Mathematical programming and electricity markets

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Volume 9, Number 1, 1-53 June 2001

REPRINT

Antonio J. Conejo and Francisco J. Prieto.

Mathematical Programming and Electricity Markets

L.F. Escudero (comment) 23 S..A. Gabriel (comment) 30 F.D. Galiana (comment) 33 A. Gómez Expósito and J.L. Martínez Ramos (comment) 34 N. Nabona (comment) 37 G.B. Sheblé (comment) 41 A. J. Conejo and F. J. Prieto (rejoinder) 45

Published by Sociedad de Estadística e Investigación Operativa

Madrid, Spain

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Volume 9, Number 1 June 2001

Editors

Marco A. LÓPEZ-CERDÁ Ignacio GARCÍA-JURADO

Technical Editor

Antonio ALONSO-AYUSO

Associate Editors Ramón ÁLVAREZ-VALDÉS Nelson MACULAN Julián ARAOZ J.E. MARTÍNEZ-LEGAZ Jesús ARTALEJO Jacqueline MORGAN Jaume BARCELÓ Marcel NEUTS Emilio CARRIZOSA Fiovarante PATRONE Eduardo CASAS Blas PELEGRÍN Laureano ESCUDERO Frank PLASTRIA Simon FRENCH Francisco J. PRIETO Miguel A. GOBERNA Justo PUERTO Monique GUIGNARD Gerhard REINELT Horst HAMACHER David RÍOS-INSUA Onésimo HERNÁNDEZ-LERMA Carlos ROMERO Carmen HERRERO Juan TEJADA Joaquim JÚDICE Stef TIJS Kristiaan KERSTENS Andrés WEINTRAUB

Published by Sociedad de Estadística e Investigación Operativa

Madrid, Spain

Sociedad de Estadística e Investigación Operativa

Top (2001) Vol. 9, No. 1, pp. 1�54

Mathematical Programming and Electricity Markets

Antonio J. Conejo

Departamento de Ingeniería Eléctrica, ETSI Industriales

Universidad de Castilla - La Mancha

Campus Universitario s/n, 13071 Ciudad Real, Spain

e-mail: [email protected]

Francisco J. Prieto

Departamento de Estadística y Econometría

Universidad Carlos III de Madrid

C/ Madrid, 126, 28903 Getafe (Madrid), Spain

e-mail: [email protected]

Abstract

The electric power industry in Europe and all over the world is undertaking ma-

jor regulatory and operational changes. The underlying rationale behind all these

changes is to move from a centralized operation approach to a competitive one.

That is, the understanding of power supply as a public service is being replaced by

the notion that a competitive market is a more appropriate framework to supply

reliable and cheap electric energy to consumers. In some cases, the aforementioned

transition process has included the privatization of power utilities. This new frame-

work requires new tools and procedures, and some of these procedures drastically

di�er from traditional ones. Therefore, new challenging mathematical program-

ming and operations research problems naturally arise in this context. This paper

provides a review of some of these problems, particularly operational problems span-

ning a time horizon from one day to one year. The approach adopted emphasizes

mathematical programming issues, describing the structure and characteristics of

these problems and suggesting appropriate solution techniques.

Key Words: Electric power, competitive markets, large-scale optimization,

mixed-integer optimization.

AMS subject classi�cation: 90B30, 90C06, 90C11, 91B26.

1 Introduction

The electric power industry in Europe and all over the world is under-

taking major regulatory and operational changes. The underlying rationale

behind all these changes is to move from a centralized operation approach

Relevant comments and suggestions were provided by Alexis Motto from McGill

University, Montréal, Québec, Canada.

2 A.J. Conejo and F.J. Prieto

to a competitive one. That is, the understanding of power supply as a pub-

lic service is being replaced by the notion that a competitive market is a

more appropriate framework to supply reliable and cheap electric energy to

consumers.

This new competitive paradigm is implemented via two market struc-

tures: (i) a power pool and (ii) a �oor to facilitate bilateral contracts among

producers and consumers.

A pool is an e-commerce marketplace where producers and consumers

submit production and consumption bids, respectively. The market oper-

ator clears the market using market rules and produces a market-clearing

price and sets of accepted production and consumption bids. Typically, the

market operator clears the market once a day. Additional markets for minor

adjustments are possible on an hourly basis. Producers, consumers and the

regulatory authority agree upon any market rules before their implementa-

tion.

In a monthly or yearly framework, the structure to allow competitive

trade among producers and consumers is the �oor for bilateral contracts.

A bilateral contract is an agreement between a producer and a consumer

so that the producer supplies electric energy to the consumer at a given

price. Physical bilateral contracts are medium-term decisions lasting from

one month to one year. Financial contracts to hedge price volatility are not

considered in this work.

