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Optimization for Sustainability of Integrated Ecological-Economic Model System of Planet

Megan SchwarzJohns Hopkins UniversityDr. DiwekarJuly 1, 2013

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What is Sustainability?• “The development that

meets the needs of the present without compromising the ability of future generations to meet their own needs” (Brundtland 1987)

• Goal: To design a simplified model of the planet to explore regulatory strategies to try to increase sustainability

Heriberto Cabezas, Christopher W. Pawlowski, Audrey L. Mayer, N. Theresa Hoagland Clean Techn Environ Policy 5 (2003) 167–180

Sustainability: A path through time

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Integrated Ecological-Economic Model System of Planet

Non-D

omestic

Dom

estic

Natural Resources

Primary Producers (Plants)

Herbivores

Carnivores

Human Households

Energy Source

Industrial Sector

Energy Producer

Biologically inaccessible resourcesInaccessible Resource Pool

Resource Pool

P1 P2

H1

P3

H2 H3

C1 C2

HHEnergy Source

IS EP

fence

fence

grazing

Model Adapted from Kotecha, P.; Diwekar, U.; Cabezas, H.. “Model-based approach to study the impact of biofuels on the sustainability of an ecological system” (2011).

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Basic Mathematics of the Model•Three general types of equations

▫Basic food web model equations▫Macroeconomic model equations▫All other algebraic equations

98 constant parameters 19 time dependent state variables described

by differential equations 61 model outputs About 2000 lines of code in Matlab

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Model Simulation•Looked at how economic and ecological

parameters changed over a time period of 200 years with and without the use of biofuel as a source of energy

•Two different scenarios▫Population Explosion▫Increase in Per Capita Consumption

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Population ExplosionDynamics of Human Population and Primary Producer 2

• The population is expected to peak to about twice today’s size in the next 50 t0 100 years

• A steady drop is then expected due to an aging population and a decrease in fertility rates

• Primary producer 2 was the only ecological compartment to reach extinction

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Increase in Per Capita ConsumptionDynamics of Carnivore 1 and Human Population

• Consumption of many resources is estimated to increase by approximately 50% in the next 50 years

• Most ecological compartments reached extinction

• Shows the catastrophe where limited resources cause loss of human life▫ Decrease in population

sooner with the use of biofuels because compartments reach extinction earlier

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Conclusions about Model Simulation•Sustainability of even a simple ecosystem

may not be intuitive▫Use of biomass as a source of energy

accelerates the extinction of species•Increasing per capita consumption is

more critical than population explosion▫The ecosystem can’t sustain high levels of

human consumption

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Numerical Optimization•Goal: increase the lifetime of dying

compartments▫Increase sustainability of the system

•Need a mathematical measure of sustainability▫Fisher Information (FI) ▫FI can be used as a measure of order of a

system Information is a fundamental quantity of a

system Able to incorporate the physics and economics

of the model

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Objective Function•Objective: develop policies so the system

FI is close to the FI of a stable system▫Base case scenario

•Objective Function: minimization of the FI variance:

▫ is the current FI profile▫ is the targeted FI for the stable base

case scenario▫T is the total time under consideration

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Numerical Optimization: Non-linear Programming (NLP)

• Initial values of the decision variable are known

• The model calculates the objective function and the optimizer tries to satisfy optimality conditions (Karush-Kuhn-Tucker conditions, KKT)

• Optimizer calculates a new value for the decision variable

• Iterative sequence continues until the optimization criteria (KKT) are met

Optimal Design Initial Values

Optimizer

MODEL

Decision Variables

Objective Function

Model Adapted from: Diwekar, Urmila M. Introduction to Applied Optimization. Norwell, MA: Kluwer Academic, 2003. Print.

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Design of Techno-economic Policies for Sustainability

• Policies (control variables) are used as the decision variables at each time step

• Governmental Policies:▫ Discharge fee charged to the

industrial sector (pISHH)▫ Amount of primary producer

2 consumed by herbivore 1 through grazing

• Policy related to Efficiency of Technology:

▫ Amount of primary producer 1 required to produce a unit of the industrial sector product

0-5

10-1

520

-25

30-3

540

-45

50-5

560

-65

70-7

580

-85

90-9

510

0-10

511

0-11

512

0-12

513

0-13

514

0-14

515

0-15

516

0-16

517

0-17

518

0-18

519

0-19

5

0.0E+00

2.0E-02

4.0E-02

6.0E-02

8.0E-02

1.0E-01

1.2E-01

1.4E-01

Controlled Efficiency Policy Variable (theta) for Discretized Time Period (no bioenergy)

Time, years

Thet

a

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Governmental PolicyDischarge fee charged to the industrial sector (pISHH)

0 20 40 60 80 100 120 140 160 180 2000.00E+00

2.00E-08

4.00E-08

6.00E-08

8.00E-08

1.00E-07

1.20E-07

Uncontrolled and Controlled Discharge Fee for Increase in Per Capita Consumption Scenario (no bioenergy)

