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Networks and Spatial Economics, 3: (2003) 97–122©C 2003 Kluwer Academic Publishers, Manufactured in the Netherlands.

Computational Experience with a Large-Scale,Multi-Period, Spatial Equilibrium Modelof the North American Natural Gas System

STEVEN A. GABRIELProject Management Program, Department of Civil & Environmental Engineering,University of Maryland, College Park, Maryland 20742-3021, USAemail: [email protected]; http://www.eng.umd.edu/∼sgabriel

JULIO MANIKSHREE VIKASEnergy Cluster, ICF Consulting, Fairfax, Virginia 22031-1207, USAemail: [email protected]: [email protected]

Abstract

In this paper we describe numerical results for a market equilibrium model for the North American natural gassystem. The model is based on the notion of maximizing total surplus less transportation costs (Takayama andJudge, 1971; Samuelson, 1952) resulting in a large-scale nonlinear program (Gabriel et al., 2000). The modelrelies on building up supply curves from the “bottom up” using a data base of some 17,000 natural gas reservoirs.This feature provides a good deal of realism in simulating the effects of technology, market forces, and policyconsiderations on the supply side of the market while making the computations challenging due to the lack of closedform supply curves. A successive linear programming strategy is employed to solve the overall nonlinear problem.We describe several mathematical algorithms that are employed in the successive LP approach to efficientlycompute market equilibrium values. These algorithms are heuristic in nature with excellent convergence results.In the numerical results section of this paper, we describe several experiments regarding schemes to acceleratethe overall convergence based on iterative smoothing (similar to a Gauss-Seidel strategy) as well as tests aimed atoptimal spatial and temporal aggregation.

Keywords: Spatial modeling, energy equilibrium, natural gas, energy sector optimization, nonlinear program-ming, successive linear programming, Gauss-Seidel

1. Introduction

The North American natural gas system is an example of a spatially diverse market connectedvia a pipeline network structure. This market includes Canada, the United States, and Mexicoand is well integrated with for example, suppliers in Canada competing against producersin the U.S. for selling natural gas.

Natural gas is used in four main consumption sectors: residential, commercial, industrial,and electric power. Its role in the electric power sector is becoming increasingly moreprominent in the greatly expanding North American electrical power market for two main

98 GABRIEL, MANIK AND VIKAS

reasons. First, gas is relatively clean burning and therefore helps to satisfy environmentalregulations imposed on the electrical power industry. Second, many new gas-fired powerplants are being built (combined cycles and combustion turbines) due to their favorableefficiencies and costs.

In this paper, we describe the Gas Systems Analysis Model (GSAM), a modular, reservoir-based model of the North American natural gas system, developed under the sponsorshipof the U.S. Department of Energy (DOE) to analyze this changing North American naturalgas market. GSAM has been used extensively in both public and private sector analyses ofthe North American natural gas market to answer questions concerning the supply, demand,and transportation sides of the market (Becker et al., 1995; Gabriel et al., 1996, 1998, 2000;Vikas et al., 1996, 1998).

As described in Gabriel et al. (2000), GSAM is a large-scale nonlinear program (NLP)that computes market equilibria based on the notion of maximizing total surplus less trans-portation costs (Takayama and Judge, 1971; Samuelson, 1952). While optimization mod-els of the natural gas market in the U.S. and other countries are certainly not new tothe modeling literature, GSAM’s contribution is novel. Before describing the special fea-tures of this model, we first present a short but significant sample of natural gas mod-els as a way of comparison; omission of other models is made only for purposes ofbrevity.

2. Description of natural gas optimization models

Natural gas optimization models have generally focused on either optimization of gasoperations for a particular entity (gas marketer, utility, etc.) or computation of marketequilibrium prices, flows and quantities. The latter is often accomplished by solving anappropriate optimization problem or sequence of optimization problems. In addition, gasmodels can be divided into those that approximate the nonlinear relationships between flowsthrough an arc, fi j and the pressures pi and p j at the terminal nodes of the arc and thosethat don’t. These relationships can be valid for segments of pipes, compressors, or valves;see O’Neill et al. (1979) and De Wolf and Smeers (1996) for details on these nonlinearrelationships and their use in optimization models. The models that do not directly treatthese flow, pressure relationships generally are more concerned with the “big picture” inthe sense of the gas flows between regions. The actual flow along a particular segment ofpipe is too detailed for such models.

One of the early formulations was the peak-load pricing and investment model for thedomestic gas market in Great Britain (Tzoannos, 1977). This model was based on maximiz-ing the social welfare function (sum of produers’ and consumer’s surplus) while makinguse of the SUMT computational procedure (Fiacco and McCormick, 1968; Mylander et al.,1972). In addition, cost and demand functions were statistically fit for Great Britain. Later,in O’Neill et al. (1979), the authors presented a network optimization model depicting theinterstate pipeline system using a linearization scheme to handle the nonlinear relation-ships between gas flows and pressure in pipelines, compressors, or valves; this model alsoconsidered several objective functions. In two papers (Guldmann, 1983, 1986) both a prob-abilistic as well as a deterministic optimization model were presented which considered

COMPUTATIONAL EXPERIENCE WITH A NATURAL GAS MODEL 99

supply, storage, and service reliability questions from the perspective of the natural gasdistribution utility (with applications to the East Ohio Gas Company). Following in thistrend was the decision support system named “Contract Analyzer” (Avery et al., 1992)which assisted gas utility planners in both strategic as well as operational planning. TheNatural Gas Transmission and Distribution Module (NGTDM), the natural gas componentof the Department of Energy’s National Energy Modeling System (Energy InformationAdministration, 2000) is an example of another model of the whole gas market. Unlike theother models listed, NGTDM, via its Interstate Transmission Submodule, simulates marketequilibrium prices, flows, and quantities via an iterative heuristic algorithm in combinationwith an acyclic hierarchical representation of the main arcs in the gas network. Previousversions of NGTDM made use of a linear programming formulation, which fed into theoverall NEMS equilibrium problem (Gabriel et al., 2001). GRIDNET (Brooks, 1999) is anexample of a generalized network optimization model for gas that contains very detaileddata on pipelines and gas transactions from the gas marketing company’s perspective. Thenetworks in this model tend to be very large and the associated problems are solved usingthe generalized network approaches found in Brown and McBride (1982) and McBride(1985). The Hydrocarbon Supply Model (HSM) produced by Energy and EnvironmentalAnalysis, Inc. (http://www.eea-inc.com/index.html) is a detailed optimization model of theNorth American natural gas market used by the Gas Research Institute (http://www.gri.org)to produce its gas forecasts; in HSM, the natural gas resource is divided by depth class suchas how much resource is in 0–5,000 feet, 5,000–10,000 feet, etc. Lastly, the Market Clear-ing Engine is a detailed gas market model developed for the state of Victoria, Australia(Pepper and Lo, 1999) which combines the detailed engineering restrictions as well asmarket considerations.

