv
LIST OF FIGURES
Figure
Page
1.1
LMP contour map of the Midwest ISO . . . . . . . . . . . . . . . . . . . . . .
6
1.2
Variation of LMP map of Midwest ISO by time . . . . . . . . . . . . . . . . .
7
2.1
Framework for Engineered Complex Adaptive Systems . . . . . . . . . . . . . 14
2.2
Example for Theorem II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3
Example for Theorem IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4
A complicated example for adaptation . . . . . . . . . . . . . . . . . . . . . . 42
3.1
Integration of the Structural Entity Model . . . . . . . . . . . . . . . . . . . . 44
3.2
Influence Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3
Decision Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4
Social layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5
Physical Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.6
Logical Flow Chart for Scheduling of the Process Overview . . . . . . . . . . . 53
3.7
Calculating the Interrelationship . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.8
Desirability Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1
Patterns of Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2
Linearized Growth Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3
Convergence of the Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4
Distribution of Convergence to Pattern i . . . . . . . . . . . . . . . . . . . . . 67
4.5
Convergence of the Average Entropies . . . . . . . . . . . . . . . . . . . . . . 68
4.6
Total Difference With and Without a Social Network . . . . . . . . . . . . . . 69
4.7
Comparison of Awareness-Threshold=0.1 and 0.3 . . . . . . . . . . . . . . . . 70
4.8
Comparison of ρmax = 1 and ρmax = 3 . . . . . . . . . . . . . . . . . . . . . . 71
4.9
The Scale-free Metric of the Network . . . . . . . . . . . . . . . . . . . . . . 72
4.10 Effects of Irrationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
v
Figure
Page
Effect
4..111
44.11
1 Ef
E
fect of saturated friendship on the topology of the network . . . . . . . . . . 74
74
Pareto
Chart
Normal
Probability
Plot
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44.12
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P
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78
vi
Chapter 1
INTRODUCTION AND LITERATURE REVIEW
Today’s engineered complex adaptive systems (ECASs) are composed of a huge number
of autonomous and heterogeneous components with myriad interrelationships. ECASs exhibit the inherent behaviors of natural complex adaptive systems (CASs) but also have the
ability to effect design and control actions. In ECASs, interacting agents act with limited information on their environment and without any central control mechanism [1, 2].
An ECAS may evolve to adapt to unforeseen dynamic conditions. Highly dynamic and
non-linear behaviors of these systems make them difficult to fully understand and thus to
effectively design, operate, and maintain. Self-organizing components readjust themselves
continuously and emerge new behaviors in interaction to other components and their environment.
Traditionally, we analyze a system by reductionism. We study behaviors of large
systems by decomposing the system into components, analyzing the components, and then
inferring system behavior by aggregation of component behaviors. However, this bottomup method of describing systems often fails to analyze complex levels and to fully describe
behavior. Current research (see literature review Sections 1.2 and 1.3) usually considers
natural systems (biological, physical, and chemical systems) where the emergence and
evolutionary behaviors can be studied by thermodynamic laws, biological rules, and their
intrinsic dynamics that are innate parts of these systems. However, in engineered systems,
decision makers or system designers develop or define rules and procedures to engineer the
outcomes and control the possibilities as needed. In engineered complex adaptive systems
(ECASs), objectives are artificially defined and interoperabilities between components can
be manipulated to achieve desired goals, while objectives and interoperabilities of natural systems are naturally embedded. This does not preclude the presence of unintended
1
complexity behaviors but does allow for a design and control aspect of the system. These
facts motivate us to propose a new framework for modeling this class of complex adaptive
systems (CASs).
Considering the US Electricity Markets as an ECAS, the lack of supporting technology and behavioral knowledge provide challenges for developing price-based Demand
Response (DR) programs. Uncertainties in likelihood of customer participation and customer responses decrease the reliability of DR. Traditional studies only focus on rational
choice of economical drivers while consumers may fail to adopt existing incentives because of their low elasticity to offered drivers. Challenges arise from the fact that the DR
assumes active retail customers participating in electricity markets by responding to dynamic prices or incentives; however, currently most of electricity customers only see flat
rates that are based on average costs. Also, most DR programs only focus on large industries and business sectors (they mainly consider passive energy efficiency more than active
DR options).
In our study an ECAS evolves on the basis of embedded normative behavior rules
related to non-convex consumer-based optimization. Individual agents adjust their decisions over time to optimize their objective functions but they may not be fully rational.
These readjustments based on individual limited observations of the environment may
cause emergence in agent’s behaviors. This micro optimized emergence in the lower level
causes optimized evolution in the higher level of the system. Behavioral-based DR in
the US power markets is used to demonstrate the applicability of the modeling approach.
Economic incentives motivate local consumers to adjust their behavior to limit maximum
system usage. Challenges to implement DR in accordance with Federal Energy Regulatory
Commission (FERC) Order 745, March 2011 and Sections 1252(e)/(f) of the US Energy
Policy Act of 2005 motivated the research and are considered in the modeling approach.
2
To study and analyze ECAS, Haghnevis and Askin [3, 4] defined emergence as “the
capability of components of a system to do something or present a new behavior in interaction, and dependent to other components that they are unable to do or present individually”,
evolution as “a process of resilience and agility in the whole system”, and adaptation as
“the ability to learn and adjust to a new environment to promote their survival”. We will
explain how to study these hallmarks of CASs in our agent-based simulation for Demand
Response complex systems.
Power markets are considered an ECAS and agent-based modeling (ABM) has been
attempted by several national laboratories and research institutes (see Section 1.3 for references). Our research builds upon previous modeling approaches to resolve weaknesses
of the current models (see comprehensive survey studies for agent-based electricity market
models, tools, and simulation in [5, 6, 7]). We employ consumer interoperability in a social
layer whereas previous research studied economic models in a business layer (see Chapter
3 for more details).
We can summarize the features of our research as follows:
• Mimicking behavior of real-world DR participant in the US Electricity Markets,
• Considering adaptive mechanisms and social learning in multi-layered power systems,
• Use of mathematical programming in modeling and solving complex decision problems,
• A careful concrete integrated agent-based simulation.
3
The key contributions of this study are:
• Improving effects of economical incentives by social education,
• Understanding effects of the social network topologies of cascading behaviors,
• Developing a platform that enables us to study DR in electricity markets as an ECAS.
The reminder of Chapter 1 provides background on electricity markets and ECASs
and discusses the motivations of this study. Chapter 2 presents the engineering framework of our method to study ECASs. Hallmarks and theoretical concepts of complexity
are considered in building this framework. Sections 2.1 and 2.2 detail the mathematical
mechanisms of features and relationships of components (step 1 of the framework). These
lead to analyzing the interoperabilities that induce emergence in Section 2.3 (step 2 of the
framework). Evolution of traits as the process of system adaptation and their response to the
changes is covered in Section 2.4 (step 3 and 4 of the framework). Chapter 3 explains our
integrated approach. Properties of the agents and their decision diagram are defined in 3.1.
We depict the layers of the environment in Section 3.1/Environment. Section 3.2/Process
Overview details the logic of agent-based process and presents mathematical equations
and definitions for the agent-based engine. Also, in Section 3.2/Decision Rules all decision
variables and rules (defined in the previous sections) are integrated into an agent-based
optimization model. Chapter 4 shows the simulation results. We define required variables
to run the model and we analyze different scenarios. Various examples demonstrate the
validity of our method in each section.
1.1
Motivations to study the US power system as an ECAS
Millions of components, their interactions, and self-organizing abilities of the components
make the US power system one of the most complex systems ever invented [8]. The US
4
electricity power system is a complex network of approximately 160,000 miles of high
voltage lines and 3200 electric distribution utilities that is fed by about 100,000 generating
units [9]. The dynamically increasing rate of total electricity consumption in the US (nearly
3,884 Billion Kilowatt-hours (kWh) in 2010, i.e. 13 times greater than 1950, and expected
to grow to 4,880 billion KWh by 2035) has a large impact on the electricity markets.
Several reasons have been presented for considering the US power system as an
ECAS in the literature. These include:
• Emerging technologies (e.g. time-based pricing, smart meters, and home-based solar
systems) serve to add a decentralized consumer-interactive role to the traditionally
producer-controlled systems,
• The diversity of people, their interdependencies (human decisions based on other
consumers), and their willingness to cooperate,
• Time dependency of the network topologies [10], and
• Scale-free or single-scale feature of these networks - their node degree distribution
follows a power-law or Gaussian distribution [11].
The US Federal Energy Regulatory Commission (FERC) formed independent System Operators (ISOs) and regional transmission organizations (RTOs) based on the Wholesale Power Market Platform [12] to coordinate, control, and monitor the operations of the
US wholesale electrical systems. They manage electricity generation and transmission
across geographic regions and keep supply and demand in balance. ISOs/RTOs serve twothirds of electricity consumers in the US based on Locational Marginal Price (LMP). This
structure increases the complexity of the US wholesale electricity markets.
5
Electricity prices vary by the maximum consumption rate and uniformity of aggregate regional demand in time that has high economic impact in our society. As an
illustration, Fig. 1.1 provides the LMP contour map of the Midwest Independent System
Operators [13]. There is a huge gap between LMP values for two neighborhoods A and B
(from +82USD in Point A to -40USD in Point B). This pattern may change totally in the
next 5 minutes. Fig. 1.2 depicts the LMP maps in different time intervals. Figures 1.2a to
1.2c show variation in short time intervals (10 minutes) while Figures 1.2d to 1.2f shows it
for long time intervals (more than one hour).
B
A
Figure 1.1: LMP contour map of the Midwest ISO
Demand Response is defined by DOE [14] as ”changes in electric usage by end-use
customers from their normal consumption patterns in response to changes in the price of
electricity over time, or to incentive payments designed to induce lower electricity use at
times of high wholesale market prices or when system reliability is jeopardized”. Pursuant
6
(a) 4:50 pm
(b) 5:00 pm
(c) 5:10 pm
(d) 5:50 pm
(e) 7:45 pm
(f) 8:50 pm
Figure 1.2: Variation of LMP map of Midwest ISO by time
7
to the Sections 1252 (e) and (f) of the US Energy Policy Act of 2005 (EPACT), the U.S.
Department of Energy (DOE) has provided a report to Congress that identifies and quantifies DR benefits and makes recommendations for achieving them. The report shows that
actual peak demand reduction was only 9,000MW in 2004 while the potential demand response was 20,500 MW [14]. FERC issued Order 745 in 2011 to meet the Congressional
direction and remove barriers to the participation of DR in organized wholesale electricity
markets [15]. A list of questions that DR studies strive to answer includes:
• When (what day) is the best time to trigger a demand response event?
• What is the event window (start and finish time)?
• How much trigger is required?
• What type of events (rewards, penalties, larger and discrete, smaller and continuous)
are more effective and efficient?
• Who are the event recipients? What is the schedule to receive their triggers?
• What percent of the recipients may answer to the triggers? What is their outcome
(e.g. total saving)?
Studies on behavior of consumers for the PJM Interconnection Regional Transmission Organization show that small shifts in peak demand would have a large effect on
savings (a 1% shift in peak demand would result in savings of $1.27 billion at the system)
[16]. Even a 5% drop in peak demand can save $3 billion a year [17]. Electric power is
generated from deferent sources of energy (coal, hydro, wind, solar, natural gas). Capacity,
flexibility, and cost of generators increase the complexity of dispatching and pricing electricity. Some types of generation are more expensive but flexible (usually work during the
8
peak demand) while some generators have lower variable costs with longer startup times.
Large-scale social science field experiments [18], academic studies [19], and a testimony
by American Council for an Energy-Efficient Economy to Congress about DOE roles on
social norms in electricity consumption suggest that social and behavioral programs can
increase the efficiency of load management programs [20].
Our model enables us to improve the ability of regulators and consumers to communicate and optimize the consumption. Here, instead of event based demand response
mechanisms, consumers can receive incentives for controlling their demand at all times.
One advantage of our model is the ability to study dynamic pricing in smart grid applications. As electricity consumption is inelastic in short time frames, social education mechanisms will increase the effect of economic incentives. Our reward approach attempts
to improve the effectiveness of triggers to motivate consumers to shift their demand. We
show that combining social drives with economical incentives enables us to reduce and control the huge gap between potential load management and actual DR. We propose a more
complex, less idealized multi-layered adaptive approach to show influences of combined
social-economical incentives on behavioral-based DR programs. We seek to change consumption patterns of end-users and load aggregators by providing economical incentives
and social education that makes them more likely to respond to economical incentives and
sustainability concerns.
1.2
Engineered Complex Systems and Complex Adaptive Systems
Difficulties to understand emergence, abstract theoretical concepts, and incomplete applicable frameworks are current challenges in the study of CASs [21]. Page [22] shows that
simple rules between components can describe an organized agent. The author uses this
fact to analyze self-organization and adaptive agents. Complex systems are classified by
Magee and Weck [23]. Who present several examples for each classification. Four reports
9
of the Defense R&D Canada-Valcartier (1- list of works, experts, organizations, projects,
journals, conferences and tools in 471 references and 713 related Internet addresses [24], 2formulations and measures of complexity [25], 3- glossary of 335 related key words [26],
and 4- An overview of theoretical concepts of complexity theory [27]) present a comprehensive survey to the study of complexity theory, chaos and complex systems.
Mathematical modeling of CASs is still fragmented and inapplicable because engineers and mathematicians focus on purposes and outcomes while complex systems are
strongly characterized by emergence in behaviors of components and evolution in behaviors of a system. Bar-Yam [28, 29] mathematically studied the structure of complex systems and the interdependence of components. Bar-Yam [30] used the complexity profile to
measure the amount of information needed to describe each level of detail.
Complex networks are among the most engineered and mathematically modeled
complex systems. Watts and Strogatz [31] quantify the dynamics of small-world networks
and Avin and Dayan-Rosenman [32] model evolutionary structure of population and components in social networks. Hanaki et al. [33] showed emergence of cooperative behavior
by combining social network dynamics and stochastic learning. The concepts of complex
networks are applied in the structure of power grids. Amaral et al. [34] studied structural
properties of the electric power grid of Southern California and Strogatz [35] considered
the complex network of the New York electric power grid.
Engineered systems involve human designers, controllers, and consumers. As such,
human decision processes are relevant and impact system behavior. The ability of changing structures and organizations to respond to the unseen challenges of the environment,
increases the complexity of engineered systems. In this study we include human decision
making and show how our proposed framework encapsulates the previous models. Lee et
al. [36] classified three developed approaches to mimic human complex decision behav-
10
iors. In more details, they employed engineering methodologies to represent such systems
in complex environments [37, 38]. Leskovec et al. [39] consider information cascades in
large social networks and investigate a large person-to-person recommendation network.
Kleinberg [40] discusses probabilistic and game-theoretic models for flow of information
or influence through social networks. Our study discusses the complex structure and behavior of a human decision network itself and how components cooperate as a result of
their connections through a network and in relationship with their environment.
