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系统动力学入门教程

学习型组织研修中心·中国学习型组织网 http://www.cko.com.cn 编辑发布，2007 1

U.S. Department of Energy's

Introduction to System Dynamics

A Systems Approach to Understanding Complex Policy Issues

Version 1.0

Prepared for:

U.S. Department of Energy,

Office of Policy and International Affairs,

Office of Science & Technology Policy and Cooperation

Adapted from

Foundations of System Dynamics Modeling

by:

Michael J. Radzicki, Ph.D.

Sustainable Solutions, Inc.

Copyright © 1997

Designed by:

Robert A. Taylor

U.S. Department of Energy

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Index

Overview ........................................................................................................................ 4

Chapter 1 Introduction .................................................................................................. 6

Origin of System Dynamics ............................................................................................... 6

System Dynamics and Energy Modeling .......................................................................... 11

Chapter 2 Why Model? ..................................................................................................25

Why Build Models in the First Place? .............................................................................. 25

Formal Models .......................................................................................................... 25

Mental Models ..........................................................................................................26

Combining Mental Models with System Dynamics ................................................. 27

Chapter 3 The Building Blocks .................................................................................... 29

Taking A Dynamic Perspective......................................................................................... 29

Time Paths ................................................................................................................ 29

Actual Data ............................................................................................................... 33

Seeing System Structure ................................................................................................... 39

Stocks and Flows ....................................................................................................... 39

Identifying Stocks and Flows ................................................................................... 40

Four Characteristics of Stocks ................................................................................. 40

Feedback ................................................................................................................... 47

Implicit and Explicit Goals ....................................................................................... 55

Archetypes ................................................................................................................ 58

Nonlinearity .............................................................................................................. 59

Implications of System Structure .................................................................................... 60

A System Problem and Its Symptoms are Separated By Time and Space ............... 60

Counterintuitive Behavior ........................................................................................ 62

Better Before Worse or Worse Before Better............................................................. 63

Policy Resistance ...................................................................................................... 64

Unpredictability ........................................................................................................ 65

Disequilibrium ......................................................................................................... 68

Chapter 4 Simple Structures ........................................................................................ 69

Exponential Growth, Exponential Decay, and Goal Seeking Behavior .......................... 69

Analytical Versus Simulated Solutions ..................................................................... 70

Exponential Growth ..................................................................................................76

Goal Seeking Behavior ............................................................................................. 90

Exponential Decay .................................................................................................... 93

System Oscillation ............................................................................................................97

S-Shaped Growth ............................................................................................................ 102

Chapter 5 The Modeling Process ................................................................................ 107

The Modeling Process .................................................................................................... 107

Management Flight Simulators ....................................................................................... 110

Chapter 6 Natural Gas Discovery Model ..................................................................... 114

Natural Gas Discovery Model: Reference Modes ............................................................ 114



Step-By-Step Re-Creation of the Natural Gas Discovery and Production Model ........... 116

Natural Gas Discovery and Production Model: First Cut ........................................ 116

Natural Gas Discovery and Production Model: Second Cut .................................... 119

Natural Gas Discovery and Production Model: Third Cut ..................................... 124

Natural Gas Discovery and Production Model: Fourth Cut ................................... 129

Natural Gas Discovery and Production Model: Fifth Cut ........................................ 133

Natural Gas Discovery and Production Model: Sixth Cut ....................................... 137

Policy Explorations with Natural Gas Model.................................................................. 142

Chapter 7 Next Steps ................................................................................................... 148

System Dynamics Resources .......................................................................................... 148



Overview

The success or failure of a particular policy initiative or strategic plan is largely dependent

on whether the decision maker truly understands the interaction and complexity of the

system he or she is trying to influence. Considering the size and complexity of systems that

public and private sector decision makers must manage, it is not surprising that the

"intuitive" or "common sense" approach to policy design often falls short, or is

counter-productive, to desired outcomes. This online book was written to introduce the

concepts and "language" that make a systems-based study of such complex problems

possible. Our intent is to provide the reader with a broad overview of the field of system

dynamics, acquaint him or her with the fundamental stock-flow-feedback structures that

determine the dynamic behavior in systems, and motivate the reader to begin analyzing

problems dynamically and holistically. Knowing how to speak and think in terms of

systems and interconnections is a critical step in effective policy design, policy

implementation, and consensus building.

In Chapter 1, Introduction, we provide some context for system dynamics by

presenting a brief history of the field, beginning with Professor Jay W. Forrester's

work at the Massachusetts Institute of Technology in the 1950's, through the

present day. In addition, work related to energy policy is presented in some detail

to provide the reader with examples of the type of issues that can be studied using

system dynamics principles. In Chapter 2, Why Model?, we continue this discussion

by exploring the question: "Why model at all?"

Chapter 3, Building Blocks, presents the basic concepts behind the study of

complex systems by first examining the patterns of behavior that real-world

systems exhibit, and then discussing the structure that causes such patterns to

emerge. This chapter can be thought of as the "language chapter" because it is here

that the reader learns the concepts and terms required to construct the "theories"

or "models" of their particular issues.

With the concepts and language of system dynamics in hand, Chapter 4, Simple

Structures, examines system behavioral types such as exponential growth or

oscillation in greater detail. In this chapter, the reader is introduced to the concept

of computer simulation. Current computer simulation technology allows

decision-makers to easily construct models of a system's structure (including

mathematical equations) to use in conducting policy experiments.

In chapter 5, Basic Modeling Process, we share some thoughts and ideas about the

process of constructing system dynamics models, from conceptual causal diagrams

through detailed computer simulation models.

Chapter 6, Natural Gas Discovery, reinforces the concepts introduced in the first

five chapters by presenting a case-study based on a well-known system dynamics

model of the U.S. natural gas industry. For this discussion, we recreate the model's

structure incrementally by presenting a series of model versions - each version

increasing in complexity as additional structure is added. Throughout the chapter,



the reader is invited to "run" a computer simulation of the particular structure

being discussed to observe the dynamic behavior first hand.

Finally in Chapter 7, Next Steps, we conclude with a brief discussion of the

resources available to the reader interested in further study of system dynamics.

Layout of Introduction to System Dynamics

The figure below shows the basic layout of Introduction to System Dynamics. Although we

present the material in what we feel is the most logical learning progression, you are free to

move to any section that interests you by using the chapter and section menus along the

top and left-hand side of the screen, respectively.



Chapter 1 Introduction

System dynamics is a powerful methodology and

computer simulation modeling technique for framing,

understanding, and discussing complex issues and

problems. Originally developed in the 1950s to help

corporate managers improve their understanding of

industrial processes, system dynamics is currently

being used throughout the public and private sector

for policy analysis and design. In this chapter, we

provide some context for this book by presenting a

general history of the field. The land-mark models

presented in this chapter will give the reader an idea

of the types of issues/problems that can be explored

using system dynamics.

Origin of System Dynamics

Jay W. Forrester and the History of System Dynamics

System dynamics was created during the mid-1950s by Professor Jay W. Forrester of the

Massachusetts Institute of Technology. Forrester arrived at MIT in 1939 for graduate study

in electrical engineering. His first research assistantship put him under the tutelage of

Professor Gordon Brown, the founder of MIT's Servomechanism Laboratory. Members of

the MIT Servomechanism Laboratory, at the

time, conducted pioneering research in

feedback control mechanisms for military

equipment. Forrester's work for the Laboratory

included traveling to the Pacific Theater during

World War II to repair a hydraulically

controlled radar system installed aboard the

aircraft carrier Lexington.The Lexington was

torpedoed while Forrester was on board, but

not sunk .



WHIRLWIND I and SAGE

At the end of World War II, Jay Forrester turned his attention to the creation of an aircraft

flight simulator for the U.S. Navy. The design of the simulator was cast around the idea,

untested at the time, of a digital computer. As the brainstorming surrounding the digital

aircraft simulator proceeded, however, it became apparent that a better application of the

emerging technology was the testing of computerized combat information systems. In 1947,

the MIT Digital Computer Laboratory was founded and placed under the direction of Jay

Forrester. The Laboratory's first task was the creation of WHIRLWIND I, MIT's first

general-purpose digital computer, and an environment for testing whether digital

computers could be effectively used for the control of combat information systems. As part

of the WHIRLWIND I project, Forrester invented and patented coincident-current

random-access magnetic computer memory. This became the industry standard for

computer memory for approximately twenty years. The WHIRLWIND I project also

motivated Forrester to create the technology that first facilitated the practical digital

control of machine tools.

After the WHIRLWIND I project, Forrester agreed to lead a division of MIT's Lincoln

Laboratory in its efforts to create computers for the North American SAGE

(Semi-Automatic Ground Environment) air defense system. The computers created by

Forrester's team during the SAGE project were installed in the late 1950s, remained in

service for approximately twenty-five years, and had a remarkable "up time" of 99.8%.

System Dynamics

Another outcome of the WHIRLWIND I and SAGE

projects that is perhaps, for the purposes of this

online book, most noteworthy, was the

appreciation Jay Forrester developed for the

difficulties faced by corporate managers.

Forrester's experiences as a manager led him to

conclude that the biggest impediment to progress

comes, not from the engineering side of industrial

problems, but from the management side. This is because, he reasoned, social systems are

much harder to understand and control than are physical systems. In 1956, Forrester

accepted a professorship in the newly-formed MIT School of Management. His initial goal

was to determine how his background in science and engineering could be brought to bear,

in some useful way, on the core issues that determine the success or failure of

corporations.

Forrester's insights into the common foundations that underlie engineering and

management, which led to the creation of system dynamics, were triggered, to a large

degree, by his involvement with managers at General Electric during the mid-1950s. At that



time, the managers at GE were perplexed because employment at their appliance plants in

Kentucky exhibited a significant three-year cycle. The business cycle was judged to be an

insufficient explanation for the employment instability.

From hand simulations (or calculations) of the stock-flow-feedback structure of the GE

plants, which included the existing corporate decision-making structure for hiring and

layoffs, Forrester was able to show how the instability in GE employment was due to the

internal structure of the firm and not to an external force such as the business cycle. These

hand simulations were the beginning of the field of system dynamics.

During the late 1950s and early 1960s, Forrester and a team of graduate students moved the

emerging field of system dynamics, in rapid fashion, from the hand-simulation stage to the

formal computer modeling stage. Richard Bennett created the first system dynamics

computer modeling language called SIMPLE (Simulation of Industrial Management

Problems with Lots of Equations) in the spring of 1958. In 1959, Phyllis Fox and Alexander

Pugh wrote the first version of DYNAMO (DYNAmic MOdels), an improved version of

SIMPLE, and the system dynamics language that became the industry standard for over

thirty years . Forrester published the first, and still classic, book in the field titled

Industrial Dynamics in 1961.

Urban Dynamics

From the late 1950s to the late 1960s, system

dynamics was applied almost exclusively to

corporate/managerial problems. In 1968,

however, an unexpected occurrence caused

the field to broaden beyond corporate

modeling. John Collins, the former mayor of

Boston, was appointed a visiting professor

of Urban Affairs at MIT. Collins had been stricken with polio during the 1950s and, as a

result, required an office in a building with automobile access to the elevator level. By

chance, Jay Forrester's office was located in such a building and the office next to his was

vacant. Collins thus became Forrester's work-day neighbor, and the two began to engage in

regular conversations about the problems of cities and how system dynamics might be

used to address the problems.

The result of the Collins-Forrester collaboration was a book titled Urban Dynamics . The

Urban Dynamics model presented in the book was the first major non-corporate

application of system dynamics . The model was, and is, very controversial, because it

illustrates why many well-known urban policies are either ineffective or make urban

problems worse. Further, the model shows that counter-intuitive policies -- i.e., policies

that appear at first glance to be incorrect, often yield startlingly effective results. As an

example, in the Urban Dynamics model, a policy of building low income housing creates a



poverty trap that helps to stagnate a city, while a policy of tearing down low income

housing creates jobs and a rising standard of living for all of the city's inhabitants.

World Dynamics

The second major noncorporate application of system

dynamics came shortly after the first. In 1970, Jay

Forrester was invited by the Club of Rome to a meeting in

Bern, Switzerland. The Club of Rome is an organization

devoted to solving what its members describe as the

"predicament of mankind" -- that is, the global crisis that

may appear sometime in the future, due to the demands

being placed on the earth's carrying capacity (its sources

of renewable and nonrenewable resources and its sinks for the disposal of pollutants) by

the world's exponentially growing population. At the Bern meeting, Forrester was asked if

system dynamics could be used to address the predicament of mankind. His answer, of

course, was that it could.

On the plane back from the Bern meeting, Forrester created the first draft of a system

dynamics model of the world's socioeconomic system. He called this model WORLD1.

Upon his return to the United States, Forrester refined WORLD1 in preparation for a visit

to MIT by members of the Club of Rome. Forrester called the refined version of the model

WORLD2. Forrester published WORLD2 in a book titled World Dynamics .

From the outset, World Dynamics drew an enormous amount of attention. The WORLD2

model mapped important interrelationships between world population, industrial

production, pollution, resources, and food. The model showed a collapse of the world

socioeconomic system sometime during the twenty-first century, if steps were not taken to

lessen the demands on the earth's carrying capacity. The model was also used to identify

policy changes capable of moving the global system to a fairly high-quality state that is

sustainable far into the future.

In response to the notoriety of World Dynamics, the Club of Rome offered to fund an

extended study of the predicament of mankind via system dynamics. As Forrester was

committed to extending his Urban Dynamics project at the time, he declined to participate.

He did, however, suggest that one of his former Ph.D. students -- Dennis Meadows --

conduct the study. The model that was created by Meadows and his associates was called

WORLD3 and published in a book titled The Limits to Growth . Although the WORLD3

model was more elaborate than WORLD2, it generated the same fundamental behavior

modes and conveyed the same fundamental messages as its predecessor. Despite the

similarities, The Limits to Growth received even more world-wide attention than World

Dynamics. Some authors have speculated that this was due to the "friendly" style of writing

that made the book accessible to nontechnical readers .



In 1991, three of the original authors of The Limits to Growth redid the study in preparation

for the twentieth anniversary of the book's publication. The results were published in a

book titled Beyond the Limits . The revised system dynamics model created for the study

was called WORLD3-91. Once again, the results presented in Beyond the Limits were

consistent with the results presented in World Dynamics and The Limits to Growth,

although Beyond the Limits included a significant amount of numerical data that did not

exist when the original studies were undertaken. Beyond the Limits also contained a

careful presentation of arguments aimed at counteracting the criticisms that were directed

at the earlier world modeling books.

The System Dynamics National Model and K-12 Education

During the last twenty years, Jay Forrester's attention has been

focused primarily in two areas: 1) the creation of a system

dynamics model of the United States economy, and 2) the

extension of system dynamics training to kindergarten through

high school education. Forrester sees the former project as

leading to a new approach to economic science and a

fundamental understanding of the way macroeconomic systems

work. He views the latter project as crucial, not only for the

future health of the field of system dynamics, but also for the

future health of human society.

Although Forrester's national economic model remains unfinished, early and intermediate

results have been published . The most noteworthy of the results is that the model

generates a forty- to sixty-year economic cycle or "long wave" that not only explains the

Great Depression of the 1930s, but also shows that deep economic slumps are a repetitive

feature of capitalist economies. At the time of this writing, Forrester's model shows that

the United States economy is just beginning to rise out of the trough of the latest long

wave downturn.

Forrester's efforts to extend system dynamics to K-12 education have, in a sense, taken him

full circle, as the story begins with his original MIT mentor Gordon Brown. Brown retired

from MIT in 1973 and began wintering in Tucson, Arizona. During the late 1980s, Brown

introduced system dynamics to teachers in the Tucson school system. The results were

remarkable. System dynamics spread, not only through the original junior high in which it

was introduced, but through the entire school district. Subjects as diverse as Shakespeare,

economics, and physics are today taught in the school district, wholly or in part, via system

dynamics. Moreover, the district itself is using system dynamics in an effort to become a

learning organization .

The future of system dynamics in K-12 education appears promising. Today, an

international clearinghouse for K-12 system dynamics materials exists and Internet and

World Wide Web sites have been created to disseminate information . Numerous K-12



teachers, in the United States and abroad, have begun to integrate system dynamics into

their classes, and attend international conferences on the subject. All of these

developments suggest that the corporate and public sector decision makers of the future

may well begin to view the problems they face through the lens of system dynamics .

System Dynamics and Energy Modeling

The World Models

System dynamics modeling has been used for strategic

energy planning and policy analysis for more than

twenty-five years. The story begins with the world modeling

projects conducted in the early 1970s by the System

Dynamics Group at the Massachusetts Institute of

Technology. During these projects the WORLD2 and

WORLD3 models were created to examine the

"predicament of mankind" -- that is, the long term socioeconomic interactions that cause,

and ultimately limit, the exponential growth of the world’s population and industrial

output .

One of the central assumptions underlying the world models is that the earth’s natural

resources are, at some level, finite and that the exponential growth in their use could

ultimately lead to their depletion and hence, to the overshoot and collapse of the world

socioeconomic system. Due to the decision to explain the "predicament of mankind" with

fairly small system dynamics models, the resource depletion dynamics were represented in

the world models with structures that aggregated all of the earth’s natural resources into a

single variable.

The decision to represent the earth’s natural resources in aggregate form did not take place

in a vacuum. As part of the WORLD3 modeling project, several disaggregated,

resource-specific/issue-specific, models were created . The conclusion drawn from these

models was that it was appropriate to lump all natural resources into a single variable in

the WORLD3 model .

The Life Cycle Theory of M. King Hubbert

One of the disaggregated, resource-specific, system dynamics analyses that was conducted

in support of the world modeling efforts was a natural gas discovery and production model

created by MIT Master’s student Roger Naill . Naill based his model on the life cycle theory

of oil and gas discovery and production put forth by petroleum geologist M. King

Hubbert .

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In formulating his theory, Hubbert took the physical structure of the fossil fuel system into

account and assumed that the total amount of oil and gas in the United States (i.e., the

amount of oil and gas "in place"), and hence the "ultimately recoverable" amount of oil and

gas in the United States, is finite . As a result, according to Hubbert, the cumulative

production of domestic oil and gas must be less than or equal to the ultimately recoverable

amount of oil and gas in the United States .

Figure 1 is a system dynamics stock-flow structure that represents Hubbert’s theory. The

most important features of the structure are that (1) there is no inflow to the

Ultimately_Recoverable stock (i.e., there is a fixed stock of oil and gas), and (2) the

resource is being produced and consumed at an exponential rate.

Figure 1: Stock-Flow Structure Representing Hubbert’s View of Oil and Gas Discovery and

Production

A direct implication of Hubbert’s theory is that a time series graph of either oil or gas

production (at either the world-wide or domestic levels) must, at a minimum, be "hump"

shaped. That is, the area beneath the production curve for oil or gas is the cumulative

production of the resource, and the cumulative production of the resource must be a finite

number. In fact, Hubbert argued that the life cycle of oil and gas discovery and production

yields a bell-shaped production curve, which describes a period of low resource price and

exponential growth in production, a peaking of production as the effects of resource

depletion cause discoveries per foot of exploratory drilling to drop and resource price to

rise, and a long period of rising costs and declining production as the substitution to

alternative resources proceeds. Figure 2 shows a graphical representation of Hubbert’s life

cycle theory of oil and gas discovery and production.

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Figure 2: M. King Hubbert’s Life Cycle Theory of Oil and Gas Discovery and Production

Before proceeding, it is important to note that Hubbert’s view of natural resource discovery

and production is not shared by all energy analysts. For example, Morris Adelman, a

world-renowned resource economist (emeritus) at the Massachusetts Institute of

Technology, believes that there is no fixed stock of oil. Indeed, Adelman’s views vis-à-vis

oil price and depletion were summarized in a 1991 column by Boston Globe reporter David

Warsh:

The price of oil is a study in monopoly, nothing more. The question of

mineral depletion doesn’t really enter into it...[I]n fact, there is no ‘fixed

stock’ of oil, says Adelman; there is only an inventory we call ‘reserves,’

which we replenish with new prospecting and lifting techniques. What we

don’t choose to find or lift remains a secret of the earth, ‘unknown, probably

unknowable, surely unimportant; a geological fact of no economic interest.’

In the endless tug of war between diminishing returns and increasing

knowledge,’ he says, technology wins out...[The] worldwide stability of the

development cost of new oil since 1955 shows that oil is no more scarce

today than it was then. ‘The great shortage is like the horizon, always

receding as you go toward it,’ says Adelman. What’s left are the monopolistic

political high-jinks .



Figure 3: Stock-Flow Structure Representing Adelman’s View of Oil and Gas Discovery and

Production

Figure 3 is a system dynamics stock-flow structure representing Adelman’s view of oil and

gas discovery and production. It is directly comparable to Figure 1. Two things about the

figure are important to note. First, the cloud-like icon on the left side of the figure

indicates that there is no limit to oil or gas discovery (i.e., there is no fixed stock of oil or

gas). Second, two influences are battling each other for control of the Discovery_Rate:

technological change and diminishing returns. Historically, technology has always

defeated diminishing returns and Adelman, as well as many energy analysts, feels it will

continue to do so.

Naill's Master's Thesis

The results of Roger Naill’s Master’s thesis study confirmed Hubbert’s life cycle hypothesis.

Indeed, Naill concluded that the production of US domestic natural gas, which peaked in

1973, will continue to decline well below the US natural gas discovery rate until depletion

halts all domestic production sometime in the late twentieth or early twenty-first century.

The results of Naill’s work brought to the forefront the following question among the

system dynamicists who were working on energy modeling problems under the umbrella

of the world modeling programs: Will US economic growth be impeded by an energy limit

similar to those suggested in the Limits to Growth? To begin answering this question, in

1972 the Resource Policy Group at Dartmouth College received a three year contract from

the National Science Foundation to study the United States’ "energy transition problem."

The US Energy Transition Problem

The "energy transition problem" refers to the set of disruptions that the US economy must

go through as it reduces its dependence on domestic gas and oil (due to depletion) and

increases its reliance on new sources of energy. Historically, the US economy has gone

through two energy transitions: (1) from wood to coal during the late 1800s, and (2) from

coal to oil and gas during the early 1900s. But, these transitions were motivated by the

availability of abundant new energy sources that were cheaper and more productive than

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the existing sources. The energy transition that the US is currently facing, however, is

being forced by depletion and rising production costs, and not by a cheaper and more

productive energy source .

The implications of the energy transition problem for the United States are quite

significant. The continued growth in US energy demand, coupled with the depletion of

domestic oil and gas resources and long delays in the development of alternative domestic

energy sources is causing a widening domestic energy gap (domestic energy demand -

domestic energy supply). This gap can only be filled, in the near term at least, through

increased dependence on foreign imports of natural gas and oil. In addition, as long as

abundant oil and gas imports are available at prices that are low relative to the marginal

cost of developing new domestic supplies (i.e., as long as it’s easier to import oil and gas

than it is to develop new domestic energy sources), US oil and gas depletion will continue,

if not accelerate .