The power pool is commonly denominated Power Exchange (PX). Usu-

ally, producers are referred to as GENerating COmpanies (GENCOs) and

consumers can be referred to as CONsumption COmpanies (CONCOs). En-

ergy Service COmpanies (ESCOs) buy energy from bilateral contracts and

from the power pool to sell it to di�erent types of customers with the pur-

pose of maximizing their own bene�ts. The market-clearing entity is often

known as the Market Operator (MO). Power transactions are carried out

through the transmission and distribution networks. The TRANSmission

COmpanies (TRANSCOs) provide the wires to materialize the power trans-

actions. Transmission companies are highly regulated entities that provide

a non-discriminatory access to their wires for a regulated fee. Analogously,

DIStribution COmpanies (DISCOs) o�er a non-discriminatory use of their

distribution wires for a regulated fee. The entity in charge of ensuring the

technical feasibility of the power transactions agreed upon at the Power Ex-

change is denominated the Independent System Operator (ISO). The ISO

Mathematical Programming and Electricity Markets 3

GENCOs

Bilateralcontract �oor

PX

CONCOsESCOs

TRANSCOs

DISTCOs

CONCOsESCOconsumers

Figure 1: Electricity market agents and structure.

has usually the authority to modify power transactions already scheduled

if system security is at stake. These modi�cations should be minimal. The

above structure is illustrated in Figure 1. Confusingly, sometimes the name

ISO is applied to the MO in the technical literature and consumption and

energy service companies are referred to as DISCOs. See Sheblé (1999) for

additional details.

This paper addresses only energy markets. Although reserve and regu-

lation markets are also relevant for the power industry, the largest share of

market trade is within the energy market.

Relevant books addressing the new competitive framework include

Sheblé (1999), Meier and Hobbs (1998), Chao and Huntington (1998), Ilic

et al. (1998) and Hobbs et al. (2000). Dozens of conference papers on the

subject can be found in the proceedings of the IEEE Power Engineering

Society (http://www.ieee.org/) conferences, and the University of Califor-

nia Energy Institute symposia and workshops, in particular the Program

on Workable Energy Regulation, POWER, (http://www.ucei.berkeley.edu).

Relevant journal papers are mostly found in the issues of the �IEEE Trans-

actions on Power Systems� (during the last ten years) and �The Electricity

Journal�, Elsevier Science, The Netherlands (http://www.elsevier.nl).


The remainder of this paper is organized as follows. In Section 2 the

producer perspective is adopted. Both bilateral contract and pool bidding

issues are analyzed. Market power topics are also treated. In Section 3 the

viewpoint of the energy service company is studied, and its decision-making

problems are analyzed. In Section 4 the point of view of the consumer is

considered for both short-term decisions and long-term ones. Section 5

presents the problems to be solved by the market operator. Three di�er-

ent market-clearing algorithms are reviewed: single-period auctions, multi-

period auctions and Walrasian auctions. Finally, Section 6 provides some

conclusions.

2 Producer viewpoint

The decisions faced by a GENCO are described in this section. The

decision making problems of a GENCO are mainly two: (i) how much

energy to allocate to bilateral contracts and (ii) how much energy to sell in

the pool. These problems are analyzed below.

2.1 Bilateral contract selection

In most electricity markets, producers and consumers are allowed to

establish physical supply contracts outside the pool. From the point of view

of a GENCO, it needs to determine if it would be more pro�table to sell a

certain amount of energy directly through one of these contracts or through

the pool. As the main parameters in the contract (prices, quantities) are

�xed in advance, a GENCO may reduce its risk signi�cantly by using these

contracts as an alternative to the pool. Consequently, any model that would

consider these decisions must also take into account some representation of

the risk associated with trading through the pool.

The decision problems associated with these bilateral contracts are of

two (related) kinds: i) how to design a contract in an optimal manner,

taking into account the peculiarities of the producer and the consumer, and

ii) to decide if a contract with a given structure is of interest to the GENCO,

as an alternative to the pool.

The details of a contract may vary signi�cantly from one case to another.

These details a�ect the speci�c structure of the mathematical models result-


ing from the preceding decision problems. In what follows we will assume

that a contract is a sequence through time of values of (demanded) energy

satisfying certain constraints, and prices related to the energy amounts;

these constraints and price functions de�ne the contract. For example, a

contract could be de�ned by a maximum amount of energy to be served

in a given time (a year), with bounds on the energy served in each period

within its time horizon, and prices depending on the total amount of energy

served.

The following discussion will concentrate on the second decision prob-

lem, that is, to determine if a given contract is acceptable for a GENCO.

The �rst problem (the design problem) could be solved in terms of this one,

once some information on the form of the contract is available.

2.1.1 Contract selection under uncertainty

One of the main di�culties when posing and solving contract selection

problems stems from the fact that the decisions that must be compared take

place on quite di�erent time frames. The outcome of the energy market (a

daily process with hourly prices) must be compared to that of the bilateral

contract under consideration (a monthly or yearly arrangement). Modeling

these di�erent time scales gives rise to problems of very large size.