1.3
Agent-based Modeling of Engineered Complex Adaptive Systems
Usually, game theoretical modelers only consider a limited number of components with
often unrealistic assumptions. Traditional equilibrium models disregard strategic behaviors and learning of components [5]. Weaknesses of traditional modeling methods make
agent-based modeling and simulation useful tools to study and analyze systems that exhibit
emergent phenomena, nonlinear dynamics, and path-dependent behavior [41, 42, 43, 44].
Recently, ABMs have been used to study the impact of new electricity technologies.
Hamilton et al. [45] consider performance of a new technology versus an old technology
and study the effects of a specific spatial externality (fashion effect). They analyzed the dynamics of technology diffusion among bounded rational agents with uncertainty by using
ABM. An ABM for a market game is presented by [46] to evaluate the effects of government strategies on promoting new electricity technologies in complex systems involving
human behavior. Athanasiadis et al. [47] used ABM to control consumer demands by
supporting interaction between consumers in a diffusion mechanism. In another approach,
advantages and disadvantages of optimization models and ABM for technological change
in different energy systems is compared in [48].
The impact of the structure of a social network on the spread of innovations has
been an actively researched issue. Montanaria and Saberi [49] considered competing al-
11
ternatives when an agent adapts to a new behavior based on its neighbors. In their model
the payoff for agents increases with the number of neighbors who adopt the same choice.
Guardiola et al. [50] use dynamic pricing in modeling diffusion of innovations in a social network. Bohlmann et al. [51] analyze network topologies and communication links
between innovator and follower market segments in the diffusion process. Rahmandad
and Sterman [52] compared agent-based and differential equation models and analyzed
the effect of individual heterogeneity and network topologies in the dynamics of diffusion.
Kempe et al. [53, 54] study spread of an innovation or behavior based on word-of-mouth
recommendations in social networks.
Agent-based modeling of complex adaptive electricity systems has been attempted
by several countries and US national laboratories. Bunn et al. [55, 56, 57, 58, 59] analyzed market power and price-formation of utilities and generators in the UK. Bagnall
and Smith [60] created a multi-agent model for the UK electricity generation market. In
Germany, Bower [61] studied bidding strategies, and an agent-based German electricity
market is presented by Sensfub [62]. Grozev et al. [63] and Chand et al. [64] present
an ABM for the CAS of interactions between human behaviors in markets, infrastructures
and environment of Australia’s national electricity market by using the National Electricity
Market Simulator (NEMSIM). In the US, the Agent-Based Modeling of Electricity Systems (AMES) is designed for computational study of wholesale power market [65, 66].
Argonne National Laboratory developed the Electricity Market Complex Adaptive System
(EMCAS) to analyze the possible impacts on the power system of various events [67, 68].
Sandia National Laboratories presented the Aspen-EE to simulate the effects of market decisions in the electric system on critical infrastructures of the US economy [69]. Honeywell
Technology Center constructed the Simulator for Electric Power Industry Agents (SEPIA)
[70]. Pacific Northwest National Laboratory also studied power systems as CASs [71].
12
Chapter 2
A FRAMEWORK FOR ENGINEERED COMPLEX ADAPTIVE SYSTEMS
Couture and Charpentier [21] and Mostashari and Sussman [72] present a framework to
study complex systems. Prokopenko et al. [73] depicted complex system science concepts.
Also, Sheard and Mostashari [74] visualized characteristics of complex systems. Frameworks for ECASs are still incomplete and fragmented. In this study we propose a more
detailed framework for ECASs (Fig. 2.1). The framework can help us focus on critical factors that change the states of an ECAS, and enables us to synthetically employ engineering
and mathematical models to analyze and measure complexity in an adaptive system without complex modeling. Four profiles of ECASs and their characteristics are presented in
component and system levels to show behavior of the three hallmarks.
In our proposed approach a preparatory step identifies adaptive complexity in an
engineered system. This step is necessary to make sure we do not spend unnecessary resources to analyze a normal system as a complex system. To identify a complex engineered
system we check [2, 75],
1. System structure:
• displays no or incomplete central organizing for the system organization (prescriptive hierarchically controlled systems are assumed to not be complex systems),
• behavioral interactions among components at lower levels are revealed by observing behavior of the system at higher level,
13
Flexibility
&
Robustness
Performance
System
level
Learning
Adaptation
Environment
Categories
System Traits
Synchronization
&
Exchangeability
Interoperability
AutonomyÏ
vs
DependencyÐ
Emergence
Components
level
Interrelationship
Complexity Hallmarks
Resilience
&
Agility
Evolution
Threshold
Self-information
Decomposability
&
Willingness
Features
DiversityÏ
vs
CompatibilityÐ
Environment
Characteristics
Profiles
Indicators
Figure 2.1: Framework for Engineered Complex Adaptive Systems
2. Analysis of system behavior:
• analyzing components fails to explain higher level behavior,
• reductionist approach does not satisfactorily describe the whole system.
Total electricity consumption grows every year affecting the topology of power
grids. Emerging technologies such as time-based pricing, smart meters, and home-based
solar systems serve to add a consumer-interactive role to the traditionally producer-controlled
systems. This decentralization results in complexity in this system by decreasing central
14
organization. Moreover, the interaction of physics with the design of the transmission links
increases its complexity as does the diversity of people, their interdependencies, and their
willingness to cooperate. Time dependency of the network [10], scale-free or single-scale
feature of these networks (their node degree distribution follows a power-law or Gaussian
distribution in long run) [11], and human decisions based on other consumers all justify
considering an electric power system as an ECAS. These factors have placed the US power
grid beyond the capability of mathematical modeling to date [35].
To take advantage of the fundamental theories of complex systems, we study and
analyze complex systems based on the framework in Fig. 2.1. Systems are composed of
components. Components possess individual features and interoperable behaviors. Systems then have traits and learning behaviors. Together these form the system profile comprised of the following aspects (we define state of each profile in parentheses),
• Features (components readjust themselves continuously): Here, dissection of features leads to decomposability (e.g., number of each component type and patterns
of individual behaviors) and willingness (e.g., growth rate of each component and
behavioral/decision rules). The environment of the system may also affect component actions. A measurable property of this profile is self-information (entropy) of
components. Entropy is increased with the diversity of components and is decreased
with their compatibility. Sections 2.1 and 2.2 mathematically model and analyze the
dissection of features and show how self-organization appears.
• Interoperabilities (components update their interdependences): In this profile, emergence as the hallmark of interoperability shows what components can do in interaction and dependent to other components that they would not do individually. Components have exchangeability and synchronization. Autonomy increases and depen-
15
dency decreases the interrelationship of components. This profile helps us to infer
the behavior of the components. Section 2.3 models this profile.
• Traits (system tries to improve its efficiency and effectiveness): In this profile, systems may evolve. The whole system applies its resilience and agile abilities to perform more effectively and efficiently. Categories of trait structures or behaviors will
be considered here. The threshold for changing the nature or perceived characteristic
of the system is the measurable property of this profile. It is discussed in Section 2.4.
• Learning (system has flexibility to perform in unforeseen situations): After evolving, the system must adapt to the new situation. Systems need to be adaptive to survive otherwise they may collapse in dynamic conditions. Flexibility and robustness
allow systems to adapt and show the performance of the system. In some studies,
adaptation is one kind of evolution while other researchers delineate a difference
between evolution and adaptation (modeled in Section 2.4).
We define complexity of a system with the measurable properties of the profiles;
Entropy (E), Interoperabilities (I), and Evolution Thresholds (τ). E measures diversity
vs. compatibility of component features (Section 2.1/Exponential Growth and 2.1/Logistic
Growth). I’s define sensitivity (autonomy vs. dependency) to other related components and
their effects (Section 2.3). τ’s are milestones for changes and adjustments in the system
performance that can differentiate trait categories (Section 2.4). In addition a system may
have a goal. In our case this is to minimize dis-uniformity of electricity demand, D, to be
formally defined later (Section 2.1/Dis-Uniformity).
The framework starts with dissection of features. First, we study dynamics of components similar to non-complex systems (Section 2.1/Exponential Growth and 2.1/Logistic
Growth). Then we define a new measure to depict the relationships (Section 2.1/Dis-
16
Uniformity). These relationships are the initial source of emergence and are defined based
on ECAS goals (in natural CASs unlike ECASs this measure is embedded to the system
and should be found by analyzing the system behavior). Then we focus on the emergence
phenomena of ECASs as the core concept of complex adaptive behaviors and the source
of dynamic evolution. We present a comprehensive section on dissection of features and
propose four detailed theorems to show controllability and predictability of the framework
at the emergence level of a system. Then we generalize the theorems in the comprehensive
theory of mechanisms of components for ECASs (Section 2.2).
To distinguish an ECAS from a pure multi-agent system (MAS) we define interoperability as the behavioral changes that are caused by interactions (Section 2.3). In MASs
components have relationships however, in CASs the interactions and behaviors evolve. Interoperability shows how components cooperate/compete based on other components and
interactions to evolve and adapt to new environments (see new measures in Section 2.3).
While either would suffice, we use the term interoperability instead of interaction to indicate information sharing and beneficial behavior coordination. Finally, the framework
shows the adaptability and learning behavior of a system at Section 2.4.
2.1
Dissection of Features
Various studies apply the concept of information theory to study system complexities. The
key point is that the required length to describe a system is related to its complexity [76].Yu
and Efstathiou define an complexity measure based on entropy and a quantitative method to
evaluate the performance of manufacturing networks [77]. A relative complexity metric in
complex system is proposed by [78]. They employed cross-entropy to analyze interaction
of subsystems. An approach to describe self-organization systems by Renyi entropy is
shown by [79]. Also, [80] describes self organization in a complex system by using Renyi
and Gibbs-Shannon entropy. Several of these studies applied the concept of entropy in
17
their research however, they do not discuss other hallmarks of CASs. Here (Step 1 of the
framework, Fig. 2.1), we start with the same idea then we extend it to the other hallmarks.
Consider System μ (e.g. a power system) comprised of a large number of components (e.g. consumer agents). We study the individual electricity consumption C(w)
of these agents which changes dynamically during a period of length w0 . For example,
C(w), 0 ≤ w ≤ 24, is the profile of daily consumptions when w0 = 24. We assume there
are q, q = 1, ..., T equal discrete periods and that Cq (w) may differ at each period. As an
illustration, to show consumption of electricity for a season, w covers the 24 hours of consumption at each day while q = 1, ..., 90. Behaviors of components are grouped into classes
based on similar consumption pattern. Assume there are n defined classes of patterns and
q
each component follows one pattern at period q. We use Xi , i = 1, ..., n to show the number
q
of components that follow class i. Here, Qq = ∑i Xi is the total number of components in
q
the system at period q. Ci (w) presents a profile of daily consumption at period q for conq
sumer that follows pattern i (Ci (w) is the consumption of electricity at time w for pattern i
in period q).
Exponential Growth
Consider a system of components with n different patterns of behavior. For example there
may be n daily electricity usage profiles for the different classes of consumers. Assume initially that agents are independent (this assumption will be relaxed later to induce complex
behavior). If population of pattern i (Xi ; i = 1, ..., n) changes exponentially with growth
rate bi at period q, q = 1, ..., T
(q+1)
Xi
q
q
= bi .Xi + Xi or
ΔXi
= bi Xi .
Δq
(2.1)
Note that generally bi can be negative or positive. To increase the readability of the formulation in the following sections all q’s are suppressed from the expressions except when
18
necessary to compare different periods. The probabilities of the patterns can be measured
by the percentage of each pattern,
Pi =
Xi
.
∑ Xi
(2.2)
We obtain the growth equation for percentage of each group,
ΔPi bi Xi ∑ Xi − Xi ∑ bi Xi
=
= bi Pi − Pi ∑ bi Pi .
Δq
Q2
(2.3)
In long run we may assume small periods of q’s as continuous intervals. In continuous
time, the exponential function (Eq. 2.4) replaces Eq. 2.1,
Xi = αi eβi q or
dXi
= αi βi eβi q ,
dq
(2.4)
and Eq. 2.3 becomes
dPi
= βi Pi − Pi ∑ βi Pi
dq
where, Pi =
αi eβi q
.
∑i αi eβi q
(2.5)
To find self-information of components, we can measure the entropy
of population by
E = − ∑ Pi log2 Pi .
(2.6)
ΔPi 1
ΔE
= − ∑[
(
+ log2 Pi )].
Δq
Δq ln 2
(2.7)
ΔE
= bi Pi (∑ Pi log2 Pi − log2 Pi ).
Δq ∑
(2.8)
So growth of entropy is
From Eq. 2.3 and Eq. 2.7,
Logistic Growth
If population Xi has limit Li , its growth follows a logistic function and Eq. 2.1 will change
to
ΔXi
Xi
= bi Xi (1 − ).
Δq
Li
19
(2.9)
Thus, Eq. 2.3 becomes
ΔPi
Xi
Xi
= bi Pi (1 − ) − Pi [∑ bi Pi (1 − )].
Δq
Li
Li
(2.10)
Define growth potential μi = 1 − XLii , then
ΔPi
= Pi (bi μi − ∑ bi μi Pi ).
Δq
(2.11)
From Eq. 2.11 and Eq. 2.7, Eq. 2.8 can be rewritten to
ΔE
= μi bi Pi (∑ Pi log2 Pi − log2 Pi ).
Δq ∑
(2.12)
Generally, without considering interoperability between components we can find
the growth of entropy in the system with a simple procedure as follows. However, without
interoperability between components the system is not complex (see the introduction of
this chapter).
1. System components has exponential (Eq. 2.1) or logistic growth (Eq. 2.9),
2. Solve these equations to find Xi at period q,
3. Define proportions Pi ’s by Eq. 2.2 at period q,
4. Calculate entropy by Eq. 2.6.
We will show how the changes in entropy may be affected or controlled by other
properties of the components in the system. The objective of this study is not simply
solving Eq. 2.8 or Eq. 2.12. We will simulate behavior of these equations when considering interoperabilities between components effects them and examine differences. The
next sections discuss sources of self-organization, emergence and adaptation in behavior
of components in a complex system. These hallmarks of complex adaptive systems modify
the behavior of the equations.
20
Growth of entropy shows how the population changes over periods by the exponential or logistic function (entropy is self-information). However it is not sufficient for
interpreting the combination of components as any combination of three components with
0.3, 0.3, and 0.4 probability leads to the same entropy. In addition, engineered systems
have a defined goal that is not shown in the entropy (we call it dis-uniformity).
Dis-uniformity
q
q
Recall Ci (w) is the consumption of electricity at time w for pattern i in period q. Let Ci =
w0 q
0
Ci (w)dw
w0
be the average daily consumption of pattern i at period q. The dis-uniformity of
pattern i in period q is:
q
Di
=
w0
0
q
q
(Ci (w) −Ci )2 dw,
i = 1, ..., n,
q = 1, ..., T.