GSAM differs from other gas models in several important ways. First, unlike the modelsthat operate from the perspective of the distribution company, GSAM models the entireNorth American natural gas system from wellhead to burner-tip. It shares the characteristicsof market equilibrium models by maximizing total surplus (less transportation costs) but itdoes this with a significant amount of detail on both the upstream as well as downstreamsides of the market.

Perhaps the most unique feature of GSAM as compared to other market equilibriummodels is its database of over 17,000 natural gas production reservoirs each with up to200 variables (porosity, permeability, thickness, etc.). This richness in detail on the up-stream side allows for a full characterization of the economic behavior of reservoir oper-ator decision-making (relative to drilling and other exploration and production activities).In effect, these micro-level data allow the model to build supply curves from the “bot-tom up” by simulating the price responses from individual reservoirs based on break-eventype calculations. This produces a much richer model than other approaches that imposea specific functional form for the supply curves or that treat only aggregate supply fac-tors such as supply by depth (e.g., the Hydrocarbon Supply Model). Indeed, this featureof micro-level detail and simulation of individual reservoir production provides a gooddeal of realism in simulating the effects of technology, market forces, and policy con-siderations on the supply side of the market while necessarily making the computationschallenging.


On the downstream side, GSAM is also detailed but not at the level of individual pipelines,compressors, and valves for which the nonlinear relationships discussed above are valid.GSAM’s time horizon spans 23 years (1998–2020) with each year segmented into four gasseasons based on the concept of a load duration curve. In addition, four demand sectorsare considered: residential, commercial, industrial, and electrical power generation. Thedemand in the first three sectors is based on econometrically derived demand equations.The electrical power demand is based on a dispatching model that selects power planttypes (differentiated by fuel and seasonal burning patterns) with the most cost effectiveplant groups selected first. A gas pipeline network composed of 46 nodes (supply, demand,other), and 79 aggregate pipeline links tie together the supply and demand sides and iscomplemented with a storage reservoir data base of some 500 reserovirs as well as regionalpeak-shaving options (LNG, propane/air).

While the extensive detail on the supply side is perhaps GSAM’s most novel feature,it makes for a computationally challenging model. Due to the lack of closed-form supplycurves as well as the size and nonlinearities involved with the GSAM optimization model,the market equilibrium problem represented by GSAM is solved by a method of successivelinear programming (SLP). Unlike the SLP methods of Zoutendijk and others (Bazaraaet al., 1979), the LP subproblem is not meant to find a feasible direction of descent. Rather,the purposes of the LP are to:

1. Approximate the nonlinear aspects of the optimization model (e.g., integrals replacedby step-functions) and

2. Generate new “guesses” at the equilibrium supply prices around which supply curveswill be generated.

In considering this second purpose of the SLP method, the GSAM approach is similarto a fixed-point algorithm (Ortega and Rheinboldt, 1970) in which one starts with a set ofsupply (and other) prices and tries to get back supply prices that are the same after solvingthe associated LP. The rationale being that if the marginal prices determined by the LP matchthe prices that generated the supply curves, then the supply curve representation was correctso that the original NLP was adequately approximated by the last linear programming prob-lem. In this SLP scheme, each GSAM iteration corresponds to solving a large-scale linearprogram on the order of 1.9 million variables and 80,000 constraints, with supply curvesupdated as well as slight modification of other variables such as storage configurations. Ournumerical experience is based on this iterative approach.

3. Overall approach of this study

In this paper, we report on the computational behavior of heuristic approaches in GSAM’ssolution methodology used to produce equilibrium values and we present encouragingnumerical results concerning two main issues:

1. Speeding up the time to convergence using an iterative smoothing scheme similar to aGauss-Seidel strategy (Ortega and Rheinboldt, 1970) and

2. Measuring the effects of spatial and temporal aggregation on computational times andthe quality of equilibrium solutions.


The first point addresses a heuristic approach that allows for using a weighted combinationof equilibrium price estimates from the last iteration with those generated in the currentiteration. Such an approach is at times necessary since strong oscillation between pricesestimates has been observed in practice. This smoothing strategy is of the Gauss-Seideltype of strategy and tends to mitigate the effect of the swings in the price estimates andtherefore more quickly produce equilibrium values. In our numerical tests, we found that aweight of 50% of the previous iteration’s price estimate plus 50% of the current iteration’svalue is the “safest” convergence strategy.

The second point addresses a concern of many mathematical models that involve regionalor spatially dispersed data as well as many years or time periods. On the one hand, it wouldbe potentially more accurate to allow for the maximum number of spatial and temporaldivisions. However, using the finest regional and time-related breakdowns can result in amodel that is computationally prohibitive. Indeed, that is our experience when all supplyregions are allowed to vary on their own and all years are modeled explicitly in terms ofexploration and production activities.

This paper introduces a heuristic approach, which allows for a smaller number of supplyregions called “mega regions” in which several supply regions are grouped together. Inaddition, the notion of “mega times” is also used in which only a subset of the time periods(“mega times”) have exact computations to produce supply quantities based on detailedpetroleum engineering principles, the other time periods being interpolated. For example,if we considered just the spatial aspects, instead of having two production regions “PacificOnshore” and “Pacific Offshore”, both of these are put in the “Pacific” mega region. Equi-librium price estimates for each individual region are still calculated separately from certaindual prices in the associated linear programs. What is different is that the regional prices ina mega region all move up or down by the same percentage albeit from a different “base”price. We believe that it is important to select these mega regions carefully to correspondto how real gas markets function together.

We tried three levels of regional and temporal aggregations with results that surprisinglydid not differ substantially between the cases. The conclusion is that one can use thecoarsest spatial and temporal aggregations resulting in much faster computational timeswith essentially the same results as if one had used the finest possible aggregations.

