modeling infrastructure and network industries: theory and ...extensions of complementarity...

Modeling Infrastructure and Network Industries: Theory and Applications*

Steven A. GabrielSteven A. GabrielProject Management Program, Dept. of Civil & Project Management Program, Dept. of Civil & EnvEnv. Engineering, University of Maryland, . Engineering, University of Maryland,

College Park Maryland, 20742 USACollege Park Maryland, 20742 USAApplied Mathematics and Scientific Computation Program, UniversiApplied Mathematics and Scientific Computation Program, University of Maryland, College ty of Maryland, College

Park, Maryland 20742 USA Park, Maryland 20742 USA Gilbert F. White Fellow, Resources for the Future, Washington, DGilbert F. White Fellow, Resources for the Future, Washington, DC USA (2007C USA (2007--2008)2008)

Visiting Scholar, LMI Research Institute, McLean, Virginia, USA Visiting Scholar, LMI Research Institute, McLean, Virginia, USA (2007(2007--2008)2008)

Presented atPresented atInfradayInfraday 20072007

Berlin, GermanyBerlin, GermanyOctober 6, 2007October 6, 2007

*National Science Foundation Funding, Division of Mathematical S*National Science Foundation Funding, Division of Mathematical Sciences, Awards ciences, Awards 0106880, 0408943

22

Outline of PresentationBriefly, My BackgroundFrom Optimization to Complementarity Problems then on to MPECs and EPECs: Why All the Fuss? – Complementarity Problem Application: Natural Gas Market

Equilibrium

Stochastic Optimization Models– Stochastic Multiobjective Optimization Application:

Telecommunications Network Reconfiguration

Conclusions and Future WorkGeneral invitation to Trans-Atlantic Critical Infrastructure Modeling Conference at Univ. of Maryland, Nov. 2, 2007

33

My Background

44

University of Maryland

My Affiliations– Department of Civil & Environmental Engineering– Applied Mathematics and Scientific Computation Program– Engineering and Public Policy Program (joint between

Engineering School and Public Policy School)

55

Overview of Research

Research: Main Topics– Mathematical modeling in engineering-economic systems usually

involving critical infrastructure using optimization and equilibrium analysis• energy market models (natural gas and electricity)• transportation/traffic• land development (Multiobjective optimization for “Smart Growth”

in land development)• wastewater treatment (Optimization and statistical modeling in

biosolids)• telecommunications (Optimization)

– Development of algorithms for solving equilibria in energy & transportation systems and other planning problems

– Development of general purpose algorithms for equilibrium models(using the nonlinear complementarity format)

66

Overview of Research

Design of Optimization/

Complementarity Algorithms

Analysis of Public

Policy Issues

MathematicalModeling

of Critical Infrastructure

77

From Optimization to Complementarity Problems then on to MPECs and EPECs: Why All the Fuss?

88

Example of an Equilibrium Problem A Variation on a Transportation Problem

10

5

4

6

2

10

1S1=20

S2=20 2

Supplies

1

2

3

D1=10

D2=10

D3=10Demands

i jcij

99


0

0

10

10

10

0

1S1=20

S2=20 2

Supplies

1

2

3

D1=10

D2=10

D3=10Demands

01 =ψ

32 =ψ

91 =θ

52 =θ

43 =θ

Solution:• flow on arcs• dual prices at

nodes

1010

ij

ij

1 13 3 13

2 23 3 23

Optimality conditions are of the formc , 1, 2, 1, 2,3

0 +c , (+ other conditions)

Example:+c 0 4 4 and 10+c 3 10 4 and 0

i j

ij i j

i j

x

xx

ψ θ

ψ θ

ψ θψ θ

+ ≥ = =

> ⇒ =

= + ≥ = >

= + > = =


1111


demand andsupply dependent -priceusing before stated conditions optimality thegeneralizeCan 2.

why?

demand)for 3,2,1j, supply,for 1,2i,( prices eappropriat theoffunction a as vary to themallowing than realistic less is this

constants, asgiven werequantities demand andsupply The .1:marksRe

ji =θ=ψ

1212


( )( )