(2.13)
We define dis-uniformity as the mean square error of difference between the current
state of components and the goal state, namely average daily consumption to level the
load. Dis-uniformity could be reduced by incentives that change one or more profiles
or rearrange class probabilities (source of self-organization). We will show interactions
between components (e.g. a social network with friendship links) and their interoperability
effects on dis-uniformity of the system.
We will illustrate how dis-uniformity can be extended to other ECASs. At first
glance, the dis-uniformity of an individual component, Eq. 2.13, looks similar to variance.
We do not use this term because the consumption is not considered as a random variable.
Furthermore, it is customary to refer to the variance as the range/noise of consumption at a
specific time w.
21
The control objective is to minimize the total dis-uniformity of the system (consumers cooperate to achieve uniform aggregate consumption at each period). Thus, we
seek to minimize D,
w0 n
q
q
q
∑i=1 (Ci (w) −Ci )Xi 2
(
) dw.
D =
q
n
q
0
∑i=1 Xi
(2.14)
Note that we remove q’s in our formula to increase readability. However D, C, and X are
functions of q.
Here we use dis-uniformity to show how the system behaves as an ECAS (we will
show how it causes dependencies between behaviors later). Concepts from information
theory are adopted to describe complexity, self-organization and emergence in the context
of our ECASs [73]. Controlling dis-uniformity is a source of self-organization in ECASs
(see Section 2.2). Shalizi [81] and Shalizi et al. [82] define a quantifying self-organization
for discrete random fields (e.g. Cellular Automata). We reinterpret these concepts to apply
them in ECASs that may have continuous states and, unlike natural physical systems, may
not have a natural embedded energy dynamic or self-directing law. Self-organization and
adaptive agents are analyzed by [22]. We will extend these concepts to all hallmarks of
ECASs. Bashkirov [80] describes self-organization in a complex system by using Renyi
and Gibbs-Shannon entropy. These studies are applicable in natural and physical systems. For example a biological application, gene-gene and gene-environment interactions,
is identified by interaction information and generalization of mutual information in [83].
2.2
Entropy vs. Dis-uniformity, Source of Self-organization
In this section (Step 1 of the framework, Fig 2.1) , we connect the concept of entropy and
dis-uniformity for patterns. We propose lemmas for a system with two components that
interact. Let C =
w0 ∑ni=1 Ci (w)Xi
0
(
w0 ∑ni=1 Xi
)dw be the total average daily consumption of patterns.
22
Definition I:
• Dominance: behavior i dominates behavior j (i j) if Di ≤ D j .
• Strict Positive Dominance: behavior i strictly positively dominates behavior j (i j)
if Di < D j , |Ci (w) −Ci | ≤ |C j (w) −C j | for all w and sgn(Ci (w) −Ci ) = sgn(C j (w) −
C j ) for all w.
• Positive Dominance: behavior i positively dominates behavior j (i j) if Di < D j ,
|Ci (w) − Ci | > |C j (w) − C j | for some w and sgn(Ci (w) − Ci ) = sgn(C j (w) − C j ) for
all w.
• Strict Negative Dominance: behavior i strictly negatively dominates behavior j (i j) if Di < D j , |Ci (w)−Ci | ≤ |C j (w)−C j | for all w and sgn(Ci (w)−Ci ) = sgn(C j (w)−
C j ) for all w.
• Negative Dominance: behavior i negatively dominates behavior j (i j) if Di < D j ,
|Ci (w) − Ci | > |C j (w) − C j | for some w and sgn(Ci (w) − Ci ) = sgn(C j (w) − C j ) for
all w.
Note that E is increasing in period (E ↑) means E ( q + 1) > E ( q) and (E ↓) means E ( q +
1) < E ( q). We use the same definition for (D ↑) and (D ↓).
Lemma I: Given two different patterns of behavior (i and j) in the population, i j,
and at period q = q0 :
I.1) Pi 0 < Pj 0 and bi > b j i f f E q0 +1 > E q0 and Dq0 +1 < Dq0 i.e. E is increasing in period
q
q
(E ↑) and D decreases in period (D ↓);
I.2) Pi 0 > Pj 0 and bi > b j i f f E q0 +1 < E q0 and Dq0 +1 < Dq0 i.e. E is decreasing in period
q
q
(E ↓) and D decreases in period (D ↓);
23
I.3) Pi 0 < Pj 0 and bi < b j i f f E q0 +1 < E q0 and Dq0 +1 > Dq0 i.e. E is decreasing in period
q
q
(E ↓) and D increases in period (D ↑);
I.4) Pi 0 > Pj 0 and bi < b j i f f E q0 +1 > E q0 and Dq0 +1 > Dq0 i.e. E is increasing in period
q
q
(E ↑) and D increases in period (D ↑).
Proof: Sufficiency of Lemma I.1:
q
q
q
q
q
q
We are given, n = 2, Pi 0 + Pj 0 = 1, and Pi 0 < Pj 0 . So, Pi 0 < 1/2 and Pj 0 > 1/2.
Also, bi > b j results in,
q
q
q
Xi 0
bi Xi 0 + Xi 0
<
q
q
q
q
q
q ,
Xi 0 + X j 0
bi Xi 0 + Xi 0 + b j X j 0 + X j 0
(2.15)
q
q +1
because bi Xi2 + Xi2 + b j Xi X j + Xi X j < bi Xi2 + bi Xi X j + Xi2 + Xi X j at q0 . Thus, Pi 0 < Pi 0
q
q +1
and similarly we can show Pj > Pj 0
. So, the probabilities are closer to a uniform distri-
bution (Pi is closer to Pj i.e., both are closer to 1/2) in q0 + 1.
Recall from [84] Page 29, the uniform distribution of Xi s (frequency of patterns)
gives the maximum entropy of the system. Suppose Pi =
1
n
is the uniform probability mass
function for Xi ; i = 1, ..., n, so the maximum entropy of the system is log2 n.
q
q
Thus, max(E) = 1 when n = 2 and Pi = Pj = 1/2 at period q, hence, E is an
increasing function of q, i.e., E q0 +1 > E q0 while Pi 0 < Pj 0 .
q
q
Furthermore, i strictly dominants j for all time intervals w and has similar sign
with j thus, increasing the portion of
q
Xi 0
q
Xj
<
q0+1
Xi
q +1
Xj 0
Xi
Xj
decreases dis-uniformity in Eq. 2.14 because,
and |Ci (w) − Ci | ≤ |C j (w) − C j | ∀w. These conditions increase the portion
of |Ci (w) − Ci | (that are smaller than |C j (w) − C j |) in the total dis-uniformity; therefore,
Dq0 +1 < Dq0 i.e., D increases.
24
We can apply the same argument to prove sufficiency of Lemma I.2, I.3, and I.4.
Note that we do not consider bi = b j or Pi = P j because, they are neutral cases and do not
have any effect. So, all four combinations of b s and P s are generated in Lemma I.
Necessity of Lemma I.1: (proof by contradiction)
Suppose E q0 +1 > E q0 and Dq0 +1 < Dq0 i.e., E increases and D decreases, but one or both
conditions of Lemma I.1 do not hold. In this case, necessary conditions for one of the I.2 or
q
q
q
q
I.3 or I.4 holds. For example, if bi > b j but Pi 0 > Pj 0 instead of Pi 0 < Pj 0 , this is Lemma
I.2 and E q0 +1 < E q0 (E decreases) which contradicts our assumptions.
Corollary I: When conditions of Lemma I hold and t → ∞:
I.1) In exponential growth Di is a lower bound for D and E ∈ (0, 1) when D decreases
(Lemma I.1 and I.2). Also, D j is an upper bound for D and E ∈ (0, 1) when D increases
(Lemma I.3 and I.4);
I.2) Consider logistic growth where f , f , g, and g are functions of the logistic limits
Li . Then max{Di , f (Li )} is a lower bound for D and E ∈ (0, g(Li )) when D decreases (Lemma I.1 and I.2). Also, min{D j , f (L j )} is an upper bound for D and
E ∈ (0, g (L j )) when D increase (Lemma I.3 and I.4).
Proof: In corollary I.1, D decreases when proportion
Xi
Xj
increases (due to the dominance
condition), so min(D) = Di when all components are i ( XXij → ∞ and E = 0). And D increases when proportion
Xi
Xj
decreases, so max(D) = D j when all components are j ( XXij → 0
and E = 0). However, max(E) = log2 n and n = 2, so max(E) = 1 and E is nonnegative.
25
When the fitness follows a logistic function (Corollary I.2), we have limits for the
number of i’s and j’s,
Xi
Xj
< ∞ if X j = 0 and
Xi
Xj
of the limit of i when
Xi
Xj
increases and max(D) is a function of the limit of j when
> 0 if Xi = 0. So min(D) is a function
Xi
Xj
decreases. Clearly, min(D) = Di when X j = 0 and max(D) = D j when Xi = 0. Using the
same argument we can find the range of E which is a function of limits. Theorem I: Given n different patterns of behavior (i = 1, ..., n) in population S,
bk ≥ 0, ∀k ∈ S and i j, for i ∈ S and j ∈ S − S :
I.1) E < − log2 Pi (∑i∈S Pi log2 Pi > log2 Pi ) and bi > b j for i ∈ S and j ∈ S − S i f f E is
increasing in period (E ↑) and D decreases in period (D ↓);
I.2) E > − log2 Pi (∑i∈S Pi log2 Pi < log2 Pi ) and bi > b j for i ∈ S and j ∈ S − S i f f E is
decreasing in period (E ↓) and D decreases in period (D ↓);
I.3) E < − log2 Pi (∑i∈S Pi log2 Pi > log2 Pi ) and bi < b j for i ∈ S and j ∈ S − S i f f E is
decreasing in period (E ↓) and D increases in period (D ↑);
I.4) E > − log2 Pi (∑i∈S Pi log2 Pi < log2 Pi ) and bi < b j for i ∈ S and j ∈ S − S i f f E is
increasing in period (E ↑) and D increases in period (D ↑).
Proof: Sufficiency of Theorem I:
This theorem generalizes Lemma I to n components. Similar to Lemma I the entropy
of the system increases when probability of components are closer to uniform distribution. It happens when for exponential growth in (2.8) or for logistic growth in (2.12),
∑i∈S Pi log2 Pi = log2 Pi . To reach this point E increases if there is a larger fitness rate for
components which have probability less that uniform distribution. In general larger fitness
rates increase the entropy if − log2 Pi > E (Theorem I.1) for the cases that we can not reach
26
the uniform distribution or if we want to compare some components where all have smaller
or larger probabilities than uniform.
Like Lemma I, increasing the number of dominant components decreases the total
dis-uniformity (2.14). The same argument will prove Theorem I.2, I.3, and I.4. We can
also prove the necessity of Theorem I by contradiction. Corollary II: When conditions of Theorem I hold and t → ∞:
II.1) Corollary I.1 can be generalized to n components in Theorem I with E ∈ (0, log2 n);
II.2) Corollary I.2 can be generalized to n components in Theorem I with different f , f ,
g, and g functions.
Note that bk > 0, ∀k ∈ S means all Xi ’s are growing over period, however some Pi s may
decrease.
Lemma II: Given two different patterns of behavior (i and j) in the population, i j,
and given period q = q0 ; Lemma I.1, I.2, I.3, and I.4 and Corollary I.1 and I.2 are valid.
Proof: This is a generalization of Lemma I to the positive dominance case. This
case allows j to dominate i in some time interval w however, the proof is still valid because
D is total dis-uniformity. Theorem II: Given n different patterns of behavior (i = 1, ..., n) in population S,
bk ≥ 0, ∀k ∈ S and i j, for i ∈ S and j ∈ S − S ; Theorem I.1, I.2, I.3, and I.4 and
Corollary II.1 and II.2 are valid.
27
Proof: This Theorem is a generalization of Lemma II to n components. We can
use the same argument which we used to generalize Lemma I to Theorem I to generalize
Lemma II to Theorem II. Example 1: (Features on Fig. 2.1) Assume there are 100 components in a complex
system which only follows three patterns i, j, and k. Fig. 2.2a shows the average daily
consumption of electricity in 24 hours for a season. The patterns repeat every season for
next 8 years (32 seasons). Choose initial populations Pi = 0.15, Pj = 0.65, and Pk = 0.2
(i.e. at season q = 1, 15% of components follow pattern i, 65% follow j, and 20% follow
k). Let the population grow according to Eq. 2.1 with bi = 0.2, b j = 0.1, and bk = 0.3. The
objective is simulating and analyzing the complex system for the next 32 seasons.
From Eq. 2.13, Di = 42.99, D j = 77.33, and Dk = 61.96. With Eq. 2.3 we can
calculate populations in different seasons in Fig. 2.2b and with Eq. 2.8 and Eq. 2.14 we
measure changes in E and D by seasons in Fig 2.2c. At q = 1 the system follows Lemma
II.1 until q = 8:
Pi (q = 1) = 0.15,
Pj (q = 1) = 0.65,
E(q = 1) = 1.28,
D(q = 1) = 65.08.
Pk (q = 1) = 0.2,
At q = 9 we have max(i) and the system follows Lemma II.2 until q = 18:
Pi (q = 9) = 0.18,
Pj (q = 9) = 0.38,
E(q = 9) = 1.49,
D(q = 9) = 59.08.
Pk (q = 9) = 0.44,
At q = 19 dis-uniformity starts increasing again (Lemma II.3) until end of the simulation:
Pi (q = 19) = 0.13, Pj (q = 19) = 0.12, Pk (q = 19) = 0.75,
E(q = 19) = 1.07, D(q = 19) = 57.45 (D(q = 18) = 57.36).
28
7
Consumption(KW)
6
5
4
i
3
j
2
k
1
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
w
(a) Patterns
1
0.9
0.8
Probability
0.7
0.6
0.5
%i
0.4
%j
0.3
%k
0.2
0.1
0
1
6
11 16 21 26 31 36 41 46 51 56 61 66 71 76
t
(b) Growth
66
1.6
64
1.4
1.2
62
1
60
0.8
58
0.6
D
E
56
0.4
54
0.2
52
0
1
11
21
31
41
51
61
71
q
(c) D vs. E
Figure 2.2: Example for Theorem II
29
Fig. 2.2b shows the probability changes and Fig. 2.2c presents the behavior of
components and simulates entropy and dis-uniformity of the system for 32 seasons. Fig.
2.2c shows the three different possible areas for Theorem II.