One possible alternative is to summarize the expected behavior of the

energy market through weekly or monthly averages. This works reasonably

well if perfect competition is assumed, but it may not be appropriate if the

GENCO has signi�cant market power.

The presence of uncertainty, mostly associated with the prices resulting

from the energy market, adds to the complexity of these models. Other

possible sources of uncertainty are those derived from the e�ective use that

will be made of the contract (that is, the speci�c amount of energy de-

manded), and the availability of renewable resources for energy generation.

The particular (but important) case of hydrogeneration will be considered

later on.

In summary, to decide if a contract would be acceptable for a GENCO,

it would have to solve two optimization problems to compare the pro�ts

generated with the contract and those without it. If we assume the GENCO

to be a price taker, each problem would maximize the pro�ts from the pool


and the contracts (with or without the one under consideration), subject to

technical constraints and those speci�c for the contracts. It would have the

form:

maxpit;8i;t;st;8t

E!

nXt

��t(!)st �

Xi

cit(pit)�o

subject toXi

pit = st +Xj

ljt(!) 8t (2.1)

pit 2 �i 8i; t

where pit is the total power output of generator i (owned by the GENCO)

in period t, st is the total power output of the GENCO that is allocated to

the pool during period t, cit(pit) is the production cost for energy generation

pit of generator i at period t (data), �t(!) is the (average) value of energy

market prices under realization of uncertainty ! for period t (data), ljt(!) is

the requirement of energy from contract j during period t under realization

of uncertainty ! (data), �i is the feasible operating region of generator i

(data), and E!f�g indicates expected value over !. It should be noted that

the above problem includes binary variables that do not appear explicitly

in its formulation.

The optimal expected pro�t should be modi�ed by the term

E!

�Xt

Xj

��jt(!)ljt(!), where ��jt(!) denotes the unit payment to be re-

ceived from contract j during period t. This value does not depend on the

decision variables and can be taken into account after problem (2.1) has

been solved.

Problem (2.1) maximizes the pro�ts under the optimal allocation of

generation between the energy market and bilateral contracts. The con-

straints are the operational restrictions on the units and the satisfaction of

the contracts. The existence of contracts implies that the problem cannot

be separated by generator or time unit, unless additional assumptions are

made in advance.

The uncertainty is modeled through the energy market prices and the

actual requirements of the contracts. Given the potential size of the prob-

lem, the periods usually considered are either weeks or months, and the

energy market prices used in the model are averages over these time pe-

riods. The uncertainty is usually discretized by introducing scenarios for

the parameters �t(!) and ljt(!). The objective function could be modi�ed


to take into account the risk exposure of the GENCO, by adding terms

penalizing this risk, for example.

In practice, this problem can be simpli�ed by ignoring the uncertainty

in the satisfaction of the contracts, and by making a priori assumptions on

the allocation of generation from units to contracts.

Problem (2.1) is a large MINLP that can be approximated either as a

MILP or as a continuous (and nonconvex) NLP. Its solution requires the

use of either a sophisticated branch and cut solver (Brooke et al. (1998),

GAMS (2000)) or a large scale NLP solver (Gill et al. (1997)).

2.1.2 Renewable energies and water value functions

The preceding model assumed a GENCO whose generating plants in-

cluded only thermal units. Energy generated from hydroelectric units, if

they are available, has very low production costs and it is a very valuable

generation resource for the GENCO. The preceding model (2.1) must be

modi�ed to take into account that the availability of water in the reservoirs

for the time horizon under consideration (several months to a few years)

may introduce signi�cant additional uncertainties, and that these reservoirs

are typically interconnected within river basins.

Using model (2.1) as a reference, the modi�cations associated with the

management of the hydro generation are: i) Hydro generation is a nonlinear

function of water released and height in the reservoir, which may change

signi�cantly in the time horizon considered for these problems; this requires

keeping track of both water released and water stored in the reservoirs. ii)

The availability of water depends not only on the actions of the GENCO,

but also on other stochastic parameters related to the climate, alternative

uses for the water and the actions of the owners of other reservoirs in the

basin. iii) The amount of water stored at the end of the planning period

is an important decision variable, and should be treated explicitly in the

model.

Regarding this last item, the storage of water in a given period, as

opposed to its usage for hydro generation, allows to delay generation to other

periods with larger expected unit pro�t. The water left in the reservoirs at

the end of the planning horizon should be treated in this same manner, that

is, it is a resource that should be stored if the expectation of future pro�ts


(beyond the planning horizon) is su�ciently large. As a consequence, an

informed decision will require an estimation of these expected future pro�ts.

An e�cient manner to do this is to introduce water value functions that

quantify the expected future income for each amount of water stored at the

end of the period in each reservoir.