These results are very encouraging since the finest level of spatial and temporal divisionresulted in computational times of around 33 hours per iteration on a Pentium III, 933Megahertz machine with 500 Megabytes of RAM. The coarsest level of spatial aggregationneeded only about 1.5 hours/iteration with no appreciable difference in the results. (Eachiteration consists of generating supply curves for each supply region and year and thensolving the associated linear program1). We anticipate that other researchers may findcomparable savings in time by applying similar aggregation schemes in their multi-regional,multi-temporal models.

The organization of the rest of this paper is as follows: in Section 4 we provide anoverview of GSAM; in Section 5 we discuss procedures for computing a market equilibriumwhile maintaining consistent supply curves; in Section 6 we discuss the specific heuristicapproach for generating supply curve points; and in Section 7 we present our numericalresults.


4. Overview of GSAM: An equilibrium model of the North American naturalgas market

GSAM is a modular, reservoir-based model of the North American natural gas system.On the upstream side of the market, GSAM has a natural gas resource base character-ized by a host of explicit reservoir geologic properties (porosity, permeability, thickness,etc.) and operational characteristics for over 17,000 individual gas reservoirs representedin 23 production supply regions as well as several regions of other supply type. Sincethe characteristics of potential North American gas supplies are specified at this level ofaggregation, GSAM is free from restrictive assumptions such as pre-specified, regional,average supply curves normally imposed by traditional gas market models. Instead, GSAMbuilds up supply curves from the “bottom up” with calls to its Exploration and Produc-tion (E&P) Module based on petroleum engineering principles such as fluid flow underporous formations (e.g., Darcy’s Law); see for example (Craft and Hawkins, 1959). Inaddition, this approach, since it computes all the supply curves for all the time periodstogether, captures the inter-temporal nature of production of natural gas. Namely, sup-ply in year y being not only a function of the price in year y but also a function ofthe prices in years 1, 2, . . . , y − 1 due to how much gas was produced in these earlieryears.

On the downstream side of the market, GSAM consists of 16 North American demandregions with 79 transportation links connecting supply and demand regions. These linksrepresent collections of pipelines serving the regions (or other intermediate nodes) in ques-tion. The demand for natural gas is characterized by four sectors: residential, commercial,industrial, and electric power generation. For the first three sectors, econometrically de-rived models have been used to forecast demand for natural gas. The electrical power sectoris represented by an aggregate dispatching model simulating fuel competition as well asenvironmental aspects of producing electricity. These four sectors are combined with fourseasons plus consideration of over 500 natural gas storage reservoirs, as well as peak-shavingoptions such as LNG and propane/air.

The upstream and downstream sides of the market are brought into balance by a large-scale, integrating nonlinear program based on the economic concept of maximizing totalsurplus in the market, less transportation costs subject to network and other constraints.The classical approach of maximizing the sum of producers’ and consumers’ surplus afterdeducting for transportation costs (Samuelson, 1952; Takayama and Judge, 1971; Labysand Pollak, 1984) is based on maximizing the following expression2

∑i

( ∫ di

0pD

i (di )dηi −∫ si

0pS

i (si ) dξi

)−

∑i

∑j

ti j xi j (1)

where

xi j = the flow of the product from region i to region j

ti j = the unit transportation costs for flow between regions i and j

pDi (di ) = the inverse demand function for a demand level of di in region i

pSi (si ) = the inverse supply function for a supply level of si in region i


Because the supply curves are generated from the “bottom up” and are thus not knownin closed form, as well as the inter-temporal linkages in gas production described above,the integrating nonlinear program is approximated using a successive linear programmingapproach; see Gabriel et al. (2000) for details on the mathematical formulation of the modelas well as the economic equilibrium theory underlying it.

Each GSAM iteration consists of generating supply curves for each region and year,passing them to the Integrating LP, and generating new estimates of market prices fromcertain dual variable values to the material balance constraints in the LP. For a supply regionn, time period t , and season s, these material balance constraints expressed in millions ofcubic feet/day (MMcf/day) state that the total gas produced, plus gas from extra supplyprojects, plus the gas transported in along pipelines equals the total gas leaving the regionby pipeline. Mathematically these constraints are of the form:

∑k

GPntsk +∑

e∈ES(n)

∑t ′≤t

ESt ′se +∑

p=( j,n)

{(1 − PLLOSSp)FFLpts − RFLpts}

=∑

p=(n,m)

{FFLpts − (1 − PLLOSSp) RFLpts} (2)

where

n, m = (node) region indicest = (time) year indexs = natural gas season index (1, 2, 3, or 4)k = supply curve increment indexp = aggregate pipeline link indexe = extra supply project indexGPntsk = gas production supply increment (MMcf/day)PLLOSSp = loss of gas along pipeline due to leakage and fueling the compressors (%)FFLpts = forward (standard3) flows of gas along pipeline (MMcf/day)RFLpts = reverse flow of gas along pipeline (MMcf/day)EStse = incremental extra supply project usage (MMcf/day)ES(n) = set of extra supply projects available at node n

GSAM includes similar material balance constraints for demand regions which take intoaccount injection and extraction from storage as well. The dual variables to the materialbalance constraints measure the marginal value to the gas system of one more unit of gasfrom the marginal source. From this point of view they can be thought of as estimates ofmarket equilibrium prices. Note that since GSAM works at the level of pipeline aggregates,only the flow between regions is modeled. The nonlinear flow-pressure relationships asdiscussed in O’Neill et al. (1979) and De Wolf and Smeers (1996) are not necessary in thissetting.

The organization of the various GSAM submodules is shown on the next page (figure 1).The main function of the Exploration and Production (E&P) module is to produce a

supply curve for each supply region in each year being modeled. There are three maincomputational issues related to generating these supply curves:


-Optimal flows-Equilibrium prices and quantities

-Investment & operating costs -Deliverability -Fuel usage

-Gas deliverability profiles-Summary economics

Resource Module

Reservoir PerformanceModule

Exploration andProductionModule

Production andAccountingModule

Demand andIntegratingModules

-Geological characterization-Drilling & production histories

-Supply curve elements (price & quantity pairs by region & year)

-Equilibrium gas prices (by region & year)-Final gas production

-Gas prices

-Final pro-forma cash flow at national, regional and/or state level

-Investment Decisions

Storage ReservoirPerformanceModule

Figure 1. Major components of the gas systems analysis model.

1. Lack of a closed-form expression for the supply curves,2. Inter-temporal dependence in producing supply curves, and3. Inter-regional dependence in producing supply curves.