The COAL1, COAL2 and FOSSIL1 Models

Roger Naill’s natural gas model represented the US gas system at a very aggregate level.

The model was not broken down by region, technology, or type of gas. It did not allow for

the substitution of fuels nor for endogenous technological change. Thus, although it

helped to motivate the study of the US energy transition problem, it was inadequate for the

study itself. A new, expanded, model was required.

For his Ph.D. dissertation at Dartmouth, again under the supervision of Dennis Meadows,

Naill expanded the boundary of his natural gas model to include all major US energy

sources (energy supply), as well as US energy consumption (energy demand) . He called

his dissertation model COAL1, because his analysis showed that the best fuel for the US to

rely on during the energy transition was coal .

After he had completed his Ph.D., Naill worked with the Dartmouth Resource Policy Group

to improve and extend COAL1 as part of the Group’s National Science Foundation grant

activities. The improved and extended version of the model was called COAL2 . In 1975,

the Energy Research and Development Administration (which later became the US

Department of Energy) provided support to further improve and extend COAL2 for use in

government energy planning. This improved and extended model was called FOSSIL1,

since it looked at the transition of an economy that is powered by fossil fuels (i.e., by oil,

gas, and coal) to one that is powered by alternative energy sources.

The FOSSIL1 model (as were its predecessors) was thus based on Hubbert’s theory of

resource abundance, depletion, and substitution, and used to analyze and design new

legislation that would enable the US economy to pass through the energy transition

smoothly. It consisted of four main sectors: (1) energy demand, (2) oil and gas, (3) coal, and

(4) electricity, and addressed, among others things, the following questions:



Is energy independence for the US feasible and, if so, when?

Should a national energy strategy emphasize conservation or increased supply?

Which transition energy source should be accelerated?

The results from using FOSSIL1 to analyze the energy transition questions were that:

Due to the momentum of past energy policies and the inherent delays before new

policies become effective, in the short term the US energy problem cannot be

solved.

Neither supply side nor demand side policies alone will ameliorate the transition

problem sufficiently.

Smoothly passing through the energy transition requires policies that both

stabilize energy demand and increase alternative energy supplies.

The FOSSIL2 and IDEAS Models

In response to the United States’ first energy crisis in 1977, the Carter Administration

created the first National Energy Plan. Shortly thereafter, the US House of Representatives

asked the Dartmouth Resource Policy Group to evaluate the Plan using the FOSSIL1 model.

After the evaluation of the Plan was completed, Roger Naill left the Resource Policy Group

to head the Office of Analytical Services at the Department of Energy and, among other

things, prepare energy projections in support of future National Energy Plans.

To prepare the energy projections for future National Energy Plans, Naill implemented

FOSSIL1 in-house at the Department of Energy and supervised a team that extensively

modified it so that national energy policy issues could be analyzed. The modified version of

FOSSIL1 was called FOSSIL2 .

From the late 1970s to the early 1990s, the FOSSIL2 model was used at the Department of

Energy to analyze, among other things:

the net effect of supply side initiatives (including price deregulation) on US oil

imports.

the US vulnerability to oil supply disruptions due to political unrest in the Middle

East or the doubling of oil prices.

policies aimed at stimulating US synfuel production.

the effects of tax initiatives (carbon, BTU, gasoline, oil import fees) on the US

energy system.

the effects of the Cooper-Synar CO2 Offsets Bill on the US energy system.

In 1989, the Congress directed DOE to conduct a study of energy technology and policy

options aimed at mitigating greenhouse gas emissions. FOSSIL2 was used for this purpose.

Some preliminary conclusions from the study were that:



Reforestation is a promising alternative to taxes or standards.

Effectively promoting cost-effective conservation measures would be worthwhile.

There needs to be a significant long-term switch from coal to advanced nuclear

power and renewables (environmentally benign) in the US electric power sector.

Due to compensating feedbacks in the US energy system, a combination of policies,

rather than any single policy, is going to be necessary to successfully combat the

global warming problem.

Policy makers should not aim policy changes at a single sector of the US energy

system because this approach ignores the ramifications of the policy changes in

other sectors of the US energy system .

In recent years, extensive improvements have been made to FOSSIL2’s transportation and

electric utilities sectors . The improved version of FOSSIL2 has been renamed IDEAS,

which stands for Integrated Dynamic Energy Analysis Simulation. The IDEAS model is now

maintained for the DOE by Applied Energy Services of Arlington, Virginia .

Sterman's Model of Energy-Economy Interactions

During the late 1970s John Sterman, an MIT Ph.D. student and former Dartmouth College

undergraduate, was hired by Roger Naill to work with a team to modify and extend the

FOSSIL1 model into the FOSSIL2 model. During this work, Sterman came to realize that

the FOSSIL2 model ignored important feedbacks and interactions between the energy

sector of the economy and the economy itself. For his Ph.D. dissertation, Sterman built a

system dynamics energy model that captured, for the first time, significant

energy-economy interactions .

To be more precise, Sterman noticed that in the COAL-FOSSIL-IDEAS family of models,

the energy sector is modeled in isolation from the rest of the economy. That is:

GDP is exogenous to the model. It is not affected by the price or availability of

energy.

Costs of unconventional energy technologies are exogenous to the model.

Investment in energy is unconstrained by the investment needs of other sectors of

the economy.

Interest rates are exogenous to the model.

Inflation is unaffected by domestic energy prices, production, or policies.

World oil prices are unaffected by domestic energy prices, production, or policies .

Sterman addressed these deficiencies through his modeling and found that:

The economic consequences of depletion are much more severe during the

transition period (extending to approximately 2030) than during the long run or

equilibrium state.

http://www.albany.edu/cpr/sds/DL-IntroSysDyn/feed.htm



The magnitude of the economic effects are substantial in absolute terms and

include reductions in economic growth; increased unemployment;inflationary

stress; higher real interest rates;reduced consumption per capita.

Energy price increases (sudden or gradual) alone cannot produce sustained

inflation. An accompanying increase in the money supply, relative to real economic

activity, is also required (or an increase in the velocity of money).

The model’s major behavior modes are remarkably robust -- i.e., insensitive to

parameter variations (uncertainties).

In the model, a large exise tax on energy coupled with offsetting income tax

reductions caused economic performance to improve; energy prices to fall; OPEC

revenues to fall; short term inflationary pressures to worsen; income taxes to be

reduced only during the transition .

Fiddaman's Model of Economy-Climate Interactions

Building on the work of his teachers, in 1997 Tom Fiddaman submitted his Ph.D.

dissertation on economy-climate interactions to the Sloan School of Management at MIT .

The dissertation included a critique of existing (non system dynamics) climate-economy

models and a new climate-economy system dynamics model called FREE (Feedback-Rich

Energy Economy model). The FREE model explicitly incorporates the dynamics of oil and

gas depletion as a "source constraint" on the energy-economy system (as do all of its

system dynamics predecessors), as well as the dynamics of a "sink constraint" (i.e., climate

change) on the energy-economy system. The FREE model is the first energy-economy

model of any kind to explicitly examine the impact of a source constraint on

energy-economy interactions.

The FREE model also explores a number of feedback processes (e.g., endogenous

technological change and bounded rational decision making with perception delays and

biases) that have not been previously explored in a climate change context. In addition, it

is constructed so that a particular parameterization will yield the results found in

neoclassical (traditional) climate-economy models.

Estimating the Amount of Oil In-Place

In the early 1980s, system dynamicist George Richardson met a British petroleum analyst

who claimed that the "amount of oil in the world is increasing." Richardson replied that,

although world oil reserves may be increasing, or that the estimate of the amount of oil in

the world may be increasing, the actual amount of oil in the world (the amount of oil

in-place) is decreasing.

Richardson was unable to persuade the British petroleum analyst to change his mind, so he

decided to build a system dynamics model that could demonstrate his point. He enlisted

the assistance of John Sterman, who had recently finished his dissertation on



energy-economy interactions and, as a starting point, turned to M. King Hubbert’s

research and Roger Naill’s natural gas model.

Richardson and Sterman produced an oil exploration, discovery, and production model

that was similar in spirit to Naill’s natural gas model, but that also had important

extensions and improvements. More precisely, their model allowed for endogenous

technological change and the substitution of synfuels for oil .

Richardson and Sterman first used their model to run a synthetic data experiment that

addressed the following question: Which method of forecasting the world’s ultimately

recoverable supply of oil is more accurate, M. King Hubbert’s life cycle method or the

geological analogy method? Since the world’s ultimately recoverable supply of oil is

currently not known, and cannot be known until all of the world’s oil has been depleted, a

synthetic data experiment was required to answer the question.

The logic of Richardson and Sterman’s synthetic data experiment was quite simple. First

build a system dynamics model that accurately replicates the exploration, discovery, and

production behavior of the world oil system and assume that it is the "real world." Second,

formally code and add the Hubbert and geologic analogy methods to the model so that

they "watch" the "real world oil system" and create forecasts of the ultimately recoverable

amount of oil in the "world." The results of the synthetic data experiment were that:

The model replicated the behavior of the actual world oil system very well.

The model replicated the actual forecasts produced by the Hubbert and geologic

analogy methods very well.

Hubbert’s method was clearly the most accurate.

over simulated time, the geologic analogy method rose to, and then overestimated,

the ultimately recoverable amount of oil in the world .

The implications of the geologic analogy method significantly overestimating the

ultimately recoverable amount of oil in the world include:

a possible reduction in oil conservation efforts.

the probable overestimation of the amount of time available to develop substitutes

for oil and the technologies, institutions and values needed to create a sustainable

energy system.

Sterman and Richardson, with the assistance of system dynamicist Pål Davidsen, went on

to apply their model and synthetic data technique to the question of the amount of

ultimately recoverable oil in the United States . As in the case of world oil, Hubbert’s

method was judged to be clearly superior and the model was able to replicate US oil

discovery and production data extremely well. Of course, unlike the case of world oil

production which has not yet peaked, US domestic oil production (in the lower 48 states)

peaked in 1970. Since Hubbert had forecast in 1956 that US oil production (in the lower 48

states) would peak between the years 1966 and 1971, his forecast is one of the most accurate



and remarkable in the history of energy forecasting . In light of this, Sterman, Richardson

and Davidsen’s synthetic data experiment for the United States is perhaps best interpreted

as supporting the argument that Hubbert’s method is the most accurate.

Other System Dynamics Modeling in the Oil and Gas Industry

System dynamics modeling has been used by numerous researchers, outside of the

Naill-to-IDEAS lineage, to examine firm-level and industry-level issues in the oil and gas

industry. Table 1 lists some of the work that has been done. Inspection of the table reveals

that topics such as the behavior of OPEC and world oil markets, business process

re-engineering in an oil and gas producing firm, international relations stemming from

world oil supply and demand relationships, and oil firms as learning organizations, have

been addressed with system dynamics. The Energy 2020 model was developed by George

Backus and Jeff Amlin to provide individual energy firms and state agencies with a

multi-fuel energy model. It is similar in design to the DOE’s IDEAS model.

Topic Area Authors

Hubbert’s method versus the

geologic analogy method

Sterman and Richardson (1985); Sterman,

Richardson and Davidsen (1988); Davidsen,

Sterman and Richardson (1990)

Hubbert’s Method applied to Mexico Duncan (1996a, 1996b)

The behavior of OPEC and world oil

markets

Powell (1990a, 1990b); Morecroft (1992);

Morecroft and van der Heijden (1992)

Business process re-engineering in a

gas and oil producing firm

Genta and Sokol (1993)

Shell Oil as a learning organization De Geus (1988)

Learning about the oil industry from

a management flight simulator

Kreutzer, Kreutzer and Gould (1992);

Morecroft (1992); Morecroft and van der

Heijden (1992); Genta and Sokol (1993)

International relations stemming

from world oil supply and demand

relationships

Choucri (1981)

Multi-fuel energy model for use by

individual firms and state agencies

Ford (1997, pp. 58-59)

Table 1: Some Well-Known System Dynamics Studies in the Oil and Gas Industry



The efforts of oil companies to become "learning organizations" through the use of

"management flight simulators" is a particularly noteworthy use of system dynamics in

energy modeling. In 1990, system dynamicist Peter Senge wrote a book that outlined a way

for organizations to become "learning organizations" through the use of system dynamics

and other tools . A learning organization is composed of employees who possess a shared,

holistic, and systemic vision, and have the commitment and capacity to continually learn,

rather than simply executing a "plan" put forth by the "grand strategist" at the top of the

organization.

One of the principal tools used by learning organizations is the "management flight

simulator." Management flight simulators are computerized learning environments that

invite decision makers to train in a simulator just like a pilot does. The flight simulator

runs an underlying system dynamics model for a number of periods, pauses, and waits for

the decision maker to make a policy change. After the policy change has been entered, the

flight simulator again simulates the model forward in time, pauses, and waits for the next

policy change. After a decision maker has finished a session in the simulator, he or she is

invited to determine why the system behaved as it did. Once the decision maker ascertains

this, he or she is invited to play again. Of course, after a number of plays the decision

maker’s understanding of the system should improve and, hopefully, he or she will apply

the lessons learned to an actual organization.

System Dynamics Modeling in the Coal Industry

Applications of system dynamics outside of the Naill-to-IDEAS lineage also exist in the coal

industry. As shown in Table 2, system dynamics has been used to study industry-level

problems, mining systems, the dynamics of small surface coal operations, international

mining ownership, and the representation of discrete events in system dynamics models.

Topic Area Authors

The dynamics of small surface

coal operations

Kinek and Jambekar (1984a, 1984b, 1983)

Industry-level studies Zahn (1981); Mendis, Rosenburg and Medville

(1979)

Mining systems Wolstenholme and Holmes (1985);

Wolstenholme (1983, 1982b, 1981); Schwarz (1978)

International mining ownership Wolstenholme (1984)

Representing discrete events in

system dynamics models

Coyle (1985)

Table 2: Some Well-Known System Dynamics Studies in the Coal Industry



System Dynamics Modeling in the Electric Power Industry

One of the studies that followed the world modeling projects, was conducted by the

Dartmouth Resource Policy Group and undertaken in a fashion parallel to Roger Naill’s

COAL1 study, was Andrew Ford’s system dynamics analysis of the future of the US electric

power industry . For his Ph.D. dissertation, Ford produced the ELECTRIC1 model, which

was the first in a series of system dynamics electric utility models known as the EPPAM

models . Modified versions of Ford’s model and its descendants were also used to build the

electricity sectors of the COAL2, FOSSIL1 and FOSSIL2 models .

Since Ford’s pathbreaking work, system dynamics has been used extensively by utility

managers for strategic planning . Table 1 lists some well-known system dynamics studies

that have addressed problems in the electric power industry, including: the effects of

regulatory policy on utility performance, the "spiral of impossibility," the effects of external

agents on utility performance, the financial performance of utilities, the effects of energy

conservation practices on utility performance, regional strategic electricity/energy

planning, national strategic electricity/energy planning, electric vehicles, deregulation in

the UK electric power industry, deregulation in the US electric power industry, and river

use and its impact on hydroelectric power.

The system dynamics work on river use and its impact on hydroelectric power is

particularly noteworthy as it involves the use of a management flight simulator in a public

policy context. More precisely, the management flight simulator is designed to allow

ordinary citizens, as well as utility managers and other stakeholders, to test policies aimed

at moving hydroelectric systems in desired directions, while taking multiple criteria into

account.

Topic Area Authors

Effects of regulatory policy on utility

performance

Geraghty and Lyneis (1983)

The "spiral of impossibility" Ford and Youngblood (1983).

Effects of external agents on utility

performance

Geraghty and Lyneis (1985)

Financial performance of utilities Lyneis (1985)

Effects of energy conservation practices on

utility performance

Ford, Bull and Naill (1989); Ford and

Bull (1989); Aslam and Saeed (1995)

Regional strategic electricity/energy

planning

Dyner et al. (1990)

National strategic electricity/energy Coyle and Rego (1983); Naill (1977,



planning 1992); Sterman (1981)

Electric vehicles Khalil and Radzicki (1996); Ford

(1996b); Ford (1995a); Ford (1994)

Deregulation in the UK electric power

industry

Bunn and Larsen (1992, 1994, 1995);

Bunn, Larsen and Vlahos (1993); Larsen

and Bunn (1994)

Deregulation in the US electric power

industry

Lyneis, Bespolka and Tucker (1994)

River use and its impact on hydroelectric

power

Ford (1996a)

Table 3: Some Well-Known System Dynamics Studies in the Electric Power Industry

Summary: Intellectual Lineage of System Dynamics Energy

Modeling

Figure 4 presents a diagram of the intellectual lineage of system dynamics energy modeling.

The lineage begins with the first book on system dynamics modeling -- Industrial

Dynamics by Jay W. Forrester . Forrester’s original work spawned various firm and

industry-level system dynamics energy models and inspired Peter Senge to write the Fifth

Discipline. Senge’s book has led to the creation of energy-related management flight

simulators and several attempts at turning energy companies into learning organizations.

Forrester was also responsible for creating the WORLD2 model and initiating the world

modeling projects at MIT. The world models, along with M. King Hubbert’s work on oil

and gas discovery and production, stimulated the creation of Roger Naill’s natural gas

model, his COAL1 model, and the improvements to COAL1 that have culminated in the

IDEAS model and its offshoots (FOSSIL79 and DEMAND81). Naill and Hubbert’s work

formed the basis for Sterman, Richardson and Davidsen’s synthetic data experiments on

analyzing techniques for forecasting the ultimately recoverable amount of oil in the world

and in the United States, while knowledge of the weaknesses in the FOSSIL2 model caused

Sterman to investigate the dynamics of energy-economy interactions during the energy

transition. Fiddaman’s recognition that, although the source constraints on the

energy-economy system had been investigated by energy modelers, sink constraints had

not, lead to the creation of the FREE model. The world modeling projects also stimulated

the study of the US electric power industry by Andrew Ford and the subsequent EPPAM

models and their offshoots.



Chapter 2 Why Model?

As we see from the brief history of

system dynamics presented in

Chapter 1, much of the emphasis of

system dynamics centers around the

use of models and modeling

techniques to study complex policy

issues. As we proceed to outline

system dynamics concepts, it is

important that we step back briefly

to consider the question, Why model

at all? From observation, we know

that modeling is something that all

human beings do, and for a variety of reasons. However, the importance of understanding

the distinction and application of models is less obvious. In this chapter, we continue to

outline the basic premise and philosophy of system dynamics by addressing this

fundamental question.

Why Build Models in the First Place?

Given that the stated goal of this online book is to teach

basic system dynamics concepts and modeling

techniques, it is legitimate to step back and ask: Why

build models in the first place? Are there some advantages

to be gained from the creation of system dynamics models?

In general what, if anything, is to be gained from the

creation of models?

Formal Models

Modeling is something that all human beings do. Children create models out of playdough,

tinker toys, legos, blocks, cards, sand, and the like. Engineers create clay models of

automobiles, metal models of aircraft, wooden models of bridges, plastic models of cities,

hand-drawn models of buildings, and computer models of most of the things they create.

Natural scientists create physical models of molecules, the human body, and the solar

system, and mathematical and/or written models of the evolution of the universe. Social

scientists create computer and written models of the mind, mathematical and computer

models of the economy, and physical models of ancient civilizations. Managers build



financial models with spreadsheet programs and database applications. Playwrights create

models capturing aspects of the human condition.

Humans create models for a variety of reasons. Models are simplifications of reality and

(usually) help people to clarify their thinking and improve their understanding of the

world. Models can be used for experimentation. A computer model, for instance, can

compress time and space and allow many system changes to be tested in a fraction of the

time it would take to test them in the real world. Further, testing changes on a model,

rather than on an actual system, is a good way to avoid "shooting yourself in the foot." That

is, if a change does not perform well in a model of a system, it is questionable as to whether

it will perform well in the actual system itself. In addition, experimenting on a model can

avoid causing harm to an actual system, even when the change being tested is successful.

For example, testing a more effective sprinkler system design does not require setting an

actual building on fire, if the testing is done on a model of the building.

Mental Models

Although it is perhaps self-evident that humans regularly create and use formal models, it

is less obvious that they regularly create and use informal models, or mental models. More

precisely, human beings do not have actual families, clubs, churches, universities,

corporations, cities, states, national socioeconomic systems, and the like, inside of their

heads, but rather mental representations of these systems. Thus, in the field of system

dynamics it is argued that policy makers should not worry about whether or not to use a

model, but rather which model to use . In other words, system dynamicists believe that

policy makers should decide whether they wish to make decisions based on results

obtained from their unaided mental models, or from results obtained from some

combination of formal and mental models.

Both formal models and mental models have many strengths and weaknesses . Mental

models are flexible, rich in detail, and constructed from the most abundant and valuable

source of information in the world - experience "data" collected in your brain. John

Sterman, a prominent system dynamics professor at Massachusetts Institute of Technology,

points out that the "great systems of philosophy, politics, and literature are, in a sense,

mental models" .

To illustrate the importance of mental models, it is useful to consider the following

thought experiment . Imagine that spacemen land in Detroit, remove every worker from

one of the city's automobile plants, and replace them with workers who are identical in

every way to the removed workers, except that they have no mental information to guide

them in making automobiles. Could the new workers build automobiles by using the

available written and numerical information? The answer is "no," and the reason is that an

overwhelming majority of the information that is crucial for automobile manufacturing is

contained in the minds of the removed workers.



Although mental models have many strengths, they also have many weaknesses. Mental

models are often fuzzy, incomplete, imprecise, and filled with unstated assumptions and

goals. During conversation and debate, different people use different mental models, and

these models can and do change -- even during the course of a single conversation or

debate . Further, the human mind is a poor dynamic simulator. Cognitive limitations

prevent humans from accurately thinking through the dynamic behavior inherent in all

but the simplest of their mental models .

To illustrate the richness and complexity of human mental models and the difficulty of

thinking through their dynamic consequences, it is useful to examine Douglas Hofstadter's

book Gödel, Escher, Bach: An Eternal Golden Braid . In his book, Hofstadter integrates

Gödel's Incompleteness Theorem, the music of Johann Sebastian Bach, and the art of

Maurits Cornelis Escher, to address the issue of whether machine reasoning and artificial

intelligence are truly possible. Hofstadter's self-made sketch of a portion of his mental

model that underlies his book is presented in Figure 1.

(click on image to see detailed view)

Figure 1: Hofstadter's self-made sketch of a portion of his mental model

After examining Hofstadter's sketch, it is perhaps easier to accept the argument that trying

to accurately think through the dynamic behavior inherent in mental models is a difficult

task indeed!

Combining Mental Models with System Dynamics



The solution to the "mental model problem," according to system dynamicists, is to have

decision makers map-out their mental models on the computer via

system dynamics, and let the machine trace through the inherent

dynamics. Then, through interaction with the computer, decision

makers can improve their mental models and learn about the

system they are trying to understand and control. Indeed, in

system dynamics modeling, the process of modeling is seen as

being more valuable than the model itself .