The resulting model would have the form:

maxpit;8i;t;St;8t;x;y

E!

n�Xi2H

vi(yiT ) +Xt

(�t(!)st �Xi2G

cit(pit))�o

subject to st +Xj

ljt(!) =Xi

pit 8i; t

pit 2 �i 8i; t 2 G (2.2)

pit = gi(xit; yit) 8i; t 2 H

Ax+By = b(!)

where pit is the total power output of generator i (owned by the GENCO)

in period t, st is the total output of the GENCO that is allocated to the

pool during period t, xit is the amount of water released through hydro

unit i (units in set H) in period t (x is the vector of all xit), yit is some

average measure of the water stored in reservoir i through period t (y is

the vector of all yit), vi(yiT ) is the water value function (in monetary units)

at reservoir i, evaluated at yiT (data), the water stored at the end of the

planning horizon T (data), cit(pit) is the production cost of generator i

at period t for the thermal generators (units in set G) (data), gi(xit; yit)

provides the power generation at hydro unit i corresponding to a water �ow

xit and a storage level yit (data), �t(!) is the (average) value of energy

market price under realization of uncertainty ! for period t (data), ljt(!) is

the requirement of energy from contract j during period t under realization

of uncertainty ! (data), and �i is the feasible operating region of generator i

(data). A and B are node-arc incidence matrices that represent the topology

of the river basins where the reservoirs are located (if they are linked),

as well as the dependence between time periods for the water stored in

the reservoirs (data), and b(!) is a vector of external in�ows and out�ows

to the reservoirs (rain, evaporation and regulated �ows) in the basins for

uncertainty realization ! (data). It should be noted that the above problem

includes binary variables that do not appear explicitly in its formulation.

The optimal expected pro�t should again be modi�ed by


E!

�Xt

Xj

��jtljt(!), the expected income from the bilateral contracts,

where ��jt is the resulting price from contract j during time period t.

To simplify the formulation we have assumed that each reservoir is as-

sociated with a single hydro unit. The main di�erences between this model

and (2.1) are the water value functions vi(�) and the balances on �ows and

stored water in the reservoirs. If water levels in the reservoirs are assumed

to be (approximately) constant, the variables representing the hydro units

(water released and stored) can be replaced by the energy generated at these

units, resulting in a model similar to that of a thermal unit, except that the

hydro units have uncertain levels of availability.

Problem (2.2) requires an estimate of the water value functions vi(�).

This estimate can be obtained from data external to the model, or it can be

generated within the model itself. An interesting and e�cient proposal to

compute an approximation for these functions within the model is given in

Pereira and Pinto (1991). In it, model (2.2) is extended beyond the planning

horizon, to cover for example several years. This extension must also incor-

porate the corresponding uncertain information for the additional periods.

The resulting model is very large, but it is not solved directly; instead it

is decomposed into the time periods corresponding to the original planning

horizon and those beyond it. A Benders decomposition approach (Benders

(1962)) is used to generate cuts from the subproblems corresponding to the

periods beyond the planning horizon. These cuts provide piecewise linear

approximations to the water value functions. The approximations are gen-

erated at the optimal values of the decision variables for a previous approx-

imation, and the procedure is repeated until the error in the approximation

of the water value function is below a certain tolerance. This procedure is

closely related to the standard procedure in dynamic programming, where

a so-called value function is approximated from its values at certain points,

but in this case it is applied to the dual of the auxiliary problems; it is often

referred to as dual dynamic programming.

The resulting model is a very large MINLP. It can be approximated by

either MILP models (by introducing piecewise linear approximations to the

functions in the model) or by large scale NLP models (by removing the

zero-one variables).


2.2 Pool response

A GENCO with no capability to alter market-clearing prices will sched-

ule its production to maximize its pro�t given a forecasted price pro�le.

Conversely, a GENCO with capability to alter market-clearing prices ad-

justs both (i) its productions and (ii) the resulting market-clearing prices

to maximize its pro�t. The determination of an optimal adjustment re-

quires a precise knowledge of how it can in�uence prices. This knowledge is

embodied in the so-called price-quota (or residual demand) curve that pro-

vides the market-clearing price as a function of the GENCO market quota.

Forecasting these price-quota curves is a challenging research topic.

Once the GENCO best production schedule is known, a bidding strategy

to achieve this production schedule should be devised. This section will

focus only on the determination of the GENCO best production schedule,

which can be formulated as a mathematical programming problem.

For the sake of clarity, hydro units are not considered in the following.

The models presented below can be extended to consider hydro units in an

analogous fashion to the preceding description.

2.2.1 Price taker

A GENCO with no capability to alter market-clearing prices can be

modeled as a number of generators that maximize their pro�ts indepen-

dently. In this case, given the market prices, the pro�t maximization prob-

lem for the GENCO as a whole decomposes directly by generator. There-

fore, a single generator is considered in the following model. The objective

of this generator is to maximize its pro�ts subject to its operational con-

straints. Therefore, its pro�t maximization problem is formulated as:

maxpt;8t

Xt

��t pt � ct(pt)

�subject to pt 2 �; 8t

(2.3)

where pt is the energy produced by the generator at hour t, ct(pt) is the

production cost at hour t (data), �t is the forecasted market-clearing price at

hour t (data), and � is the feasible operating region of the generator (data).