( )( )( ) 333

222

111

222

111

D14DD5.010D

D19DDemand

1S2.0S20SS

Supply:functions demand andsupply (inverse) following theAssume

−=θ−=θ−=θ

−=ψ−=ψ

( )( )jj

ii

DS

θψ

iS

jD

( )ii Sψ

( )jj Dθ

1313


( ) ( )( ) ( )

ij

ij

3

1

2

1

Complete Optimality Conditions

c , 0, 1, 2, 1,2,3

0 c

, 1, 2

, 1, 2,3

This is an example of a complementarity problem(Spatial Price Equilibrium)

i i j j ij

ij i i j j

i ijj

j iji

S D x i j

x S D

S x i

D x j

ψ θ

ψ θ

=

=

+ ≥ ≥ = =

> ⇒ + =

= =

= =

∑

∑

1414

NLP

QPLP

Complementarity Problems vis-à-vis Optimization and Game Theory Problems

Other non-optimization based problems

e.g., spatial price equilibria, traffic equilibria, Nash-Cournot games, zero-finding problems

Complementarity Problems

1515

Complementarity Problems and Variational Inequalities

Complementarity Problems

VariationalInequality Problems

But, when polyhedral constraints, VI is a special case of the mixed complementarity problem

1616

Optimization vs. Complementarity Problems

Complementarity problems are more general covering:– Zero-finding problems– Optimization problems (via Karush-Kuhn-Tucker conditions)– Game Theory problems (e.g., Bimatrix or Nash-Cournot games)– Host of other interesting problems in engineering and economics

Thus, theorems and algorithms designed for CPs can be applied to a wide variety of applicationsSome problems have no natural optimization counterpart (e.g., via Principle of Symmetry), therefore, can only use CPs in this contextCPs very useful for solving policy-related network infrastructure problems (cf. SPE)– Can include some network participants having market power– Can include other players as price-takers

1717

Optimization vs. Complementarity Problems (con’t)Complementarity problems can also include problems in which prices (Lagrange multipliers) appear in the primal formulation– PIES energy infrastructure model of the 1970s– More generally infrastructure models whose modules might represent a

detailed sector (e.g., power production) and for which subsets of prices and quantities (and other variables) are passed between these modules, e.g., National Energy Modeling System

Source: http://Source: http://enduse.lbl.gov/Projects/NEMS.gifenduse.lbl.gov/Projects/NEMS.gif

S. A. Gabriel, A. S. Kydes, P. Whitman, 2001. "The National Energy Modeling System: A Large-Scale Energy-Economic Equilibrium Model," Operations Research, 49 (1), 14-25.

1818

Extensions of Complementarity Problems: MPECs and EPECsStackelberg Games or More Generally MPECs– What if two-level problem where top level is a dominant company or

the government and bottom level is the rest of the market– This is no longer a complementarity problem since all the players are

not at the same level– Instead it’s an example of a mathematical program with equilibrium

constraints (MPEC)

1919

Extensions of Complementarity Problems: MPECs and EPECsStackelberg Games or More Generally MPECs

– x upper-level planning variables, y lower-level variables, S(x) solution set of lower-level problem (e.g., Nash-Cournot game or optimization)

– Lately a number of research papers on MPECs in energy infrastructure planning, transportation planning, etc.

min ( , ). .

( )

f x ys t xy S x

∈Ω∈

2020

Extensions of Complementarity Problems: MPECs and EPECs

EPECs– Can also make the top level a game to get equilibrium problems with

equilibriuim constraints (EPEC)

MPECs and EPECs are hard problems for several reasons– Feasible region not generally known in closed form (can use KKT

conditions though)– Instance of global optimization problem

Advantages for regulators– Can more accurately reflect market behaviors when both strategic

players exist in combination with non-strategic ones– Can allow regulators to see what effects for certain potential regulations

or policies might be on the market with better feedback mechanisms

2121

Example of Complementarity Problem for Natural Gas Infrastructure Planning

2222

The Natural Gas Supply Chain

INDUSTRIAL

CITY GATE STATION

COMMERCIAL

RESIDENTIAL

DISTRIBUTION SYSTEM

UNDERGROUND STORAGE

TRANSMISSION SYSTEM

Cleaner

Compressor Station

GAS PROCESSING PLANT

GAS PRODUCTION

Gas Well Associated Gas and Oil Well

Impurities Gaseous Products

LiquidProducts

ELECTRIC POWER

From well-headto burner-tip

2323

Recent Complementarity Modeling and Natural Gas Markets: Gabriel et al.