Lemma III: Given two different patterns of behavior (i and j) in the population,
i j, and given period q = q0 :
III.1) Pi 0 < Pj 0 and bi > b j i f f E q0 +1 > E q0 and Dq0 +1 < Dq0 i.e. E is increasing in
q
q
period (E ↑) and D decreases in period (D ↓) until D = 0 (Xi (Ci (w) − Ci )dw =
X j (C j (w) −C j )dw) afterward D increases in period (D ↑);
III.2) Pi 0 > Pj 0 and bi > b j i f f E q0 +1 < E q0 and Dq0 +1 < Dq0 i.e. E is decreasing in
q
q
period (E ↓) and D decreases in period (D ↓) until D = 0 (Xi (Ci (w) − Ci )dw =
X j (C j (w) −C j )dw) afterward D increases in period (D ↑);
III.3) Pi 0 < Pj 0 and bi < b j i f f E q0 +1 < E q0 and Dq0 +1 > Dq0 i.e. E is decreasing in
q
q
period (E ↓) and D increases in period (D ↑);
III.4) Pi 0 > Pj 0 and bi < b j i f f E q0 +1 > E q0 and Dq0 +1 > Dq0 i.e. E is increasing in period
q
q
(E ↑) and D increases in period (D ↑).
Proof: To prove this lemma we should consider different sgn(Ci (w) −Ci ) between
dis-uniformity of i and j, for all w. So, the total dis-uniformity decreases until 0 and increases after that (because of power of 2 in (2.14)). D = 0 when the weighted dis-uniformity
for all components i is equal to weighted dis-uniformity for all components j. When the
total dis-uniformity increases (Lemma III.3 and III.4) we do not need to consider any minimum point, because the function is non decreasing. 30
Corollary III: When conditions of Lemma III hold and t → ∞:
III.1) In exponential growth ∃ε > 0 where, D < ε (ε is a lower bound for D) and E ∈ (0, 1)
when D decreases (Lemma III.1 and III.2). Also, D j is an upper bound for D and
E ∈ (0, 1) when D increases (Lemma III.3 and III.4);
III.2) In logistic growth max{0, f (Li )} is a lower bound for D and E ∈ (0, g(Li )) when D
decreases (Lemma III.1 and III.2). Also, min{D j , f (L j )} is an upper bound for D
and E ∈ (0, g (L j )) when D increase (Lemma III.3 and III.4).
Proof: Proof is similar to Corollary I, however for a specific w = w0 where, Xi (Ci (w0 )−
Ci )dw0 ≈ X j (C j (w0 ) − C j )dw0 , we have D ≈ 0. This point may happen before all components become similar to i’s so min(D) = 0 where, E = 0 and E = 0 where, D = 0. Theorem III: Given n different patterns of behavior (i = 1, ..., n) in population S,
bk ≥ 0, ∀k ∈ S and i j for i ∈ S and j ∈ S − S :
III.1) E < − log2 Pi (∑i∈S Pi log2 Pi > log2 Pi ) and bi > b j for i ∈ S and j ∈ S−S i f f E is in
creasing in period (E ↑) and D decreases in period (D ↓) until D = 0 (∑ Xi (Ci (w) −
Ci )dw = ∑ X j (C j (w) −C j )dw) afterward D increases in period (D ↑);
III.2) E > − log2 Pi (∑i∈S Pi log2 Pi < log2 Pi ) and bi > b j for i ∈ S and j ∈ S −S i f f E is de
creasing in period (E ↓) and D decreases in period (D ↓) until D = 0 (∑ Xi (Ci (w) −
Ci )dw = ∑ X j (C j (w) −C j )dw) afterward D increases in period (D ↑);
III.3) E < − log2 Pi (∑i∈S Pi log2 Pi > log2 Pi ) and bi < b j for i ∈ S and j ∈ S − S i f f E is
decreasing in period (E ↓) and D increases in period (D ↑);
31
III.4) E > − log2 Pi (∑i∈S Pi log2 Pi < log2 Pi ) and bi < b j for i ∈ S and j ∈ S − S i f f E is
increasing in period (E ↑) and D increases in period (D ↑).
Corollary IV: When conditions of Theorem III hold and t → ∞:
IV.1) Corollary III.1 can be generalized to n components in Theorem III with E ∈ (0, log2 n);
IV.2) Corollary III.2 can be generalized to n components in Theorem III with different f ,
f , g, and g functions.
Lemma IV: Given two different patterns of behavior (i and j) in the population,
i j, and given period q = q0 ; Lemma III.1, III.2, III.3, and III.4 and Corollary III.1 and
III.2 are valid.
Theorem IV: Given n different patterns of behavior (i = 1, ..., n) in population S,
bk ≥ 0, ∀k ∈ S and i j, for i ∈ S and j ∈ S − S ; Theorem III.1, III.2, III.3, and III.4 and
Corollary IV.1 and IV.2 apply.
Example 2: Assume we modify Example 1 to three components with negative dominance, (Fig. 2.3a, Di = 42.99, D j = 77.33, and Dk = 59.96).
Fig. 2.3b shows the behavior of the complex system and Fig. 2.3c shows the
different possible cases of Theorem IV.
Summary: We summarize the results of Theorem I, II, III, and IV in Table 2.1 and
conclude Theorem V as a general theorem to control decomposability and indicate results
of interactions between components of a complex system in all dominance cases.
32
7
Consumption(KW)
6
5
4
i
3
j
2
k
1
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
w
(a) Patterns
1
0.9
0.8
Probability
0.7
0.6
0.5
%i
0.4
%j
0.3
%k
0.2
0.1
0
1
6
11 16 21 26 31 36 41 46 51 56 61 66 71 76
t
(b) Growth
70
1.6
60
1.4
1.2
50
1
40
0.8
30
0.6
20
0.4
10
0.2
0
0
1
11
21
31
41
51
61
71
q
(c) D vs. E
Figure 2.3: Example for Theorem IV
33
D
E
Table 2.1: Summary of emergence
i
i
i
i
j
j
j
j
− log2 Pi > E
− log2 Pi < E
bi > b j
bi < b j
bi > b j
bi < b j
E ↑∧D↓
E ↓∧D↑
E ↓∧D↓
E ↑∧D↑
E ↑∧D↓
E ↓∧D↑
E ↓∧D↓
E ↑∧D↑
E ↑ ∧ D ↓ (1), D ↑ (2) E ↓ ∧ D ↑ E ↓ ∧ D ↓ (1), D ↑ (2) E ↑ ∧ D ↑
E ↑ ∧ D ↓ (1), D ↑ (2) E ↓ ∧ D ↑ E ↓ ∧ D ↓ (1), D ↑ (2) E ↑ ∧ D ↑
(1) if ∑ Xi (Ci (w) −Ci )dw > ∑ X j (C j (w) −C j )dw,
(2) if ∑ Xi (Ci (w) −Ci )dw < ∑ X j (C j (w) −C j )dw.
Theorem V (mechanisms of components): If i j, i.e., i’s dominate j’s, dis-uniformity
of the system is decreasing in period if the entropy increases in period when − log2 Pi > E
or if the entropy decreases in period when − log2 Pi < E while, ∑ Xi (Ci (w) − Ci )dw <
∑ X j (C j (w) −C j )dw for both conditions.
We can apply this theorem to control or at least predict the complex behaviors in
large ECASs. Here, we provide incentives to motivate the components to decrease the
dis-uniformity by adjusting their patterns (this adjustment changes the growth rates bi ’s
dynamically). This heterarchical rearrangement with external changes to the environment
but without central organization is a source of self-organizing in components. As an illustration, assume n patterns of consumption in a system. When n is large (e.g. patterns of
consumers in large metropolitan area), it is impossible to control and predict all behaviors
and their relationships. We can focus on a few groups (pattern i where − log2 Pi > E) and
increase the entropy by motivating other consumers to adjust to this pattern (migrate to
this pattern or increase its growth portion). This phenomena makes non-linear complex
dynamic growth rates i.e. bi = K(R(D); E). Here, K is a function of R(D) and population
of other patterns (i.e. E). R(D) shows the motivations based on D (e.g. rewards that consumers receive by cooperating to reduce the dis-uniformity). These changes in bi ’s make
Xi dependent on each other. To predict the behaviors at each period, we can map the system
34
conditions (dominance, entropy, and growth rates) to an appropriate theorem. In the next
section we will show how we can control the interoperability between patterns by using a
third pattern (catalyst) i.e. indirectly utilize Theorem V to decrease the dis-uniformity.
2.3
Emergence as the Effect of Interoperability
In Step 2 of the framework (Fig. 2.1), we study the engineering concept of emergence in
ECASs. Bar Yam [85] conceptually and mathematically shows the possibility of defining a
notion of emergence and described four concepts of emergence. Conceptual classification
for emergence is proposed by Halley and Winkler [86]. Prokopenko et al. [73] interpret
concepts of emergence and self-organization by information theory and compare them in
CASs. We borrow concepts from information theory to analyze and predict emergence
behaviors of ECASs and show the applicability of Theorem V.
Consider Xi to be an integer variable that describes the number of components that
follow pattern i in a population with total Q components. Assume there are certain defined
states for proportion
Xi
Q
(e.g. low, medium, and high proportion). Let mi , mi = 1, ..., Mi ,
present number of defined states for pattern i. We define Pmi m j to be the joint probability to
find simultaneously pattern i and pattern j in state mi and m j . These probabilities can be
found by statistical analysis of historical information about the population. For example in
Table 2.2a we consider pattern i has low state when it is 0%-20% and pattern j has medium
state when it is 0%-15% out of the population. Here, Pmi m j = P(mi = low, m j = medium) =
0.05 shows the joint probability of 0 ≤
Xi
Q
< 0.2 and 0 ≤
Example 2 for more details).
35
Xj
Q
< 0.15 is equal to 0.05 (see
Emergence cannot be defined by properties and relationships of the lower component level [81]. Assume there is an interaction between pattern i and j. Then Eq. 2.6
becomes,
E(i, j) = −
Mi
Mj
∑ ∑
mi =1 m j =1
Pmi m j log2 Pmi m j ,
(2.16)
We measure the interoperability between i and j which is the amount of information
that i and j share and reduce the uncertainty of each other by
p
I (i; j) =
Mi
Mj
∑ ∑
mi =1 m j =1
Pmi m j log2
Pmi m j
,
Pmi Pm j
(2.17)
where, Pmi is the marginal probability for State mi . This equation is interaction information
(mutual information) of i and j in information theory. There can be other interoperabilities
in a system such as interoperability between class of agents (I c , to be formally defined later
in Section 3.2) so, I generally shows the interoperabilities between properties of agents.
We can obtain [84],
E = E(i, j) = E(i) + E( j) − I p (i; j),
(2.18)
where, E(i) = I p (i; i) is the self-information of i.
From Eq. 2.18 when I p (i; j) increases (I p ↑), E decreases (E ↓). For the case of
only two groups of patterns in the system, the mutual information is a positive number with
maximum of one, 0 ≤ I p ≤ 1 (from Eq. 2.17). E is minimal when i and j are identical, I p =
1 (one group follows the other one) and E is at its maximum when i and j are independent,
I p = 0 (groups are completely autonomic). We can use this property to control the entropy
in Lemma I, II, III, and IV.
The generalization of Eq. 2.18 to three pattern cases is
E = E(i, j, k) = −[E(i) + E( j) + E(k)] − I p (i; j; k) + E(i, j) + E(i, k) + E(k, j),
36
(2.19)
where, interoperability I p , can be negative. Let I p (i; j|k) define conditional interoperability
between i and j conditioned on k,
I p (i; j|k) =
Mi
Mj
Mk
∑ ∑ ∑
mi =1 m j =1 mk =1
Pmi m j mk log2
Pmk Pmi m j mk
.
Pmi mk Pm j mk
(2.20)
Here, we define Pi jk to be the joint probability to find simultaneously patterns i, j and k in
the State mi , m j and mk , respectively. Then, we obtain,
I p (i; j; k) = I p (i; j) − I p (i; j|k).
(2.21)
Positive I p means k supports and increases the interoperability between i and j.
However, negative I p shows k inhibits and decreases the interoperability.
Definition II:
• Catalyst: Pattern k is a positive catalyst for other patterns in the system if k supports
their interoperability and is a negative catalyst if inhibits their interoperability.
Let I p = I(μ), μ = {in0 |n0 = 1, ..., n} represents interoperability between patterns
in a system where, in0 shows its pattern types that each of them can be appear in mi states.
We can generalize Eq. 2.19 and Eq. 2.21 to n patterns [83, 87],
E(μ) =
∑
ν⊆μ,ν=μ
(−1)(|μ|−|ν|−1) E(ν) − I(μ),
(2.22)
μ = {in0 |n0 = 1, ..., n}.
I p (i1 ; ...; in ) = I p (i1 ; ...; in−1 ) − I p (i1 ; ...; in−1 |in ).
(2.23)
Generally for multiple catalyst (k number of catalyst),
I p (i1 ; ...; in ) = I p (i1 ; ...; in−k ) − I p (i1 ; ...; in−k |i(n−k+1) ; ...; in ).
37
(2.24)
In Theorem V instead of increasing or decreasing the entropy we can change the
interoperability. We add catalyst(s) to control (inhibit or support) the interoperability. We
define I p (μ|k) as the conditional interoperability of System μ conditioned on existing new
pattern k.
Definition III:
• Catalyst-Associate Interoperability (CAI): Consider set of patterns that have interoperability I p (μ). Then we show the interoperability between patterns when a new
pattern k exists in System μ by I p (μ|k),
CAI = I p (μ|k) − I p (μ).
(2.25)
• Effect of Catalyst (EOC): Consider set of patterns that give entropy E(μ) then entropy of the system conditioned on existing new pattern k is E(μ|k),
EOC =
E(μ|k) − E(μ)
,
CAI
(2.26)
where, E(μ|k) and E(μ) are entropy in period q after and before applying the catalyst(s), respectively.
Example 3: (Interoperability in Fig. 2.1) Define the state of population of pattern
i to be low if 0 ≤ Pi < 0.2, medium if 0.2 ≤ Pi < 0.4, and high if Pi ≥ 0.2. Also the state
of population of pattern j is low if 0 ≤ Pj < 0.1, medium if 0.1 ≤ Pj < 0.15, and high if
Pj ≥ 0.15. Assume Table 2.2a is the joint probabilities for states of i and j in Example 1
where population of other patterns and their effects are negligible.
38
Table 2.2: Prior and Posterior joint probabilities
(a) Prior Probabilities for k ≈ 0
P(mi , m j ) low medium
low
0.20
0.05
medium 0.15
0.15
high
0.05
0.05
(b) Posterior Probabilities for k > 0
P(mi , m j |k) low medium
low
0.23
0.03
medium
0.15
0.19
high
0.02
0.03
high
0.02
0.15
0.18
high
0.02
0.13
0.20
From Equations 2.16, 2.17, and 2.18:
E(i) = 1.56, E( j) = 1.54, E(i, j) = 2.90, I p (i; j) = 0.20.
If adding Catalyst k, updates Table 2.2a to Table 2.2b (users k affect the interrelationships
between i’s and j’s),
E(i) = 1.56, E( j) = 1.53, E(i, j) = 2.73, I p (i; j) = 0.36.
So we increase the entropy by increasing the interoperability which decreases the
dis-uniformity in Example 1.