Instead of relying on a specific functional form for regional supply curves, GSAM cre-ates them by repeated calls to the E&P module. In particular, the curves are generated byfirst taking estimates of market prices in all the regions and for all years and then passingthese prices to the E&P module. Given market prices, this module simulates the explo-ration and production activities of the reservoir operators and determines from the “bottomup”, the production in each supply region. In effect, the market price for a given regionand year is compared to the break-even price faced by reservoir operators. If the marketprice is less than the break-even price, no exploration or production activity results; oth-erwise, there is E&P activity. This approach assumes an economically rational reservoiroperator.

An example of four price values and their corresponding supply quantities for a typicalregion and year are shown in figure 2.

A typical GSAM iteration consists of the following steps in which a “price vector” is oneset of natural gas prices for each supply region and year.


price

quantity

p1

p2

p3

p4

q1 q2 q3 q4

price

quantity

p1

p2

p3

p4

q1 q2 q3 q4

Figure 2. A sample supply curve for a given region and year.

Algorithm 1: GSAM Algorithm For Computing Equilibrium Price and QuantityValues (Overview)

Step 1. [Initialization of gas price vectors]Having a “base case” set of price values, generate the next natural gas prices for eachsupply region in each year for, i.e., the next price vector. (The first time through the“base case” prices are themselves used).

Step 2. [Generate individual points on the supply curves]Using the current price vector, invoke the E&P module to simulate the productionresponse in all supply regions and for all years. If desired number of price vectorsmet, go to Step 3, otherwise go to Step 1.

Step 3. [Generate supply curve representation for the Integrating LP]Combine the (price, quantity) values to form supply curves in each region and year.Interpolate additional points to complete the supply curve representation in the Inte-grating LP.

Step 4. [Solve the Integrating LP and check for convergence]Using the supply curves created in Step 3, solve the associated LP. Check for con-vergence in prices. If convergence criteria met, then STOP. The dual variables to thematerial balance constraints of the LP provide estimates of market equilibrium prices.Otherwise, create new base case prices from these dual variables and go to Step 1.

We see that Algorithm 1 has the following mathematical form:At iteration i , having the vector of costs ci approximating (1), we solve the linear program

min(ci )T x

s.t A1x = b1

A2x = bi2 (3)

x ≥ 0

Since the new supply curves that are generated in Step 3 of the algorithm affect theobjective function via the step-function approximation to (1), the cost vector ci can change


each iteration. Additionally, the matrix A2 and vector bi2 represent the coefficients for the

other constraints such as link capacity restrictions, storage balance, storage extraction limitsby season, etc.; also bounds on the variables are included in this format. The vector bi

2 is notconstant since, for example, market price estimates which can change each iteration havean impact on the storage extraction profiles.

In determining market price estimates, the values to the dual variables to the materialbalance constraints for supply region n and time t , and season s are volume-weighted toproduce annual price estimates {π i

nt }. These latter annual prices are then compared to theprevious iteration’s values. If there is not significant change, than the set of prices {π i

nt } aretaken as annual market equilibrium values with associated supply and demand quantitiesas computed by the integrating LP. If there is significant change, then these prices form the“base case” values around which other prices are generated. The new set of prices is sent tothe E&P module to produce consistent supply quantities and the whole process starts againwith ci , A1, b1, A2, bi

2 replaced by ci+1, A1, b1, A2, bi+12 . The values to the dual variables

to the material balance constraints can be viewed as the market prices in the sense that thisis how much the market would pay for one more unit of gas; this follows directly from LPduality theory.

From the discussion above, Algorithm 1 can be construed as a fixed point algorithm(Ortega and Rheinboldt, 1970), i.e., for a function F : Rn → Rn ,

find a point x̄ ∈ Rnsuch that F(x̄) = x̄ . (4)

Algorithm 1 starts with a vector of supply prices π = {πnt }, for which supply quantitiesare computed in the E&P module. Then, it is desired to have the same vector of pricesoutput as the (annualized) optimal dual values to the dual variables of the material balanceconstraints. Thus, the function F is the composition of generating supply quantities from theprices (F1), solving the associated linear program (F2) and finally converting the seasonaldual prices to volume-weighted annual values (F3). Thus, GSAM’s overall algorithm canbe considered as finding the fixed point of the function F(π ) = F3(F2(F1(π ))).

5. Procedures for computing an equilibrium in GSAM using regional supplycurves in the integrating LP

5.1. Inter-temporal and inter-spatial dependence of supply curves

One of the main computational complications in GSAM is the fact that supply curves needto be generated for each iteration; that is they are not known in closed form. Given a marketprice for natural gas in a particular region and year, GSAM simulates the exploration andproduction activities of reservoir operators to arrive at supply quantities. Thus, at eachiteration, these regional supply curves are built from the “bottom up”.

Adding to this computational complexity is that these supply curves are intertemporallyand to some extent, inter-regionally dependent. The dependence over time arises since inpart, the price of gas in year y is influenced by the price of gas in years 1, 2, . . . , y − 1 dueto the amount of resource that was extracted from the reservoir in these preceding years.The inter-regional dependence arises since regional production depends on available rigswhich migrate between regions based on demand.


price

years1998 1999 2000

price

quantity

high price(H)

base price (B)

low price (L)

fixed price forearlieryear

supply curvefor the year 1999

L

B

H

price

years1998 1999 2000

price

quantity

high price(H)

base price (B)

low price (L)

fixed price forearlieryear

supply curvefor the year 1999

L

B

H

Figure 3. Generation of the supply curves.

What this means is that the computation of the supply curves must be carefully doneto produce “consistent” price, quantity relationships in the integrating LP. By consistent,we mean that higher prices give higher (or the same) supply quantities. As described inGabriel et al. (2000), we employ a method, which for each region and time period, startswith a base price value, and an associated supply quantity is calculated. This base price isthen increased or decreased by a factor, just for that region and time with values from allother regions and years being held at their base levels; the corresponding supply quantitiesare then calculated. As shown in the figure above, the result is a consistent supply curve. Itis important to note that if the previous year’s prices as well as prices from other regionsare not held constant in this fashion, what may result is a supply curve which exhibitsa typical behavior such as supply decreasing as a function of price; this would adverselyinfluence the equilibrium calculations. Figure 3 shows an example of three supply prices andquantities.

5.2. Aggregating individual supply regions into mega regions

An additional complication is that for computational reasons, it is necessary to considera limited number of combined or “mega” supply regions. If all individual supply regionsare considered separately, the computational times to calculate equilibrium values wouldbe prohibitively high. See Section 7.2 for details on computational times using differentregional and temporal aggregation strategies. Note that there is an assumption that megaregions are selected to have the prices in the individual regions within them all move upor down together by the same factor. However, the base values that each individual regionstarts from can be different.