North America Natural Gas Markets1. S.A. Gabriel, S. Kiet and J. Zhuang (2005), A Mixed Complementarity-Based Equilibrium

Model of Natural Gas Markets, Operations Research, 53(5), 799-818. 2. S.A. Gabriel, J. Zhuang and S. Kiet (2005), A Large-Scale Complementarity Model of the North

American Natural Gas Market, Energy Economics, 27, 639-665.3. S.A. Gabriel, J. Zhuang and S. Kiet (2004), A Nash-Cournot Model for the North American

Natural Gas Market, IAEE Conference Proceedings, Zurich, Switzerland, September.European Union Natural Gas Markets1. R. Egging and S.A. Gabriel (2006), Examining Market Power in the European Natural Gas

Market, Energy Policy, 34 (17), 2762-2778. 2. R. Egging, S.A. Gabriel, F.Holz, J. Zhuang, A Complementarity Model for the European

Natural Gas Market, November 2006, in review.

General Natural Gas Markets and Algorithms1. S.A. Gabriel and Y. Smeers (2006), Complemenatarity Problems in Restructured Natural Gas

Markets, Recent Advances in Optimization. Lecture Notes in Economics and Mathematical Systems, Edited by A. Seeger,Vol. 563, Springer-Verlag Berlin Heidelberg, 343-373.

2. J. Zhuang and S.A. Gabriel (2006), A Complementarity Model for Solving Stochastic Natural Gas Market Equilibria Energy Economics, in press.

3. S.A. Gabriel, J. Zhuang, R. Egging, Solving Stochastic Complementarity Problems in Energy Market Modeling Using Scenario Reduction, November 2006, in review.

2424

Global Aspects of Natural Gas MarketsPreviously, natural gas was more of a continental market

– Pipeline access issues– Market structures

Now more or less a global market– Importance of natural gas for more environmentally-friendly power

generation– Greater activity in LNG transport– Market restructuring (at least in the US and the EU)

Main result is that there is a “domino” effect relative to supply security– Supplier in one country cuts back production or transportation of natural

gas– This effects downstream customers who then need more gas from a

second supply source– The customers who rely on the second supply source also affected, etc.

Issues of geopolitical market power being exerted

2525

Description of Complementarity Model for Global Natural Gas Markets (R. Egging, S.A. Gabriel, F.Holz, J. Zhuang, "A Complementarity Model for the European Natural Gas Market," November 2006)

Players– Producers– Traders (marketing aspects of production companies)– Pipeline operators– Storage operators– Marketers– LNG Liquefiers– LNG Regasifiers– Consumers

Multiple seasonsTraders (e.g., producers) allowed to have varying degrees of market power

2626

Overall picture

T11

C1

K1,2,3

S1

M1

C3

K1,2,3

S3

M3

R3

L1

Producer

Trader

SectorsMarketer

LNG Liquef

Storage LNG Regasif

Country 1 Country 3

Country 2

T31T31

T32

T12T13

•Traders are “producer specific contract agents”•Marketers and storage operators can by from any traders•Liquefier only buys from transmitter from domestic producer

2727

Complementarity Aspects

Take major players’ economic behavior consistent with maximizing net profit subject to economic and engineering constraintsCollect all the resulting optimality conditions along with market-clearing onesResulting set of conditions is a nonlinear complementarity problem (variational inequality)

2828

Maximize production revenues less production costss.t.– bounds on production rates– bounds on volume of gas produced in time-window of analysis

Decision Variables– How much to produce in season and year (cubic meters/day)

Market Clearing– Producers’ sales must equal Trader’s purchases from Producer

Producer’s Problem

2929

How Will Cutting of Gas from Russia to Ukraine Affect Other Countries?

Consider the realistic “domino” effect from one of our recent models (details on model later)Ukraine Disruption Scenario from what actually happenedHow does Japan get affected?