CAI = 0.36 − 0.2 = 0.16,
EOC =
2.73−2.90
0.16
= −1.06.
We can use the concept of EOC to select an appropriate catalyst. For example
assume n patterns of consumption in a social population where, i1 and i2 have the majority
of population and thus the largest effect on the dis-uniformity of the consumption. We
are planning to decrease the dis-uniformity with a limited amount of resources (e.g. some
rewards to give to cooperative consumers). Instead of distributing the reward between a
large group (say i1 ) to cooperate with the other group which is not so effective (because the
portion of each individual is too low), we can reward a small group of catalyst (say i3 ) to
improve the interoperability between i1 and i2 . This idea is similar to finding and investing
39
on hubs in a social network (based on power-law the numbers of components with higher
relationships decrease exponentially [34]). The next step is to show how this emergence
phenomena causes evolution in the system.
2.4
Evolution Because of Updates in the Traits
Here (Step 3 of the framework, Fig. 2.1), we analyze the evolution process. Then, in the
last step of the framework (Step 4) we depict the adaptation and learning in the system.
Some measures are developed for the complexity threshold parameter of physical complex
systems in previous studies[88]. Erdos and Renyi[89] study probability threshold function
and evolution in random graphs. We borrow the concept of threshold [89].
q
Let Mλ ; λ = 1, ..., λ0 be the number of components which have the trait λ at period
q (such as willingness to adjust consumption pattern for a specific cost saving). The probability that a system possesses trait λ at period q is
q
Mλ
Q .
q
Here, φλ is a binary variable that
shows the system possesses Attribute λ at period q.
⎧
q
Mλ
⎪
⎨ 1,
if
Q ≥ τλ ,
q
φλ =
q
Mλ
⎪
⎩ 0,
if
Q < τλ ,
(2.27)
where, τλ is the threshold for trait λ .
q
Let Φq = (φλ ; λ = 1, ..., λ0 ) be a vector of 0 and 1’s for λ0 traits where, its λ th
q
position is 1 if φλ = 1. Let Ψq be a finite set of Φ’s at period q. Based on the definition, the
/ Ψ) & Φq0 ∈
/ Ψ(or Φq0 ∈ Ψ). Where, Ψq is
system evolves when ∃q > q0 , Φq ∈ Ψ(or Φq ∈
a predefined finite set of Φ’s at period q.
Definition IV:
• Stagnation: systems are stagnant when they are not evolvable i.e., Φq ∈ Ψ(or Φq ∈
/
Ψ)∀q.
40
Example 3: (Traits in Fig. 2.1) Assume three traits i, j, and k with τi = 0.2, τ j = 0.4,
τk = 0.3, and Ψ = {[0 1 1], [1 1 1]} in Example 1:
⎫
q
Mi
⎪
26
q = 4 : Qq = 156 ≤ 0.2 ⎪
⎪
⎪
⎬
q
Mj
87
⇒ Φ(4) = [0 1 0],
q = 4 : Qq = 156 ≥ 0.4
⎪
⎪
q
⎪
⎪
M
44
q = 4 : Qqk = 156
≤ 0.3 ⎭
⎫
q
Mi
⎪
31
q = 5 : Qq = 183 ≤ 0.2 ⎪
⎪
⎪
⎬
q
Mj
95
⇒ Φ(5) = [0 1 1],
q = 5 : Qq = 183 ≥ 0.4
⎪
⎪
q
⎪
⎪
M
57
q = 5 : Qqk = 183
≥ 0.3 ⎭
⎫
q
Mi
⎪
55
q = 9 : Qq = 367 ≤ 0.2 ⎪
⎪
⎪
⎬
q
Mj
139
⇒ Φ(9) = [0 0 1].
q = 9 : Qq = 367 ≤ 0.4
⎪
⎪
q
⎪
⎪
M
⎭
≥
0.3
q = 9 : Qqk = 163
367
So related to the set Ψ = {[0 1 1], [1 1 1]} the system evolves at q = 5 and q = 9. If
we assume set Ψ contains only [1 1 1] (i.e. the system evolves only when it possesses all
traits) then Φq is never in Ψ and this system is stagnant.
In this example the system is adjusted by two evolutions. This adapting situation
can be non-stationary. In later sections we simulate a case where components adjust their
behaviors several times to increase their objectives. For example, Fig. 2.4a shows the case
that, i, j, and k compete to get more rewards by reducing the dis-uniformity. However,
to reduce the dis-uniformity they should cooperate by adjusting their behaviors (changing
growth rates, bi ’s). Adding a learning procedure (do not adjust to previous tried states)
omits the non-stationary evolution and causes faster adaptation in Fig. 2.4b. See Section
4.2 for detailed analyze of this behavior and more information about the simulation.
41
1
0.9
0.8
0.8
07
0.7
07
0.7
0.6
0.5
i
0.4
j
0.3
k
Fitne
ess
Fitne
ess
1
0.9
0.6
0.5
i
0.4
j
0.3
k
0.2
0.2
0.1
0.1
0
0
1
5
9
13
17
21
25
29
33
1
5
9
Time
13
17
21
25
29
33
Time
(a) Non-stationary adaptation
(b) Fast adaptation
Figure 2.4: A complicated example for adaptation
To extend this framework, the concept of dissection of features can extend to other
ECASs easily. Entropy of components is a general concept for all systems and dis-uniformity
can be interpreted in different ECASs. For example, reducing demand fluctuations in
wholesale marketing, resource allocations in supply chain management, and synergism
of commands to reduce the distances to a target in AI or defense sectors are other types of
dis-uniformity. Decision makers may assign different objectives to ECASs based on their
requirement and they are not limited to dis-uniformity. However, any ECAS of the class
being addressed needs at least one minimizing/maximizing measure to study dissection of
features other than component entropy. Other hallmarks (Evolution and Adaptation) are
driven from the emergence concept (dissection of features and the interactions) and their
mathematical calculation is not limited to electricity usage.
42
Chapter 3
AN AGENT-BASED OPTIMIZATION PLATFORM
We propose a methodology that advances our ability to model and understand engineered
complex adaptive system behaviors through computational intelligence. The development
of this methodology has significant intrinsic merit as it will explore fundamental issues
in multi-layered ECASs and presents new optimization actions and descriptive paradigms.
This ABM presents a novel platform and toolkit that creates a dynamic laboratory to study
properties and behavior of DR in the US Electricity Market as an ECAS. We integrate concepts from social networks (e.g. friendship, scale-free and single-scale networks), social
science (e.g. opinion exchange, media impact, and irrationality), complexity theory (e.g
emergent behavior and adaptation), diffusion theory (e.g. adaptability and innovativeness),
and decision theory (e.g. utility) to build our comprehensive ABM. This model enables
us to integrate optimization rules and build a comprehensive social layer to analyze and
simulate details of social behaviors in ECASs. We justify our model by illustrating its
applicability on the US power system.
In this study, a multi-layer descriptive modeling approach (Fig. 3.1) composed
of conceptual behavior and fundamental entity aspects is considered as the whole system
structure. This approach integrates layers by cross-layer effects, effect of stochastic events
(e.g. system failure and extreme weather condition), and rules of standards and procedures.
Usually, physical layer(s) model networks, their characteristics, and physical components
[90]. In this layer effects of topology and structure of networks have been studied in the
past [49, 50, 51]. Decision/control layer(s) represent different business strategies, optimization models and rules for decision makers (e.g. regulators). Those models are used for
short-term, mid-term, or long-term analysis. Previous modeling approaches such as AMES
[65, 66] are useful for modeling behaviors of this layer. Some models (e.g. EMCAS [67])
43
present a more comprehensive view on different layers. In this study, Social/ Swarm Layers(s) include the interrelationship between agents and their effects on each other. Also
behavioral properties and attributes of decision makers are considered in this layer (see
Section 3.1/Environment for details and examples of each layer).
Figure 3.1: Integration of the Structural Entity Model
To show cascading of decisions from Decision/Control Layer(s) to Physical Layer(s),
we detail the Social/ Swarm Layer(s). This integration helps us depict emergence in behavioral patterns in the lower level of the system and their effects on evolution in the higher
levels (e.g. predict the result of dynamic pricing and prizes in future). Moreover, we
formulate a consumer-based optimization model and integrate it with the ABM. Then we
include effects of externalities and subjective behaviors in the integrated agent-based optimization model. We study the strategies that have critical effects on the decision/control
and social/swarm layers. Useful managerial insights are derived from executing the model.
44
Electricity regulators adjust service attributes (e.g. price and reward functions)
in response to demand fluctuations to motivate the consumer agent to cooperate in balancing workload. Consumer agents make decisions to maximize their quantitative and
qualitative utilities. They change consumption patterns in response to incentives and to
social education provided by the regulators. Based on the individual and total patterns
of the electricity consumption, agents may receive individual and cooperation rewards.
Moreover, they may have dissatisfaction to change their behavior (Section 3.2/Process
Overview presents details for calculating utilities, rewards, and dissatisfaction). We study
cooperativeness or competition in the consumers game environment by embedding a nonconvex consumer-based optimization model with the agent-based simulation (see Section
3.2/Decision Rules). We study the behavior of consumers under different control and incentive strategies. Then, we include intrinsic environment and control factors to the model
dynamics.
Instead of reducing nonlinear systems to a set of causal variables and error terms,
our ABM shows how complex adaptive outcomes flow from simple phenomena and depend on the way that agents are interconnected. Rather than aggregating outcomes to find
a total equilibrium, our ABM presents the evolution of outcomes as the result of the efforts of agents to achieve better fitness. This method allows the study and analysis of the
complexities arising from individual actions and interactions that exist in the real world by
bottom-up iterative design methodologies.
Applying this integrated agent-based optimization model has the following benefits
for the US power system and may lead to a reduction in investment on the power grid
infrastructure:
45
• Motivates consumers to balance the total workload by providing incentives and social
education,
• Encourages agents to cooperate with the grid regulators in high stress times and
environments by communicating energy information,
• Increases the grid’s security and reliability by analyzing its behavior during accidents
or system faults,
• Allows studying complex system response to dynamic pricing and other control
strategies,
• Predicts and controls emergent behavior of the agents and system evolution by mathematical modeling in respect to economic incentives and social interactions.
3.1
Agent-based Structure
This section presents an overview of agents’ properties and their environment. We present
more details and mathematical modeling of the framework in Section 3.2. Outcomes of
running the simulation are discussed in Chapter 4.
Decision Makers/Agents
1- Pattern of Consumption: Consider a electricity market system comprised of a large
number consumer agents. We study these agents by a measurable outcome of their behavior
q
where Ci (w) presents profile of daily consumption (0 ≤ w ≤ 24 hours) at period q for a
consumer that follows pattern i (formally defined in Section 2.1).
2- Properties of Agents: Agents have the capability to present new behaviors by interaction and dependent to other agents. They can switch to new patterns and change their
attributes based on their current properties and neighborhood relationships. We represent
the influence on an agent from another agent by interoperability (will be defined math-
46
ematically in the next section). We initiate a social network of consumer agents where,
Xi (0) is the initial size of the population for each pattern and the population of each pattern
(Xi ; i = 1, ..., n) can grow with growth rate, bi , exponentially. Agents may switch from one
pattern to another based on the attributes of the patterns (namely; price, convenience, and
glamour) or through influences of these in their social network. Price defines the total cost
of consumption in a 24 hour cycle time for the related pattern. Convenience shows how
fast and easy consumers can make a decision to switch to this pattern. Glamour shows the
effects of advertisement or other fashion attributes of the patterns. These micro interrelationships make new networks and update the structure of the population in the macro level
of the system.
2- Influences on Agent Decisions: Fig. 3.2 shows how agents are influenced in the
system. Ovals present uncertain variables. Rectangles show the variables that the agents
have the power to modify or select. Hexagons are measurable outcomes. Arcs denote direct influences. Dashed arcs indicate there are some influences; however, the values can be
calculated from other variables with direct influences. Agents consume electricity based on
the environment conditions (e.g. seasonal weather conditions and time of the day) and consumption attributes (e.g. price of electricity consumption and advertisements). This consumption shapes their pattern of behaviors. The entropy of the system, E = − ∑ Pi log2 Pi ,
is defined by the frequency of these patterns and the interrelationships between the agents
(i.e. emergent behavior of agents, interoperabilities, and willingness to effect or follow
each other). This entropy of the patterns in a system, yields the dis-uniformity (Eq. 2.14)
of the whole system. Cooperation reward (Eq. 3.11) can be measured based on the regulator’s decisions. Consumers measure their profit (Eq. 3.10) on the bases of the cooperation
reward, their individual rewards, their dissatisfaction, and qualitative utilities. Individual
rewards (Eq. 3.12) and dissatisfaction (Eq. 3.13) are calculated based on the patterns of
47
individual behavior. Qualitative utilities (Eq. 3.9) are functions of the consumption attributes. This total profit makes the agents change their consumptions in the cooperative
game model to adapt to the new environment.
Consumption
attributes
Regulator’s
decisions
Qualitative
values
Interoperability
betweentheagents
Environment
conditions
Agent
consumptions
Individual
Rewards
Patternsof
behaviors
Dissatisfaction
Disuniformity
ofthesystem
Entropyof
thesystem
Cooperation
Rewards
Consumer
profit
Figure 3.2: Influence Diagram
We can characterize the agents as follows (they will be discussed and defined by
details in the next sections):
• Each agent runs a pattern that has three attributes (price, convenience, and glamour),
• Agents connect to each other based on a complex network,
• An agent has desirability coefficients and importance weights for each attribute,
• Agents belong to interoperability classes based on their node degrees,
48
• An agent (partially or fully) optimizes its profits based on optimization rules in cooperation with other agents,
Environment/Patches
Figures 3.3, 3.4, and 3.5 present an overview of a three-layer image of our modeling study.
The Decision/Control Layer (Fig. 3.3) includes strategies (e.g. dynamic pricing), optimizing (e.g. maximum profits and peak reduction), plans (e.g. new generators), investments
(transmission and distribution infrastructure), and information sharing (e.g. communicate
energy information). In the US energy system Independent System Operators (ISOs) and
Regional Transmission Organizations (RTOs) work in this layer of the complex system.
Information sharing (e.g.
communicate energy
information)
Plans (e.g. new generators)
Investments
Inve
estments (e.g. tra
transm
tr
transmission
nsmiission
nsm
n
and distribution infrastructure))
Optimizing (e.g. maximum
um
profits and peak reduction)
reductiiion)
on)
SStrategies
Strategie
ies (e.g. dynamic pricing)
Regulator(s)
Figure 3.3: Decision Layer
The Social/ Swarm Layer (Fig. 3.4) considers interrelationships between agents and
preferences. We can consider bounded rationality for agents and externalities (e.g. fashion
49
effects) in this layer. Non-physical attributes of agents (e.g. glamour) and their utility
function can be studied and analyzed here. We will discuss how the behavior of agents as
a group emerges in this layer and converges to a steady pattern. Learning algorithms (e.g.
reinforcement learning) and adaptation scenarios can be considered in this layer.