It should be noted that the above problem includes binary variables that

do not appear explicitly in its formulation.


The objective function of problem (2.3) includes two terms: revenues

and costs. Their di�erence provides the pro�ts for the GENCO. The con-

straints state that the generator should work within its feasible operating

region. A detailed description of the operating region of a generator using

MILP is provided in Arroyo and Conejo (2000).

The solution of this problem provides the optimal production of the

generator every hour. The generator should bid in the market so that its

optimal production plan is scheduled by the MO.

Model (2.3) is a MILP problem. Its size is small and it can be solved

using a simple branch and bound solver.

2.2.2 Price maker

A GENCO with market power usually owns a signi�cant number of

generators. Its objective is to maximize its pro�t subject to the operation

constraints of the generator. To that end, the GENCO modi�es its hourly

productions with the purpose of altering market-clearing prices to achieve

the highest possible pro�ts. This requires a coordinated action from all

generators of the GENCO.

The above problem is formulated as:

maxpit;8i;t;qt;8t

Xt

��t(qt) qt �

Xi

cit(pit)�

subject to pit 2 �i 8i; t

qt =Xi

pit 8t

(2.4)

where pit is the power output of generator i (owned by the GENCO) at hour

t, qt is the GENCO market quota at hour t, cit(pit) is the production cost

of generator i at hour t (data), �t(qt) is the GENCO price-quota function

at hour t (data) (Sheblé (1999)), and �i is the feasible operating region

of generator i (data). It should be noted that the above problem includes

binary variables that do not appear explicitly in its formulation.

The objective function of problem (2.4) represents the pro�ts for the

GENCO. The �rst block of constraints expresses the GENCO market quota

as a function of the production of its generators. The second block of

constraints enforces the operating restrictions of the generators belonging


to the GENCO.

The solution of problem (2.4) provides the optimal production of every

generator of the GENCO. The GENCO should bid in the market in such a

way that its generators are allocated their optimal productions.

Problem (2.4) is a medium-size MINLP problem. Through the use of

additional binary variables it can be converted into a MILP problem. Its

solution requires the use of a sophisticated branch and cut solver.

3 Energy service company viewpoint

An energy service company obtains energy from bilateral contracts, from

the pool and from its own production plants and sells it to di�erent cus-

tomers. The ESCO target is to maximize its own pro�t.

An ESCO must decide which are the most favorable bilateral contracts

to sign in the medium term. In the short term, it buys in the pool any ad-

ditional energy needed to supply its contractual obligations with its clients.

If the ESCO has self-production capability, it can use it to protect itself

against high prices in the pool. The contract selection and the pool opera-

tion problems are analyzed below.


For an ESCO, the choice of a portfolio of contracts is a similar problem

to that of a GENCO, analyzed in Section 2.1. An important di�erence is

that the ESCO must select both energy purchase and energy sales contracts.

The decisions on purchase and sales contracts involve both the design of

the contracts and their evaluation versus alternatives (purchases from the

pool). The remainder of the section will consider only the evaluation of al-

ternatives. A model for this evaluation would estimate the pro�ts associated

with the optimal operation of the system under each of the alternative sit-

uations on a time horizon de�ned by the duration of the contract (typically


one year). The model would have the following form:

maxpit;8i;t;st;8t;bt;8t;rkt;8k;t

E!

nXt

��t(!)(st � bt)�

Xi

cit(pit)�Xk

�kt(rkt)rkt

�o

subject to st +Xj

ljt =Xi

pit +Xk

rkt + bt 8t

pit 2 �i 8i; t

rkt 2 �k 8k; t

(3.1)

where pit is the total energy output of generator i (owned by the ESCO)

during period t, st is the total energy output of the ESCO that is sold

through the pool during period t, bt is the total amount of energy that the

ESCO purchases through the pool during period t, rkt is the amount of

energy that the ESCO purchases from contract k in period t, cit(pit) is the

production cost of generator i at period t (data), �t(!) is the (average) value

of energy market price under realization of uncertainty ! for period t (data),

ljt is the amount of energy that the ESCO sells to contract j in period t

(data), �kt(rkt) is the unit price associated with a purchase r from contract

k during period t (data), �i is the feasible operating region of generator

i (data), and �k is the set of constraints associated with the speci�cation

of purchase contract k (data). It should be noted that the above problem

includes binary variables that do not appear explicitly in its formulation.

For simplicity, the preceding model has been formulated ignoring hydro

generation. When comparing di�erent alternatives, the optimal objective

function should be modi�ed by adding the term E!f

Pt

Pj��jt(!)ljt(!)g,

that is, the expected income from the bilateral contracts, independent of

the variables. Note that ��jt is the resulting price of contract j during time

t.

This model is a large scale MINLP that can be transformed into an

large MILP (by adding binary variables for example). It can be solved

using e�cient branch and cut algorithms.