3030

Analysis of Strategic LNG Flows From Model

Basic Map from: www.insectzoo.msstate.edu/Curriculum/Activities/WorldMap.html

13.1

LNG modeled as “spot market”Flows in BCM/year

55.5

6.5

Japan:75.1

SEA

ARBNIG

T&TALG

EU

UKR

RUS

JAP

AUS

KOR

US

3131

Disruption Ukrainian Pipeline

Basic Map from: www.insectzoo.msstate.edu/Curriculum/Activities/WorldMap.html

+0.1

Changes in LNG flows. (BCM/year)

-3.9

Japan: -3.8to 71.3

+4.3-4.3

+3.9

-0.9

+0.9

-1.9

ALG pipes more to EUR, less LNG to

USA

T&T

EU

ALG

NIG

JAP

KOR

SEA

AUS

UKR

RUS

ARB

US

3232

Stochastic Optimization Models

3333

Stochastic Optimization Background

( )( )

( )s.t

0, 1, ,

0, 1, ,

::

:

i

j

n

ni

nj

Min f x

g x i m

h x j n

f R Rg R R

h R R

≤ = …

= = …

→

→

→

Nonlinear programming problem, objective and constraint functions usually assumed deterministic

What if some aspects of these functions (e.g., coefficients) are not known with certainty?This is then a stochastic(nonlinear) programming problem

3434

Stochastic Optimization BackgroundMany method to solve such a stochastic problem, some examples ofapproaches– Decomposing the problem (e.g., L-shaped method)– Using a sampling approach– Using a scenario tree for the finite (but usually large) number of

realizations, then approximating it with a reduced tree

Römisch, Dupačová, Gröwe-Kuska, Heitsch (2003)

3535

Stochastic Optimization Background (J.R. Birge and F. Louveaux, “Introduction to Stochastic

Programming,” Springer, New York, 1996)

Stochastic Optimization allows for endogenous handling of riskThis is NOT the same as running a number of different scenarios, why?Two important notions:– Expected Value of Perfect Information (EVPI)– Value of Stochastic Solution

3636

Stochastic Optimization Application: Autonomous Near-Real Time Reconfiguration in Telecommunications

Networks

3737

Free Space Optical Communications

Emerging Communications Technology – A high-speed bridging technology to current fiber optics network– A valuable technology in commercial and military backbone

network

Advantage of FSO communications – Optical wireless (no fibers)– Directional (no frequency interference)– High-speed data rate (~Gbps) 1.25Gbps Optical Transceiver

(Canobeam DT-130-LX)

3838

Topology ControlMain challenge in FSO networking– Autonomous Physical Reconfiguration: Pointing, Acquisition,

and Tracking– Autonomous Logical Reconfiguration: Topology Optimization

• For example, in the event of a hurricane wiping out links• Need to reconfigure network quickly

Challenge – Responding quickly to a sudden change in link or traffic demand

to provide robust quality of serviceResearch questions– How to steer narrow laser beams between two remote optical

transceivers automatically and precisely ? – Autonomous Physical Reconfiguration

– How to get the optimal topology with respect to physical layercost or network layer congestion in near-real time? – Autonomous Logical Reconfiguration

3939

Autonomous Reconfiguration

1

2

4

3

Free Space Optical Network

2

23

4

1

1

4

33

22

3

44

11

= FSO transceiver

1

2

3

4 = =

sudden change in trafficdemand or link loss

4040

Generating Cost Matrices: Cloud Model (courtesy: Jaime Llorca, Univ. of Maryland)

4141

Complexity of Autonomous Logical ReconfigurationCandidate solutions are all the

permutations (topologies) of a set of nodes N=1,…,n

– Number of possible topologies = (n-1)!/2

– e.g., n=12→20 million topologies, n=14 → 3 billion topologies

For each topology, – Number of OD pairs = n(n-1)– For each OD pair,

• Two possible routings (clockwise or counter-clockwise)

• Number of possible routings = 2^n(n-1) (e.g. n=12→5.4×1039, n=14→6.1×1054)

This complexity makes it hard to get an optimal topology in real-time.

1

7

5

3

82

4 6

destinationnode

originnode

4242

Stochastic Multiobjective Optimization Problem (SMOP)

Jointly minimize cost and congestion to obtain Pareto optimal topologies

Objective Function

– Link cost– Uncertainty in traffic demands

• K: # of scenarios (i.e. number of possible traffic demands)• pk: probability in the realization of the k-th scenario

– Constant weight on each single objective function• Weight = user’s preference to cost or congestion

1, 2

( , ) ( , ) ( , ) 1min (1 ) ( )

Kk k k

y f ij ij iji j o d i j k

w c y w p r f SPω ωω= =

+ −∑ ∑ ∑∑

4343

SMOP Formulation

bi-connectivityconstraints

directional flowconstraints

binary link variablebinary flow variable

4444

Traffic Matrices: Examples

4545

SMOP Swapping Heuristic: Near-Pareto Optimum (n=10)