Bounded rationality
Externalities (e.g. fashion effects)
Learning algorithms (e.g. reinforcement learning)
Figure 3.4: Social layer
The Physical Layer (Fig. 3.5) includes features, mechanism of components of the
system, and their physical network. In the energy system example this layer includes generators (e.g. main power plant, wind forms, solar panels, and small generators), distributors
and transmission, and consumers (e.g. homes, businesses, and industries). The topology of
networks is considered in this layer.
The Social /Swarm layer helps us to cascade effects of the Decision/ Control layer(s)
to the Physical layer(s) by modeling the emergence and adaptation of the whole swarm.
50
Components
Features
Networks
Figure 3.5: Physical Layer
The topology of connectedness and protocols for interaction in the network are important
properties of the network for our study. The topology of interactions defines who is, or
could be, connected to whom. The protocols of interactions are the mechanisms of the dynamics of the interactions. Both specify the local information accessible to an agent. The
structure of the network (e.g. scale-free or single-scale properties) helps us to calculate the
class of interoperability for each agent and find its influencer nodes. We assign the class of
an agent based on the size of the network and the degree of the node. Also, we consider
the pattern of behavior for each consumer. The distribution of patterns of behaviors and
the way the agents are seeded with patterns are important. We present the value of interoperability between two agents as the weights of the edges. In summary, three parts of the
network are used in our ABM (they are mathematically defined in the next section):
51
1. Patterns of nodes (consumer agents),
2. Degree of agents that is related to the structure of the network,
3. Weights of edges that show the interoperability of agents.
We assume the social network G between agents with preferential attachment and
growth can be represented by a scale-free network [35, 49, 50, 91]. Where |E(G)| shows
the number of links in the social network G. The nodes of the network depict autonomous
consumer agents while the edges symbolize the influences or interoperability between the
agents. The node degree distribution of the network follows a power law i.e., the probability
a node has κ edges is cκ −λ where, c is a normalization constant and λ defines the shape
of the distribution [92, 93]. We will relax this assumption to study single-scale behavior
of a Gaussian network later. The size of the network may grow by period while its scalefree/single-scale characteristic remains valid.
3.2
Agent-based Optimization Engine
This section presents the detail procedure of decision rules to show how agents optimize
their actions and change their behaviors.
Process Overview and Scheduling
Fig. 3.6 shows the logical flowchart of the agent-based process and its scheduling. The
flowchart includes three main sections: setup, switch, and optimize.
1- Setup section initiates the scale-free network of interrelated consumers and seeds
them with the patterns of consumption. Assume initially that agents are independent (this
assumption will be relaxed later to induce complex behavior). Population of pattern i
(Xi ; i = 1, ..., n) changes exponentially with growth rate bi at period q, q = 1, ..., T (Eq.
2.1). For pattern probability Pi =
Xi
∑ Xi
then the growth of entropy follows Eq. 2.8. Here, at
52
Start
Settheagentpopulationandthescalefreenetwork
Seedthepopulationwiththepattern ratios
Assigndesirabilitycoefficientsandattribute
weightsto eachagent
Setup
Growthenetworkbasedonthe
fitnessrates
Update disuniformity andEntropyofthesystem
Incrementthetimeperiod
Findtheinteroperabilityclassofagents
basedonthenetworkprotocol
Switch
Computetheinterrelationshipof
agentsbasedon interoperabilities
Findtheagentwithfirst/second
highestinterrelationship
Switchtotheagentwith
first/secondhighest
interrelationship
Calculatetheagentutilities basedonthe
desirabilities
Y
NewȺ>
OldȺ
Calculatetheagentdissatisfactionsbasedon
differenceofnewandswitch patterns
Calculatethevalue forthetotal rewardȺ
Figure 3.6: Logical Flow Chart for Scheduling of the Process Overview
53
Optimize
N
Compute thecooperative rewardbasedonthe
systemdisuniformityandindividualagent
rewardsbasedeachagentdisuniformitychanges
each time increment, new agents create ρ connections with previous agents based on the
scale-free property where ρ is a random number between 1 and ρmax . Here ρmax defines the
max link generation. We assign desirability coefficients and attribute weights to the agents
that will be used to compute optimization decisions (to be formally defined later).
We measure the agents’ level of cooperation by the dis-uniformity of aggregated
pattern consumption at period q. The dis-uniformity of pattern i is shown in Eq. 2.13.
The regulators strive to motivate the decision makers (consumer agents) to cooperate and
achieve in such a way that the aggregate consumption in each period is uniform over time.
Thus, we seek to minimize the total dis-uniformity, Eq. 2.14.
The total Dis-uniformity, Eq. 2.14, could be reduced by incentives that change one
or more profiles or rearrange probability of patterns. We will show interactions between
agents and their interoperability effects on dis-uniformity of the system.
2- Switching section defines a set of rules for finding a switching target based on interoperability of agents. We define four sets of classified interoperability for the social layer
population on the basis of the network protocol. This classification shows the agent’s ability to connect and effect each other where, δυ shows the class of agent υ, υ = 1, ..., ∑ni=1 Xi .
Influencers(INF) have the highest number of connection links (interoperability). They
maintain central positions in the social network and their numbers are small. Based on
behavioral profiles, we classify agents by type INF, EF, LF, and ISL. Early Followers(EF)
are more localized and have less interoperability compared to the influencers. Late Followers(LF) are classified between the Early Followers and Isolated categories. Isolated(ISL)
have low interoperability. They do not have a big influence on other individuals and are
insensitive to others. The idea of this classification is similar to the term innovativeness categories in diffusion theory by Rogers [94]. However, innovativeness categories consider
speed of adaptation while we consider influences and interoperability in a network.
54
The network node degree distribution follows a scale-free power-law and the node
degrees depend on the network size. To measure the interoperability classification based
on this protocol we need to linearize the power-law functions. We use Eq. 3.1 as the
linearization function and Eq. 3.3 shows how the power-law functions are linearized for
each classification.
Lin(αδυ , βδυ ) = αδυ .ln(|E(G)|) − βδυ ,
(3.1)
where,
δυ ∈ {INF, EF, LF, ISO},
αINF > αEF > αLF > αISO ,
(3.2)
βINF > βEF > βLF > βISO ,
and, |E(G)| shows the number of links in the social network G (total number of interoperability links). Degυ is the degree of agent υ in the network i.e. the number of interoperability connections that each consumer makes in the system.
⎧
⎪
⎪
INF,
if Degυ ≥ Lin(αINF , βINF ),
⎪
⎪
⎪
⎪
⎪
⎨ EF,
if Lin(αEF , βEF ) ≤ Degυ < Lin(αINF , βINF ),
δυ =
⎪
⎪
LF,
if Lin(αLF , βLF ) ≤ Degυ < Lin(αEF , βEF ),
⎪
⎪
⎪
⎪
⎪
⎩ ISL,
o.w.
(3.3)
In summary, to classify the agents we chose α and β (see an example in Chapter
4). Then the size of the network and the degree of the node determine the agent classes.
We will use these classes to assign interoperability between agents and calculate their interrelationship.
For agent υ consider all other agents that are connected to υ. The influence on
agent υ from agent z is based partly on the classes of the two agents. Let Iδcυ δz represent
interoperability between classes of agents. Here, Iδcυ δz is a monotonic function of the influence that a agent of Class δz has on a agent of Class δυ . Interoperability is a positive
55
number with maximum of one where, Iδcυ δz = 0 shows autonomic (independent) class of
agents and Iδcυ δz = 1 when they follow each other (identical). Table 3.1 shows an example
of an interoperability classification matrix. Chapter 4 discusses how to measure realistic
values for this table.
Table 3.1: Interoperability between consumer agents
Iδcυ δz
INF
EF
LF
ISL
INF
0.8
0.6
0.4
0.3
EF
0.6
0.5
0.3
0.2
LF
0.4
0.3
0.2
0.1
ISL
0.3
0.2
0.1
0.001
Self-preference θυz , lets agents vary their individual interoperability i.e. it lets two
agents have different interoperability with a specific agent even if all three of them are in the
same set of classification. We assume that the interoperability matrix defines a maximum
possible value for all pairs of classification sets and 0.5 ≤ θυz ≤ 1 may vary this value
to half of its maximum. θυz = 1 shows agent υ does not have any self-preference and
follows the interoperability matrix. Note if θυz = 0, the model will be modified to a system
without the network. To simplify the formulation we use θυδz =
∑z∈δz θυz
Xz∈δz ,
average self-
preference, in Eq. 3.4. For example, assume consumer 1 has two neighbors (consumers 2
and 3) and all three of them are INF. Based on Table 3.1, consumer 1 can have maximum
interoperability 0.8 with both neighbors. However she/he may has less willingness to be
affected by consumer 3. Then consumer 1 reduces its interoperability to 0.4 by selecting
self-preference equal to 0.5 (See more examples in Figures 3.7a and 3.7b).
The interrelationship ℜ, between agents will be assigned based on the interoperability, I c , matrix and their classification. Assume Xδzi is the number of connected agents
(neighbors) to agent υ in class δz with pattern i. To calculate interrelationship of agent υ
with its neighbors in pattern i we used Eq. 3.4,
56
ℜυi =
∑δz θυδz .Iδcυ δz .Xδzi
∑δz ∑i Xδzi
,
i = 1, ..., n, υ = 1, ..., Q,
(3.4)
where, Iδcυ δz is the interoperability of the agent in class δυ with agent in class δz where
δz ∈ {INF, EF, LF, ISO}.
To select an appropriate pattern as a switching target, we use Eq. 3.5 or Eq. 3.6
allowing the agent to consider the strongest or top two alternatives held by connected agents
q
where, switchυ is the pattern that agent υ chooses as the switching target at period q. The
agents have two ways to select their target. Eq. 3.5 lets an agent compare its objective with
the objective of its highest weighted interrelationship neighbor. Here the switching target
of agent υ is the pattern of an interconnected agent with the highest interrelationship in its
neighborhood. To consider the agent tendency to stay in her current pattern, we add a fixed
amount of the awareness-threshold ϒυ (a user-defined variable) divided by total number
ϒυ
of agents to her interrelationship with her current pattern i.e., ℜυi = ℜ0υi + TotalNeighbors
if
i =Pattern(υ), where ℜ0υi is her current interrelationship.
q
switchυ = argi {max(ℜυi )},
∀i.
(3.5)
To avoid similar patterns when the agent with the highest interrelationship has the same
pattern with the target, we may use Eq. 3.6 to let agents look at the second highest interrelationship.
q
switchυ = argi {max(ℜυi )},
∀i = Pattern(υ).
(3.6)
Example 1: Fig. 3.7a presents an example for calculating interrelationships. Assume consumer υ is an agent trying to decide whether to switch her consumption pattern.
Agent υ is an influencer agent, INF and six other agents are connected to this agent. Two of
57
them follow pattern i (1 EF, 1 LF), three of them follow pattern j (1 INF, 1 EF, 1 LF), and
one of them follows pattern k (INF) at this time. Assume the awareness-threshold ϒυ = 0
and the agent υ does not have any self-preference i.e., θυz = 1. Then ℜυi = 0.6+0.4
= 0.17,
6
ℜυ j =
0.6+0.4+0.8
6
= 0.3, and ℜυk =
0.8
6
= 0.13. Hence, if agent υ has the pattern j the
q
q
switching target switchυ = i from Eq. 3.6 and switchυ = j otherwise.
Example 2: Fig. 3.7b continues Example 1 by adding a self-preference to the agent
υ for one of its neighbors (underlined number in the arc). This means agent υ prefers to
weigh the interoperability of the EF class of pattern i by half (θυi = 0.5). We calculate
ℜυi =
0.5∗0.6+0.4
6
q
= 0.12 changing switchυ = k if we use Eq. 3.6 and Pattern(υ)= j.
i
i
i
EF
0.4
0.6
K
INF
0.4
v
J
J
0.6
INF
0.4
v
J
LF
INF
0.8
EF
0.8
K
LF
INF
0.6
LF
0.4
0.6
INF
0.8
i
0.5
EF
LF
J
J
(a) Without self-preference
0.8
EF
INF
J
(b) With self-preference
Figure 3.7: Calculating the Interrelationship
3- Optimizing section: once a target pattern is defined, the agent has to decide
whether to keep her current pattern or to switch based on the pattents’ utility functions,
rewards, and dissatisfactions. The total utility value of a pattern is a weighted average
of the desirability score Ωυs that is a number between 0-1 for each attribute [95]. Recall
that the attributes are price, convenience, and glamour. Here, Ωυs shows the desirability
score of agent υ for attribute s. When the agents target the maximum value of an attribute
58
(e.g. glamour), to calculate the desirability, we use Eq. 3.7 where y shows the value of
the attribute and L and U are its lower and upper bound in Fig. 3.8a. We apply Eq. 3.8
when the target is a minimum value, Fig. 3.8b (e.g. time to make decisions or cost of
changing behaviors). The Desirability coefficient γυs , parameterize the desirability score
Ωυs . Desirability functions showing (γ < 0) characterizes risk-averse agents and (γ > 0)
characterizes risk-seeking agents if the agent targets a maximum and vise versa if the agent
targets a minimum.
Ωυs =
Ωυs =
⎧
⎪
⎨
⎪
⎩
⎧
⎪
⎨
⎪
⎩
s
1−exp(γυs ∗ Uyss−L
−Ls )
,
1−exp(γυs )
if
γυs = 0,
υ = 1, ..., Q, s = 1, 2, 3,
ys −Ls
Us −Ls ,
if γυs = 0,
υ = 1, ..., Q, s = 1, 2, 3.
Us −ys
1−exp(γυs ∗ U
)
s −Ls
,
1−exp(γυs )
if
γυs = 0,
υ = 1, ..., Q, s = 1, 2, 3,
Us −ys
Us −Ls ,
if γυs = 0,
υ = 1, ..., Q, s = 1, 2, 3.
(3.7)
(3.8)
1
1
Riskaverse
<0
Riskseeking
<0
RiskNeutral
=0
RiskNeutral
=0
Riskseeking
>0
L
U
>0
Riskaverse
y
(a) targeting maximum
L
U
y
(b) targeting minimum
Figure 3.8: Desirability Functions
The Utility Value Uυ , is the weighted average of the desirability scores of its pattern
attributes. Agents can assign their own importance weight Wυs , to different attributes to
have individual value functions (Eq. 3.9).
59
Uυ =
∑s Wυs Ωυs
,
∑s Wυs
υ = 1, ..., Q, s = 1, 2, 3.