In system dynamics modeling it is also typical to take a model that is insightful and turn it

into a game or management flight simulator. The argument for doing this is that a decision

maker should train in a simulator just like a pilot does and, in doing so, greatly improve his

or her mental model of the system he or she is trying to understand and control. Creating

an effective system dynamics game from an insightful system dynamics model is a

nontrivial task and a research area in and of itself.

http://www.albany.edu/cpr/sds/DL-IntroSysDyn/flsim.htm



Chapter 3 The Building Blocks

System dynamics provides the basic

building blocks necessary to construct

models that teach us how and why

complex real-world systems behave the

way they do over time. For us, the goal

is to leverage this added understanding

to design and implement more efficient

and effective policies. This chapter

introduces the concept of time paths

and how to begin looking at the world in terms of changing patterns over time rather than

as static snap-shot events. This chapter also introduces the concept of system stocks, system

flows, and system feedback. To understand the dynamic behavior of a system, its key

physical and information stocks, flows and feedback structures must be identified.

Taking A Dynamic Perspective

System dynamics is concerned with the behavior of a system over time. A critical step in

examining a system or issue is to identify its key patterns of behavior - what we often refer

to as "time paths." In this section, we examine the meaning of a time path and consider the

general types commonly found in dynamic systems. To provide a "real life" grounding to

this discussion, several time path examples reproduced from books, newspapers, and

magazines are presented.

Time Paths

System dynamics modeling is concerned with the dynamic behavior of systems - that is,

the behavior of systems over time. In system dynamics modeling, the modeler attempts to

identify the patterns of behavior being exhibited by important system variables, and then

build a model that can mimic the patterns. Once a model has this capability, it can be used

as a laboratory for testing policies aimed at altering a system's behavior in desired ways.

Figure 1 is a straight line with a positive slope and an example of a dynamic time path.

Although actual systems can exhibit a variety of time paths (often simultaneously), the

good news is that the number of distinct time path "families" that exists is relatively small.



Figure 1: Example of time path

Figures 2 through 5 present a collection of fifteen distinct time paths which occur

frequently in the real world. These paths have been grouped into five distinct families, one

of which is shown in each of the figures. Other, more complicated, time paths are

observable, but they are almost always combinations of the paths presented here.

Linear Family

The first identifiable family of time paths is the linear family, which is presented in Figure 2.

The linear family of paths includes equilibrium, linear growth, and linear decline. Given

the simplicity and intuitive appeal of these paths, it is important to point out some facts

which can place them in the proper perspective, vis-à-vis system dynamics modeling.

Figure 2: Linear Family of Time Paths

The first thing to note is that the average person, not trained in dynamic modeling, tends

to think that actual systems grow or decline in a linear fashion - that is, as shown in second

and third time paths of Figure 2.The truth, however, is that most systems do not grow and

decline along linear time paths, but rather along exponential time paths (as shown in

Figure 3) . Pure linear time paths are usually generated by systems devoid of feedback - a

crucial building block of both actual systems and system dynamics models.

A second thing to note is that the equilibrium time path presented in Figure 2 is a dynamic

behavior that few actual systems exhibit. It is a state of perfect balance in which there are

no pressures for change. In fact, from a system dynamics point of view, equilibrium implies

that all of a system's state variables reach their desired values simultaneously - a very



artificial situation . Curiously, much of modern economics and management science uses

models based on the concept of equilibrium . System dynamicists, on the other hand,

believe that the most interesting system behaviors (i.e., problems) are disequilibrium

behaviors and that the most effective system dynamics models exhibit disequilibrium time

paths. This is not to say, however, that the concept of equilibrium is useless. On the

contrary, system dynamics models are often placed in an initial state of equilibrium to

study their "pure" response behavior to policy shocks .

Exponential Family

The second distinct family of time paths is the exponential family, shown in Figure 3. The

exponential family consists of exponential growth and exponential decay. As previously

mentioned, real systems tend to grow and decline along exponential time paths, as

opposed to linear time paths.

Figure 3: Exponential Family of Time Paths

Goal-seeking Family

The third distinct family of time paths is the goal-seeking family as shown in Figure 4. All

living systems (and many nonliving systems) exhibit goal-seeking behavior. Goal-seeking

behavior is related to exponential decay. This can be seen by comparing the second time

path in Figure 3 with the second time path in Figure 4. The only difference between the

two is that in Figure 3, the time path is seeking a goal of zero, whereas in Figure 4, the time

path is seeking a nonzero goal.

Figure 4: Goal Seeking Family of Time Paths



Oscillation Family

The fourth distinct family of time paths is the oscillation family. Oscillation is one of the

most common dynamic behaviors in the world and is characterized by many distinct

patterns. Four of these distinct patterns are shown in Figure 5: sustained, damped,

exploding, and chaos.

Figure 5: Oscillating Family of Time Paths

Oscillations that are sustained can have periodicities (the number of peaks that occur

before the cycle repeats) of any number. Sustained oscillations are characterized by a

periodicity of one. Damped oscillations are exhibited by systems that utilize dissipation or

relaxation processes. Examples of dissipation or relaxation processes include friction in

physical systems and information smoothing in social systems. Exploding oscillations either

grow until they settle down into a sustained pattern or grow until the system is torn apart. .

As a result, exploding oscillations usually do not occur very frequently in the real world,

nor last very long when they do. Chaotic behavior is an oscillatory time path that unfolds

irregularly and never repeats (i.e., its period is essentially infinite). Chaos is a unique type

of time path because it is an essentially random pattern that is generated by a system

devoid of randomness .

S-shaped Family

The fifth distinct family of time paths is the s-shaped family, shown in Figure 6. Close

inspection of the s-shaped growth time path pattern reveals that it is really a combination

of two time paths - exponential growth and goal-seeking behavior. More precisely, in the

case of s-shaped growth, exponential growth gives way to goal-seeking behavior as the

system approaches its limit or carrying capacity (indicated by the green line).



Figure 6: S-Shaped Family of Time Paths

Sometimes, however, a system can overshoot its carrying capacity. If this occurs and the

system's carrying capacity is not completely destroyed, the system tends to oscillate around

its carrying capacity. On the other hand, if the system overshoots and its carrying capacity

is damaged, the system will eventually collapse. This is referred to as an "overshoot and

collapse" system response. A third possible outcome from an overshoot event is that a

system will simply reverse direction and approach the system capacity in a reverse s-shaped

pattern. As with a "normal" s-shaped pattern, a reverse s-shaped pattern is a combination

of two time paths -- exponential decay and a self-reinforcing spiral of decline.

Actual Data

A story that is frequently recounted by Jay Forrester is that a student once told him that he

"learned to read the newspaper differently," after taking a class in system dynamics.

Although it is possible to interpret the student's comment in a number of ways, one can

speculate that, among other things, he began to see the graphs of actual time series data,

presented in the media, in a different light after studying system dynamics.

Figures 1 through 10 present graphs of a variety of actual time series data reproduced from

books, newspapers, and magazines, which appeared during the last few years. An

examination of the figures reveals that it is possible to identify, in actual data, many of the

time paths shown in the previous section.

Figure 1, for example, is a graph of actual Medicare costs from 1967 to 1994 and Medicare

costs from 1967 to 1994 as projected in 1965. Two things about this graph are noteworthy.

The first is that Medicare costs were projected to grow linearly from 1967 to 1994 -- a

common mistake among persons not familiar with the dynamics of systems. The second is

that actual Medicare costs grew exponentially from 1967 to 1994. This time path is clearly a

member of the exponential family of time paths.



Figure 1: Actual and Projected Medicare costs from 1967 to 1994

Figure 2 is a graph of the number of cases of the measles reported in the United States

from 1960 to 1993. The time shape exhibited by these data is clearly exponential decay.

Figure 2: Measle cases from 1960 through 1993

Figure 3 below is a graph of National Basketball Association attendance during the years

1988 to 1993 and Figure 4 is a graph of the average U.S. selling price of a 486 desktop

computer. These data are clearly following non-zero goal-seeking time paths.



Figure 3: National Basketball Association attendance from 19988 through 1993

Figure 4: Average U.S. selling price of a 486 desktop computer.

Figure 5 is a graph of the number of lynx trapped in Canada (in log form) from 1821 to 1933.

Although slightly "noisy" or choppy, the time path can certainly be classified as a sustained

oscillation.



Figure 5: Lynx trapped in Canada (in log form) from 1821 to 1933.

Figure 6 is a graph of Norwegian pulp inventory, production, and sales, for the years 1957

to 1978. Clearly, the time path of pulp inventory is an exploding oscillation .

Figure 6: Norwegian pulp inventory, production, and sales, for the years 1957 to 1978.

Figure 7 is a time series graph of the number of third class mail pieces (in billions)

delivered by the United States Postal Service, during the years 1981 to 1991. The path is

clearly s-shaped.



Figure 7: Number of third class mail pieces (in billions) delivered by US Postal Service, 1981

to 1991.

Figure 8 is a time series graph of materials consumption in the United States for the years

1900 to 1992. Although a bit noisy, the data exhibit an overshoot and oscillate time path.

Figure 8: Materials consumption in the US from 1900 to 1992.

Figure 9 presents a time series graph of United States bank, savings and loan, and total

financial institution, failures, during the years 1980 to 1992. All three series exhibit an

overshoot and collapse time path.



Figure 9: United States bank, saving and loan, and total financial institution failures, 1980

to 1992.

Figure 10 is a graph of grainland in Japan for the years 1950 to 1994. The time path

exhibited by this data is clearly a reverse s-shape.

Figure 10: Grainland in Japan, 1950 to 1994.

The general conclusion that the reader should draw from these graphs is that real systems

often generate clearly identifiable time patterns and that system dynamic models can be

built to mimic the patterns.



Seeing System Structure

In the previous section, we defined the concept of time paths and considered how these

system behavioral patterns can be placed into one or a combination of five distinct

categories. Now, we consider the source of these characteristic patterns. Specifically, we

discuss the concept of cumulative (stock) and rate-of-change (flow) system variables. As

the elemental building blocks of system feedback networks, "stocks" and "flows" virtually

ensure that a dynamic system will behave in a difficult-to-understand or counterintuitive

way. In addition, the role of nonlinear relationships in determining a system's limits, the

strength or dominance of its feedback loops, and the complexity of its behavior,will be

discussed.

Stocks and Flows

In system dynamics modeling, dynamic behavior is thought to arise due to the Principle of

Accumulation. More precisely, this principle states that all dynamic behavior in the world

occurs when flows accumulate in stocks.

Figure 1: Example of a simple stock and flow structure

In terms of a metaphor, a stock can be thought of as a bathtub and a flow can be thought

of as a faucet and pipe assembly that fills or drains the stock as shown in Figure 1. The

stock-flow structure is the simplest dynamical system in the world. According to the

principle of accumulation, dynamic behavior arises when something flows through the

pipe and faucet assembly and collects or accumulates in the stock. In system dynamics

modeling,both informational and noninformational entities can move through flows and

accumulate in stocks.

Figure 2 is an example of a system dynamics stock and flow structure that has two flows --

an inflow and an outflow. In this case, the dynamic behavior of the system arises due to the

flows into, and out of, the stock (the inflow minus the outflow). Clearly, if the inflow

exceeds the outflow the number of entities in the stock will increase. If the outflow exceeds

the inflow, on the other hand, the number of entities in the stock will decrease. Finally, if

the outflow equals the inflow, the number of entities in the stock will remain the same.

This last possibility describes a state of dynamic equilibrium in a system dynamics model.



Figure 2: Example stock and flow structure with an inflow and outflow

In principle, a stock can have any number of inflows and outflows. In practice, however, a

system dynamics model usually contains stocks with no more than four-to-six inflows

and/or outflows. The principle of accumulation holds regardless of the number of inflows

and outflows that work to change the number of entities accumulating in a stock.

Identifying Stocks and Flows

One of the fundamental skills a system dynamics modeler must learn is how to identify the

stocks and flows in the system experiencing the problem that he or she is trying to model.

This is a nontrivial task and one that people often find difficult. For example, it is not at all

unusual for people, not trained in system dynamics modeling, to confuse stocks and flows.

They'll say "deficit" (a flow) when they mean "debt" (a stock), or they'll say that "inflation (a

flow into a stock) is lower so therefore the general level of prices (a stock) is falling" In fact,

decreasing inflation to a lower value means that prices are rising but at a slower rate.

In order to identify stocks and flows, a system dynamics modeler must determine which

variables in the system experiencing the problem define its state (its stocks), and which

variables define the changes in its state (its flows). That said, the following guidelines can

be used to help identify stocks and flows:

Stocks usually represent nouns and flows usually represent verbs.

Stocks do not disappear if time is (hypothetically) stopped (i.e., if a snapshot were

taken of the system); Flows do disappear if time is (hypothetically) stopped.

Stocks send out signals (information about the state of the system) to the rest of

the system.

Four Characteristics of Stocks

Stocks possess four characteristics that are crucial in determining the dynamic behavior of

systems. More specifically, stocks:

Have memory,

Change the time shape of flows,

Decouple flows,

Create delays.



Each of these characteristics will be discussed in turn.

Stocks have memory

The first key characteristic of stocks is that they have memory (i.e., persistence or inertia).

An easy way to see this is to recall the simple stock-flow structure presented in Figure 1. If

the inflow to the stock is shut-off, the number of entities in the stock will not decrease, but

rather stay at the level it is at when the inflow stops. An outflow in excess of the inflow is

required to decrease the number of entities in the stock. The importance of this

characteristic should not be underestimated. This is because people often believe that

shutting-off an inflow to a stock will cure a problem being created by the number of

entities in the stock. Regrettably, nothing could be further from the truth.

Figure 3: US federal budget deficit, 1930 - 1993.

Consider the problem that the United States is currently having with its federal deficit and

debt. Figure 3 presents a time series graph of the U.S. federal deficit and Figure 4 presents a

time series graph of the U.S. federal debt. The deficit is the yearly amount that the U.S.

federal government spends in excess of its revenues, while the debt is the accumulation of

all the previous deficits (less the debt that has been retired). From an inspection of these

figures, it is clear that both the U.S. federal deficit and the U.S. federal debt are growing

exponentially.



Figure 4: US Federal Debt from 1930 - 1993.

In terms of a simple system dynamics model, the deficit is a flow and the debt is a stock, as

shown in Figure 5 . Therefore, if the U.S. federal government were to balance its budget

(i.e., make the deficit inflow zero), the debt would not decrease at all. This fact is

surprising to many people!

Figure 5: Structure of US Debt/Deficit problem.

A similar case involves the depletion of the earth's ozone layer. In 1990 the nations of the

world agreed to a ban on the production (and hence on the eventual release into the

atmosphere) of ozone-killing chlorofluorocarbons (CFC's).

Figure 6: System dynamics structure of CFC accumulation in atmosphere.

From a system dynamics point of view, this agreement caused the flow of

chlorofluorocarbons into the atmosphere to stop, but did nothing to remove any of the

chlorofluorocarbons already in the atmosphere and destroying ozone (a stock -- see Figure

6). The removal of these pollutants can only be accomplished via the earth's natural

cleansing processes .



Stocks Change the Time Shape of Flows

A second important characteristic of stocks is that they (i.e., the accumulation process)

usually change the time shape of flows . This can be seen by simulating the simple

stock-flow structure shown in Figure 1, with different time shapes for the flow. Figure 7 for

example, presents the time shape of the stock when the flow is at a constant level of 5

units/time. An examination of the figure reveals that the accumulation process changes the

horizontal time shape of the flow into a linear growth shape for the stock .

(Click on image to see animation)

In a similar way, Figure 8 through Figure 13 show the time shape of the stock for different

time shapes exhibited by the flow.




In Figure 8, we see that if the inflow valve is "opened" or increases in a linear fashion, (i.e.,

a linear growth shape that is at all times positive), the "level" of the stock variable will grow

at an increasing rate.


Figure 9 shows that a linear decline shape for the inflow, which is at all times positive,

causes the stock to grow at a decreasing rate. Recall the example above in which a decrease

in inflation reduced the rate of increase in price. In this case, price would be analogous to

the stock variable and rate of inflation would be the analogous to the inflow variable.


In figure 10, we see a case in which the inflow (valve) is at all time negative. This is

equivalent to saying that the inflow valve is "drawing down the level" (i.e., it's a drain). In

this case, the time path for the inflow valve is at all times negative and is declining linearly.

This causes the stock to decline at an increasing rate. .




The case shown in Figure 11 is similar to that of Figure 10 in that the inflow valve is always

negative, indicating that the stock is being drawn down. However, in this case the flow

valve is closing, i.e., becoming less negative. As show in the figure, a linear growth shape

for the flow, which is at all times negative, causes the stock to decrease at a decreasing rate.


Figure 12 illustrates case in which "shape" of the inflow time path is not changed by the

accumulation process . In this case, an inflow that increases exponentially causes the stock

(the accumulation ) to increase in a similar fashion . In terms of real-world stock and flow

data, the National deficit and debt data shown in Figure 3 and Figure 4 illustrate this case.




In Figure 13, we see the result of having the inflow oscillating in a sinusoidal manner

between a value of 1 and -1, i.e., the stock is being filled and drained repeatedly. As one

would expect, the stock mimics the inflow's time path character (as in Figure 12). It is

important to note, however, that maximum value of the stock is reached after the inflow's

maximum .

Stocks decouple flows

A third important characteristic of stocks is that they "decouple" or interrupt flows. A stock

thus makes it possible for an inflow to be different from an outflow and hence for

disequilibrium behavior to occur . In addition, the decoupling of flows by stocks makes it

possible for inflows to be controlled by sources of information that differ from those

controlling outflows. Figure 14 shows a number of examples in which flows are decoupled

by stocks.

Figure 14: Examples where flows are decoupled by stocks

Stocks create delays

A fourth important characteristic of stocks is that they create delays. This can be seen by

re-examining Figure 13 -- the response of a stock to a sinusoidal inflow. In this example, it



is clear that the stock reaches each of its peaks and troughs after the flow reaches each of

its corresponding peaks and troughs, during each repetition of the cycle. Henri Bergson

once remarked that "time is a device that prevents everything from happening at once" .

Bergson's point of course is that, despite human impatience, events in the world do not

occur instantaneously. Instead, there is often a significant lag between cause and effect. In

system dynamics modeling, identifying delays is an important step in the modeling process

because they often alter a system's behavior in significant ways. The longer the delay

between cause and effect, the more likely it is that a decision maker will not perceive a

connection between the two. Figure 15 presents some examples of stock-flow structures

specifying significant system delays.

Figure 15: Examples of Stock-Flow Structures Specifying Significant System Delays

Feedback

Although stocks and flows are both necessary and sufficient for generating dynamic

behavior, they are not the only building blocks of dynamical systems. More precisely, the

stocks and flows in real world systems are part of feedback loops, and the feedback loops

are often joined together by nonlinear couplings that often cause counterintuitive

behavior.

From a system dynamics point of view, a system can be classified as either "open" or

"closed." Open systems have outputs that respond to, but have no influence upon, their

inputs. Closed systems, on the other hand, have outputs that both respond to, and

influence, their inputs. Closed systems are thus aware of their own performance and

influenced by their past behavior, while open systems are not .

Of the two types of systems that exist in the world, the most prevalent and important, by

far, are closed systems. As shown in Figure 1, the feedback path for a closed system

includes, in sequence, a stock, information about the stock, and a decision rule that

controls the change in the flow . Figure 1 is a direct extension of the simple stock and flow

configuration shown previously with the exception that an information link added to close

the feedback loop. In this case, an information link "transmits" information back to the

flow variable about the state (or "level") of the stock variable. This information is used to

make decisions on how to alter the flow setting.



Figure 1: Simple System Dynamics Stock-Flow-Feedback Loop Structure.

It is important to note that the information about a system's state that is sent out by a

stock is often delayed and/or distorted before it reaches the flow (which closes the loop

and affects the stock). Figure 2, for example, shows a more sophisticated

stock-flow-feedback loop structure in which information about the stock is delayed in a

second stock, representing the decision maker's perception of the stock (i.e.,

Perceived_Stock_Level), before being passed on. The decision maker's perception is then

modified by a bias to form his or her opinion of the stock (i.e., Opinion_Of_Stock_Level).

Finally, the decision maker's opinion is compared to his or her desired level of the stock,

which, in turn, influences the flow and alters the stock .

Given the fundamental role of feedback in the control of closed systems then, an important

rule in system dynamics modeling can be stated: Every feedback loop in a system dynamics

model must contain at least one stock. .



Figure 2: More Sophisticated Stock-Flow-Feedback Loop Structure

Positive and Negative Loops

Closed systems are controlled by two types of feedback loops: positive loops and negative

loops . Positive loops portray self-reinforcing processes wherein an action creates a result

that generates more of the action, and hence more of the result. Anything that can be

described as a vicious or virtuous circle can be classified as a positive feedback process.

Generally speaking, positive feedback processes destabilize systems and cause them to "run

away" from their current position. Thus, they are responsible for the growth or decline of

systems, although they can occasionally work to stabilize them .

Negative feedback loops, on the other hand, describe goal-seeking processes that generate

actions aimed at moving a system toward, or keeping a system at, a desired state. Generally

speaking, negative feedback processes stabilize systems, although they can occasionally

destabilize them by causing them to oscillate.

Causal Loop Diagramming

In the field of system dynamics modeling, positive and negative feedback processes are

often described via a simple technique known as causal loop diagramming. Causal loop



diagrams are maps of cause and effect relationships between individual system variables

that, when linked, form closed loops.

Figure 3, for example, presents a generic causal

loop diagram. In the figure, the arrows that

link each variable indicate places where a

cause and effect relationship exists, while the

plus or minus sign at the head of each arrow

indicates the direction of causality between the

variables when all the other variables

(conceptually) remain constant. More

specifically, the variable at the tail of each

arrow in Figure 3 causes a change in the

variable at the head of each arrow, ceteris

paribus, in the same direction (in the case of a

plus sign), or in the opposite direction (in the

case of a minus sign) .

Figure 3: Generic causal loop diagram

The overall polarity of a feedback loop -- that is, whether the loop itself is positive or

negative -- in a causal loop diagram, is indicated by a symbol in its center. A large plus sign

indicates a positive loop; a large minus sign indicates a negative loop. In Figure 3 the loop

is positive and defines a self reinforcing process. This can be seen by tracing through the

effect of an imaginary external shock as it propagates around the loop. For example, if a

shock were to suddenly raise Variable A in Figure 3, Variable B would fall (i.e., move in the

opposite direction as Variable A), Variable C would fall (i.e., move in the same direction as

Variable B), Variable D would rise (i.e., move in the opposite direction as Variable C), and

Variable A would rise even further (i.e., move in the same direction as Variable D).

By contrast, Figure 4 presents a generic

causal loop diagram of a negative

feedback loop structure. If an external

shock were to make Variable A fall,

Variable B would rise (i.e., move in the

opposite direction as Variable A), Variable

C would fall (i.e., move in the opposite

direction as Variable B), Variable D would

rise (i.e., move in the opposite directionas

Variable C), and Variable A would rise

(i.e., move in the same direction as

Variable D). The rise in Variable A after

the shock propagates around the loop,

acts to stabilize the system -- i.e., move it

back towards its state prior to the shock.