For example, consider the following sample designation of mega regions shown in Table 1,used to reduce the overall computational time yet retain realistic accuracy (this is the coarsestaggregation as explained in Section 7):


Table 1. Crosswalk between GSAM supply regions and GSAM mega regions.

GSAM supply region finest level Mega region name coarsest level

Pacific offshore Pacific

Pacific onshore Pacific

San Juan Rocky Mt.

Rockies Foreland Rocky Mt.

Williston Rocky Mt.

Permian Southwest, Mid-Continent, Appalachia

Arkla-East Texas Southwest, Mid-Continent, Appalachia

Mid-Continent Southwest, Mid-Continent, Appalachia

Appalachia Southwest, Mid-Continent, Appalachia

Mid-West Southwest, Mid-Continent, Appalachia

Texas Gulf Coast Gulf Coast

Mexico Gulf Coast

Gulf of Mexico-West Gulf Coast

Gulf of Mexico-Central Gulf Coast

Gulf of Mexico-East Gulf Coast

Norphlet Gulf Coast

South Louisiana Gulf Coast

MAFLA onshore Gulf Coast

Alberta Western Canada

British Columbia Western Canada

North Alaska Non-producing

MacKenzie Delta Non-producing

Atlantic offshore Non-producing

5.3. Aggregating individual time periods into mega times

Besides the spatial aggregation scheme discussed in the previous section, another step usedto limit the computations is to select a key set of years for which the supply quantities are gen-erated using the E&P module and then to interpolate for the remaining years in the study. Forconsistency with the supply region terminology, we call these selected years “mega times”.

The last step before solving the integrating LP is to insert additional supply points betweentime periods and existing price-quantity pairs and then write these points to the supply price,quantity file. For example, if four time periods (years) were selected, 1998, 2005, 2012,2020, then these years would have exact calculations using the E&P module with interveningyears interpolated for prices and quantities. For adding price vectors, if three were chosen,representing a high, base, and low case, but a total of 20 were desired, then 17 interpolatedprice, quantity points would be generated.

Inserting price and quantity data between time periods is necessary to provide the integrat-ing LP with yearly supply information. Adding more price, quantity pairs to the calculated


supply data is optional. The purpose of interpolation for price, quantity pairs is to providegradual stepwise changes between main supply points in the supply curves. In this paper, alinear interpolation technique was used to calculate the additional supply points.

6. Methodology used to update supply curves

In what follows, we present more formally, the heuristic described above for generating eachGSAM iteration’s data. Some of these data are user-defined to allow the model to allowflexibility in the computations. The overall flow of the process is given below in figure 4.

CONVERGED ?

YES

Generate Linear Program Data

NO

STOP

Generate Consistent Supply Curve Elements for Each Region and Time Period

Solve Linear Program

Create Reports & Check for Convergence

Figure 4. Flowchart of overall GSAM algorithm.

We describe two algorithms below which provide the details of this whole process. Forthe algorithms described below, we use the following notation.

Times

Time PeriodsY = {1998, 1999, 2000, . . . , 2020} the model years


y = generic year for the model, y ∈ Yyb = beginning year for the model [user-defined], example: 1998ye = ending year for the model [user-defined], example: 2020 where yb ≤ ye

Indices for Time PeriodsT = {1, 2, . . . , 23}t = index for year, t ∈ Twhere t = 1 ↔ y = 1998, t = 2 ↔ y = 1999, . . . , t = 23, ↔ y = 2020

Original Years and Indices Selected by the User for Use in the E&P Module (“mega times”)Y ′ ⊆ Ywhere Y ′ contains {yb ye} and usually Y ′ contains additional years as well. Example: Y ′ =

{1998, 2005, 2017, 2020}T ′ ⊆ Twhere T ′ contains the indices for Y ′

Years and Indices Not Selected by the User for Use in the E&P ModuleY ′′ ⊆ Y where Y ′′ = Y/Y ′

T ′′ ⊆ T where T ′′ contains the indices for Y ′′

Supply Curve Price Increments

Supply Curve Price IncrementsK = {1, 2, . . . , nk}, where nk is user-definedexample: K = {1, 2, . . . , 20}Supply Curve Price Increments Selected by the User for Use in the E&P ModuleK ′ ⊆ K , |K ′| ≥ 3, where K ′ is user-definedexample: K ′ = {1, 2, . . . , 5}Supply Curve Price Increments Not Selected by the User for Use in the E&P ModuleK ′′ ⊆ Kwhere K ′′ = K\K ′

By convention, the values in K ′ and K ′′ are ordered asK ′ = {1, 2, . . . , |K ′|} and K ′′ = {|K ′| + 1, |K ′| + 2, . . . , nk}

Supply Regions

Supply RegionsSR = {Pacific Offshore, Gulf of Mexico-West, Alberta, British Columbia, . . .}Indices for the Supply Regions, S = {1, 2, . . . , ns}“Mega” Supply Regions, example: MSR = {Pacific, Gulf Coast, Western Canada, . . .}

[user-defined]

Indices for the Mega Supply RegionsM = {1, 2, . . . , nm}, SRm = Set of supply region indices in mega region mwhere


∪nmm=1SRm = S and SRm ∩ SRm ′ = ∅ for m �= m ′,

i.e., the mega regions are a partition of the set of supply regions

Prices and Supply Quantities

pricestk = natural gas price for supply region s, time t , increment k ($/Mcf)4

quantitystk = natural gas supply quantity for supply region s, time t , increment k (Bcf)5

Algorithm 1: GSAM Algorithm For Computing Equilibrium Price and QuantityValues (Detailed View)

Step 1Generate supply price, quantity pairs for each supply region s, time t , & supply curveincrement k ′ ∈ K ′.