(3.9)
Agents receive two types of rewards when they switch to new patterns. The Cooperation reward R̂, is a function of the total dis-uniformity. An agent may receive this
reward without any switching when their current pattern cooperates with others to reduce
the dis-uniformity i.e. cooperation reward will be payed to all agents based on the total disuniformity that they create. The Individual reward Rυ , reflects the controllers individual
rewards to the agent based on the contribution of the agent pattern to the central objectives. Agent will receive more individual reward when they choose to change their pattern
of behavior in the way that minimizes their dis-uniformity. Dissatisfaction ϕ, is the only
variable that has a negative effect on the decisions. Dissatisfaction will be measured based
on the amount of difference between new and old pattern of each agent. We assume the
more each agent changes its pattern the more dissatisfaction with its decision (see Section
3.2 for the formulations).
At the end of each period the pattern of the agents are updated as follows:
1. In a growing network, the new network expanded from the old one,
2. Agent classes are redefined based on the network topology,
3. Interoperabilities are reassigned to the network (weight of edges),
4. Self-preference of agents are added to the edges,
5. Interrelationships between agents are recalculated,
6. A target pattern will be selected based on the chosen switching rule,
7. The agent applies a decision rule based on value of perceived choices.
60
Decision Rules
In the previous section we described how an agent determines the utility value of a possible
switching target. A consumer-based optimization will be executed in order to determine
if switching actually occurs. Decision makers (consumer agents) desire to maximize their
total reward πυ . The total reward is a normalized sum of the cooperation reward R̂, the individual reward Rυ , the utility value Uυ , and dissatisfaction ϕ of the agents. The consumerq+1
based optimization compares the anticipated total reward πυ
q+1
reward obtained continuing the current pattern, πυ
q
q
q
(Cυ (w),Cswitch (w)) with the
q
(Cυ (w),Cυ (w)). If
q+1
q
q
(Cυ (w),Cswitch (w))
q+1 q
q
πυ (Cυ (w),Cυ (w))
πυ
switching threshold ϒυ , the agent switches her pattern to the pattern given by the possible
switch in the period q + 1. A general model for agent υ at period q would be,
q
q
q
πυ = kr̂ R̂q + kr Rυ + kuUυ − kϕ ϕ q ,
(3.10)
q
where, Uυ is defined by Eq 3.9 and
R̂q = f0 (Dq ),
∀q,
(3.11)
Rυ = f0 (Dυ ),
∀υ, q,
(3.12)
∀υ, q, w.
(3.13)
q
q
q
q
q
ϕυ = f2 (Cυ (w),Cswitch (w)),
The maximization is to be taken relative to the decision that agent υ can make in period
q. To be specific, we choose the cooperation reward R̂q a decreasing exponential function
of the total dis-uniformity (defined in Eq. 2.14) at period q, f0 (Dq ) = λ0 eD
qλ
00
, and the
q
individual rewards Rυ a decreasing exponential function of the individual dis-uniformity
q
(Eq. 2.13) of each agent at period q, f0 (Dυ ) = λ0 eDυ λ00 . Values of λ0 , λ00 , λ0 , and λ00
q
q
are constants and will be assigned later in this study. We choose the dissatisfaction ϕυ
to be a logarithmic function of the differences between current and previous pattern of
consumption behavior,
q
q
f2 (Cυ (w),Cswitch (w))
=
61
ln(
w0
0
q
q
|Cυ (w)−Cswitch (w)|dw)
.
λ2
The value of λ2
>
will be assigned later in this study (see Chapter 4 for an example about the functions).
Note that this presents the agents decision model. Above this sits the master problem
of the controlling entity who sets prices and rewards in order to induce desired behavior
(consumption).
Termination Rules
The simulation continues the flow described in Fig. 3.6 until one of the following termination rules apply:
• The patterns do not change any more,
• A maximal period (simulation time) has been reached.
62
Chapter 4
SIMULATION OF ELECTRICITY MARKETS AND ANALYZING BEHAVIORS
Here is an example to show how our integrated Agent-based optimization model simulates and analyzes the emergent behavior of consumption patterns. This study enables the
electricity regulators to examine the effect of social interactions topologies with more realistic cognitive behaviors and improves the efficiency and effectiveness of demand response
and peak reduction programs via behavioral-based incentives (e.g. social education and
advertisement).
4.1
Specifying the Parameters
To illustrate the dynamic behavior of our system we assume three patterns of consumption.
Fig. 4.1 shows the behavior of those patterns in 24 hours (see [96, 97] for more patterns).
7
Consumption(KW)
6
5
4
i
3
j
2
k
1
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
w
Figure 4.1: Patterns of Consumption
Table 4.1 presents the variables for running the agent-based simulation model. We
assume convenience of a pattern is a qualitative variable between 2 to 5. Lower values
indicate more convenience (it is a scale of the time that agents should spend to feel conve-
63
nient). Glamour is a qualitative variable with higher values implying more fashionable or
advertised patterns.
Table 4.1: State Variables for Running the Simulation
Parameter
Initial Population
Initial Patterns
Growth Rate
Max Link Generation
Price
Convenience
Glamour
Desirability Coefficient
Attribute Weight
Average Self-preference
Variable
υ = 1, ..., ∑i Xi
Xi
, i = 1, 2, 3
∑i Xi
bi , i=1,2,3
ρmax
S1
S2
S3
γυs
Wυ
θυδz
Range
100- inf
0 to 1
0 to 1
1 to 10
3-8
2-5
1-10
∼Normal (0, 4)
∼Normal (1, 0.2)
∼Uniform (0.5, 1)
Units
# of consumers
% of total population
% of pattern population
number
Currency (cents/KW h)
Qualitative
Qualitative
N/A
N/A
N/A
Default
500
0.15,0.65,0.20
0, 0, 0
1
5, 5, 5
4, 4, 4
3, 3, 3
N/A
N/A
N/A
Area*
E
E
E
E
C
C
C
A
A
A
* Here E, C, and A stand for Environment, Consumption and Agents respectively.
Consumers are autonomous and receive γυ , Wυ , and θυδz identically from sampling random Normal and Uniform distributions ignoring negative samples. To classify
the consumer agents based on their interrelationships, we assume αINF = 4.5, αEF = 2.5,
αLF = 1.5, βINF = 17, βEF = 9, and βLF = 5. These assumptions are created by fitting
a linear function to the growth of degree of the nodes in the network. Fig. 4.2 shows
how these assumptions are used to make linear growth for the degree of nodes at each
class of interoperability. Moreover, we assume f0 (Dq ) = 100e
q
q−1
and f2 (Cυ (w),Cυ (w)) =
ln(
w0
0
q
q
|Cυ (w)−Cswitch (w)|dw)
2
D(t)
10
, f0 (Dυ ) = 100e
q
Dυ (t)
10
,
. These parameters are defined to pro-
vide proper scaling and their impetus is discussed in previous sections.
To define interoperability between agents we use the results of empirical studies
on spread of happiness [98], dynamics of smoking [99], and formal/informal influence in
adoption [100]. Fowler and Christakis [98] and Christakis and Fowler [99] classify individuals in large social networks based on their influence and study probability of changing
behaviors (smoking habits and happiness) in agents when their connected agents adopt to a
new behavior. Based on both studies we conclude that the probability to influence a participant in the same class is approximately 50%, 35%, 25%, and 10% for the classes of close
64
60
D
Degreeofagents
50
40
30
INF
20
EF
LF
10
0
1
11
21
31
41
51
61
71
81
91
Period
Figure 4.2: Linearized Growth Functions
friend, friend, spouse, and colleague receptively. Tucker [100] showed that authority structure like the one between managers and workers will add to or subtract to the influences
determined in the previous studies. Based on these studies we choose the interoperability
between the two classes of agents as shown in Table 4.2. These interoperabilities are selected for consistency with the finding of the three mentioned comprehensive studies but
are not uniquely determined.
Table 4.2: Measuring the Interoperabilities
Iδcυ δz
INF
EF
LF
ISL
4.2
INF
0.79
0.65
0.41
0.28
EF
LF
ISL
0.4 0.1 0.002
0.51 0.25 0.03
0.35 0.29 0.06
0.2 0.15 0.1
Analyzing Behaviors
We study the behavior of the integrated agent-based optimization system under a variety
of different circumstances. We examine how the agents’ tendency to stay on their own
pattern (awareness-threshold) effect the distribution of entropy of the system. We define a
baseline awareness-threshold based on previous studies. Then we show how the regulators
65
can improve the effect of a social network. Also we build more realistic results by considering limited rationality in cognitive capabilities of agents. Finally, we show the effect of
saturated friendship behavior in the system.
The Effect of The Awareness-Threshold
To study behavior of the system we run the model with the default values in Table 4.1
while we modify the awareness-threshold ϒ from 0.00 to 0.32 in 0.02 increments (for
larger thresholds the system will not converge in 400 time periods). Fig. 4.3 shows the
results of 300 runs. This figure shows the distribution of the time to converge as a function
of awareness-threshold ϒ. This time is defined as the time when the entropy of the system
is less than 0.3364 i.e. when at least 95% of agents converge to a pattern (here is Pattern i).
450
400
350
300
TimePeriod
Min
10%
250
1stQua
200
Median
3rd Qua
3rdQua
150
90%
100
Max
50
0
0.32
0.3
0.28
0.26
0.24
0.22
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
AwarenessThreshold
Figure 4.3: Convergence of the Entropy
Fig. 4.4 depicts a vertical slice from Fig. 4.3. We see how the distribution of
convergence to pattern i shifts from left to right by increasing the awareness-threshold ϒ.
The convergence time increases by increasing the awareness-threshold i.e. the agents have
66
more tendency to stay in their current pattern and converge slower. Fig. 4.5 compares
the average entropy of systems with different awareness-thresholds. Larger awarenessthresholds cause higher entropy in the system because the system converges slower and
has more agent diversity by period. For very large awareness-thresholds (e.g. 0.3), the
system does not converge in 400 periods. At the beginning of the simulation, the entropy
may increase because the agents may start to switch from a high populated pattern to a low
16
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Frequ
uencyofC
Converge
ence
Frequ
uencyofC
Converge
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populated one.
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TimePeriod
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(a) Awareness-Threshold=0
(b) Awareness-Threshold=0.1
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uencyofC
Converge
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uencyofC
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385
0
Ti
TimePeriod
P i d
Ti
TimePeriod
P i d
(c) Awareness-Threshold=0.2
(d) Awareness-Threshold=0.3
Figure 4.4: Distribution of Convergence to Pattern i
Establishing a BaseLine
It has been observed that only 5% of end users are currently empowered to react to the realtime or day-ahead locational marginal price (LMP) and participate in DR Programs [17].
In order to calibrate our agent-based simulations, we will determine a awareness-threshold
that generates a change in pattern over time that is as close as possible to an simulation
67
Th=0
Th=0.1
Th=0.2
Th=0.3
1
21
41
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81
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01
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01
Entropyy
502058
1.8 1.387334 1.306998
526618
1.6 1.433907 1.332431
521885 1.463869 1.354272
1.4
05751 1.485003 1.372586
1.2
486321 1.50028 1.388672
4672491 1.510736 1.402339
0.8 1.517728 1.415224
44456
0.6 1.521325 1.426525
420105
39821
0.4 1.522141 1.436724
0.3364
74004
0.2 1.52085 1.446221
348360 1.518426 1.454407
22206 1
1.515625
515625 1
1.46199
46199
298723 1.511883 1.468582
TimePeriod
274652 1 506561 1 474632
Figure 4.5: Convergence of the Average Entropies
without a network and 5% participation rate. We quantify the closeness of two simulations
(1)
(2)
by registering the average percentage of population Piq and Piq of pattern i in the system
as function of time period q. The difference H (Eq. 4.1) is the absolute sum of difference
pattern over time:
(1)
(2)
H = ∑ ∑ |Piq − Piq |
i
i = 1, 2, 3;
q = 1, ..., 400.
(4.1)
q
Fig. 4.6 shows the total difference H between an agent-based simulation without
network (5% participation) and an agent-based simulation with a social network as a function of the awareness-thresholds.
Based on Fig. 4.6 awareness-threshold between 0.08 to 014 yields the most similar behavior with current systems without networks. For example Fig. 4.7 compares the
emergence in the pattern of behavior in the first ten runs for awareness-threshold=0.1 and
0.3.
68
0.9
0.8
0.7
T
TotaldifferenceH
0.6
Min
10%
0.5
1stQua
0.4
Median
3 dQ
3rdQua
0.3
90%
0.2
Max
0.1
0
0.32
0.3
0.28
0.26
0.24
0.22
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
AwarenessThreshold
Figure 4.6: Total Difference With and Without a Social Network
Improving Effect of the Network
We can assume a well designed social network can increase the number of friendships
and/or decrease awareness-threshold of agents (i.e. decrease the tendency to stay in their
initial behavior). Here we show the behavior of the system when we increase max link
generation ρmax . Fig. 4.8 compares the results of networks with ρmax = 1 and ρmax = 3
(i.e. average link generation equal to 2) when awareness-threshold is equal to 0.1. This
figure shows how the social network increases the effects of economical incentives.
We checked the behavior of the topology of the network by log-log plots [34, 101].
Fig. 4.9 compares the scale-free behavior of the network for ρmax = 1 and ρmax = 3 and
depicts the scale-free metric λ increases from 1.47 to 1.81 when we increased ρmax . This
shows how regulators can improve the convergence time by sharing information between
agents or by encouraging them to increase their friendships via social electricity networks.
69
1.2
PerccentageofP
Population
1
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04
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(a) Awareness-Threshold=0.1
Perce
entageoffPopulattion
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0
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TimePeriod
P i d
(b) Awareness-Threshold=0.3
Figure 4.7: Comparison of Awareness-Threshold=0.1 and 0.3
Externalities and Irrationality
Some characteristics of agents (such as irrationality) and the effects of externalities (such
as a fashion effect) are not considered in most of the previous studies on ABM of electrical
CASs. We analyze the impact of these factors on the social/swarm layer by incorporating
them into the agent decisions and study their effect on objectives and control strategies.
Considering irrationality helps us to create more realistic cases. We propose discrete choice
models (Eq. 4.2) based on the total reward π and the interoperability classification to study
70
450
400
TimePerio
T
od
350
300
250
Th=0.1,_max=1
200
Th=0.1,_max=3
150
100
50
0
Min
10%
1stQua
1st
Qua Median 3rdQua
3rd Qua 90%
(a) Convergence of the Entropy
Max
Freque
encyofConvergence
18
16
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12
10
8
Th=0.1,_max=1
6
Th=0
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0.1,_max
1 max=3
3
4
2
1
24
47
70
93
116
139
162
185
208
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254
277
300
323
346
369
392
0
TimePeriod
(b) Distribution of convergence to Pattern i
Figure 4.8: Comparison of ρmax = 1 and ρmax = 3
the behavior of irrational agents. We assume two types of irrationality (namely: Irr-I and
Irr-II). The probability that agent υ does not switch to the new pattern even if the π value
of the new pattern is higher than the π value of the old pattern (i.e. Irr-I) is given by,
Probυ =
exp(kI ∗ πNew )
exp(kI ∗ πNew ) + exp(kI ∗ πOld )
71
(4.2)
1
0.5
0
P
Probability
0.5
0
0.5
1
1.5
1
y=1.8061x+0.4304
1.5
2
_max=1
max=3
_max
3
2
2.5
y=1.4698x 0.4154
3
3.5
NodeDegree
Figure 4.9: The Scale-free Metric of the Network
where, πNew and πOld are the new and old values for the total reward of agent and kI shows
the sensitivity of the agents to differences between the π values (the higher the kI , the more
sensitive agents). The probability that agent υ switches to a new pattern even if the π values
do not satisfy the switching threshold (i.e. Irr-II) can be measured by similar formulation
with a new sensitivity factor kII .