The shock is thus counteracted by the

system's response.

Figure 4: Generic causal loop diagram of a

negative feedback loop structure



Occasionally, causal loop diagrams are

drawn in a manner slightly different from

those shown in Figure 3 and Figure 4.

More specifically, some system

dynamicists prefer to place the letter "S"

(for Same direction) instead of a plus sign

at the head of an arrow that defines a

positive relationship between two

variables. The letter "O" (for Opposite

direction) is used instead of a minus sign

at the head of an arrow to define a

negative relationship between two

variables. To define the overall polarity of

a loop system dynamicists often use the

letter "R" (for "Reinforcing") or an icon of

a snowball rolling down a hill to indicate

a positive loop. To indicate a negative

loop, the letter "B" (for "Balancing"), the

letter "C" (for "Counteracting"), or an icon

of a teetertotter is used . Figure 5

illustrates these different causal loop

diagramming conventions.

Figure 5: Alternative Causal Loop Diagramming

Conventions

In order to make the notion of feedback a little more salient, Figure 6 to Figure 17 present a

collection of positive and negative loops. As these loops are shown in isolation (i.e.,

disconnected from the other parts of the systems to which they belong), their individual

behaviors are not necessarily the same as the overall behaviors of the systems from which

they are taken.

Positive Feedback Examples

Population Growth/Decline: Figure 6 shows the feedback mechanism responsible for the

growth of an elephant herd via births. In this simple example we consider two system

variables: Elephant Births and Elephant Population. For a given elephant herd, we say that

if the birth rate of the herd were to increase, the Elephant Population would increase. In

this same way, we can say that if - over time - the Elephant Population of the herd were to

increase, the birth rate of the herd would

increase. Thus, the Elephant Birth rate

drives the Elephant Population that drives

Elephant Birth rate - positive feedback.

Figure 6: Positive Loop Responsible for the

Growth in an Elephant Herd via Births



National Debt: Figure 7 is a positive loop that shows the growth in the national debt due

to the compounding of interest payments. First, we note that that an increase in the

amount of interest paid per year on the national debt (itself a cost within the federal

budget ) will cause the overall national debt to increase. In this same way, an increase in

the level of national debt will increase the

amount of the interest paid each year.

Figure 7: Positive Loop Showing Growth in the

National Debt Due to Compounding Interest

Payments

Arms Race: Figure 8 shows a generic arms

race between Country A and Country B. In its

simplest form, an "arms race" can be

described as a self-sustaining competition for

military superiority. An arms race is driven

by the perception that one's adversary has

equal or greater military strength. If Country

A moves to increase its military capability,

Country B interprets this as a threat and

responds in-kind with its own increase in

military capability. Country B's action, in turn, causes Country A to feel more threatened.

Thus, Country A moves to further increase its military capability.

Figure 8: Arms Race is a Positive Feedback Process.

Bank Panic: A common scene during the Great Depression in the 1930s was that of a panic

stricken crowd standing outside their local bank waiting to withdraw what remained of

their savings. Figure 8 shows the feedback mechanism responsible for the spiraling decline

of the banking system during this period.

From the diagram, we see that the frequency of

bank failures increases public concern and the

fear of losing their money. In this case, we say

that the two system variables "move" in the same

(S) or positive (+) direction. The relationship

between the "fear of not being able to withdraw

money" and the rate at which bank withdrawals

are made is also positive.

Figure 9: Bank panic is a positive feedback

process.

The relationship between withdrawals and bank health is negative (-) or opposite (O). This

means that if the rate of bank withdrawals increases, the health of the bank decreases as



capital reserves are drawn down. The relationship between the banking industry's health

and the rate of bank failures is also negative. This means that if the health of the banking

industry increases, the number of bank failures per year will decrease.

This vicious cycle was clearly seen during the 1930s. An overall economic downturn caused

the rate of bank failures to

increase. As more banks

failed, the public's fear of not

being able to withdraw their

own money increased. This,

in turn, prompted many to

withdraw their savings from

banks, which further reduced

the banking industry's capital

reserves. This caused even

more banks to fail.

Figure 10 depicts three

interacting positive feedback

loops that are thought to be

responsible for the growth in students taking drugs in high school .

Figure 10: Feedback structure responsible for growth high school drug use

Negative Feedback Examples

Population Growth/Decline:In Figure 6, we saw how an elephant population and its

corresponding birth rate form a positive feedback loop. Now, we consider the other half of

the equation, that is, the feedback structure between Elephant Population and Elephant

Death rate. Figure 11 shows the negative feedback process responsible for the decline of an

elephant herd via deaths. If the Elephant Death rate increases, the Elephant Population will

decrease. A negative sign indicates this counteracting behavior. The causal influence of

Elephant Population to Elephant Death rate is just the opposite. An increase in the number

of elephants in the herd means that a

proportionally larger number of elephants will

die each year, i.e., an increase in the herd's death

rate. A plus sign indicates this complimentary

behavior. These two relationships combine

together to form a negative feedback loop.

Figure 11: Elephant population negative feedback

loop.

Figure 12 and Figure 13 are two simple and familiar examples of negative feedback processes.

Figure 12 shows the negative feedback process responsible for the dissipation of Itching due



to Scratching. Figure 13 considers the negative feedback involved in Eating to reduce

Hunger. An increase in one's Hunger causes a person to eat more food. Increasing in the rate

food consumption, in turn, reduces Hunger.

Figure 12: Scratching an itch and negative

feedback

Figure 13: Dissipation of hunger

Law Enforcement:Figure 14 depicts a negative feedback process that maintains a balance

between the number of drug dealers and the number of police officers in a neighborhood.

An increase in the number of drug dealers in a

neighborhood will prompt local officials to

increase the number of law enforcement

persons as a counter measure. As the number

of police officers increase, more arrests are

made and the number of drug dealers is

reduced.

Figure 14: Neighborhood drug intervention

negative feedback

Car Pools:Figure 15 shows a negative feedback

process that maintains a balance between car

pools and gasoline consumption. An increase in

gasoline consumption increases gasoline price

(supply reduction). A higher gasoline price

pushes many individual motorists to join

carpools, which reduces the total number of

vehicles on the road. This, in turn, reduces

gasoline consumption.

Figure 15: Gasoline consumption negative

feedback



Implicit and Explicit Goals

The negative feedback loops presented in Figure 11 through Figure 15 are, in a sense,

misleading because the goals they are seeking are implicit rather than explicit. For example,

the implicit goal of the loop in Figure 11 is zero elephants. That is, if the loop were to act, in

isolation, for a substantial period of time, eventually all of the elephants would die and the

population would be zero. The same sort of logic applies to Figure 12 and Figure 13, in

which the loops implicitly seek goals of zero itching and zero hunger respectively.The logic

gets even murkier in the case of Figure 14 and Figure 15. In Figure 14, there is an implicit

goal of an "acceptable" or "tolerable" level of drug dealers in the neighborhood, which may

or may not be zero. In Figure 15, there is an implicit goal of an acceptable or tolerable

gasoline price, which is certainly a lower price rather than a higher price, but is also

(realistically) not zero.

Figure 16: Generic negative feedback structure with explicit goal

An alternative and (often) more desirable way to represent negative feedback processes via

causal loop diagrams is by explicitly identifying the goal of each loop. Figure 16, for

example, shows a causal loop diagram of a generic negative feedback structure with an

explicit goal. The logic of this loop says that, any time a discrepancy develops between the

state of the system and the desired state of the system (i.e., goal), corrective action is called

forth that moves the system back into line with its desired state.

A more concrete example of a negative feedback structure with an explicit goal is shown in

Figure 17. In the figure, a distinction is drawn between the actual number of elephants in a

herd and the desired number of elephants in the herd (presumably determined by a

knowledge of the carrying capacity of the environment supporting the elephants). If the

actual number of elephants begins to exceed the desired number, corrective action -- i.e.,

hunting -- is called forth. This action reduces the size of the herd and brings it into line

with the desired number of elephants.



Figure 17: Example of negative feedback structure with an explicit goal

Examples of Interacting "Nests" of Positive and Negative Loops

In system dynamics modeling, causal loop diagrams are often used to display "nests" of

interacting positive and negative feedback loops. This is usually done when a system

dynamicist is attempting to present the basic ideas embodied in a model in a manner that

is easily understood, without having to discuss in detail.

As Figure 18 and Figure 19 show, when causal loop diagrams are used in this fashion, things

can get rather complicated. Figure 18 is a causal loop diagram of a system dynamics model

created to examine issues related to profitability in the paper and pulp industry. This figure

has a number of features that are important to mention. The first is that the authors have

numbered each of the positive and negative loops so that they can be easily referred to in a

verbal or written discussion. The second is that the authors have taken great care to choose

variable names that have a clear sense of direction and have real-life counterparts in the

actual system . The last and most important feature is that, although the figure provides a

sweeping overview of the feedback structure that underlies profitability problems in the

paper and pulp industry, it cannot be used to determine the dynamic behavior of the

model (or of the actual system). In other words, it is impossible for someone to accurately

think through, or mentally simulate, the dynamics of the paper and pulp system from

Figure 18 alone.



Figure 18: Causal Loop Diagram of a Model Examining Profitability in the Paper and Pulp

Industry

Figure 19 is a causal loop diagram of a system dynamics model created to examine forces

that may be responsible for the growth or decline of life insurance companies in the United

Kingdom. As with Figure 18, a number of this figure's features are worth mentioning. The

first is that the model's negative feedback loops are identified by "C's," which stand for

"Counteracting" loops. The second is that double slashes are used to indicate places where

there is a significant delay between causes (i.e., variables at the tails of arrows) and effects

(i.e., variables at the heads of arrows). This is a common causal loop diagramming

convention in system dynamics. Third, is that thicker lines are used to identify the

feedback loops and links that author wishes the audience to focus on. This is also a

common system dynamics diagramming convention . Last, as with Figure 18, it is clear that

a decision maker would find it impossible to think through the dynamic behavior inherent

in the model, from inspection of Figure 19 alone.



Figure 19: Causal Loop Diagram of a Model Examining the Growth or Decline of a Life

Insurance Company .

Archetypes

An area of the field of system dynamics or, more precisely, of the much broader field of

"systems thinking," that has recently received a great deal of attention is archetypes .

Archetypes are generic feedback loop structures, presented via causal loop diagrams, that

seem to describe many situations that frequently appear in public and private sector

organizations. Archetypes are thought to be useful when a decision maker notices that one

of them is at work in his or her organization. Presumably, the decision maker can then

attack the root causes of the problem from an holistic and systemic perspective . Currently,

nine archetypes have been identified and cataloged by systems thinkers, including:

Balancing Process with Delay,

Limits to Growth,

Shifting the Burden,

Eroding Goals,

Escalation,

Success to the Successful,

Tragedy of the Commons,

Fixes that Fail, and

Growth and Underinvestment .

Recent efforts, however, have suggested that the number can be reduced to four:

Growth Intended-Stagnation/Decline Achieved,

Control Intended-Unwanted Growth Achieved,



Control Intended-Compromise Achieved, and

Growth Intended At Expense to Others .

No matter what the true number archetypes is or will be, however, the central question

remains unanswered: How successful are archetypes in helping decision makers solve

problems in their organizations?

Problems with Causal Loop Diagrams

Causal loop diagrams are an important tool in the field of system dynamics modeling.

Almost all system dynamicists use them and many system dynamics software packages

support their creation and display.

Although some system dynamicists use causal loop diagrams for "brainstorming" and

model creation, they are particularly helpful when used to present important ideas from a

model that has already been created . The only potential problem with causal loop

diagrams and archetypes then, occurs when a decision maker tries to use them, in lieu of

simulation, to determine the dynamics of a system .

Causal loop diagrams are inherently weak because they do not distinguish between

information flows and conserved (noninformation) flows. As a result, they can blur direct

causal relationships between flows and stocks. Further, it is impossible, in principle, to

determine the behavior of a system solely from the polarity of its feedback loops, because

stocks and flows create dynamic behavior, not feedback. Finally, since causal loop

diagrams do not reveal a system's parameters, net rates, "hidden loops," or nonlinear

relationships, their usefulness as a tool for predicting and understanding dynamic behavior

is further weakened. The conclusion is that simulation is essential if a decision maker is to

gain a complete understanding of the dynamics of a system .

Nonlinearity

A large part of the system dynamics modeling process involves the application of common

sense to dynamic problems. A good system dynamic modeler is always on the look-out for

model behaviors that do not make sense. Such behaviors usually indicate a flaw in a model,

and the flaw is often that a crucial piece of system structure has been left out of the model.

A common error that novice system dynamicists make is to build models with stocks

whose values can either go negative, or run off to infinity. Common sense, of course,

dictates that no real system can grow infinitely large, and hence that no model of a real

system should be able to grow infinitely large. Similarly, common sense suggests that, since

many real world variables cannot take on negative values, their model-based counterparts

should not be able to take on negative values .



When a system dynamicist looks for relationships in an actual system that prevent its

stocks from going negative or growing infinitely large, he or she is usually looking for the

system's nonlinearities. Nonlinear relationships usually define a system's limits .

Nonlinear relationships play another important role in both actual systems and system

dynamics modeling. Frequently, a system's feedback loops will be joined together in

nonlinear relationships. These nonlinear "couplings" can cause the dominance of a

system's feedback loops to change endogenously. That is, over time, a system whose

behavior is being determined by a particular feedback loop, or set of loops, can (sometimes

suddenly) endogenously switch to a behavior determined by another loop or set of loops.

This particular characteristic of nonlinear feedback systems is partially responsible for their

complex, and hard-to-understand, behavior . As a result, system dynamics modeling

involves the identification, mapping-out, and simulation of a system's stocks, flows,

feedback loops, and nonlinearities.

Implications of System Structure

One of the reasons that mental models are so complex is simply that the real-world

systems that humans are trying to understand are highly complex. Over the years, system

dynamicists have identified characteristics that seem to appear again and again in

real-world systems -- particularly social systems . They have found that: (1) symptoms of a

problem are often separated from the actual problem by time and space; (2) complex

systems often behave counter to human intuition; (3) policy intervention in complex

systems can frequently yield short-term successes but long-term failure, or the reverse; (4)

internal system feedback often counters external policy intervention;(5) it is better to

structure a system to withstand uncertain external shocks than to try to predict those

external shocks; (6) real-world complex systems are not in equilibrium and are continually

changing.

These characteristics arise due to the nonlinear stock, flow, feedback structures of social

systems.

A System Problem and Its Symptoms are Separated By Time and

Space

One of the reasons that problems are often hard to solve in social systems is that they are

frequently separated from their causes by both time and space. This is a result of both

system delays (due to stocks) and system interconnectedness (due to feedback loops).

Consider once again, Figure 19, shown below. Investment performance in an insurance

company (shown in the top right of the figure) is shown to be directly influenced by

portfolio mismatch, but only after a significant delay, and indirectly influenced by many



factors that are far removed (spatially) from the firm's financial performance (e.g., sales

person skills). Managers in actual insurance companies may or may not be aware of these

delays and connections, but their existence ensures that, if a problem arises with an

insurance firm's investment performance, deciding what to do to permanently correct it

will be difficult indeed. Mentally keeping track of the various interactions and delays, over

time, is next to impossible in such a system.

Figure 19: Causal Loop Diagram of a Model Examining the Growth or Decline of a Life

Insurance Company .

The fact that system problems and system symptoms are separated by time and space

implies the need for a holistic or systems approach to problem solving. Although this is

perhaps obvious to someone reading this book, it is the exact opposite of the traditional

approach to problem solving practiced in most of science and in most organizations. The

traditional approach to problem solving is the reductionist approach, which involves

breaking down a system experiencing a problem into "manageable" pieces and then

analyzing each piece in isolation. The reductionist approach ignores the connections

between a system's pieces, because the behavior of an entire system is thought to be

merely the sum of the behaviors of its parts. This view is implicitly a "linear" view of the

world as the behavior of a linear system is, indeed, merely the sum of the behavior of its

parts. The behavior of a nonlinear system, however, is more than just the sum of its parts.

A nonlinear system can only be analyzed in its entirety, with the connections between its

parts being as important as the parts themselves. An old Sufi allegory known as the "Blind

Ones and the Matter of the Elephant" illustrates the folly of the reductionist approach to

problem solving, and the usefulness of a "nonlinear" approach to problem solving, quite

nicely.



Beyond Ghor was a city.

All its inhabitants were

blind. A king with his

entourage arrived nearby;

he brought his army and

camped in the desert. He

had a mighty elephant,

which he used in attack and to increase the people's awe.

The populace became anxious to learn about the elephant, and some sightless

from among this community ran like fools to find it. Since they did not know

even the form or shape of the elephant, they groped sightlessly, gathering

information by touching some part of it. Each thought that he knew something

because he could feel a part.

When they returned to their fellow-citizens, eager groups clustered around them,

anxious, misguidedly, to learn the truth from those who were themselves astray.

They were asked about the form, the shape, of the elephant, and they listened to

all they were told.

The man whose hand had reached the ear said, "It is a large, rough thing, wide

and broad, like a rug." One who had felt the trunk said, "I have the real facts

about it. It is like a straight and hollow pipe, awful and destructive."One who had

felt its feet and legs said, "It is mighty and firm, like a pillar."

Each had felt one part out of many. Each had perceived it wrongly.

Idries Shah. 1969. "The Blind Ones and the Matter of the Elephant," p. 25. In: Tales of the Dervishes. New York: E. P.

Dutton.

Counterintuitive Behavior

Nonlinear stock-flow-feedback systems frequently behave in ways that are counterintuitive

or different from that which a decision maker's unaided mental model would suggest --

particularly in response to the dynamics of a policy change. Forrester notes, for example,

that both the places where "high leverage points" (i.e., places at which a policy change can

permanently alter a system's behavior) are located, and the direction in which a decision

maker must "push" a system to change it's behavior, are counterintuitive .



Frequently, the counterintuitive behavior of social systems is due to the existence of

negative feedback loops. As an example, consider a small sailboat with a rudder in the

stern. If the sailor operating the boat wishes to turn the bow (front) to the starboard (right),

he or she must turn the rudder, located in the stern, to port (left). In other words, he or she

must intervene with a policy change (turn the rudder) in a place (the stern) different from

where the change will manifest itself (the bow) and in a direction (port) opposite from the

one desired for the entire system (starboard). A sailboat is a negative feedback system in

the sense that there is a desired direction the sailor wishes to point the bow of the boat and

an actual direction the bow of the boat is pointing. If a discrepancy develops between the

desired direction and the actual direction, corrective action (moving the rudder) is taken.

A second example of a simple negative feedback loop system that can exhibit

counterintuitive behavior is a thermostat. Consider the following thought experiment. If a

person is given a cigarette lighter (i.e., a small heat source) and an ice cube (i.e., a small

source of cold) and asked to apply one to the sensor on the thermostat to make a room

warmer, which one should he or she choose? The answer, of course, is the ice cube because

applying a cold source to the thermostat's sensor would cause the furnace to turn on and

heat the room.

Better Before Worse or Worse Before Better

Another characteristic of complex feedback systems is that policy changes can frequently

make them better before making them worse, or worse before making them better. Again,

this is due to the long run effects of feedback. Ignoring a system's long run feedback effects

can lead to policies that yield unintended consequences (see Figure 20 below).

Figure 20: Ignoring the Long Run Effects of Feedback Can Lead to Unintended

Consequences



Consider the example of congestion in a city's expressway system. If the policy response is

to significantly expand the capacity of the system, the short run result is a lessening of

congestion. But, as information about the improved, even pleasant-to-use, expressway

system begins to affect longer run decisions such as where people choose to live (e.g.,

people move to the suburbs and commute to work via the improved expressway system),

the congestion can return and even be worse than before. Of course, the effect can occur in

the opposite direction as, closing down some of a city's expressway to create congestion

can, in the long term, cause people to resettle near the city and/or to build and use public

transportation.

Policy Resistance

Frequently, a nonlinear feedback system will respond to a policy change in the desired

manner for a short period of time, but then return to its pre-policy-change state. This

occurs when the system's feedback structure works to defeat the policy change designed to

improve it.

Policy resistance is caused by a system's negative feedback processes. Consider the example

of the percentage of white students attending Boston schools with nonwhite students,

before and after mandatory busing was instituted as a policy change. The data for the years

1968 - 1992 is shown in Figure 21. Inspection of the figure reveals that the percentage of

white students attending Boston schools with nonwhite students shot up immediately after

the busing policy was instituted in 1974, but then gradually declined so that by 1982 it had

return to its pre-busing level.

Figure 21: Percentage of White Students Attending School with NonWhite Students in

Boston



Policy resistance occurs when a policy is applied to a system dominated by negative

feedback processes and the policy change does not alter the desired states of the negative

loops. In the case of the percentage of white students attending Boston schools with

nonwhite students, the busing policy did not change the desire of white parents to have

their children attend school primarily with other white students. In other words, it did not

change the parents' desired percentage of white students in the schools to which they sent

their children. Thus, after busing was instituted, many white families gradually moved to

the suburbs and enrolled their children in primarily white schools.

Unpredictability

Many decision makers spend enormous amounts of time and money trying to develop

models to precisely predict or forecast the future state of a system. From a system

dynamics point of view, however, this is a poor use of a decision maker's resources. There

are two reasons for this. The first is that it is impossible, in principle, to precisely predict

the future state of a nonlinear feedback system, except in the very short term. The second

is that, even if it were possible to predict the future state of a nonlinear feedback system, a

decision maker's resources are better spent trying to predict the behavior mode of a system

in response to a proposed policy change, and in trying to redesign the stock-flow-feedback

structure of a system so that it behaves well, regardless of what happens in the future.

Figure 22: A System Continuously Shocked by External Forces

Any real system, whether it is a physical system, biological system, or social system, is

continuously shocked or buffeted by external (exogenous) forces. An aircraft, for example

is shocked by bursts of wind. The human body is shocked by changes in temperature (e.g.,

stepping into a cold shower). A firm is shocked by sudden changes in the demand for its

product due to, say, changes in customer tastes and preferences. Figure 22 depicts a system

that is being continuously shocked by external forces.

When a decision maker attempts to forecast the future state of a system, he or she is

essentially trying to forecast the timing, magnitude, and direction of the incoming shocks

that are perturbing the system, so that he or she can respond to them. The implied

mind-set of the decision maker is that the shocks, rather than the stock-flow-feedback

structure of the system, are responsible for the behavior of the system. The decision



maker's search for the causes of the system's problems is outward or towards the shocks,

rather than inward or towards the stock-flow-feedback structure of the system. From a

system dynamics point of view, the decision maker's resources would be better spent trying

to redesign the stock-flow-feedback structure of the system so that it responds well to

shocks, regardless of when they arrive, how large their magnitude, or in what direction

they push the system. The system dynamics perspective is an inward or endogenous point

of view.