Step 2Insert additional price, quantity points between time periods:

for all s ∈ S, t ∈ T, k ′ ∈ K ′, set(pricestk ′

quantitystk ′

)=

(w1 ∗ pricest ′k ′ + w2 ∗ prices(t ′+1)k ′

w̄1 ∗ quantityst ′k ′ + w̄2 ∗ quantitys(t ′+1)k ′

)

where t ′ ∈ T ′, w1, w2 ≥ 0; w1 + w2 = 1, w̄1, w̄2 ≥ 0; w̄1+w̄2 = 1, t ∈ T, t ∈ [t ′, t ′+1]Step 3

Insert additional price, quantity points between original price, quantity points:

for all s ∈ S, t ∈ T, k ′ ∈ K ′, set(pricestk

quantitystk

)=

(ν1 ∗ pricestk ′ + ν2 ∗ pricest(k ′+1)

ν̄1 ∗ quantitystk ′ + ν̄2 ∗ quantityst(k ′+1)

)

where k ′ ∈ K ′; ν1, ν2 ≥ 0; ν1 + ν2 = 1, ν̄1, ν̄2 ≥ 0; ν̄1 + ν̄2 = 1, k ∈ [k ′, k ′ + 1]Step 4

Generate LP dataStep 5

Solve LPStep 6

Check for ConvergenceIf convergence achieved, then STOP.Otherwise, generate new prices from dual variable values to the material balanceconstraints and return to Step 1.

The next algorithm describes the methodology used in Step 6 when new prices needto be generated. Having converted the seasonal dual prices associated with the materialbalance constraints in the integrating LP into annual values (e.g., by volume-weighting),these annual estimates of market prices are then used as “base prices” from which valueshigher and lower are generated. The base prices taken together with the other values that


are generated in this fashion will form the set of prices sent to the E&P module in order tocompute associated production quantities. These price, quantity pairs will thus provide thebasis of the supply curves in the next iteration.

The choice of these new prices is critical to the convergence of the overall algorithm.In essence, the method assumes that the latest dual price values are close to the eventualmarket equilibrium prices. These extra points above and below the dual values are meant toensure that the computed supply curves give a sufficiently large range so that the equilibriumsolution is contained within the range of the prices and quantities. In essence, the methodis guessing where the equilibrium prices should be and tries to cover that range with newlygenerated prices.

As described in Algorithm 2 below, the choice of where new prices (above or below thebase price) are generated is decided upon by considering the largest gap between successiveprices generated up to that point. If there is a large gap in two successive prices, this couldadversely affect the representativeness of the supply curve to be subsequently generated.Thus, Algorithm 2 seeks to fill in with new points, exactly where there are the biggest gaps.

Algorithm 2: Generating New Prices in Step 6 of Algorithm 1

Having the input price πst (the annualized dual price for supply region s and year t), wegenerate the remaining nk − 1 prices for region s in year t as follows:Note:

1. minfactor, maxfactor are positive user-defined values with minfactor < maxfactor.2. We assume that |K ′| ≥ 3

Step 0: Calculate and Assign Prices for Price Vectors 2 and 3Set basepricest = πst .Set current set of prices asP = {basepricest , basepricest ∗ minfactor, basepricest ∗ maxfactor}.Sort P in increasing order to get the set P̂ = {pricest1, pricest2, pricest3}.Set number of current prices as j = 3.

Step 1: Compute GapsIf number of current prices j > nk , then STOP. The set P̂ contains the new prices.Otherwise, compute the gaps {gapst1, gapst2, . . . , gapst( j−1)}where gapstk = |pricest(k+1) − pricestk |

Step 2: Generate New PointGenerate the new point pricest( j+1) as the midpoint of the largest gap.6

Update the set P = P ∪ {pricest( j+1)}Sort P to obtain P̂ .Update number of current prices: j = j + 1.Go to Step 1.

Clearly many other numerical schemes for generating the new prices are possible.Algorithm 2 tends to favor filling in gaps in the price ranges so as to minimize the “uncov-ered” price range when computing associated supply quantities in the E&P module.


7. Numerical results

In this section, we discuss numerical results concerning two computational issues withGSAM:

1. Speeding up calculations using an iterative smoothing scheme similar to a Gauss-Seidelstrategy, and

2. Measuring the effects of spatial and temporal aggregation on computational times andthe quality of equilibrium solutions.

To know when the GSAM iterative procedure outlined in Algorithm 1 has converged toa price equilibrium, we can use the following convergence tests:7

Convergence is achieved if

1. max n,t

{∣∣pist − pi−1

st

∣∣∣∣pi−1st

∣∣}

≤ d (Maximum percent change) or

2.

{ ∑n

∑t

( |pist −pi−1

st ||pi−1

st |)}

N ∗ T≤ d (Average percent change) or

where

pist = the price in supply region s, time t, for GSAM iteration i (in $/Mcf)

d = convergence tolerance (percentage)N = number of nodes (regions)T = number of time periods

Clearly method number 1 is a stricter method than number 2. However, for purposes ofexamining the convergence behavior for both options, we ran the method for seven iterations.As can be seen from Table 2, seven iterations was generally quite sufficient for averagepercent change convergence and was adequate for the stricter maximum convergence test.In this maximum convergence test, one region for one year can throw off the convergenceand so in practice, method number 2 is probably more reasonable to use to measure if ingeneral, the process is converging.

7.1. Speeding up calculations with an iterative smoothing scheme

The first issue we examined involved using a weighted combination of equilibrium priceestimates from the last iteration with those generated in the current iteration and as suchrepresents a Gauss-Seidel type of strategy (Ortega and Rheinboldt, 1970). The Gauss-Seideltype of relaxation scheme we have employed uses the following iterative relationship

pist = ω ∗ (

pi−1st

) + (1 − ω) ∗ (π i

st

)(5)

where

ω = the relaxation or smoothing parameter in [0, 1]π i

st = volume-weighted annual dual price to the material balance constraints for supplyregion n, time t, GSAM iteration i ($/Mcf)


Table 2. Summary of GSAM tests 1–12 for iterative smoothing.

Smoothing No. of mega No. of mega Avg.% change Max. % changeTest no. Starting point parameter ω times regions after 7 iter. after 7 iter.

1 $1/Mcf 0.20 5 6 1.8 7.5

2 $1/Mcf 0.50 5 6 0.9 3.7

3 $1/Mcf 0.95 5 6 0.6 1.7

4 $1/Mcf 0.0 5 6 2.4 10.1

5 $3/Mcf 0.20 5 6 1.8 7.5

6 $3/Mcf 0.50 5 6 0.9 3.7

7 $3/Mcf 0.95 5 6 0.7 1.7

8 $3/Mcf 0.0 5 6 2.5 10.1

9 $5/Mcf 0.20 5 6 1.8 7.5

10 $5/Mcf 0.50 5 6 1.0 3.8

11 $5/Mcf 0.95 5 6 0.7 1.7

12 $5/Mcf 0.0 5 6 2.5 10.1

Such an approach was at times necessary to dampen observed oscillatory behavior of theequilibrium price estimates. To determine the most efficient value of the smoothing param-eter, it was necessary to select representative values and observe the associated convergencebehavior. We used four smoothing values (0.0, 0.2, 0.5, and 0.95) in conjunction with a va-riety of starting points to determine which smoothing value was “best”. It was believed thatthese four values were sufficiently spread out to cover the entire interval of [0, 1] from whichthe smoothing parameter should be chosen. In our numerical tests, we found that a weightof 50% of the previous iteration’s price estimates plus 50% of the current iteration’s valuesproduced the “safest” convergence strategy. Tests 1–12 shown above were used to examinethis first issue. Note that “starting point” is the price value for all regions in all years.