3.2 Pool response and self-operation

For simplicity, the energy allocated to bilateral contracts is not ac-

counted for below. In this case, the ESCO target is to maximize its pro�ts

from the sale of energy to its consumers. This energy is either self-produced


or bought in the pool. This pro�t maximization problem is formulated as:

maxpt;8t;bt;8t

�

Xt

�ct(pt) + �t bt

�

subject to pt + bt = dt 8t (3.2)

pt 2 �

where pt is the energy self-produced at hour t, bt is the energy bought from

the pool at hour t, dt is the forecasted total customer demand at hour

t (data), ct(pt) is the production cost of energy self-produced at hour t

(data), and �t is the forecasted price of the energy bought in the pool at

hour t (data). It should be noted that the above problem includes binary

variables that do not appear explicitly in its formulation.

The objective function of problem (3.2) includes two terms: costs from

self-producing energy and costs from buying energy from the pool. After

solving the problem, the objective function should be modi�ed by adding

a term that represents the revenues from selling energy to the customers,Pt�tdt, where �t is the customer selling price of the energy during hour t.

This term does not depend on the optimization variables. The �rst block

of constraints establishes that the customer demand should be supplied in

every period. The last constraint requires that the generators belonging to

the ESCO should work within its feasible operating region.

The solution of this problem provides the amount of power to buy from

the pool and to self-produce in every period of the production horizon.

Problem (3.2) is a medium size MILP problem that can be easily solved

using an e�cient branch and cut solver.

4 Consumer viewpoint

The general case of a consumer with self-production capability is ana-

lyzed below. If the consumer has no self-production capability, the formu-

lation below can be simpli�ed in a straightforward manner. Two decision

making problems faced by the consumer are addressed: (i) how much en-

ergy to buy from bilateral contracts and (ii) how much energy to buy from

the pool.



The objective of the CONCO in the medium-term horizon is to select

the best bilateral contracts to sign among an array of available alternatives.

If the bilateral agreements are adequate, the CONCO may decide not to buy

from the pool. Conversely, if the portfolio of contracts is not competitive,

the CONCO may decide to buy all its required energy from the pool.

The problem of a CONCO is similar to that of an ESCO, (3.1), except

that the corresponding model would not include energy sales to other parties

or to the pool.

The resulting model is:

maxpit;8i;t;dt;8t;bt;8t;rkt;8k;t

E!

�Xt

(ut(dt)� �t(!)bt �Xi

cit(pit)�Xk

�kt(rkt)rkt)

subject to dt =Xi

pit +Xk

rkt + bt 8t

pit 2 �i 8i; t

rkt 2 �k 8k; t

(4.1)

where pit is the self-produced energy from generator i (owned by the

CONCO) during period t, dt is the total energy consumption of the CONCO

during period t, bt is the total amount of energy that the CONCO pur-

chases through the pool during period t, rkt is the amount of energy that

the CONCO purchases from contract k in period t, cit(pit) is the production

cost of generator i at period t (data), ut(dt) is the CONCO utility function

(in monetary units) at period t (data), �t(!) is the (average) value of energy

market price under realization of uncertainty ! for period t (data), �kt(rkt)

is the unit price associated with a purchase r from contract k during period

t (data), �i is the feasible operating region of generator i (data), and �k

is the set of constraints associated with the speci�cation of purchase con-

tract k (data). It should be noted that the above problem includes binary

variables that do not appear explicitly in its formulation.

The resulting model is again a large scale MINLP that can be solved

by transforming it into a large MILP and using an e�cient branch and cut

algorithm.


4.2 Pool response

From the pool perspective, the target of a CONCO is to maximize its

consumer utility minus its self-production costs, subject to satisfying its

own demand. For the sake of clarity, and without loss of generality, it is

assumed that there is no bilateral contracts.

This model is formulated as:

maxpt;8t;bt;8t

Xt

�ut(dt)� ct(pt)

�subject to bt + pt = dt 8t (4.2)

pt 2 �

where bt is the energy bought in the pool by the consumer at hour t, pt is

the energy self-produced by the consumer at hour t, dt is the CONCO own

demand at hour t (data) , ut(dt) is the consumer utility function at hour t

(data), and ct(pt) is the consumer production cost at hour t (data).

The objective function of problem (4.2) is the di�erence between the

utility of the consumer and its self-production costs. The �rst block of

constraints states that the demand of the consumer should be satis�ed at

every period. The last constraints establish that the generators owned by

the consumer should work within their feasible operating region.

The solution of problem (4.2) provides the amounts of energy the con-

sumer should buy from the pool or self-produce in each time period.

Problem (4.2) is a small-size MILP problem that is easy to solve.