In order to illustrate how it is impossible, in principle, to forecast the future state of a

nonlinear feedback system, consider the following experiment . Figure 23, below, is a

simple system dynamics model depicting the interaction between elephants and hunters.

In the experiment, this model is defined to be the "real world system." Next, an exact copy

of the "real world system" is made. The "model" is perfect in the sense that its nonlinear

stock-flow-feedback structure, its parameters, its distribution of random variates, and its

initial values, are identical to those of the "real world system." The "model" is thus more

perfectly specified than any actual social system model could ever be in the true real

world.



(click on figure to run simulation)

Figure 23: Elephant-Hunter Model

The experimental simulation of the "model" and "real world system" is set up so that, in

period twenty, the "model" begins utilizing a random number stream that has been

initiated by a seed value different from the one that initiated the random number stream

that is being used by the "real world system" . In other words, before period twenty, the

"model" and "real world system" are completely identical. After period twenty, the "model"

and "real world system" are identical in every way except for the seed values that initiate

the streams of random numbers exciting their behaviors.



Figure 24: Simulated Time Series Plot of Hunters from the "Model" and "Real World

System"

Figure 24 is a time series plot of the simulated stock of hunters from both the "model" and

"real world system." Before period twenty, the two curves are clearly identical and overlay

perfectly. After period twenty, however, they begin to diverge significantly. Indeed, from

about period thirty forward, the perfectly specified "model" of the "real world system"

predicts the correct number of hunters in the system, only by chance. The conclusion is

thus that, even a perfectly specified model cannot predict the future state of a nonlinear

feedback system, except in the very short term. Point prediction is thus impossible in

principle.

Disequilibrium

As was pointed out earlier, all actual social systems exist in a state of disequilibrium. This

implies that actual social systems possess on-going pressures for change, and stocks that

decouple flows and allow inflows to differ from outflows. It also implies that using

equilibrium-based modeling techniques, such as many of those used in economics and

management, will frequently not yield insights that directly relate to the dynamic behavior

of actual social systems.



Chapter 4 Simple Structures

In the previous chapter,

Building Blocks, we

presented the basic

"concepts" and "language"

used to describe the

structure underlying

complex system behavior.

We put forth the case that

a well-known set of time

paths can describe,

individually or in

combination, the behavior

of virtually all dynamical

systems. We introduced the idea that all dynamic behavior in the world occurs due to the

interaction between stock and flow variables, and the interaction between sets of

stocks/flows within larger feedback loops.

In this chapter, we use these ideas to explore the underlying system structure responsible

for the basic families of time paths including: exponential growth and decay, goal-seeking

behavior, system oscillation, and s-shaped behavior. To do this, we develop a sequences of

small simulation models that explain the major modes of systems behavior, while

presenting some the basic mathematics behind computer simulation.

Exponential Growth, Exponential Decay, and

Goal Seeking Behavior

In this section, we will examine the dynamics of

three fundamental behavior modes: exponential

growth, exponential decay, and goal seeking, with

the intention of illustrating that these well-known

time paths can be mimicked, and hence

understood and explained, by simple system

dynamics structures. We will pull together the

building blocks presented in the previous chapter,

and for the first time, introduce the reader to some

of the fundamentals of system dynamics computer

simulation modeling. Before this can be



accomplished, however, it is necessary to draw a distinction between analytical solutions

and simulated solutions to dynamical systems of equations.

Analytical Versus Simulated Solutions

Recall that Chapter 3 described the characteristics of stocks: they have memory, (usually)

change the time shape of flows, decouple flows, and create delays. Now, let us consider

more specifically how a computer calculates the behavior of stock and flow structures. The

subject of computer simulation is critical as we move beyond the discusssion of simple

linear feedback systems and delve into situations where there are many stocks, and their

corresponding flows are complicated and continuously changing over time, due to the

(often nonlinear) interactions between them. In other words, this question is of primary

importance when the limitations of human cognition are reached and computer simulation

is necessary to reveal the dynamics of a system.

Figure 1: Example of a Stock and a Flow

Figure 1 above is a copy of the simple stock-flow structure that was first presented in

Chapter 3 (Stocks and Flows section, Figure 1). Given this structure, Figure 2 below shows

the time path of the flow which, in this instance, is linear and set at an arbitrarily

determined constant positive value. In other words, the flow valve is opened to some

positive value and does not change from this position over time.




Figure 2: The Amount that Flowed into the Stock is the Area of the Rectangle (DT = 1)

Knowing that flows cause stocks to fill or drain, looking at the time path of the flow

variable in Figure 1 we can say that the height to which the stock will fill due to its rate of

flow over the (arbitrarily determined) time period from t=0 to t=1, is the area beneath the

time path of the flow from period t=0 to t=1. This is shown by the shaded rectangle in

Figure 2. The area of the rectangle is easy to calculate, of course, as it is merely the height

of the rectangle (i.e., the value of the flow at any point from t=0 to t=1) multiplied by the

length of the rectangle (i.e., the time span from t=0 to t=1). If we define the time span from

t=0 to t=1 to be "DT" then the:

Exact Amount that Flowed into Stock from Time Period 0 to Time Period 1

= Area of Rectangle

= Value (Height) of the Flow * Change in Time

= f(1) * DT

Equation 1

Simulated Solutions

The situation gets a little more complicated, however, when the flow variable in Figure 1 is

not constant but changes over time. Figure 3 is a graph of a quadratic inflow to the stock

(i.e., one that is nonlinear with one "bend" or "curve"). Applying the same logic that was

used in the last example -- i.e., finding the area of a rectangle "beneath" the time path of

the flow -- yields an overestimate of the amount that has flowed into the stock from t=0 to

t=1. The "wedge" area above the curve in Figure 3 is the amount by which the rectangle

overestimates the amount that has flowed into the stock.



Figure 3: The Approximate Amount that Flowed into the Stock is the Area of the Rectangle (DT

= 1)

But what if the change in time, DT, is made smaller? What if DT is, say, cut in half? Figure

4 replicates the time path from Figure 3, but shows a DT of .5 -- i.e., half as large as before.

Figure 4: Approximating the Amount that Flowed into the Stock (DT = .5)

This time, the approximate amount that has flowed into the stock from t=0 to t=1 is the

sum of the areas of two rectangles. That is the:

Approximate Amount that Flowed into Stock from Time Period 0 to Time Period 1

= Area of Rectangle 1 + Area of Rectangle 2

= [f(.5) * DT ] + [f(1) * DT]

= [f(.5) + f(1)] * DT

= t=0t=1

f(t) * DT

Equation 2

Further, the amount by which the sum of the area of the rectangles overestimates the

amount that has flowed into the stock is now the sum of the two wedges above the time

path of the flows.

There are two significant implications of reducing DT to .5 -- i.e., of cutting it in half. The

first is that it doubles the number of areas (rectangles) that have to be calculated (from one

to two). The second is that it reduces the error in the estimate of the amount that flowed

into the stock from t=0 to t=1. The second implication can be seen by comparing Figures 4

and Figure 3. The sum of the areas of the two wedges above the time path of the flow in

Figure 4 is less than the area of the single wedge above the time path of the flow in Figure

3.




In fact, the error in the estimate of the amount that flowed into the stock from t=0 to t=1

can be reduced even more by systematically reducing the size of DT. Figure 5 shows what

happens when DT is cut in half again (to a value of .25) and Figure 6 shows what happens

when DT is given a value of .125. In each case, the overestimate of the amount that flowed

into the stock is reduced and the number of areas (rectangles) that must be calculated is

doubled. For example, in the case of Figure 5 the:

Approximate Amount that Flowed into Stock from Time Period 0 to Time Period 1

= Area of Rectangle 1 + Area of Rectangle 2 + Area of Rectangle 3 + Area of Rectangle 4

= [f(.25) * DT ] + [f(.5) * DT ] + [f(.75) * DT ]+ [f(1) * DT]

= [f(.25) + f(.5) + f(.75) + f(1)] * DT

= t=0t=1

f(t) * DT

Equation 3




The trade-off between accuracy and number of calculations should be clear at this point.

The more accurate the estimate of the amount that has flowed into a stock, the more areas

(rectangles) that have to be calculated.

Most modern-day system dynamics software packages allow a system dynamicist to create

a model by placing icons (stocks, flows, feedback links, etc.) on an electronic piece of paper

displayed on a computer screen, and connecting them together with the computer’s mouse.

While this is occurring, the software automatically generates equations that correspond

directly to each of the icons. The stock icon in Figure 1, for example, corresponds directly

to a mathematical equation that tells a computer how to calculate the amount that has

flowed into it during a particular period of time. In general terms, this equation is:

Stockt = Stock(t-DT) + DT * Flow(t-DT) -> t

Equation 4

In english, Equation 4 says that: "The amount that has flowed into the stock up to some

time, t, is approximately equal to the sum of the areas of the rectangles prior to the time

that the most recent time step (DT) was taken (Stock(t-DT)), plus the area of the rectangle

defined by the height of the Flow during the most recent time step (Flow(t-DT) -> t) . This

step-by-step process of determining the areas of rectangles and adding them up -- i.e., of

determining the amount that has flowed into a system’s stocks over time or the system’s

time path -- is called numerical simulation.

In order to more accurately locate the place of system dynamics in the more general world

of dynamical modeling, a simulated solution to a system’s equations of motion must be

contrasted with an analytical solution to a system’s equations of motion.



A simulated solution to a dynamical set of equations has some drawbacks. It cannot reveal

the exact amount that has flowed into a system’s stocks at any time period, but only the

approximate amount that has flowed into a system’s stocks at short time intervals (DTs)

between successive computations. Further, to determine the approximate amount that has

flowed into a system’s stocks at a future period of time, a simulated solution requires the

calculation of the amounts that flowed into the system’s stocks in all earlier periods of time.

Lastly, a simulated solution is a particular solution (i.e., a particular time path) that cannot

reveal anything about the general nature of the system’s dynamics.

Analytical Solutions

In principle, a dynamical system’s time step, DT, could be cut in half forever. If this were to

occur, the error in approximating the amount that flowed into the system’s stocks would

be systematically reduced until it became infinitesimally small. In more traditional

mathematical notion this means that the:

Exact Amount that Flowed into Stock from Time Period 0 to Time Period 1

= lim DT -> 0 [ t=0t=1

f(t) * DT ]

= t=0t=1

f(t) * dt

Equation 5

Equation 5 represents (in general terms) the continuous time version of the dynamical

system presented in Figure 1. In this version of the model, DT is infinitesimally small and

the solution to the equation reveals exactly how much has flowed into the system’s stock

from t=0 to t=1 . The solution to Equation 5, if it exists, is an exact analytical solution .

In contrast to simulated solutions, analytical solutions are general solutions to dynamic

systems of equations. Rather than revealing the approximate amounts that have flowed

into a system’s stocks at short time intervals (DTs) between successive computations, they

reveal the exact amounts that have flowed into a system’s stocks at any period of time. In

other words, for any system possessing an analytical solution, any time period can be

substituted into the solution and it will yield the exact amounts that have flowed into the

system’s stocks. The solution does not require the calculation of the amounts that flowed

into the system’s stocks in all earlier periods of time. In addition, the form of a system’s

analytical solution reveals much about the general nature of its dynamics, even without

any numerical computation.

Given all of the apparent advantages offered by exact analytical solutions, why do modelers

bother to use simulated solutions at all? The answer is quite simple: only linear systems

have exact analytical solutions. This is because: (1) they can be broken down into parts, (2)

the behaviors of the parts can be determined, and (3) the behavior of the entire system can



be found by adding together the behaviors of the parts. In other words, the behavior of a

linear dynamical system is merely the sum of the behaviors of its parts.

Nonlinear systems, on the other hand, cannot be broken down into pieces and solved

analytically. They must be analyzed as a whole. The connections between the parts of a

nonlinear system are as vital to its behavior as the pieces themselves. Indeed, the behavior

of a nonlinear system is more than merely the sum of the behaviors of its parts. Since

nonlinear dynamical systems do not have exact analytical solutions, their behavior can

only be determined through simulation.

In sum, if the real world is nonlinear (i.e., has limits), if humans are incapable of thinking

through all of the dynamic implications of their mental models, and if nonlinear dynamical

models have no analytical solutions, simulation modeling is necessary for improved

dynamical thinking and learning.

Exponential Growth

Exponential growth is one of the most common time paths exhibited by systems in the

world. It has two characteristics that are very important to note. The first characteristic is

its power. Exponential growth can occur very rapidly. The second characteristic is its

insidiousness. Exponential growth can "sneak up" on a person!

Consider the case of repeatedly folding a standard piece of paper in half, doubling its

thickness with each fold . After five or six folds, the paper is not particularly thick. But how

thick would it be after 36 more folds? As shown in Figures 7a and 7b, after 42 folds a

standard piece of paper would be approximately 280,000 miles thick -- more than the

distance from the Earth to the moon! Further, the shape of the plot shows that the big

explosion in thickness occurs at the end of the plot. For most of the simulation run, the

system’s growth is hardly distinguishable from the horizontal axis . The last few doublings,

however, really make the thickness grow .

Figure 7a: Folding a Piece of Paper

Figure 7b: Exponential Growth of the Thickness of a

Piece of Paper



In terms of a system dynamics model that can produce exponential growth, consider

Figure 8 -- a "first cut" stock-flow structure of an elephant herd system. The figure presents

the model’s stock and flow icons, the corresponding system dynamics equations, and the

dimensions of each equation. As will become apparent throughout the remainder of this

book, the specification of the dimensions of every equation in a system dynamics model is

an important part of the modeling process. It helps the modeler both document a model

and conceptualize its structure. Stocks are dimensioned in terms of "units" ("Elephants")

and flows are dimensioned in terms of "units/time" ("Elephants/Year"). Other variables in a

system dynamics model must possess dimensions that correspond algebraically to its

stock-flow-feedback loop structure . Examples of system dynamics models that have been

properly dimensioned will be presented ahead.

Figure 8: "First Cut" Stock-Flow Structure of an Elephant Herd Model

Figure 9 presents a simulation of the "first cut" model, when the initial number of

elephants in the herd is set at ten and a constant number (two) of baby elephants are born

each year. Clearly the model exhibits a linear, rather than an exponential, growth.

(Click on image to run simulation)

Figure 9: Simulation of the First Cut of the Elephant Herd Model (Initial Elephants = 10)

Before moving directly to a structure that exhibits the exponential time shape, it is perhaps

better to "think like a modeler" for the moment. A good system dynamicist will test each of



the structures he or she creates for "robustness." This means that he or she will test to see

if their model behaves "properly" under a variety of circumstances, both normal and

extreme. If a model behaves properly in response to robustness testing, the system

dynamicist can be confident that it will behave properly when interfaced with other model

structures.

Figure 10: Simulation of the First Cut of the Elephant Herd Model (Initial Elephants = 0)

Figure 10 shows the results of a robustness test done on the "first cut" elephant herd model.

For the purpose of the test, the initial number of elephants in the herd (stock variable) was

changed to zero. Clearly the simulation makes no sense. The number of elephants in the

herd grows even though there are initially no elephants to produce baby elephants. The

conclusion is that the model does not behave properly in response to this extreme test and

is therefore "nonrobust."

The next cut of the model should be aimed at correcting the robustness problem. The first

cut of the model generated an implausible result because elephants are needed to produce

more elephants. This biological fact implies that a feedback link from the stock of

elephants to the flow of elephant births is required so that the computer can calculate the

number of baby elephants produced by the elephant herd each year.

Figure 11 presents the "second cut" of the elephant model. It contains a feedback link from

the stock of elephants to the birth rate flow, an additional icon (the birth rate coefficient

(BirthRateCoef)), and an additional information link.



Figure 11: "Second Cut" of the Elephant Herd Model (Initial Elephants = 0)

Figure 12 shows the results of a simulation of the model when the initial number of

elephants is set at zero. Clearly, the new formulation of the model has cured the robustness

problem revealed in Figure 10. When there are initially no elephants in the herd, no baby

elephants are born and the size of the herd remains at zero.

(click on image to run simulation)

Figure 12: Results of a simulation using "Second Cut" of the Elephant Herd Model (Initial

Elephants = 0)

Figure 13, on the other hand, is a simulation of the model when the initial number of

elephants is ten. In this case, the positive feedback loop created between elephants and

baby elephants causes the exponential growth path of the elephant herd.




Figure 13: Simulation of the "Second Cut" of the Elephant Herd Model (Initial Elephants = 10)

Generic Structure

Over the years, system dynamicists have identified combinations of stocks, flows and

feedback loops that seem to explain the dynamic behavior of many systems. These

frequently occuring stock-flow-feedback loop combinations are often referred to as

"generic structures" .

Figure 14 shows a generic structure responsible for exponential growth and its

corresponding causal loop diagram. It is the generic version of Figure 11 -- a generic first

order positive feedback loop system . Most system dynamicists think about this structure

when modeling a system that grows exponentially; the contents of the stock produces

more of the contents of the stock.

Figure 14: A Generic Structure Responsible for Exponential Growth

Figures 15 and 16 illustrate two other specific instances of this generic structure. Figure 15

shows positive exponential feedback in an interest compounding savings account. Figure 16

shows how knowledge accumulates in a particular scientific field of study.



Figure 15: Structure for a simple interest compounding savings account

Figure 16: Structure for the growth of knowledge in a particular flield of study

Exact Analytical Solution for a First Order, Linear, Positive Feedback

Loop System

The first order positive feedback loop system of Figure 14 is a linear system. As a result, in

addition to its simulated solution, it possesses an exact analytical solution. This solution is:

Stockt = Stock0 * e Coef * t

Equation 1



For the more mathematically inclined reader, the derivation of this solution is shown in

the box below. Although the case has already been made that simulated solutions to

nonlinear systems are preferable to analytical solutions to linear systems, Equation 1 can be

used to generate some insights that are useful in system dynamics modeling.

Derivation of the Analytical Solution

Therefore, the exact analytical solution is:

Stockt = Stock0 * eCoef * t

For example, at time = 50, with a Coef of .02 and an initial Stock of 100:

Stock50 = 100 * e 0.02 * 50 = 271.8281828

Using the Exact Analytical Solution to Test Numerical Integration

Methods

Equation 1 can be used to determine the accuracy of various simulated solutions to the

generic first order positive feedback loop system shown in Figure 14. Generally speaking,



simulated solutions to system dynamics models differ by step size (DT) and/or numerical

integration method employed. As was previously discussed, reducing DT makes a

simulated solution more accurate, but forces the computer to do more work. In a similar

way, numerical integration methods vary by accuracy and required number of calculations.

Most system dynamics software packages go through the following sequence of events to

simulate a model :

Initialization Phase

1. Create a list of all equations in required order of evaluation.

2. Calculate initial values for all stocks, flows and auxiliaries (in

order of evaluation).

Iteration Phase

3. Estimate the change in stocks over the interval DT.

4. Calculate new values for stocks based on this estimate.

5. Use new values of stocks to calculate new values for flows and

auxiliaries.

6. Add DT to simulation time.

7. Stop iterating when Time >= StopTime

Step 1 of the Iteration Phase is the place where most numerical integration methods differ.

More specifically, different numerical integration methods estimate the change in a

model's stocks over the interval DT in different ways.

Although many numerical integration methods exist, the two that are most frequently

used in system dynamics modeling are Euler's method and a second order Runge-Kutta

method. Euler's method is the simplest and least accurate (for a given DT) numerical

integration method available . Indeed, due to its conceptual simplicity, it is the method

upon which the nomenclature of the generic system dynamics stock equation is based.

In Step 1 of the Iteration Phase of the simulation of a system dynamics model, Euler's

method estimates the change in a model's stocks over the interval DT as:

Stock = DT * Flow at Beginning of DT

Thus, in Step 2 of the Iteration Phase, the new values for a model's stocks are calculated in

the following way:

Stockt = Stock(t-DT) + Stock



Stockt = Stock(t-DT) + DT * (Flow(t-DT) => t)

Of course, this is precisely the same formulation as Equation 4 of Analytical-vs-Simulated

Solutions.

A second-order Runge-Kutta method differs from Euler's method in that it uses two flow

calculations within a given DT to create an estimate for the change in a model's stocks over

the interval DT. More precisely, during Step 1 of the Iteration Phase of a simulation, a

second order Runge-Kutta method estimates the change in a model's stocks over the

interval DT as:

F1 = DT * Flow at Beginning of DT [same as in Euler's method]

F2 = DT * Flow at End of DT = DT * Flow(Stock(t) + F1)

Stock = 1/2 * (F1 + F2)

Stockt = Stock(t-DT) + Stock

Stockt = Stock(t-DT) + 1/2 * (F1 + F2)

In other words, a second-order Runge-Kutta method estimates the change in a model's

stocks by taking the mean of the flows that occur at the beginning and at the end of each

time step.

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Figure 17: Comparison of Exact Analytical Solution to Simulated Solution (Euler's Method; DT

= .5)

Figure 17 compares the results of simulating the generic first order positive feedback loop

system shown in Figure 14 via Euler's method with a DT of .5, against the exact analytical

solution to the system given by Equation 1. In each case, the initial value of the stock is set

equal to 100 units and the value of the growth coefficient is set at .02 units/unit/time.

Athough each solution starts at the same place, the simulated solution quickly diverges

from its exact counterpart. Indeed by period 10, the exact amount in the stock is 122.14

units, while the simulated solution indicates that it contains only 122.02 units.

In a similar way, Figure 18 compares the exact analytical solution to the generic first order

positive feedback loop system against a simulated solution using Euler's method and a DT

of .25. Clearly, cutting DT in half improves the accuracy of the simulated solution, as the

simulated value of the stock at period 10 is now 122.08.

Figure 18: Comparison of Exact Analytical Solution to Simulated Solution (Euler's Method; DT

= .25)

Finally, Figure 19 compares Equation 1 to a simulated solution utilizing a second order

Runge Kutta method and a DT of .25. Inspection of this figure reveals that the simulated

solution is now equal to the exact solution, to two places to the right of the decimal point,

throughout the entire simulation.



Figure 19: Comparison of Exact Analytical Solution to Simulated Solution (Second Order Runge

Kutta Method; DT = .25)

Measures of Exponential Growth

The exact analytical solution to a first order, linear, positive feedback system (Equation 1)

can be used to derive two important measures of exponential growth. The first of these

measures is the system's time constant. The second is sytem's doubling time.

Time Constant

Figure 20 depicts a system dynamics representation of a first order, linear, positive

feedback loop system with an exact analytical solution represented by Equation 1. The time

constant of this system, Tc, measured in units of time, is defined to be 1/Coef.

Figure 20: First Order, Linear, Positive Feedback Loop System



Given the definition of the system time constant, the following question can be asked: How

much exponential growth occurs during one time constant's worth of time? To answer this

question, the system's time constant must be inserted into equation 1. Thus, the amount of

exponential growth that occurs during one time constant of time is:

Stockt = Stock0 * eCoef * t

Stockt = Stock0 * eCoef * Tc

Stockt = Stock0 * eCoef * (1/Coef)

Stockt = Stock0 * e1

Stockt = Stock0 * 2.71828

In other words, the amount of exponential growth that occurs during one time constant is

2.71828 times the initial amount in the system's stock. Pushing this logic to the next step,

the amount of exponential growth that occurs during two time constants is 7.3890 (i.e.,

2.71828 * 2.71828) times the initial amount in the system's stock.