It is important to mention that while the 0.95 value for smoothing produced price estimatesthat didn’t change much from one iteration to the next (1.7% for the maximum percentagechange), this should not be confused with convergence, only a slowly changing set of prices.As shown below, the prices at the end of the seventh iteration for the other smoothing valueswere more indicative of the equilibrium value.

The primary conclusions from the numerical experiments using iterative smoothing areas follows:

1. By the end of 4th iteration, all regions converged within a 4% tolerance (around $0.10/Mcf) of their associated “equilibrium” values8 irrespective of the ω value (except forω = 0.95). Using the smoothing parameter values of 0.0 and 0.20 convergence wasachieved faster than for ω = 0.5. See figure 5, which graphs the percentage of region andyear combinations whose prices have converged at each iteration.

2. While smoothing values of 0.0 and 0.20 are “faster” relative to convergence in the sensedescribed above, these values can lead to price oscillations from one iteration to thenext. A smoothing value of 0.5 appears to be a “safer” strategy. The phenomenon of


Figure 5. Rate of convergence of all producing regions as a function of the smoothing value.

Absolute Value of % Deviation From Equilibrium

Alberta, year 2010 (3$ starting price)

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7

Iteration

%

w=0.0

w=0.2

w=0.5

Figure 6. Rate of convergence as a function of the smoothing parameter for the Alberta supply region in 2010.

oscillating price estimates is shown in figure 6 for the Alberta supply region in the year2010 at a starting price of $3.0/Mcf. The vertical axis shows the absolute percentagedeviation from the “equilibrium” value by iteration. Note that figure 6 does not show aconvergence pattern for ω = 0.95 because it did not yet converge within seven iterations.

3. The rate of convergence of ω = 0.95 is quite slow and does not yet converge in seveniterations. It appears, that if the process were continued, ω = 0.95 might eventuallyconverge but in an enormously large number of iterations.

4. Using our heuristic approaches, the results appear to be independent of the startingpoints. Indeed, using three different starting prices, $1.0/Mcf, $3.0/Mcf and $5.0/Mcf


for all regions and time periods, we see that the process converges at more or less thesame rate for each of these starting points.

7.2. Speeding up calculations with spatial and temporal aggregation

The second issue addressed a common concern of many mathematical models that involveregional or spatially dispersed data as well as many years or time periods. All else beingequal, it would be more accurate to allow for the maximum number of spatial and temporaldivisions in GSAM (or similar models). However, using the finest level of regions andyears can be computationally prohibitive. Indeed, the finest level of spatial and temporalaggregation resulted in GSAM iterations of approximately 33 hours each! This is a largeamount of time given that a fast computer was used; a Pentium III, 933 Megahertz machinewith 500 Megabytes of RAM. The integrating linear program that is solved sequentially hasapproximately 1.9 million variables and 80,000 constraints and is solved using XPRESS-MP©R (Version 12) from Dash Optimization.

The medium level of aggregation averaged about 12 hours/iteration and the coarsest levelaggregation took about 1.5 hours/iteration. Note that each iteration consisted of generatingsupply curves for each supply region and year and then solving the associated linear program.As shown below in Table 3, the vast amount of time is devoted to preparing the supply curveswith the LP solution time a small piece of the total time. In fact, the solution times for the LPare almost constant at 20 minutes. The reason for this is because the structure of the LP doesnot change from one iteration to the next. It is only the data on the supply curve elementsthat change depending on the spatial and temporal aggregation. When less aggregation isused, the supply curve price, quantity pairs more accurately reflect the supply function forthe region and time period involved than compared to the coarser aggregation levels.

When the supply regions were collapsed into the fewest number of “mega regions” and“mega times”, the computational times decreased dramatically but with essentially the sameresults as the finest level. In all, three aggregation schemes were run as follows:

1. The finest level had 23 mega regions, i.e., each supply region separate and 23 megatimes in which each year from 1998–2020 was included in explicit supply calculations(Test 13)

2. The medium level had 14 mega regions and 13 mega times (Test 14) selected to fairlyrepresent the natural gas market

Table 3. Division of computational time.

Approximate % of iteration’s Approximate % of iteraions’stime devoted to producing time devoted to preparing

Aggregation level supply curves and solving linear program

Coarsest 80 20

Medium 98 2

Finest 99 1


Table 4. Spatial aggregation.

GSAM Supply region finest level Mega region name medium level Mega region name coarsest level

Pacific Offshore Pacific Pacific

Pacific Onshore Pacific Pacific

San Juan Rocky Mt. Rocky Mt.

Rockies Foreland Rocky Mt. Rocky Mt.

Williston Williston Rocky Mt.

Permian Southwest Southwest, Mid-Continent, Appalachia

Arkla-East Texas Southwest Southwest, Mid-Continent, Appalachia

Mid-Continent Mid-Continent Southwest, Mid-Continent, Appalachia

Appalachia Appalachia Southwest, Mid-Continent, Appalachia

Mid-West Mid-West Southwest, Mid-Continent, Appalachia

Texas Gulf Coast Gulf Coast Gulf Coast

Mexico Gulf Coast Gulf Coast

Gulf of Mexico-West Gulf of Mexico Gulf Coast

Gulf of Mexico-Central Gulf of Mexico Gulf Coast

Gulf of Mexico-East Gulf of Mexico Gulf Coast

Norphlet Gulf of Mexico Gulf Coast

South Louisiana South Louisiana Gulf Coast

MAFLA Onshore MAFLA Onshore Gulf Coast

Alberta Alberta Western Canada

British Columbia British Columbia Western Canada

North Alaska Non-producing Non-producing

MacKenzie Delta Non-producing Non-producing

Atlantic Offshore Non-producing Non-producing

Table 5. Temporal aggregation.