5 Pool operation viewpoint

The market operator should clear the market using an appropriate pro-

cedure, agreed in advance by all market participants. Three market-clearing

procedures are considered in this section:

1. Single-period auctions.

2. Multi-period auctions.

3. Walrasian auctions.


5.1 Single-period auctions

The objective of a single-period auction is to maximize, for a single

time period, the net social welfare subject to meeting the demand and the

operating constraints of the producers. Therefore, periods are considered

one at a time and inter-temporal constraints are neglected. As a result

of ignoring these constraints, heuristics are needed to modify the auction

solution in each time period, in order to ensure that it is technically feasible.

To clear the market, 24 hourly auctions are carried out successively. This

procedure is performed usually one day in advance. GENCOs, ESCOs and

CONCOs submit their respective bids and the MO solves for every time

period the problem below:

maxdi;8i;pj;8j

Xi

Æi di �Xj

�j pj

subject to 0 � pj � pj 8j

0 � di � di 8iXj2m

pj 2 �m 8m (5.1)

Xi2n

di 2 �n 8n

Xi

di =Xj

pj

where di is the demand bid i, pj is the production bid j, di is the size of

demand bid i (data), pj is the size of production bid j (data), Æi is the

price of demand bid i (data), �j is the price of production bid j (data),

�n is the feasible operating region of demand n (data), �m is the feasible

operating region of producer m (data), i 2 n indicates the set of demand

blocks belonging to consumer n (data), and j 2 m indicates the set of

generation blocks belonging to producer m (data). It should be noted that


in its formulation.

The objective function of problem (5.1) is the consumer surplus plus the

producer surplus, i.e. the net social welfare. It is computed as the di�erence

of two terms: the �rst term is the sum of accepted demand bids times

their corresponding bidding prices; the second term is the sum of accepted


production bids times their corresponding bidding prices. It should be noted

that if the producers do not bid at their respective marginal costs, the second

term of the objective function is not actually the producer surplus but the

�declared� producer surplus. However, in this paper it will be considered,

without loss of generality, that producers do bid at their actual marginal

costs. The �rst block of constraints limits the sizes of the production bids.

The second block of constraints speci�es the sizes of the demand bids. The

third block of constraints ensures that the set of bids from every producer

should meet its production constraints. The fourth block of constraints

enforces that the set of bids of every consumer should meet its consumption

constraints. The �fth constraint states that the production should be equal

to the demand, so that the market clears.

The solution of problem (5.1) provides the accepted production and

demand bids and the market-clearing price, usually de�ned as the most

expensive accepted production bid. Other de�nitions are also possible.

The above problem is a medium size MILP problem that can be easily

solved.

5.2 Multi-period auctions

The objective of a multi-period auction is to maximize the net social

welfare over the auction horizon subject to meeting, in every hour, the

demand and the operation constraints of the producers. The same consid-

erations on the net social welfare made for single-period auctions are also

valid for multi-period ones. Inter-temporal constraints are explicitly taken

into account (Arroyo and Conejo (2000)). GENCOs submit productions

bids, ESCOs and CONCOs submit consumption bids and the MO solves

the problem below:

maxdit;8i;t;pjt;8j;t

Xt

�Xi

Æit dit �Xj

�jt pjt

�

subject to 0 � pjt � pjt 8j; t

0 � dit � dit 8i; tXj2m

pjt 2 �m 8m; t (5.2)


Xi2n

dit 2 �n 8n; t

Xi

dit =Xj

pjt 8t

where dit is the demand bid i at time t, pjt is the production bid j at

time t, Æit is the price of demand bid i at time t (data), �jt is the price of

production bid j at time t (data), dit is the size of the demand bid i at time

t (data), pjt is the size of production bid j at time t (data), Æit is the price

of demand bid i at time t (data), �jt is the price of production bid j at time

t (data), �n is the feasible operating region of demand n (data), and �m is

the feasible operating region of producer m (data). It should be noted that


in its formulation.

The objective function of problem (5.2) is the net social welfare over the

whole planning horizon (consumer surplus plus producer surplus). The �rst

block of constraints provides limits for production bids, while the second

block limits demand bids. The third block of constraints establishes that

the set of bids belonging to every producer should meet its production

constraints. Analogously, the fourth block of constraints states that the

set of bids of every consumer throughout the time horizon should meet

its consumption constraints. The third and fourth blocks of constraints

allow enforcing all types of inter-temporal constraints. The �fth block of

constraints enforces the balance of production and demand in every period.

The solution of problem (5.2) provides the accepted production and de-

mand bids and the market-clearing price in every time period. The market-

clearing price in each hour is de�ned as the price of the most expensive

accepted production bid that hour. Note that other de�nitions of market-

clearing prices are possible.

The above problem is a large-scale MILP problem. A state-of-the-art

branch and cut solver is required to solve it in a reasonable amount of time.

5.3 Walrasian auctions

A Walrasian auction (tâtonnement) is a multi-round auction based on

price modi�cations (Walras (1954), Galiana et al. (2000)). Note that pre-

vious auctions are not multi-round but just single-round.


This auction is described in the steps below:

Step 1. The MO broadcasts hourly trial prices, �t; 8t.