Controlled Discharge Fee Uncontrolled Discharge Fee

Time, years

Disc

harg

e Fe

e

0 20 40 60 80 100 120 140 160 180 2000.00E+00

2.00E-08

4.00E-08

6.00E-08

8.00E-08

1.00E-07

1.20E-07

Uncontrolled and Controlled Discharge Fee for Increase in Per Capita Consumption Scenario (bioenergy)

Controlled Discharge Fee Uncontrolled Discharge Fee

Time, years

Disc

harg

e Fe

e

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Governmental PolicyDischarge fee charged to the industrial sector (pISHH)

No Bioenergy

Bioenergy

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Policy Related to Efficiency of TechnologyAmount of primary producer 1 to produce a unit of the industrial sector product

0 20 40 60 80 100 120 140 160 180 2000.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1.00E-01

1.20E-01

1.40E-01

Uncontrolled and Controlled Theta for Increase in Per Capita Con-sumption Scenario (no bioenergy)

Controlled Theta Uncontrolled Theta

Time, years

Thet

a

0 20 40 60 80 100 120 140 160 180 2000.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

6.00E-01

7.00E-01

Uncontrolled and Controlled Discharge Fee for Increase in Per Capita Consumption Scenario (bioenergy)

Controlled Theta Uncontrolled Theta

Time, years

Thet

a

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Policy Related to Efficiency of TechnologyAmount of primary producer 1 to produce a unit of the industrial sector product

No Bioenergy

Bioenergy

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0 20 40 60 80 100 120 140 160 180 2000.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1.00E-01

1.20E-01

1.40E-01

Uncontrolled and Controlled Khat for Increase in Per Capita Consumption Scenario (no bioenergy)

Controlled khat Uncontrolled khat

Time, years

Khat

Governmental PolicyThe amount of primary producer 2 consumed by herbivore 1 through grazing

0 20 40 60 80 100 120 140 160 180 2000.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1.00E-01

1.20E-01

Uncontrolled and Controlled Khat for Increase in Per Capita Consumption Scenario (bioenergy)

Controlled khat Uncontrolled khat

Time, years

Khat

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Governmental PolicyThe amount of primary producer 2 consumed by herbivore 1 through grazing

Bioenergy

No Bioenergy

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Conclusions• Fisher Information is an indicator of

sustainability of a system▫The discharge fee charged to the industrial

sector is most effective in delaying the extinction of dying compartments

▫The amount of primary producer 1 required to produce a unit of the industrial sector product leads to small delays in the extinction of dying compartments

▫does not lead to any significant improvement in the model dynamics

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Future Work•Optimization using different control

variables•Multi-variable control•Further model enhancement

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Acknowledgments•The financial support from the National

Science Foundation, EEC-NSF Grant # 1062943 is gratefully acknowledged

•Dr. Diwekar•Kirti Yenkie•Pahola Thathiana Benavides•Professor Takoudis, Professor Jursich,

REU program

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ReferencesCabezas, H., C. W. Pawlowski, A. L. Mayer, and H. W. Whitmore. "On the Sustainability of Integrated Model

Systems with Industrial, Ecological, and Macroeconomic Components." Resources, Conservation and

Recycling 50.2 (2007): 122-29. Elsevier B.V. Web.Cabezas, Heriberto, N. Theresa Hoagland, Audrey L. Mayer, and Christopher W. Pawlowski. "Simulated Experiments with

Complex Sustainable Systems: Ecology and Technology." Resources, Conservation and Recycling 44 (2005): 279-91.

Elsevier B.V. Web.Diwekar, Urmila M. Introduction to Applied Optimization. Norwell, MA: Kluwer Academic, 2003. Print. "Finite Difference Schemes." Computational Fluid Dynamics. Brown University, n.d. Web.Kotecha, Prakash, Urmila Diwekar, and Heriberto Cabezas. "Model-based Approach to Study the Impact of Biofuels on the

Sustainability of an Ecological System." Clean Technology and Environmental Policy 15.1 (2013): 21-33. Springer

Verlag. Web.Meadows, Donella H., Dennis L. Meadows, and Jørgen Randers. Beyond the Limits: Confronting Global Collapse, Envisioning

a Sustainable Future. Vermont: Chelsea Green, 1992. Print."Report of the World Commission on Environment and Development Our Common Future."Brundtland Report 1987. United

Nations, n.d. Web.Shastri, Y., and U. Diwekar. "Sustainable Ecosystem Management Using Optimal Control Theory: Part 1 (Deterministic

Systems)." Journal of Theoretical Biology 241 (2006): 506-21. Elsevier B.V. Web.Shastri, Yogendra, Urmila Diwekar, and Heriberto Cabezas. "Optimal Control Theory for Sustainable Environmental

Management." Environmental Science and Technology 42.14 (2008): 5322-328. American Chemical Society. Web.Shastri, Yogendra, Urmila Diwekar, Heriberto Cabezas, and James Williamson. "Is Sustainability Achievable? Exploring the

Limits of Sustainability with Model Systems." Environmental Science and Technology 42.17 (2008): 6710-716. American

Chemical Society. Web.United States Census Bureau. U.S. Department of Commerce, n.d. Web. 01 July 2013.

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Questions?

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•Time average FI for a system with n species:

• = cycle time•

• and are the velocity and acceleration terms of the ecosystem