Mega time period finest level Mega time period medium level Mega time period coarsest level

1998–2020, All years included 1998, 2000, 2002, 2004, 2005, 1998, 2005, 2010, 20202006, 2008, 2010, 2012,2014, 2016, 2018, 2020

3. The coarsest level had 6 mega regions and 4 mega times (Test 6) selected to fairlyrepresent the natural gas market.

Table 4 shows the crosswalk between the GSAM supply regions (the finest level of spatialaggregation) and the two coarser levels. Table 5 depicts the temporal aggregation.

To understand the importance of spatial aggregation, we performed three simulation runswith $3.0/Mcf starting prices and a smoothing parameter value of 0.5. The first test (i.eTest #13) is considered to be the “base case” since it represented the finest level of division(indeed no aggregations at all) and the other two tests (Test #6 and Test #14) are comparedto this “base case”.


Figure 7. Convergence pattern as a function of spatial aggregation for gas production.

Figure 8. Convergence pattern as a function of spatial aggregation for gas price.

Figures 7 and 8 show gas production and gas price convergence behavior for the mediumand coarse levels of aggregation. The vertical axis shows the % of observations (regionsand time periods) that converged within the corresponding tolerance value in the x-axis.Convergence is calculated off of the “base case” values (i.e., Test #13).

In considering both prices and production, for the coarse aggregation (Test #6), over 90%of the values converged within 5% of the base case values and the remaining convergedwithin 25%. For the medium level aggregation (Test #14), over 98% of the values convergedwithin 5% of the base values and the remaining converged within 15%. Over 99% ofthe values converged within 15% tolerance in the coarse case reflecting a sound solutiontechnique. This indicates that the coarse level of aggregation as designed in the study resultsin an acceptable level of accuracy with a significant improvement in the run time (1.5 hoursvs. 33 hours per GSAM iteration).

As shown in Table 6, one can see that the % change in the prices from one iterationto the next was the same at the seventh iteration no matter what level of aggregation


Table 6. Summary of GSAM tests 6, 13, 14 for spatial and temporal aggregation.

Starting Smoothing No. of mega No. of mega Avg. % change max. % changeTest # point parameter ω times regions after 7 iter. after 7 iter.

6 $3/Mcf 0.50 5 6 0.9 3.7

13 $3/Mcf 0.50 23 23 0.9 3.7

14 $3/Mcf 0.50 13 14 0.9 3.7

was used. This is additional information supporting that these three aggregations levelsdid not significantly affect the quality of the computations. Also, it is important to notethat we believe that the manner in which the regions are grouped into mega regions ishowever important. Based on our knowledge of the natural gas industry, we have groupedtogether regions with similar geographical and market characteristics into the same megaregion.

In addition to supply in different regions and mega regions, we have also investigatedflows out of the production regions Rockies Foreland and Alberta in the year 2020 for thethree levels of spatial and temporal aggregation. Tables 7 and 8 show the percentage of theflow out of Rockies Foreland and Alberta to different regions for the three cases. Thesetables indicate that aggregation as used in the coarse case (i.e. Test #6) provided resultsfairly close to the finest level of aggregation. For example, in the year 2020, the flow out of

Table 7. Percentage of flow from Rockies Foreland to various regions in the year 2020.

Origin Destination Coarse (Test #6) Medium (Test #14) Fine (Test #13)

Rockies Foreland California 6 6 6

Rockies Foreland Mountain North 62 59 57

Rockies Foreland Pacific Northwest 20 23 25

Rockies Foreland San Juan 8 8 8

Rockies Foreland West North Central 4 4 3

Table 8. Percentage of flow from Alberta to various regions in the year 2020.

Origin Destination Coarse (Test #6) Medium (Test #14) Fine (Test #13)

Alberta Alliance-supply 0 1 1

Alberta BC-demand 4 4 4

Alberta Canada-East 16 16 16

Alberta East North Central 11 11 11

Alberta Pacific Northwest 2 1 1

Alberta West North Central 53 55 55

Alberta Western Canada 14 12 12


Rockies Foreland to the Pacific Northwest region is 25% of the total flow out of RockiesForeland in the fine case, compared to 23% in the medium aggregation case and 20% inthe coarse case. Similarly, in the year 2020, flow out of Alberta into the West North Centraldemand region is 55% for the fine and medium aggregation cases; the coarse case predictsthe flow to be 53%. It should be noted that before the year 2020, the flow patterns out ofRockies Foreland and Alberta were exactly same for the coarse, medium and fine cases.

8. Summary

In this paper, we have presented the Gas Systems Analysis Model (GSAM), a large-scaleoptimization model for determining the market equilibrium prices, quantities, and flows forthe North American natural gas market. This model builds supply curves from the “bottomup” by simulating the price response for over 17,000 natural gas reservoirs. This superiormodeling feature however, makes for challenging computations since supply curves arenot known in closed form. To overcome these and other computational issues a successivelinear programming methodology is employed.

Also, we have provided numerical results concerning two tests with GSAM. The firsttest concerned adoption of a Gauss-Seidel type method (Ortega and Rheinboldt, 1970) forimproving convergence. Namely, what is the “best” value of the smoothing parameter, usedto weight the equilibrium price estimates from the previous and the current iteration. Ourresults indicate that if the weights of 50% for the current price estimate and 50% for the lastiteration’s estimate are used, this strategy is the safest and ends up with the best convergencetimes overall.

The second test was aimed at determining how the coarseness of the aggregation ofregions and time periods affected the overall quality of the solution. Our results indicatedthat the coarsest level tested provided high quality answers with a very significant speedupof the computations.

Acknowledgments

We would like to thank the anonymous referees for their helpful suggestions.

Notes

1. Solving the LP includes writing out relevant reporting files.2. GSAM uses a discounted version of the objective function in (1).3. The directions are needed to correctly apply the fuel loss factor PLLOSSp (applied at the destinating node of

the link). At optimality, at most one of the flow directions should have a positive value.4. Mcf = thousand cubic feet.5. Bcf = billion cubic feet.6. Ties are broken by choosing the point closest to the associated LP dual value.7. In practice, a small positive term e, representing machine epsilon can be added to the price in the denominator

to avoid dividing by zero.8. “Equilibrium” prices are defined for these numerical experiments to be the average values at the end of the

seventh iteration for smoothing values of 0.0, 0.2, and 0.5. The prices values at the seventh iteration for eachof these three smoothing values were generally close to each other.


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