Figure 21: Exponential Growth After One and Two Time Constants have Elapsed

Figure 21 shows the amount of exponential growth that has taken place in a first

order, linear, positive feedback loop system, with a Coef of .1 and an initial

value of 100, after one and two time constants have elapsed. After one constant,

the stock has grown from 100 to 271.83 and after two time constants it has

grown to a value of 738.90.

Doubling Time



The second measure of how quickly exponential growth is proceeding in a first

order, linear, positive feedback system is its doubling time. A system's doubling

time is simply the amount of time it takes for the contents of its stock to

double.

The doubling time of a first order, linear, positive feeback loop system, Td, is

equal to seventy percent of its time constant. That is: Td = .7 * Tc . For the more

mathematically inclined reader, the derivation of this result is provided in the

shaded box below.

Derivation of the Doubling Time for a First Order, Linear, Positive Feedback Loop System

The doubling time for a first order, linear, positive feeback loop system is

calculated using Equation 1. If the doubling time, Td, is defined to be the time it

takes to double the initial value of a stock, then:

2*Stock0 = Stock0 * eCoef * Td

Since the system's time constant, Tc, is 1/Coef, then the system's Coef = 1/Tc.

Thus:

2 * Stock0 = Stock0 * e 1/Tc * Td

Dividing both sides by Stock0 yields:

2 = eTd

/ Tc

Taking the natural logarithm of both sides yields:

ln(2) = ln(eTd

/Tc)

Or:

.69 = Td / Tc

Or:

.7 = Td / Tc

Or:

Td = .7 * Tc

Thus, the doubling time of a first order, linear, positive feeback loop system is



seventy percent of its time constant.

Figure 22 shows the same simulation run presented in Figure 21. This time the

figure shows that the system's stock doubles from a value of 100 to a value of

200 in (approximately) the seventh period (.7 multiplied by the time constant

of 10 periods).

Figure 22: Doubling Time for a First Order, Linear, Positive Feedback Loop System

The concept of doubling time leads directly to a heuristic known as the Rule of

70. This rule says that, if you know, or can estimate, how quickly an

exponentially growing system is growing (i.e., if you know, or can estimate,

Coef in Equation 1), you can calculate its doubling time as Td = (70 / Coef) * 100.

Consider, for example, the growth of the United States economy. As shown in

Figure 23, U.S. nominal GDP grew exponentially at a rate of 7 percent per year,

from 1959 to 1995 (i.e., Coef is estimated to be .07055). At this rate of growth,

the Rule of 70 says that the nominal size of U.S. economy will double in (70

/ .07055)*100 = 9.86 years.



Figure 23: U.S. Nominal GDP (1959-1995)

Goal Seeking Behavior

Chapter 3 discussed the two types of feedback loops that exist in the world:

positive loops and negative loops. Positive loops generate exponential growth

(or rapid decline) and negative loops produce goal-seeking behavior. This

section introduces and analyzes a first order negative feedback loop system.

Figure 1: Generic First Order Negative Feedback Loop System



Figure 1 presents a generic, first order, linear, negative feedback loop system.

The model is initialized in equilibrium by setting the initial value of the stock

equal to its goal, and then "shocked" out of equilibrium (at time=1) by a step

function that changes the system's goal from ten to fifteen. This procedure --

ie., starting a system dynamics model in equilibrium and then shocking it out --

is widely used in system dynamics because it allows the modeler to see the

"pure" behavior of the system in response to the shock. Moreover, it illustrates

another important idea in system dynamics modeling: A system containing

negative feedback loops will be in equilibrium only when all of its stocks are equal

to all of its goals simultaneously.

The implication of this idea is very important. In the real world it is rare, if not

impossible, to find actual systems (particularly actual social systems) that exist

in states of equilibrium because actual systems contain many negative feedback

loops that do not reach their goals simultaneously. Thus, system dynamicists

often start their models in states of equilibrium, not because they feel that this

reflects reality, but rather because it is a useful reference point for analyzing the

"pure" behaviors of their models .

Figure 2: Simulation of the Generic First Order Negative

Feedback Loop System in response to a step increase at time=1

The behavior exhibited by the negative feedback loop system as shown in

Figure 2 is goal seeking. When the goal is increased from ten to fifteen, a

discrepancy between the stock and its goal is created. In response to this

discrepancy, the system initiates corrective action by increasing its flow and

smoothly raising its stock to a value of fifteen . The time it takes the system to

reach its new goal is determined by the size of TimeCoef. The bigger TimeCoef

is, the longer it takes for the system to adjust to its new goal. This is illustrated

in Figure 3, where the negative feedback loop system of Figure 1 is simulated

with three different values of TimeCoef (1, 2, and 3) in response to a reduction

in the goal (opposite the case shown in Figures 1 and 2) from ten down to five

with a step function at time=1.




Figure 3: Three Simulations of the Generic First Order Negative

Feedback Loop System give a step reduction at time=1

Exact Analytical Solution

To more precisely determine how long it takes a linear first order negative

feedback loop system to reach its goal, the exact analytical solution to the model

presented in Figure 1 must be studied. This solution is:

Stockt = Goal + (Stock0 - Goal) * e-(1 / TimeCoef)*t

Equation 1

To study Equation 1, it is helpful to look at the special case when the system is

seeking a goal of zero. Equation 2 presents this special case. It found by simply

substituting goal values of zero into Equation 1.

Stockt = Stock0 * e-(1 / TimeCoef)*t

Equation 2

One very important thing to note about Equation 2 is that it is very similar to

Equation 1 of Exponential Growth, the exact analytical solution to a linear

positive feedback loop system. Indeed, Equations 1 of Exponential Growth and

Equation 2 differ only by the signs of their exponents. Equations 1 of Exponential

Growth has an exponent of Coef * t and Equation 2 has an exponent of

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-(1/TimeCoef) * t. Since Coef and 1/TimeCoef are both numbers, the exponents,

and hence the equations, differ only by sign.

The reason that Equation 2 has a minus sign in front of its exponent is that it

represents a system seeking a goal of zero. Thus, under the assumption that the

system's stock contains positive values, the system will steadily lose units via its

outflow until there is nothing left . This process is known as exponential decay.

For the more mathematically inclined reader, the derivation of Equation 1 is

presented in the shaded box below.

Derivation of

the

Analytical

Solution to

the First

Order,

Linear,

Negative

Feedback

Loop System

Exponential Decay

Figure 1 below is a system dynamics representation of a linear first order

negative feedback loop system with an implicit goal of zero. The figure includes

the model's equations, a causal loop diagram, and a simulation of the model's

structure. The simulation is an example of exponential decay behavior that

corresponds to the exact analytical solution to the system presented in

Equation 2 of Goal Seeking Behavior.


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Figure 1: First Order, Linear, Negative Feedback Loop System with an Implicit Goal of Zero

Measures of Exponential Decay: Half Life

One measure of how long it will take for a first order, linear, negative feedback

loop system to reach its goal of zero is its half life. The half life of such a system

is the time it takes to drain half of the contents from the system's stock.

Analogous to the case of the doubling time for a first order positive feedback

loop system above, the system's time constant must be determined before its

half life can be found.

As was previously shown, the analytical solutions to Equations 1 of Exponential

Growth and Equation 2 of Goal Seeking Behavior differ only by sign. It should

not be surprising therefore that the equation for the outflow in Figure 1:

Flow = Stock / TimeCoef

or:

Flow = Stock * (1/TimeCoef)

is very similar to the equation for the inflow in the Figure 20 of Exponential

Growth:

Flow = Stock * Coef

Given that the time constant for the positive feedback loop system of Figure 20

of Exponential Growth is Tc = 1/Coef, and applying the same logic to the

negative feedback loop system presented in Figure 1, yields a time constant of




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Tc = 1/(1/TimeCoef) = TimeCoef. This result can be used to answer a question

similar to that asked about the positive feedback loop system: How much

progress toward its goal will the negative feedback loop system have made after

one time constant has elasped? The answer can be found by substituting the

time constant into the Equation 2 of Goal Seeking Behavior:

Stockt = Stock0 * e-(1 / TimeCoef) *Tc

Stockt = Stock0 * e-(1/TimeCoef)*TimeCoef

Stockt = Stock0 * e-1

Stockt = Stock0 * (1/e)

Stockt = Stock0 * .37

This result indicates that the system loses 63% of the contents of its stock after

one time constant has elapsed. Extending this logic reveals that the system

loses 86% of the contents of its stock after two time constants have elasped

(Stockt = Stock0 * .37 * .37 = Stock0 * .14) and 95% of the contents of its stock

after three time constants have elapsed (Stockt = Stock0 * .37 * .37 * .37 = Stock0

* .05). Figure 2 repeats the simulation from Figure 1 and confirms this result

graphically.

As in the case of a first order positive feedback loop system, the negative loop

system's half life is 70% of its time constant. That is: Th = .7 * Tc. Once again,

the derivation of ths result is presented in the shaded box below for the more

methematically inclined reader. A graphical presentation of the result is also

shown in Figure 2.





Figure 2: Behavior of a First Order, Linear, Negative Feedback Loop System after Three Time

Constants have Elapsed

The analysis of the first order negative feedback loop system with an implicit

goal of zero can be extended to the more general negative feedack loop model

originally presented in Figure 2 of Goal Seeking Behavior, giving the following

insight: A first order, negative, feedback loop system reaches approximately 95%

of its goal after three time constants have elapsed.

Derivation of the Half Life for a First Order, Linear, Negative Feedback

Loop System

The half life of a first order, linear, negative feeback loop system is calculated

using Equation 2 of Goal Seeking Behavior. If the half life, Th, is defined to be

the time it takes to cut the initial value of a stock in half, then:

1/2 * Stock0 = Stock0 * e-1/Coef * Th

Since the system's time constant, Tc, is 1/Coef, then the system's Coef = 1/Tc.

Thus:

1/2 * Stock0 = Stock0 * e-Tc * T

h

Dividing both sides by Stock0 yields:

1/2 = e - Th/T

c

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Taking the natural logarithm of both sides yields:

ln(1/2) = ln(e -Th/ T

c)

Or:

ln(1) -ln(2) = ln(e-Th

/Tc)

Or:

0 - .69 = - Th / Tc

Or:

.7 = Th / Tc

Or:

Th = .7 * Tc

Thus, the half life of a first order, linear, negative feeback loop system is

seventy percent of its time constant.

System Oscillation

System oscillation is the characteristic symptom of negative feedback structures

in which the information used to take goal-seeking action is delayed (See

Figure 1). In such cases, a control action is not based on the current state of a

system but on some previous state or value. As you can imagine, using dated

information to control the approach to a target is likely to cause the system to

miss or overshoot its goal. Specific structural characteristics of a system, such as

delay duration and delay order, will determine whether the resulting oscillation

will diminish over time, reach some sustained amplitude and frequency, or

experience an increasing amplitude.



Figure 1: System oscillation is created when decision information in negative feedback

structures is delayed.

Figure 2 shows a simple system in which the information about the current

"level" of the system variable first passes through a single stock/flow structure

before being used to take a controlling action (information path is shown in

red). Thus, the decision to take action by changing the variable, Action, is based

not on the current state of the system but on some previous value.

Figure 2: Simple negative feedback structure with control information delay.

Given this structure, Figure 3 shows what happens if the goal,

DesiredSystemLevel, is increased by 5 units. Because of the delay, the system

overshoots its target at about time=6, reaches a maximum of 22 units at time=10,

then undershoots at about time=12, and so on. Eventually, the system does find

its target 50 time units after the goal changed!



Figure 3: A step increase in goal value causes a damped oscillation response in system.

Elephant-Hunter Model Example

Figure 4 presents a more complex example of a system that exhibits oscillatory

behavior. In this Elephant-Hunter Model, the conservation and health of the

elephant herd is directly dependent on the ability of the park manager to keep

the herd population below the carrying capacity of the land. Too many

elephants will result in long-term damage to the park's feed production

capacity and ultimately reduce the number of elephants that can be sustained.




Figure 4: Elephant-Hunter system structure.

One of the primary herd control "leverage points" in this system is to change

the rate at which elephants are "harvested" by increasing or decreasing the

number of hunters. Keep in mind that the elephant population level is a

function of the elephant Net Birth Rate and the Harvest Rate. Thus, by

increasing the hiring rate of hunters, the total number of hunters will increase.

An increase in the number of hunters will increase the elephant harvest rate.

An increase in harvest rate will "draw down" the level of elephants in the herd.

Consider a scenario in which the net birth rate of the elephant herd increases

from 8 elephants per year to 12 elephants per year. Due to inherent delays in

herd data gathering and analysis methods, this increase is not immediately

detected. After some delay, the birth rate trends are revealed and the manager

decides to hire more hunters. However, because of the delays in hiring, more

time passes before the hunters are actually onboard. As the hunter population

increases, the harvest rate increases, which causes the herd population growth

to slow, stop, and eventually turn downward. Since the system delays are still in

place, however, the herd level continues to drop and eventually undershoots

the target, prompting the manager to reduce the number of hunters. This "trial

and error "oscillating process continues until the equilibrium target is reached,

almost 20 time units from the initial step increase in birth rate! Figure 5 and 6

shows the results of this scenario.



Figure 4: Results of a step increase in elephant birth rate.

Types of Oscillations

Information delays are prevalent in the real world so it's no surprise that

oscillations are the most common dynamic behaviors. There are four basic

types:

Sustained

Oscillation

Characterized by periodicity of one

Constant amplitude

No stable fixed equilibrium is reached

Damped

Oscillation

Amplitude of oscillation dissipates over time

Stable equilibrium is reached

Exploded

Oscillation

Not common in the real world



Amplitude increases over time

Growth either leads to a sustained pattern or

deterioration of the system

Chaos

Oscillation

Random time path that never repeats

Irregular amplitude and infinite period

Chaos oscillations may be found in many natural systems

such as weather patterns

S-Shaped Growth

S-shaped growth is the characteristic behavior of a system in which a positive

and negative feedback structure fight for dominance but result in long-run

equilibrium. As illustrated in Figure 1, the positive feedback exponential growth

loop (shown in blue) initially dominates the system, causing the system

variable to increase at an increasing rate. However, as the system approaches its

limit or "carrying capacity", the goal-seeking negative feedback loop (shown in

red) becomes the dominant loop. In this new mode, the system approaches

stasis asymptotically. This "shifting dominance" event from one loop to another

is characteristic of nonlinear behavior and is found in a variety of physical,

social, and economic systems.

Figure 1: S-Shaped structure and characteristic time path.

For this discussion, we revisit the elephant herd scenario to illustrate this

behavior. Figures 2 and 3 present our elephant population model in causal

diagram and simulation model form, respectively. Looking at Figure 2, we see

that the elephant herd system consists of one positive growth feedback loop

and two negative feedback loops.



Figure 2: Causal diagram of the elephant population model.


Figure 3: Simulation model of elephant herd growth model.

The positive Birth Rate loop, shown in blue, is responsible for the exponential

growth experienced during the early stages of herd growth. As the population

of the herd increases, the birth rate - given some Fecundity (fertility) constant -

increases. Increasing the birth rate further promotes the growth of the herd. In

balance, the negative Death Rate loop, shown in red, is responsible for drawing

down the herd population. On its own, the death rate of the elephant herd

implicitly reduces the population to zero. As elephants die, the population

decreases until an equilibrium population of zero is reached wherein no more

elephants die. The negative Carrying Capacity loop, shown in green, considers

the effect of an increasing population density on the average life span - and

thereby the Death rate - of the elephant herd. Here, the explicit goal of the

system is determined by the carrying capacity.

Given this structure, Figures 4 and 5 show the time series results from

simulating this model from time=0 to time=100. In Figure 4 ,we see the

exponential behavior of the birth rate from time=0 to about time=40. At



time=40, the influence of an increasing population density on the average life

span begins to show with an "upturn" in the rate of Deaths within the herd.

After this point, the gap between Birth and Death rate gradually closes as the

population density increases. At about time=58, the Birth and Death rate

become equal and the system achieves equilibrium.

Figure 4: Time series plot of elephant herd Birth rate and Death rates.

To illustrate this behavior another way, Figure 5 shows a normalized time series

plot of the Elephant herd population and the Net Growth rate for the herd. Net

Growth is defined as the number of elephant births minus the number of

elephant deaths. As we expect, initially the Net Growth increases exponentially

due to the dominance of the Birth rate loop. However, we see a very specific

shift at time=40 when the Net Growth peaks and begins to fall. As the rate of

death increases, the Net Growth decreases. The continued increase in Death

rate eventually brings Net Growth to zero, causing the system to reach steady

state.

Figure 5: Normalized time series for Elephant population and Net Growth

indicates that shift occurs at maximum growth point.

Overshoot and Collapse



So far, we have considered a system in which the carrying capacity is constant.

Now, let us examine the case in which a system overshoots its limit, causing the

carrying capacity itself to change. In such cases, three outcomes are possible

depending upon the specific structure of the system. The system may:

o oscillate about its carrying capacity;

o overshoot and collapse in which case the carrying capacity is destroyed

and the system crashes; or

o reverse direction and approach its carrying capacity in an s-shaped

declining fashion.

In order to demonstrate overshoot and collapse behavior in our elephant model,

we have to add some structure. As shown in Figure 6, we assume that the

system's carrying capacity is actually a stock of resource, such as acres of grass

and shrubbery, with some characteristic yearly average output. If the herd

population is below the land's ability to replenish itself, then the carrying

capacity is constant. However, if the consumption rate of the elephant herd

exceeds its steady state carrying capacity, then land's ability to regenerate is

eroded by some amount. The Erosion Rate in this case is defined as the number

of elephants in the herd multiplied by the yearly erosion rate per elephant.


Figure 6: Elephant population model with variable carrying capacity.

Figure 7 shows the results of a simulation run from time=0 to time=100 with an

initial herd population of 50 elephants. In this case, the herd exceeds the initial

Carrying Capacity at about time=20. After some delay, this excess number of

elephants begins to erode the Carrying Capacity. Since the Carrying Capacity

serves as the target in this negative feedback system, the Elephant population



responds in kind by reversing its exponential growth and follows the Carrying

Capacity downward. Note that even when the population drops to 200 animals,

the initial Carrying Capacity, it remains greater than the now decreasing

Carrying Capacity, thus the process continues until the system crashes and the

Carrying Capacity of the land becomes zero.

Figure 7: Results from Elephant population overshoot scenario.



Chapter 5 The Modeling Process

There is more to modeling

than just being able to

build a model on a

computer. Ultimately the

goal of building a system

dynamics model is to

improve the way that

people understand systems

and improve their decision

making. In this chapter, we

share some thoughts and

ideas about the process of

creating system dynamics

models and how these

models can be used to test options through computer simulation

"experiments."

The Modeling Process

The goal of any modeling effort is to understand and explain the behavior of a

complex system over time. As we discussed in the previous chapters, knowing a

system's patterns of behavior speaks volumes about its structure. For a decision

maker, understanding the structure of a system is a critical first step in

designing and implementing effective policy.

There are two ways in which a system dynamics model can aid in

understanding complex problems. A completed model can be used to test

alternative policy options (i.e. conduct experiments). However, it should be

noted that there is great benefit to participating in the model building process

itself. Practitioners have found that long-term learning tends to come from the

model building process, so it is important to involve the decision maker from

the beginning.

With this in mind, we can think of the modeling process as having the

following steps:

1. Identify Problem

2. Develop Hypothesis

3. Test Hypothesis



4. Test Policy Alternatives

Identify "SD" Problem

One of the early lessons learned in

building system dynamics models is the

importance of modeling a problem

rather than an entire system. Focusing

on a particular problem provides a

boundary to the modeling process and

forces the modeler to consider only

system variables that relate specifically

to the problem in question. Although

your understanding of the real problem

might change as the process unfolds, it

is still critical to begin with a focused

problem statement.

Develop Hypothesis (mapping

mental models)

With a clearly defined problem statement in hand, the first model building step

is to develop a theory of why the system is behaving the way that it is. Tools

such as causal loop diagrams and stock and flow networks can be used to map

out a set of assumptions about what is causing the "reference modes" of the

system to behave a particular way. Although the actual process of developing

this theory (causal diagram) varies from person to person, we summarize the

basic steps (See box below) in the process as a general guide.

2. List system variables that are directly relevant to the problem

statement. For example, if the problem statement is related to a

manufacturing firm's loss in profits, one might list variables such

as: unit cost, sales, number of employees, product quality,

inventory, order backlog, competitor price, etc.

3. Link the variables listed in step 1 by using the casual

diagramming convention discussed in Chapter 3. For each

connection, note whether the relationship between the variable

pair is positive or negative. As the diagramming process unfolds,

feel free to add or drop variables as needed. However,

throughout the process, hold the problem statement in clear



view.

4. As the diagram evolves, study the feedback "loop" structures that

form. Identify and label each feedback loops as a reinforcing or

balancing loop.

This is an important stage to collect information about the problem through

brainstorming with groups, data, literature, and of course, personal experience.

Test Hypothesis (challenging mental models)

Once you have developed a pen and paper

theory about the system it is time to

develop a computer model of your theory

to see if it will re-create the behavior over

time seen in the reference modes. The

mathematical representation requires a

deal of precision around the relationships

between different elements in the system.

Being forced to shape a consistent

mathematical model of a system often

challenges and evolves our own mental

model assumptions of how we thought it

worked. Before each simulation run of the

model, think through how you expect the

model to behave. If it

behaves differently, it either

means something is wrong in

the model or you have just

discovered an opportunity to

challenge your mental

model!

Test Policies (improving

mental models)

With a basic model that

seems to be able to explain

why the problem is behaving

the way that it is, you can look for policy interventions that lead to more



desirable long term system behaviors. A good way to start this process is to first

identify the key decisions (critical decisions/policies that the organization

makes), indicators (what you need to see from the system to assess the

decision), and uncertainties (most fragile assumptions about the relationships

or outside world) associated with the problem. Then try out different possible

sets of decisions under different assumptions about the uncertainties. Look for

tradeoffs between short term behavior and long term behavior.

Management Flight Simulators

Through the process of developing a system dynamics model, the modeler has

created a complex map of interrelationships and a better understanding of how

the system will behave over time under different policy scenarios. One of the

great challenges of developing models is in trying to communicate the insights

gained to people who did not develop the model. Presenting a report to a

manager of all the things he or she should do differently from what someone

else has learned in a model works only if the conclusions of the model were

what they were planning to do in the first place.

But a model can provide more than just bullet point conclusions, it can also be

used to create a simulated "practice field" where these managers and other

people can try managing the system by making decisions themselves. We call

these simulated worlds management flight simulators. Through these

simulators they can learn about the model of the system and improve their own

assumptions about the short and long term effects of different decisions.