4646

Significant Advantage of MOP HeuristicNumber of nodes = 20Number of weights = 39, i.e. w∈0.025, 0.05, 0.75, …, 0.975Number of traffic demand scenarios, K = 10, w/ p1=p2=…=p10=1/K=0.1MATLAB heuristic code = 39 points × 5 minutes/point = 195 min Expected enumeration time = 9,767,520 days (3.6GHz Intel P4)

4747

Numerical Results: SMOP for n=20

4848

Conclusions

Infrastructure planning involves simultaneous consideration of many important factors, for example:– Market participants– Regulated/unregulated aspects– Engineering and economic constraints– Benefits to society– Uncertainty in key elements (e.g., demand, weather)

Need for sophisticated models to take all these factors into account and provide regulators with accurate tradeoffs for example between– Level of infrastructure investments– Incentives for socially beneficial directions – Acceptable levels of risk that the infrastructure will be degraded

4949

Future Work

Continue development of sophisticated engineering-economic models taking into account– Strategic behavior– Mixture of regulated/unregulated elements– Endogenous treatment of uncertainty

Resulting models will be:– Stochastic complementarity problems (several efforts on-going

now)– Stochastic MPECs– Etc.

5050

Trans-Atlantic Infraday Conference

General invitation to Trans-Atlantic Critical Infrastructure Modeling Conference at Univ. of Maryland, Nov. 2, 2007Jointly hosted with German colleagues from Berlin and DresdenFocus on modeling and policy for networked industries: energy, transportation, telecommunications, waterWebsite: http://tai.ee2.biz/

5151

Back-Up SlidesNatural Gas Market Equilibrium Problem

5252

Major natural gas trade movements (BP Stats)

5353

j)(i, arc along flow theis xwhere

3,2,1j,2,1i,0x10xx10xx10xx

20xxx20xxx

.t.s10x2x6x4x5x0x1min

program ation)(transportlinear following theolveS

ij

ij

2313

2212

2111

232221

131211

232221131211

==∀≥=+=+=+

≤++≤++

+++++


(Harker)

5454

Equilibrium Problems:Pure Nonlinear Complementarity Problem

( )

( )

( )

( ) ( )?equivalent forms two theseare why

0xxF,0x,0xF

formin vector or tarity)(complemen i0xF xiii.

i 0 xii. i, 0xF i.

s.t. Ran x find

RR:Ffunction a Having

NCP(F) Problemarity Complement Nonlinear Pure

T

ii

i

i

n

nn

=≥≥

∀=∀≥∀≥∈

→

5555

Equilibrium Problems:Mixed Nonlinear Complementarity Problem

( )

( )( )

n

n

n

i i

i i

Mixed Nonlinear Complementarity Problem MNCP(F,l,u)

Having a function F:R , respectively, lower, upper bound vectors l,u R - ,+ with l u

find an x R s.t. ii. x F x 0 ,

ii. x F x 0

ii

n

i

i i

R and

l

l u

→

∈ ∞ ∞ <

∈ ∀

= ⇒ ≥

< < ⇒ =

U

( )i ii. x F x 0 iu= ⇒ ≤

5656

Max revenues from sales minus production costs, s.t. constraints for production rate and production ceiling are met, and nonnegativity

Producer’s Problem

5757

Producer’s Karush-Kuhn-Tucker (KKT) Conditions

Marginal Producer Profit– Marginal profit complementary to sales:

if sales > 0, marginal profit = 0

Marginal Production Rate• If sales < PR, dual of prod cap a = 0

Production Ceiling• If tot sales < PROD, dual of prod ceiling b = 0

5858

Market Clearing in the Production Market

Producer sales must equal purchases by it’s marketing arm & it’s liquefaction plants.

5959

Back-Up SlidesStochastic Multiobjective Telecommunications

Planning Problem

6060

Numerical Results: Optimum vs. MOP Heuristic Solution (I)

6161

Numerical Results: Optimum vs. MOP Heuristic Solution (II)

6262

Numerical Results: Optimum vs. MOP Heuristic Solution (III)

6363

Numerical Results: Optimum vs. MOP Heuristic Solution (I)

6464

Cost Matrices: Examples

modeling infrastructure and network industries: theory and ...extensions of complementarity...

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