Fig. 4.10 compares behaviors of completely rational agents (Irr=0) with strongly
irrational/non-sensitive agents (kI = 0.1; kII = 0.1) and weak irrational/sensitive agents
(kI = 1; kII = 1) scenarios. The weak irrational scenario does not converge to any pattern.
The strongly irrational scenario converges with an almost similar temporal distribution to
rational agents. However, this simulation of the system does not converge to a pattern of
low dis-uniformity but it converges to the pattern that had the highest initial population.
Consumers with the same patterns may create a community in the network. These
communities can become dominant and restrict the influence of other patterns even if they
have higher π values. Generally we can observe that sometimes the system does not evolve
72
450
400
TiimePeriod
350
300
250
Th=0.1,Irr0
200
Th=0.1,Irr0.10.1
150
Th=0.1,Irr11
100
50
0
Min
10%
1stQua Median 3rdQua
90%
Max
Figure 4.10: Effects of Irrationality
to the situations with lower dis-uniformity. It is because of the effect of emergent behavior
of consumers in interrelationship with each other in the network.
Effects of Saturated Interrelationships
Recent studies show the degree distribution for friendship in growing networks may not be
scale-free in the long run [91] (its tail decays faster than power-law with increasing size of
links) and can converge to a Gaussian distribution. Here, when an agent has more than a
critical number of links (capacity constraints), new edges cannot connect to it. To create
this environment (see Section 3.1/Environment) we group agents to active or inactive. All
new agents are created active. An agents becomes inactive when it reaches a maximum
number of links. Inactive agents cannot receive new links. This constraint leads to a
single-scale network where the degree of nodes follows a truncated Gaussian distribution.
We study the effect of saturation by comparing Gaussian Distribution (with a maximum number of 4 friends/influencer) to an unbounded scale-free network (power-law distribution) in Fig. 4.11. This shows regulators should try to avoid Gaussian distribution
node degrees by motivation agent to do not saturate their friendships or keep the size of the
73
network small to do not converge to a truncated Gaussian distribution. From Fig. 4.11b,
we note the convergence of entropy occurs faster in scale-free networks. However, the
scale-free network has a longer right tail and in some cases never converges.
450
400
TiimePeriod
350
300
250
Th=0.1,Max=inf
200
Th=0.1,
Th
0.1,Max
Max=4
4
150
100
50
0
Min
10%
1stQua Median 3rdQua 90%
(a) Convergence of the Entropy
Max
8
7
6
5
4
Th=0.1,Max=inf
3
Th=0
Th
0.1,Max
1 Max=4
4
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1
0
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45
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89
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177
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331
353
375
397
Freque
encyofConvergence
9
TimePeriod
(b) Distribution of convergence to Pattern i
Figure 4.11: Effect of saturated friendship on the topology of the network
74
Significant Factors
To examine importance and effect of the parameters of the social network on the system
q
outputs, we design two full factorial experiments. We study the system responses by πυ (the
average total rewards of agents), Dq (the total dis-uniformity of the system), and E q (the
entropy of the system). An Exponential Weighted Moving Average (EWMA) with discount
factor equal to 0.005 is used for weighting the data points at older periods where the number
of periods q = 1 to 400. We study effect of the social network size by considering two
levels (200 and 1000) for the initial population. Surrogate for modeling load aggregates
to impact attention to the goal, we assume two levels for the weight of individual disuniformities (kr = kr̂ and kr = 2kr̂ ) i.e. consider it equal to the weight of total dis-uniformity
and twice this weight. Also, we level attributes of pattern i from a normal utility (Price=5,
Convenience=4, Glamour=3) to a weak utility (Price=8, Convenience=3, Glamour=2). The
P-values of experimental results for the 23 full factorial experiment with 30 replications are
presented in Table 4.3. The results are created by MiniTab and 0.000 means the P-value
is less than 0.05. Note that the normality test for residuals was not convincing. We tried
several transformations for responses (e.g. Box-Cox power including square root and Log).
Response−minResponse
A logistic transformation, Log( maxRresponse−Response
), improved the results of the normality
test.
Table 4.3: Estimated effects and coefficients of ∑ Xi , Kr , and U
Response
Constant
Population
Kr
Utility
Population ∗ Kr
Population ∗Utility
Kr ∗Utility
Population ∗ Kr ∗Utility
P Reward
0.000
0.268
0.000
0.000
0.169
0.534
0.004
0.260
75
P Dis-uniformity
0.000
0.677
0.447
0.003
0.205
0.709
0.004
0.262
P Entropy
0.000
0.174
0.018
0.000
0.174
0.728
0.000
0.096
Figures 4.12a, 4.12b, and 4.12c depict the effect of ∑ Xi , Kr , and U on the responses.
The results shows that the population size is not a significant factor for the defined responses of the system; however, the utility and the weight of individual dis-uniformities
should be considered as significant factors. Kr has the largest significant effect and U has
q
the second largest significant effect on the total rewards πυ . Effect of U and interaction
effect of U and kr on the total dis-uniformity Dq and entropy E q are significant.
In another experiment we study the topology of the network by using the similar
responses and leveling the node degree distribution to scale-free and single-scale. Also, we
consider two levels for the max link generation (ρmax = 1 and ρmax = 6). The P-values of
experimental results for the 22 full factorial experiment are presented in Table 4.4.
Table 4.4: Estimated effects and coefficients of topology and max link generation
Response
Constant
Topology
Link Generation
Topology*Link Generation
P Reward
0.000
0.000
0.002
0.000
P Dis-uniformity
0.000
0.000
0.000
0.000
P Entropy
0.000
0.000
0.121
0.000
Figures 4.13a, 4.13b, and 4.13c depict the effect of topology and max link generation on the responses. The results show that all of them are significant factors. Topology,
max link generation, and their interaction have almost a similar size of effect on the total
q
rewards πυ and the total dis-uniformity Dq (topology has a slightly larger effect in comparison to max link generation). However, max link generation has a smaller effect on entropy
in comparison to topology of the network and the interaction effect of max link generation
and topology.
4.3
Expansion to other ECASs
This study examine how consumption behavior is diffused in a population. Managerial
insights about the qualities that make the diffusion successful are offered by this study.
Also the importance of social networks and peer to peer conversations are examined here.
76
Pareto Chart of the Standardized Effects
Normal Probability Plot of the Standardized Effects
(response is Trans-Reward, Alpha = .05)
(response is Trans-Reward, Alpha = .05)
99
1.97
Factor
A
B
C
B
Name
Population
K_r
Utility
Effect Type
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Significant
95
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90
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80
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Term
BC
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15
(a) Effects on Total Reward
Pareto Chart of the Standardized Effects
Normal Probability Plot of the Standardized Effects
(response is Trans-DisUniformity, Alpha = .05)
(response is Trans-DisUniformity, Alpha = .05)
99
1.970
Factor
A
B
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K_r
Utility
Effect Type
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Significant
95
BC
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Percent
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Utility
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2.5
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-3
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-1
0
1
Standardized Effect
2
3
(b) Effects on Dis-Uniformity
Pareto Chart of the Standardized Effects
Normal Probability Plot of the Standardized Effects
(response is Trans-Entropy, Alpha = .05)
(response is Trans-Entropy, Alpha = .05)
99
1.970
Factor
A
B
C
BC
Name
Population
K_r
Utility
Effect Type
Not Significant
Significant
95
BC
90
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80
Percent
Term
B
ABC
A
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60
50
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30
Name
Population
K_r
Utility
B
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AB
Factor
A
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C
10
C
5
AC
1
0
1
2
3
Standardized Effect
4
5
-5.0
-2.5
0.0
2.5
Standardized Effect
5.0
(c) Effects on Entropy
Figure 4.12: Pareto Chart and Normal Probability Plot for Experiment 1
The S-shape convergence in the pattern of behavior and bell-shape curves of frequency of
convergence depict the similarity of the behavior of our ECAS and Bass Diffusion Model
[102, 103]. A comprehensive discussion on comparison of CASs and diffusion models is
presented by Rogers et al. [104].
77
Pareto Chart of the Standardized Effects
Normal Probability Plot of the Standardized Effects
(response is Trans-Reward, Alpha = .05)
(response is Trans-Reward, Alpha = .05)
99
1.981
Factor
A
B
Name
Topology
Link Generation
90
Percent
A
Term
Effect Type
Not Significant
Significant
95
AB
80
A
70
60
50
40
30
AB
Factor
A
B
Name
Topology
Link Generation
B
20
10
B
5
1
0
1
2
3
Standardized Effect
4
5
-4
-3
-2
-1
0
1
2
Standardized Effect
3
4
5
(a) Effects on Total Reward
Pareto Chart of the Standardized Effects
Normal Probability Plot of the Standardized Effects
(response is Trans-DisUniformity, Alpha = .05)
(response is Trans-DisUniformity, Alpha = .05)
99
1.981
Factor
A
B
Name
Topology
Link Generation
Effect Type
Not Significant
Significant
95
Percent
80
Term
Factor
A
B
90
A
AB
B
70
60
50
40
30
Name
Topology
Link Generation
AB
A
20
10
B
5
1
0
1
2
3
Standardized Effect
4
5
-5.0
-2.5
0.0
Standardized Effect
2.5
5.0
(b) Effects on Dis-Uniformity
Pareto Chart of the Standardized Effects
Normal Probability Plot of the Standardized Effects
(response is trans-Entropy, Alpha = .05)
(response is trans-Entropy, Alpha = .05)
99
1.981
Factor
A
B
Name
Topology
Link Generation
Effect Type
Not Significant
Significant
95
Factor
A
B
90
AB
Percent
Term
80
A
70
60
50
40
30
Name
Topology
Link Generation
A
AB
20
10
B
5
1
0
1
2
3
4
Standardized Effect
5
6
-6
-5
-4
-3
-2
-1
0
Standardized Effect
1
2
3
(c) Effects on Entropy
Figure 4.13: Pareto Chart and Normal Probability Plot for Experiment 2
The presented platform can be applied to other ECASs easily. As an illustration
in a health care system, a distance between current state of independent agents (patients)
with an objective (quality of new products/procedures or public awareness of a disease)
can be measured by the dis-uniformity. This platform enables the regulators (hospitals,
78
governments, and other stakeholders) to cascade the effects of their decisions to the lover
level of the system by considering the social interactions and social education in the patient
layer. Some interesting questions that can be answered are ”where is the best place to
put the money?” and ”how to spend money?”. Major parts of agent-based structure (e.g.
topology/protocol of the network, interoperability, interrelationships, and utility functions)
will stay similar to this study. Our structure enables stakeholders in an engineered complex
adaptive health care system to empower patients to evolve the system by influencing their
commitment instead of trying to control their behavior. Patient agents redesign themselves
and emerge new behavior by cooperating or competing in a game environment that portrait
their future life.
79
Chapter 5
CONCLUSION
This study takes a mathematical and computational approach to a novel socio-technical
model combining social behavior, economics and technology. In this study, agents use a
consumer-based optimization model to emerge instead of using stochastic learning algorithms (e.g. reinforcement learning). Here, we integrate the agent decision rules in an
agent-based optimization model to let the ECAS evolve dynamically. The ECAS evolves
on the basis of agent observations, their subjective behavior, and reward regulations. We
propose a modeling and solution methodology that addresses shortcomings in previous
approaches and advances our ability to model and understand ECAS behaviors through
computational intelligence. The model integrates mathematical analysis and human behavioral approaches. Our integrated model facilitates reduction of electricity consumption
variability referred to as dis-uniformity, in electricity.
In the first part of the study, we present a framework that helps us to employ engineering and mathematical models to analyze certain ECASs. We can apply this framework
to study and predict the hallmarks of complex heterarchical (non-hierarchical) engineered
systems. We employ information theory in our mathematical model. Conditioned on parameterization of the framework, all possible dominance cases in complex systems are
defined and four theorems are presented to calibrate current situation and predict future behaviors of an ECAS. Theorem V (mechanisms of components) can be employed to study
self-organization in ECASs. Catalyst Associate Interoperability (CAI) and Stagnation of
the system are new concepts that can help us measure or scale the emergence and evolution
behaviors. Researchers may control the interoperability of components with CAI. Also,
they can measure evolvability or stagnation of a complex system by a threshold function.
80
In the second part of this study, agent-based modeling and simulation support and
amplify the mathematical findings of this research. We formulate a consumer-based optimization model and integrate it with the agent-based model, then we include effects of
externalities and subjective behaviors in the integrated agent-based optimization model.
Extensions are also made to agent-based optimization modeling techniques to deduce emergent and evolutionary behavior. Furthermore, we develop an integrated agent-based optimization model for social dynamics of the electricity consumption CAS.
Unlike traditional stochastic learning methods that are based on limited memory of
past experiences and utility optimization models that are built by rationality and decision
theory, our approach combines both backward and forward looking. This combination
enables us to assume individuals with more concrete cognitive capabilities. This ABM
enables us to model agents to be adaptive (can learn based on social education), goaldirected (adjust themselves or their interactions based on a goal), and heterogeneous (their
attributes and behaviors may vary and change dynamically). Moreover, complex behaviors
such as altruistic versus selfish and cooperation versus competition can be studied by our
integrated approach.
We develop methods to enable changing consumption patterns of end-users and
load aggregators through incentives and social education. We have advanced understanding
of the interaction between social behaviors and economical incentives for load management
and impacting peak demand. Furthermore, we have provided a comprehensive toolkit for
regulators and managers to help them predict behaviors of the ECASs based on the status of
variables. We have enabled them to find more effective and efficient strategies to motivate
decision makers to change their behavior. We illustrate how we can extend the concept of
our framework and platform to other ECASs.
81
Finally, we build a baseline to study emergence behavior of electricity consumers.
This study shows how topology of social network can motivate consumers to change their
patterns of behavior. Load aggregates and system operators can apply this study to examine
the value of their investment on developing a system to improve social interactions between
electricity consumers. Also, energy managers are enabled to examine different degrees of
irrationality and awareness based on different types of societies. We show how degree of
irrationality in micro level of a system affects the convergence in macro level.
Our baseline, scenario analysis, and illustrations in this study are made based on
the residential electricity consumption in the U.S. electricity Markets. However, the framework and platform can be used to study business sectors and/or industrial sectors in future
research.
82
83
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