In a management flight simulator, users of the model are placed in the position

of managing the system. Each year (or month, or quarter) they must make

decisions to try to meet some goal in the system. To help with this, a snazzy

interface is typically run "on top of" the system dynamics model that makes the

information more accessible. Generally, there are four kinds of information

presented:

o Decisions - There are typically four to six key management decisions

the user must make, using the knobs, dials, and/or input boxes

presented on the screen.

o Reports - They provide the critical "snap-shot in time" to inform the

users about the state of the system.

o Graphs - During a model simulation, time series graphs allow the user

to study system trends. As discussed earlier, studying the shape of a

system's time paths is an important part of understanding the structure

that lies beneath.

o Initial Conditions - Because of the nature of system dynamics models,

there are usually one or more key uncertainties that significantly impact



the behavior of the simulation run. It is important to allow the user to

set the value of these uncertainties as part of the experimentation

process. An important part of model design and use is making hidden

assumptions and parameters visible!

Management Flight Simulation Example

The Product Development Flight Simulator was developed by Daniel Kim and

Don Seville at the Organization Learning Center at MIT to teach users the

dynamics and highly complex nature of the product development cycle. The user

had three primary objectives or requirements:

5. The product must be finished on schedule.

6. The product must be developed using a limited pre-defined budget.

7. The product must be of sufficient quality to compete in a competitive

market.

For this simulator, the product development project was separated into two

categories of tasks: product engineering tasks (100 total tasks to complete), and

process engineering tasks (100 total tasks to complete). Each category

represented a stock of work (tasks) that would be drawn down at a particular

work rate. The product engineers (PE) were responsible for product design and

testing. Process engineers (PcE) were responsible for designing and building

the manufacturing process to turn the design into a marketable product.

Figure 1 shows the basic layout of the product development flight simulator.

During the simulation, each stock of work was gradually drawn down as the

engineers worked. Regardless of whether all the engineering design and process

work was completed, the product was manufactured and released into the

market on the scheduled day. If the all or a majority of the work was completed,

the quality of the product was high. However, if a significant amount of tasks

remained, then the quality of the product was low on the day of release.

Product sales depended heavily on whether the product was released by the

target release date and was of a competitive quality.



Figure 1: Basic Layout of the Product Development Flight Simulator

As the "pilot" of this simulator, each week of the simulation the user was

expected to decide how many engineers to hire or layoff, the number of work

hours in a week, the scheduled completion date for each team, the amount of

time spent coordinating between the two teams, and the quality "target" for the

product. The choice for each of the system variables influences the rate at

which the engineers accomplish their work, the resulting quality level, and the

rate of engineer attrition due to stress and fatigue.

In this case, the flight simulator served as a laboratory in which a wide variety

of different management strategies could be tested. More importantly, the

consequences of various strategies could be explored systematically without

risking "trial and error" learning on the real firm!

Just for fun

In the box below, we provide you an opportunity to try out a sample simulator

yourself. After reading the brief introduction to our Rocket Simulator, feel free

to run the simulator by clicking on the animated figure.



Rocket Simulator is a

simple system

dynamics model that

simulates the motion

of a rocket being

launched from a

planet's surface. We

used a system

dynamics stock and

flow structure to describe a body in motion such as a rocket or an

automobile in terms of its mass, acceleration, velocity, distance traveled,

and the forces acting on it. The figure on the right shows the basic structure

of the rocket simulation model. As the pilot, you control two of the

simulation model's parameters: the throttle and direction. The THROTTLE

controls the rocket's thrust and fuel burn rate. The ROCKET ANGLE

controls what direction the rocket is pointing. This simulation uses simple

equations of motion to teach concepts of accumulation and system delay as

well as demonstrate how even simple systems can behave in a complex and

non-intuitive way.

(click on image below to run simulation)



Chapter 6 Natural Gas Discovery Model

As we discussed in the

System Dynamics and

Energy Modeling section,

one of the system dynamics

analyses conducted in

support of the world

modeling efforts was a

natural gas discovery and

production model created

by MIT Master's student

Roger Naill. Naill based his model on the life cycle theory of oil and gas

discovery and production put forth by petroleum geologist M. King Hubbert.

Like Hubbert, Naill assumed for his analysis that the total amount of oil and gas

in the United States (i.e., the amount of oil and gas "in place"), and hence the

"ultimately recoverable" amount of oil and gas in the United States, is finite.

In this chapter, we recreate Naill's original Natural Gas Discovery Model as a

case study in how to construct and experiment with system dynamics models.

To make the explanation of the model more manageable, we present a series of

model versions, each version incrementally more complex with the addition of

new structure. In each case, the reader is encouraged to run the model to see

how the additional structure affects its behavior. In the final section of this

chapter, we present a "flight simulator" version of the completed model. The

reader is encouraged to conduct policy experiments by changing decision

variables or model uncertainty variables to see how the behavior of the system

changes in response.

Natural Gas Discovery Model: Reference Modes

It is often argued that system dynamics modeling is mis-named because system

dynamicists model problems, not systems. The reason that problems, rather

than systems, are focused on is because they direct the system dynamicist away

from the "kitchen sink approach" to modeling, which usually yields a model

containing so much detail that it is no more easy to understand than the actual

system itself.

The principle way that system dynamicists force themselves to focus on

problems is through the use of reference modes. Reference modes are time

series graphs of important system variables that exhibit patterns of behavior



(e.g., exponential growth, exponential decay, oscillation, s-shaped growth,

overshoot and collapse ) that are either problematic or not well understood.

They also define the time horizon of a model, which involves the specification

of both its time units (years, quarters, months, weeks, days, etc.) and time span

(the model’s start time and stop time). The task of the system dynamicist is to

identify, and include in the model, only those variables that help to explain (i.e.,

help the model to mimic) the reference modes.

The reference modes originally used by Roger Naill in the construction of his

natural gas model were drawn from the oil and gas life cycle theory of M. King

Hubbert. Hubbert based his theory on the knowledge of the physical structure

of the oil and gas system and hence on the assumption that there is a finite

amount of oil and gas in the earth. According to Hubbert’s theory, the

discovery and production flows of natural gas, as well as the stock of proven

reserves of natural gas rise, peak and fall over time, and the stock of unproven

resources falls monotonically due to depletion.

Although a theory can be used to provide reference modes for a system

dynamics model, actual numerical time series data, if available, can also be

used. Figure 1 presents some actual data from the U.S. natural gas system. It

includes the actual discovery rate, usage (production) rate, proven reserves, and

reserve-production ratio. Clearly, these data do not span the entire life cycle of

the natural gas system (roughly the years 1900 to 2050) a la Hubbert, so they

are of limited use in specifying reference modes for the natural gas depletion

problem. On the other hand, they can be used to complement the reference

modes resulting from Hubbert’s theory and to calibrate the model.


http://www.albany.edu/cpr/sds/DL-IntroSysDyn/exdc.htm

http://www.albany.edu/cpr/sds/DL-IntroSysDyn/oscl.htm

http://www.albany.edu/cpr/sds/DL-IntroSysDyn/sshp.htm

http://www.albany.edu/cpr/sds/DL-IntroSysDyn/sshp.htm#overshoot

http://www.albany.edu/cpr/sds/DL-IntroSysDyn/energy.htm#hubbert




Figure 1: Actual Data from the U.S. Natural Gas Discovery and Production System

Step-By-Step Re-Creation of the Natural Gas

Discovery and Production Model

A good way to gain an understanding of a system dynamics

model created by another person, is through a step-by-step

replication of its structure. In this section, Roger Naill's

model of U.S. natural gas discovery and production will be

recreated through a series of increasingly sophisticated

system dynamics models. To augment the discussion, you

can access and run each model version directly. Simulating

the models will allow you to see the effect of adding

complexity to the base model structure.

Natural Gas Discovery and Production Model: First Cut

Figure 1 is the stock-flow diagram for the first step in the recreation of Naill’s

model of U.S. natural gas discovery and production. The model consists of two

stocks: UnprovenReserves and ProvenReserves, and two flows: DiscoveryRate

and UsageRate. The UsageRate is roughly equivalent to the rate of production

of U.S. natural gas.



One feature of the model that is particularly noteworthy is that there is no

inflow to the stock of UnprovenReserves. This is because natural gas is created

in geologic time, yet used on a human time scale of a hundred years or so. Thus,

for all practical purposes, the amount of natural gas in the United States can be

treated as fixed. Naill made this assumption because he based his model on the

resource life cycle theory of M. King Hubbert. A fixed stock of gas or oil is at the

core of Hubbert’s theory.

Figure 1: First Cut of Naill’s Model of U.S. Natural Gas Discovery and Production

Note: Click on rotating icon to access first cut of

the Natural Gas Discovery Model.

Figure 2 lists the system dynamics equations that directly correspond to the

icons shown in Figure 1. Three features of these equations are important to note.

The first is that the variables on the left hand side of each equal sign have their

measurement units enclosed in brackets {}. In system dynamics modeling

stocks are measured in "units" and flows are measured in "units/time." In

Figure 2, each of the stocks is measured in "Cubic Feet" of gas and each of the

flows is measured in "Cubic Feet (of gas)/Year."

In system dynamics modeling, each of the equations in a model must be

"dimensionally correct." This means that the arithmetic of the units of

measurement must be correct. For example, if a flow {units/time} is equal to a

stock {units} divided by a constant, the constant must be measured in {time}.

Inspection of the equations in Figure 2 reveals that they are all dimensionally

correct.

The second feature of the equations in Figure 2 is that the stocks each have

initial values. More specifically, the initial amount of gas in the stock of

UnprovenReserves is 6.4*e12 cubic feet and the initial amount of gas in the stock

of ProvenReserves is 1.28434*e15 cubic feet. The initial values of stocks in system

dynamics models can be determined in a variety of ways. The values are often

known or knowable and the analyst needs only to investigate available data

sources to find out. In the case of Naill’s model presented here, the initial

values were estimated by using the system dynamics software to calculate the




"best fit" to actual U.S. natural gas discovery and production data. In the case of

UnprovenReserves, a value estimated by Hubbert’s method could also have

been used.

The third feature of the equations in Figure 2 is that both of the flows have first

order control. This means that a negative or balancing feedback loop has been

created between each stock and its outflow to prevent the stocks from ever

taking on negative values (because "negative UnprovenReserves" and "negative

ProvenReserves" would not make sense). An examination of the equations

reveals that, when either UnprovenReserves or ProvenReserves reach a value of

zero, their outflows are shut off (i.e., take on values of zero).

[Stock] ProvenReserves = dt*(DiscoveryRate - UsageRate

{Cubic Feet}

[Initial] ProvenReserves = 6.4e12

{Cubic Feet}

[Stock] UnprovenReserves = dt*(-DiscoveryRate)

{Cubic Feet}

[Initial] UnprovenReserves = 1.28434e15

{Cubic Feet}

[Flow] DiscoveryRate = UnprovenReserves*DiscoveryCoef

{Cubic Feet/Year}

[Flow] UsageRate = ProvenReserves*UsageCoef

{Cubic Feet/Year}

[Constant] DiscoveryCoef = .0075

{1/Years}

[Constant] UsageCoef = .02

{1/Years}

Figure 2: Equations from the First Cut of Naill's Model of U.S. Natural Gas Discovery and

Production

Figure 3 shows time series plots from a simulation of the model presented in

Figures 1 and 2. Of note is that, even at this very early stage in the model’s

development, it produces dynamic behavior consistent with the real system.

UnprovenReserves fall monotonically due to depletion and ProvenReserves rise

during the early part of the century, peak during the mid-1970s, and then

http://www.albany.edu/cpr/sds/DL-IntroSysDyn/feed.htm#loops



decline thereafter. The DiscoveryRate falls monotonically and the UsageRate

(essentially the production rate) rises from 1900 to the mid-1970s, peaks, and

then declines thereafter.

Figure 3: Time Series Plots from a Simulation of Naill’s Model of U.S. Natural Gas Discovery and

Production

Natural Gas Discovery and Production Model: Second Cut

Figures 4(a) and 4(b) illustrate the stock-flow and causal diagrams for the

second step in the recreation of Naill’s model of U.S. natural gas discovery and

production. The model is merely an extension of the model presented in Figure

1.

Figure 4(a): Second Cut of Naill’s Model of U.S. Natural Gas Discovery and Production



Figure 4(b): Causal Diagram the the Second Cut of Naill's Model

of U.S. Natural Gas Discovery and Production.

Note: Click on rotating icon to access second

cut of the Natural Gas Discovery Model.

A comparison of Figure 4(a) to Figure 1 reveals that the right-hand side of the

model (ProvenReserves, UsageRate, UsageCoef) has not changed. The left-hand

side of the model, on the other hand, has had a significant piece of system

dynamics structure added to it. More precisely, the simple discovery rate

equation has been replaced by a balancing feedback loop (negative feedback)

that works its way through the fraction of unproven reserves remaining and the

cost of exploration.

Refering to Figure 4(b), the logic of the balancing loop is as follows: a fall in

unproven reserves (UnprovenReserves), ceteris paribus, causes a fall (i.e., a

move in the Same direction) in the fraction of unproven reserves remaining

(FracRemaining). A fall in the FracRemaining, ceteris paribus, causes a rise (i.e.,

a move in the Opposite direction) in the effect of the fraction remaining on the

cost of exploration (EffFracRemainCost). A rise in the EffFracRemainCost,

ceteris paribus, causes a rise in the cost of exploration (CostOfExploration). A

rise in the CostOfExploration, ceteris paribus, causes a fall in the indicated

discovery rate (IndicDiscovRate) which, in turn, ceteris paribus, causes a fall in

the discovery rate (DiscoveryRate). A fall in the DiscoveryRate closes the loop

and ensures, ceteris paribus, that unproven reserves will be higher than they

otherwise would have been.

Three other features of the model presented in Figure 4a are important to note.

The first is that the balancing loop preserves first order control in the model.

That is, although the feedback from the stock of unproven reserves to its

outflow (DiscoveryRate) is not direct as it was in the first cut of the model, the




equations in the loop still causes the computer to shut-off the discovery rate

when the stock of unproven reserves is zero.

The second feature of the model presented in Figure 4a that is important to

note is the delay symbol that appears in the discovery rate icon. The purpose of

embedding a delay in the discovery rate is to account for the lag that exists

between the time a decision is made to invest in gas exploration and the time

actual gas discoveries occur. This delay could also have been modeled explicitly

with a simple stock and flow stucture.

The last feature of the model that is important to note is its table functions.

Represented by the zig-zag-like symbols, table functions allow the modeler to

externally impose relationships(linear and nonlinear) between system variables.

Figure 5 lists the system dynamics equations that directly correspond to the

icons shown in Figure 4. Three characteristics of these equations are

particularly noteworthy. The first is that, in addition to the stocks(rectangles),

flows (pipe and faucet assemblies) and constants (diamonds) presented in the

first cut of the model, there are now auxiliary equations (circles) in the

model.

The second noteworthy characteristic is that the model is again dimensionally

correct. In particular, the measurement units of each variable in the balancing

loop correspond to the arithmetic embodied in each equation in the loop. To

provide just one example, the indicated discovery rate (Cubic Feet/Year) is

equal to investment in exploration (Dollars/Year) divided by the cost of

exploration (Dollars/Cubic Foot).

The third noteworthy characteristic is that two of the auxiliaries contain table

functions (EffFracRemainCost and InvestInExplor) in nongraphical form. The

graphical versions of these functions are shown in Figure 6.

[Stock] ProvenReserves = dt*(DiscoveryRate - UsageRate)

{Cubic Feet}

[Initial] ProvenReserves = 6.4e12

{Cubic Feet}

[Stock] UnprovenReserves = dt*(-DiscoveryRate)

{Cubic Feet}

[Initial] UnprovenReserves = 1.28434e15

{Cubic Feet}




[Flow] DiscoveryRate = DELAYINF(IndicDiscovRate,4.5,3)

{Cubic Feet/Year}

[Flow] UsageRate = ProvenReserves*UsageCoef

{Cubic Feet/Year}

[Aux] CostOfExploration = NrmCostExploration*EffFracRemainCost

{Dollars/Cubic Foot}

[Aux] EffFracRemainCost =

GRAPH(LN(10*FracRemaining),-3.5,0.5,[13000,6000,2700,

1000,545,245,110,50, 22,9.98,4.48,2.02,0.91"Min:0;Max:13000;Zoom"])

{Dimensionless}

[Aux] FracRemaining = UnprovenReserves/INIT(UnprovenReserves)

{Dimensionless}

[Aux] IndicDiscovRate = InvestInExplor/CostOfExploration

{Cubic Feet/Year}

[Aux] InvestInExplor =

GRAPH(TIME,1900,15,[100000000,220000000,350000000,470000000,

600000000,720000000,820000000,900000000,950000000,990000000,

1000000000"Min:0;Max:1000000000;Zoom"])

{Dollars/Year}

[Constant] NrmCostExploration = 1.34775e-5

{Dollars/Cubic Foot}

[Constant] UsageCoef = .02

{1/Years}

Figure 5: Equations from the Second Cut of Naill’s Model of U.S. Natural Gas Discovery and

Production

An examination of Figure 6 reveals that the table functions both represent

nonlinear relationships in the model and help to define the system’s limits. The

effect of the fraction of gas remaining on the exploration cost function

(EffFracRemainCost) indicates that as gas depletion occurs, the cost of gas

exploration increases monotonically. The investment in exploration function

(InvestInExplor) indicates that, over time, the number of dollars per year

allocated to gas exploration increases and then saturates. In future cuts of the

model, this relationship will be made endogenous.



Figure 6: Table Functions in the Second Cut of Naill’s Model of U.S. Natural Gas Discovery and

Production


Figures 4 and 5. As with the previous version of the model, this cut produces

dynamic behavior consistent with the real system. That is, the UsageRate again

rises from 1900 to the mid-1970s, peaks, and then declines. UnprovenReserves

again fall monotonically due to depletion and ProvenReserves again rise during

the early part of the century, peak during the mid-1970s, and then decline.

The only behavior in this version of the model that is new involves the

DiscoveryRate. A comparison of Figures 3 and 7 reveals that the DiscoveryRate

now rises and falls whereas, in the previous version of the model, it simply fell

monotonically. The reason for this difference is that, in this cut of the model,

the InvestInExplor table function exogenously increases investment in

exploration over time. Early in the system’s evolution, the increasing amount of

investment dollars causes a rise in natural gas discovery. Later on, however, the

effects of depletion overwhelm the investment dollars and the discovery rate

falls. An interesting simulation experiment that could be done is to

systematically alter the InvestInExplor function (its shape, its saturation level,

etc.) and resimulate the model to ascertain the results of the changes.



Figure 7: Time Series Plots from a Simulation of the Second Cut of Naill’s Model of U.S. Natural

Gas Discovery and Production

Natural Gas Discovery and Production Model: Third Cut

Figure 8(a)and 8(b) correspond to the stock-flow and causal diagram for the

third step in the re-creation of Naill’s model of U.S. natural gas discovery and

production. The model is a direct extension of the model presented in Figures 4

and 5. Four significant changes were made to the second cut of Naill's model in

order to arrive at this version of the natural gas system:

a balancing feedback loop was added;

a positive or reinforcing feedback loop was added;

an exogenous exponentially growing demand for natural gas was

added; and

an exogenously growing price of natural gas was added.



Figure 8(a): Third Cut of Naill’s Model of U.S. Natural Gas Discovery and Production



Figure 8(b): Causal Diagram of additional balancing and reinforcing loops.

Note: Click on rotating icon to access third cut

of the Natural Gas Discovery Model.

The logic of the balancing loop, shown in red, is as follows. A fall in

UnprovenReserves, ceteris paribus, causes a fall in FracRemaining. A fall in the

FracRemaining, ceteris paribus, causes a rise in the EffFracRemainCost. A rise

in the EffFracRemainCost, ceteris paribus, causes a rise in the

CostOfExploration. A rise in the CostOfExploration, ceteris paribus, causes a

rise in the total cost of natural gas (TotalCost). A rise in the TotalCost, ceteris

paribus, causes a fall in the ratio of gas price to gas cost (PriceCostRatio). A fall

in the PriceCostRatio, ceteris paribus, causes a fall in the effect of return on

investment in exploration (EffROIExplor). A fall in EffROIExplor, ceteris

paribus, causes a fall in InvestInExplor. A fall in InvestInExplor, ceteris paribus,

causes a fall in the IndicDiscovRate which, in turn, ceteris paribus, causes a fall

in the DiscoveryRate. A fall in the DiscoveryRate closes the loop and ensures,

ceteris paribus, that unproven reserves will be higher than they otherwise

would have been.

The logic of the reinforcing loop,shown in blue, is as follows. A rise in proven

reserves (ProvenReserves), ceteris paribus, causes a rise in the natural gas usage

rate (UsageRate). A rise in the UsageRate, ceteris paribus, causes a rise in



revenue from the sales of natural gas (SalesRevenue). A rise in SalesRevenue,

ceteris paribus, causes a rise in InvestInExplor. A rise in InvestInExplor, ceteris

paribus, causes a rise in the IndicDiscovRate. A rise in the IndicDiscovRate,

ceteris paribus, causes a rise in the DiscoveryRate which, in turn, ceteris

paribus, closes the loop and causes a rise in ProvenReserves.

An examination of Figure 9 reveals that the equations are dimensionally correct

and that the exogenous rate of exponential growth in gas demand

(GrowthConstant) is approximately 4.5% per year. Further, the average

wellhead price of gas (AveWellHeadPrice) is determined exogenously via a

table function loaded with year-by-year data. Although these data are synthetic,

actual price data could have been used to determine its effect on the natural gas

system.

[Stock] ProvenReserves = dt*(DiscoveryRate - UsageRate) {Cubic Feet}

[Init] ProvenReserves = 6.4e12 {Cubic Feet}

[Stock] UnprovenReserves = dt*(-DiscoveryRate) {Cubic Feet}

[Init] UnprovenReserves = 1.28434e15 {Cubic Feet}

[Flow] DiscoveryRate = DELAYINF(IndicDiscovRate,4.5,3) {Cubic Feet/Year}

[Flow] UsageRate = UnRegUsageRate {Cubic Feet/Year}

[Aux] AveWellHeadPrice = GRAPH(TIME,1900,1,[0.00008,0.0000825, 0.000162,0.000335,0.000431,

0.000458, 0.000461,0.000455,0.000446,0.000436, 0.000425,0.000412,0.000399,0.000383,0.000366,

0.000346,0.000323,0.000297,0.00027, 0.000245,0.000226,0.000214,0.000208,0.000207,

0.000209,0.000211,0.000213,0.000213, 0.000213,0.000212,0.000209,0.000205,0.000201, 0.000196,

0.000191,0.000186,0.000182, 0.000179,0.000178,0.000179,0.000181,0.000185,

0.00019,0.000196,0.000202,0.000207, 0.000212,0.000216, 0.000219,0.000221,0.000223,

0.000224,0.000225,0.000226,0.000226, 0.000228,0.00023, 0.000233,0.000237,0.000243, 0.00025,

0.000259,0.00027,0.000281, 0.000294,0.000307,0.000321,0.000334,0.000348, 0.000362,0.000375,

0.000387,0.000399, 0.00041,0.000422, 0.000434,0.000447,0.000463, 0.00048, 0.000499,

0.000521,0.000545, 0.000569,0.000593, 0.000619,0.000646,0.000677,

0.000714,0.00076,0.000815,0.000881, 0.000952,0.001027, 0.001103,0.001177,0.001246, 0.00131,

0.001369,0.001423,0.001476, 0.001529,0.001585, 0.001649,0.001723,0.001815,

0.001931,0.002074,0.002241,0.00242, 0.002594,0.002757, 0.002909,0.003051, 0.003186,

0.003319,0.003455, 0.003597,0.003751, 0.003918,0.004101,0.0043,0.004519,

0.004761,0.00503,0.005327,0.005646, 0.005974,0.006294, 0.006593,0.006867,

0.007122,0.007365,0.007602,0.007842, 0.008093,0.008367,0.008673,0.009018,

0.009406,0.009828,0.010258,0.010669,0.011039, 0.011362,0.011652, 0.011916,0.012156, 0.012375,

0.012582,0.012791,0.013028"Min:0;Max:0.015;Zoom"]) {Dollars/Cubic Foot}



[Aux] CostOfExploration = NrmCostExploration*EffFracRemainCost {Dollars/Cubic Foot}

[Aux] EffFracRemainCost = GRAPH(LN(10*FracRemaining),-3.5,0.5,[13000,6000,

2700,1000,545,245,110,50,22, 9.98,4.48,2.02,0.91"Min:0;Max:13000;Zoom"]) {Dimensionless}

[Aux] EffPriceDemand = GRAPH(LN(AveWellHeadPrice*1e5),1,0.5,

[2.1,1.59,1.21,0.9,0.69,0.5,0.24,0.14,0.067,0.031,0.014,0.0055,

0.0015,0.0002,0.000017,0"Min:0;Max:2.1;Zoom"]) {Dimensionless}

[Aux] EffROIExplor = GRAPH(PriceCostRatio,0,0.2, [0,0.08,0.25,0.44,0.55,0.67,0.76,0.82,0.88,0.92,

0.96,1"Min:0;Max:1;Zoom"]) {Dimensionless}

[Aux] FracRemaining = UnprovenReserves/INIT(UnprovenReserves) {Dimensionless}

[Aux] IndicDiscovRate = InvestInExplor/CostOfExploration {Cubic Feet/Year}

[Aux] InvestInExplor = SalesRevenue*PctInvestExplor*EffROIExplor {Dollars/Year}

[Aux] PotentialUsageRate = InitUsageRate*EXP(GrowthConstant*(TIME-StartYear)) {Cubic

Feet/Year}

[Aux] PriceCostRatio = AveWellHeadPrice/TotalCost {Dimensionless}

[Aux] SalesRevenue = UsageRate*AveWellHeadPrice {Dollars/Year}

[Aux] TotalCost = CostOfExploration*CostMargin {Dollars/Cubic Foot}

[Aux] UnRegUsageRate = PotentialUsageRate*EffPriceDemand {Cubic Feet/Year}

[Constant] CostMargin = .967118 {Dimensionless}

[Constant] GrowthConstant = .0463497 {1/Years}

[Constant] InitUsageRate = 1.06331e12 {Cubic Feet/Year}

[Constant] NrmCostExploration = 1.34775e-5 {Dollars/Cubic Foot}

[Constant] PctInvestExplor = .2 {Dimensionless}

[Constant] UsageCoef = .02 {1/Years}

Figure 9: Equations from the Third Cut of Naill’s Model of U.S. Natural Gas Discovery and

Production


Figures 8 and 9. As with previous versions of the model, this cut produces



dynamic behavior that is roughly consistent with the real system. The only

behavior in this version of the model that is new is that of ProvenReserves. This

time they rise and plateau, instead of rising, peaking, and falling. This behavior

is clearly not correct and must be fixed in later versions of the model. It occurs

in this version because, during the last third of the simulation, the exogenously

rising price of natural gas reduces the UsageRate to the (very low and falling)

DiscoveryRate. This can be seen by inspecting the top-right plot of Figure 10.

Figure 10: Time Series Plots from a Simulation of the Third Cut of Naill’s Model of U.S. Natural


Natural Gas Discovery and Production Model: Fourth Cut

Figures 11(a) and 11(b) illustrate the stock-flow and causal diagrams for the

fourth step in the re-creation of Naill’s model of U.S. natural gas discovery and

production. Figure 12 lists the corresponding equations for Figure 11(a) can be

accessed by clicking directly on the diagram. The model has been extended

through the addition of a balancing loop and a reinforcing feedback loop.



(click on diagram to view equations)

Figure 11(a): Fourth Cut of Naill’s Model of U.S. Natural Gas Discovery and Production.



Figure 11(b): Causal Diagram of the Fourth Cut of Naill's Model

of U.S. Natural Gas Discovery and Production.

Note: Click on rotating icon to access fourth cut


The logic of the balancing loop (shown in red) is as follows. A rise in

ProvenReserves, ceteris paribus, causes a rise in the reserve-production ratio

(ResProdRatio), which is the number of years proven reserves will last, given

the usage rate. A rise in the ResProdRatio, ceteris paribus, causes a rise in the

relative reserve-production ratio (RelResProdRatio), which is a ratio of the

reserve-production ratio to its desired value. A rise in the RelResProdRatio,

ceteris paribus, causes a fall in the percent of sales revenue devoted to

investment in gas exploration (PctInvestExplor). A fall in the PctInvestExplor,

ceteris paribus, causes a fall in InvestInExplor. A fall in InvestInExplor, ceteris

paribus, causes a fall in the IndicDiscovRate. A fall in the IndicDiscovRate,

ceteris paribus, causes a fall in the DiscoveryRate which, after a significant

delay, closes the loop and ensures, ceteris paribus, that proven reserves will be

lower than they otherwise would have been.



The logic of the reinforcing loop (shown partially in blue) is as follows. A rise

in ProvenReserves, ceteris paribus, causes a rise in the natural gas usage rate

(UsageRate). A rise in the UsageRate, ceteris paribus, causes a rise in the

ChgAveUsageRate, which is a flow that changes the average usage rate

(AveUsageRate). A rise in the ChgAveUsageRate, ceteris paribus, causes a rise

in the AveUsageRate. A rise in the AveUsageRate, ceteris paribus, causes a fall

in the ResProdRatio. A fall in the ResProdRatio, ceteris paribus, causes a fall in

the RelResProdRatio. A fall in the RelResProdRatio, ceteris paribus, causes a

rise in the percent of sales revenue devoted to investment in gas exploration

(PctInvestExplor). A rise in the PctInvestExplor, ceteris paribus, causes a rise in

InvestInExplor. A rise in InvestInExplor, ceteris paribus, causes a rise in the

IndicDiscovRate. After a significant delay, a rise in the IndicDiscovRate, ceteris

paribus, causes a rise in the DiscoveryRate, which closes the loop and ensures,

ceteris paribus, that proven reserves will be higher than they otherwise would

have been.


Figures 11. A comparison with Figure 10 reveals that the model now exhibits two

new behaviors. The first is an oscillation that is most apparent in the plots of

proven reserves, the discovery rate, and the usage rate. The second is that

proven reserves now rise, peak and fall, instead of merely rising and peaking as

in the last version of the model.

The model’s oscillatory behavior can be readily explained. A rule of thumb in

system dynamics modeling is that oscillation is caused by a balancing feedback

loop with delayed corrective action. The balancing loop (negative feedback) that

was added to this version of the model provides precisely this type of structure.

That is, the model adjusts its investment in exploration by watching its

reserve-production ratio. When proven reserves drop and the

reserve-production ratio falls below its desired level, more revenue is invested

in exploration. When proven reserves rise and the reserve-production ratio

exceeds its desired level, investment in exploration falls. Because of the

significant delay between the time a monetary investment in exploration is

made and the time natural gas discoveries occur, by the time new proven

reserves are added to ProvenReserves, they are no longer the amount necessary

to bring the reserve-production ratio into line with its desired value. As a result,

the model must again adjust its investment in exploration via the balancing

loop. The system ends-up out of sync -- over and undershooting its desired

reserve-production ratio.

http://www.albany.edu/cpr/sds/DL-IntroSysDyn/oscl.htm




Figure 13: Time Series Plots from a Simulation of the Fourth Cut of

Naill’s Model of U.S. Natural Gas Discovery and Production

Natural Gas Discovery and Production Model: Fifth Cut

Figure 14(a) and 14(b) illustrate the fifth step in the recreation of Naill’s model

of U.S. natural gas discovery and production. Figure 15 shows the equations

corresponding to the simulation model. To view Figure 15, click directly on the

stock-flow diagram. The main additions to the model are pieces of system

dynamics structure that enable the system to determine the price of natural gas

endogenously, rather than exogenously, as in past versions of the model. At

present, this pricing structure assumes that natural gas prices are not regulated.

The way that this constraint can be relaxed will be addressed in the next

version of the model.



(click on figure to view equations)

Figure 14(a): Fifth Cut of Naill's Model of U.S. Natural Gas Discovery and Production



Figure 14(b): Causal Diagram of additional balancing and reinforcing loops for Cut 5 model.

Note: Click on rotating icon to access fifth cut


The addition of an endogenous pricing structure creates four new feedback

loops in the model. Two of the loops are reinforcing loops (positive feedback)

and two are balancing loops (negative feedback). These loops can be

identified in Figure 14, wholly or partially, via information links of different

colors.

The first new loop (shown with red information links) is a reinforcing loop. This

loop works through gas price and hence sales revenue to keep gas discoveries

rising as depletion occurs.

The second new loop (shown partially in blue) is also a reinforcing loop. In a

way similar to the previous loop, this loop works through gas price and return

on investment in exploration to keep gas discoveries rising as depletion occurs.





The third new loop is a balancing loop. It is shown partially in Figure 14 with

purple information links. This loop works to keep the reserve-production ratio

at its desired level by changing gas price and hence new gas discoveries. In

other words, this loop represents the effects of gas price on the supply side of

the gas system.

The fourth new loop shown partially in Figure 14 with teal information links is

also a balancing loop. This loop works to keep the reserve-production ratio at

its desired level by changing gas price and hence the gas usage rate. In other

words, this loop represents the effects of gas price on the demand side of the

gas system.

Figure 16 shows time series plots from a simulation of the model presented. A

comparison with Figure 13 reveals that the model’s behavior is essentially the

same as before, although its oscillations are less severe. This damping of the

system’s oscillations, of course, is due to the feedback loops that were added to

the system. Movements in the price (market) system take away some of the

pressure on the system to change via the percent of revenues invested in gas

exploration and discovery.



Figure 16: Time Series Plots from a Simulation of the Fifth Cut of Naill’s Model of U.S. Natural


Natural Gas Discovery and Production Model: Sixth Cut

Figure 17 illustrates the stock-flow diagram for the sixth step in the recreation

of Naill’s model of U.S. natural gas discovery and production. To view equations

that correspond to model (Figure 18), click direct on the stock-flow diagram.

The main additions to the system are pieces of structure that enable a model

user to easily change policies, such as gas price regulation, during the

simulation run.



(click on diagram to view equations)



Figure 17: Sixth Cut of Naill's Model of U.S. Natural Gas Discovery and Production.

Note: Click on rotating icon to access sixth cut


Price regulation is turned on in the model by changing a "switch"

(SwitchForReg), shown in blue, from a zero to a one. When the switch is

flipped, gas price is determined from both endogenous (TotalCost) and

exogenous pressures (a price ceiling whose value is determined outside of the

model (presumably by the government)). If, on the other hand, SwitchForReg

remains set at zero, gas price is again determined endogenously via pressures

from both TotalCost and the EffRelResProdRatio.

Figure 19 shows time series plots from a simulation of the model with no price

regulation. This run is directly comparable to previous model runs. When

examined relative to Figure 16, it is clear that the model’s (unregulated)

behavior has essentially not changed.

On the other hand, Figure 20 shows time series plots from a simulation of the

model under the assumption of price regulation during the years 1955-1980; the

simulation is fairly similar to the one presented in Figure 19, although there is a

large "blip" upward in the discovery rate, usage rate, and in proven reserves in

1980, when gas price deregulation occurs. Overall, however, the basic behavior

mode of growth, peaking and decline has not changed.



Figure 19: Time Series Plots from a Simulation of the Sixth Cut of Naill’s Model of U.S. Natural

Gas Discovery and Production: Unregulated Scenario



Figure 20: Time Series Plots from a Simulation of the Sixth Cut of Naill’s Model of U.S. Natural

Gas Discovery and Production: Regulated Scenario

Finally, in the Reference Modes section of this chapter, we presented some

actual data from the U.S. natural gas system including actual discovery rate,

usage (production) rate, proven reserves, and reserve-production ratio. Although

these data do not span the entire life cycle of the natural gas system (roughly

the years 1900 to 2050) a la Hubbert, they were used to complement the

reference modes resulting from Hubbert's theory and to calibrate the model.

Figure 21 shows the time series plot comparing the actual data to simulated

data between the years 1975 to 1995.



Figure 21: Time series plot comparing the actual data to simulated data between the years 1975

to 1995.

Policy Explorations with Natural Gas Model

Now that you have been introduced to the underlying assumptions of the

system dynamics model for the natural gas industry, it is time to use the model

to conduct your own policy experiments. The figure below illustrates the basic

layout of the flight simulator version of the Natural Gas Discovery Model. To

launch the "flight simulator" version of the Natural Gas Discovery model, click

on the rotating icon . Then try some of the experiments suggested below.

Once you have gained some familiarity with using the model, you can begin to

develop your own experiments.



Figure 1:Layout of the Natural Gas Discovery Simulator

Establishing the base run

For these policy experiments, we first want to establish a base run. A base run is

our first estimate of how we think the system works today. By establishing this

as a baseline, we then can compare the results of each change in the model

back to the base. Follow the steps listed below:

1. Click on the run button at the top of the flight simulator.

2. Note that clicking on run button also forces the pause button down.

This initial pause is designed to allow the user to change variables and

initial conditions particular to the desired run. However, for the base

run, the model is "preset" with the correct settings.

3. Click on the pause button to release it and begin the simulation. Your

output graphs should look like Figure 2 below:



Figure 2: Example time series plot of base run Proven Reserves and Discovery Rate variables.

After checking through the different graphs, you should be ready to start

changing parameters in the model to see how the simulation can unfold

differently under different policies and scenarios. The control panel contains

two kinds of parameters for you to control: Policy Levers and Uncertainties

variables.

o Policy Levers: These are variables that represent decisions people make

based on the information that they are receiving from the system. For

example, the Wellhead Price Tax is a government control lever.

o Uncertainties: Things that influence the system from the outside that we

cannot predict. We want to test different policies against many scenarios

of the uncertainties to see what are the best policies. For example, the

initial value for the Unproven Reserves variable is only an estimate.

Therefore, this simulator allows the user to vary this initial value to the

sensitivity of the model to this uncertainty.

We will walk you through one experiment in more detail to illustrate the steps

of changing the policy and explane the results, and then set you loose to

experiment with the rest.

Guided Experiment



What if the unproven reserves of natural gas were 50% greater than we first

estimated? This is an experiment in which we are going to alter one of the

"uncertainties" in the model to see how much difference it will make in the

behavior of the natural gas industry:

6. To start the run, press the run button as in the base run.

7. Scroll to the Unproven Reserves graph and look for the gage titled

"INITIAL." This gage will allow you to enter different assumptions about

the initial amount of unproven reserves. Click on the right arrow button

under this gauge to increase the % percent of baseline value. Since we

want to increase the initial value of the unproven reserves by 50% (2e15

ft^3), the gauge should be set to 150% of the baseline value as shown in

Figure 2.

8. Now press the pause button to resume the simulation. When the

simulation is complete, you can move over to the graphs to see what

happened as a result.

Figure 3: Setting initial uproven reserves to 50% greater than base case

One of the sets of graphs are called comparative graphs and allow the same

variable to be shown on the same graph for multiple experiments, as shown in

Figure 3 below:



Figure 4: Comparison of base run and results of increasing base unproven reserves by 50%.

The first run in the red line was the base run. The green line is the new run

with the new assumption about the amount of unproven runs. Look through

the other graphs. What happened? For how much longer did the significantly

higher amount of unproven reserves allow the industry to continue? Why do

you think this happened? These are the kinds of questions you should be asking

yourself as you go through different experiments using the slide bars in the

control panel.

To clear the results of a particular simuation experiment, simply pull down the

simulate menu and select the "clear results" option as shown in Figure 5.

Figure 5: Clearing Experiment Results

Suggested Experiments:



Conservation policy 1- What would happen if the government

(or industry) set up a policy to encourage the conservation of

natural gas whenever the proven reserves fall? We can simulate

this policy through the natural gas Demand variable in the

control panel. This variable can be changed at any time during

the simulation.

Conservation policy 2 - Use a simple tax on

the wellhead price to discourage use of natural gas. Use the

Wellhead Price Tax box to tax.

Exploration Investment - What would happen if industry

was more aggressive in their efforts to discover new sources

of natural gas? We can model this policy by changing the

base fraction of revenues invested in exploration.

Uncertainties

Amount of unproven reserves - One of the

fundamental uncertainties of natural

resources is that we don't know how much is

underground that we have not yet discovered.

An important test of the natural gas system is

to find out how much difference it would

make if there were twice as much as we first estimated. What would happen if

there were ten times (1000% of baseline) as much?

Base usage growth rate - Another important uncertainty in

the future of the natural gas industry is the growth rate of the

demand for energy. This variable is the percent increase in the

base demand for natural gas.

Technology Investment - The technology multiplier is a

multiplier against the cost of exploration. It represents

advances in technology that make it cheaper to access the

unproven natural gas. Change this multiplier to a fraction of

the base cost to see the impact of a technology advance at any time during the

simulation.



Chapter 7 Next Steps

Now that you've read this book, and

"surfed" through some examples,

what is your next step? One of the

difficulties is that while the software

used in system dynamics modeling

has made it easy to build models, it

takes some discipline and practice to

make good system dynamics models.

A good model is a model that focuses

on examining a problem or behavior,

rather than trying to model an entire

system. A good model is one that is a documented, understandable, and

concise representation of a system. And most importantly, a good model is one

that can help people better understand the world in which they are trying to act

and enable them to make more effective long-term decisions.

Cautious Enthusiasm: The goal of this book is to share the philosophy and

tools of system dynamics. Your ultimate success relies, of course, on actually

practicing yourself. Each of the systems dynamics software packages comes

with excellent manuals to get you started. We would also recommend working

with someone more experienced at some point to help you accelerate the

learning process.

Opportunities include:

o courses at some universities,

o workshops through either the software manufacturers or consultants, or

o modeling projects with more experienced practitioners, either in your

own organization or outside consultants.

System Dynamics Resources

System Dynamics Internet Related Sites



MIT System Dynamics Group

The System Dynamics Group was founded in the early 1960s by

Professor Jay W. Forrester at MIT. Professor John D. Sterman is the

current director of the System Dynamics Group. The Group has three

main areas of research.The National Model Project strives for a better

understanding of how the U.S. economy works. The System Dynamics in

Education Project was established in 1990 to write a "do it yourself" workbook

for learning system dynamics. With funding from an NSF grant, the Group

also studies the Improvement Paradox: Designing Sustainable Quality

Improvement Programs.

System Dynamics Society - The System Dynamics Society is an

international, nonprofit organization devoted to encouraging the

development and use of system dynamics around the world. It is run

solely on a volunteer basis by practitioners of System Dynamics in

academics and industry. With members in over 35 countries, the

System Dynamics Society provides a forum in which researchers can

keep up to date with applications, methodologies and tools.The

System Dynamics Society is responsible for the publication of The

System Dynamics Review, a refereed journal containing articles on

methodology and application. A President's letter provides an informal forum

to bring members up to date on society activities and business.

System Dynamics in Education Project

The System Dynamics in Education Project (SDEP) is a group of students

and staff in the Sloan School of Management at the Massachusetts

Institute of Technology, working under the guidance of Professor Jay W.

Forrester, the founder of system dynamics.

Society for Computer Simulation International

Society for Computer Simulation is an international,

multidisciplinary forum dedicated to research, development and

applications of simulation. Established in 1952, the Society for

Computer Simulation is a non-profit, volunteer-driven corporation

(Simulation Councils Inc.). SCS is the only technical society

dedicated to advancing the use of computer simulation to solve real world

problems. SCS membership represents a world-wide network of working

professionals whose work requires them to be current with advancements in

simulation technology. Regional councils from the United States and Canada,

the European Economic Community, Pacific Rim and China and Mexico meet

frequently and allow members the opportunity to network informally, while

conferences offer a forum for formal presentations of research breakthroughs.

http://sysdyn.mit.edu/sd-group/home.html

http://sysdyn.mit.edu/sd-group/home.html

http://news.std.com/vensim/SDSOCIET.HTM

http://sysdyn.mit.edu/

http://sysdyn.mit.edu/

http://www2.ncsu.edu/unity/lockers/dept/pubweb/jenna/simulate/



The Santa Fe Institute

The Santa Fe Institute is a private, independent, multidisciplinary

research and education center, founded in 1984. Since its founding,

SFI has devoted itself to creating a new kind of scientific research

community, pursuing emerging science. Operating as a small,

visiting institution, SFI seeks to catalyze new collaborative, multidisciplinary

projects that break down the barriers between the traditional disciplines, to

spread its ideas and methodologies to other individuals and encourage the

practical applications of its results.

Publications and Research Project Information for John Sterman, Director,

MIT System Dynamics Group and J. Spencer Standish Professor of

Management

o Written Material

Productivity Press - publisher of all system dynamics academic

books - P.O.Box 13390, Portland, OR 97213. Phone (503) 235-0660.

Call to get a catalog.

Pegasus Communications - publisher of The Systems Thinker

newsletter, an excellent source on causal loop diagrams and the

conceptual modeling portion of the field - P.O. Box 120, Kendall

Square, Cambridge, MA 02142. Phone (617) 576 1232

The System Dynamics Society - publishers of the journal The

System Dynamics Review - Phone: (617) 259-0969,

[email protected]

o System Dynamics Software

Powersim - is available from Powersim AS (Norway), P.O. Box

206, N-5100 Isdalstø, Norway, 47-56-342-400 (Phone),

47-56-342-401 (Fax)

ithink/Stella -distributed by High Performance Systems, 45 Lyme

Road, Hanover, NH 03755. Phone (603) 643 9636

Vensim - distributed by Ventana Systems, Inc, 60 Jacob Gates

Road, Harvard, MA 01451. Phone (508) 456-3069

Bibliography

http://www.santafe.edu/

http://web.mit.edu/jsterman/www



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