Step 2. Producers determine their productions to maximize their prof-

its subject to their respective operation constraints. Therefore, each

GENCO solves problem (2.3) or problem (2.4) and communicates to

the MO the production schedule it is willing to carry out.

Step 3. ESCOs determine the energy to buy from the pool to maximize

their respective pro�ts. Thus, each ESCO solves problem (3.2) and

informs the MO of its desirable consumption schedule.

Step 4. Consumers determine the amounts of energy that maximize their

respective utilities. Therefore, each CONCO solves problem (4.2) and

sends to the MO the consumption schedule it is willing to accept.

Step 5. The MO calculates hourly load imbalances.

Step 6. If hourly prices are unchanged in two consecutive rounds, they

produce the market-clearing prices, and the auction stops; else the

MO modi�es prices aiming at balancing the load, broadcasts new

hourly prices, and the auction continues in Step 2.

It should be noted that the above algorithm guarantees that each par-

ticipant maximizes its individual pro�ts. In fact, it corresponds to the La-

grangian relaxation solution of the dual problem of a centralized minimum

cost operation problem (with perfect information). If this primal problem

has a duality gap, the Walrasian auction may get trapped into an oscilla-

tory behavior. If there is no duality gap, the Walrasian auction converges to

the optimal solution of both the primal and dual problems. The oscillatory

behavior is not so relevant in terms of the attained primal solution because

it a�ects typically only a few units. However, changes in market-clearing

price may be relevant. A challenging research problem is how to modify

the original primal (cost minimization) problem so that its optimal solution

does not change signi�cantly but the duality gap is removed (Galiana et al.

(2000)), and therefore it can be solved using a Walrasian auction.


6 Conclusions

This paper reviews relevant mathematical programming problems that

arise in a competitive electric energy framework, such as the ones arising in

Europe and in many other places all over the world. The di�erent perspec-

tives of the producer, the consumer, the energy service company, and the

pool operator are analyzed, and the associated mathematical programming

problems are formulated and characterized. Many of the resulting models

are large-scale MILPs. Improvements in the computation of solutions for

these problems are of clear interest for the power industry. Other signi�cant

research challenges are related to:

� modeling decision making problems using stochastic programming,

MILP and MINLP techniques,

� �nding appropriate solution procedures, including decomposition tech-

niques, and

� shortening required solution times.

References

Arroyo J.M. and Conejo A.J. (2000). Optimal Response of a Thermal Unit

to an Electricity Market. IEEE Transactions on Power Systems 15,

1098-1104.

Benders J.F. (1962). Partitioning Procedures for Solving Mixed Variables

Programming Problems. Numerische Mathematik 4, 238-252.

Brooke A., Kendrick D., Meeraus A. and Raman R. (1998). GAMS. A

User's Guide, GAMS Development Corporation (http://www.gams.com/).

Chao H.-P. and Huntington H.G. (1998). Designing Competitive Electricity

Markets. Fred Hillier's International Series in Operations Research &

Management Science. Kluwer Academic Publishers.

Galiana F.D., Motto A.L., Conejo A.J. and Huneault M. (2001). Decen-

tralized Nodal-Price Self-Dispatch and Unit Commitment. In: The Next

Generation of Unit Commitment Models. Fred Hillier's International Se-

ries in Operations Research & Management Science. Kluwer Academic

Publishers.


GAMS Development Corporation (2000). GAMS - The Solver Manuals,

GAMS Development Corporation (http://www.cplex.com/).

Gill P.E., Murray W. and Saunders M.A. (1997). User's Guide for SNOPT

5.3: a Fortran Package for Large-Scale Nonlinear Programming. Report

NA 97-5, Department of Mathematics, University of California.

Hobbs B.F., Rothkopf M.H., O`Neill R.P. and Chao H.-P. (2001). The Next

Generation of Unit Commitment Models. Fred Hillier's International Se-

ries in Operations Research & Management Science. Kluwer Academic

Publisher.

Ilic M.D., Galiana F.D. and Fink L.H. (1998). Power System Restructuring:

Engineering and Economics. Kluwer Academic Publishers.

Meier P. and Hobbs B.F. (1998). Energy Decisions and the Environment

- A Guide to the Use of Multicriteria Methods. Fred Hillier's Interna-

tional Series in Operations Research & Management Science. Kluwer

Academic Publishers.

Pereira M.V.F. and Pinto L.M.V.G. (1991). Multi-stage Stochastic Opti-

mization Applied to Energy Planning. Mathematical Programming 52,

359-375.

Sheblé G.B. (1999). Computational Auction Mechanisms for Restructured

Power Industry Operation. Kluwer Academic Publishers.

Walras L.M.-E. (1954). Éléments d'Économie Politique Pure; ou la Théorie

de la Richesse Sociale. First Edition, 1874. English translation: Ele-

ments of Pure Economics or The Theory of Social Wealth, Homewood.

Published for the American Economic Association and the Royal Eco-

nomic Society, by R. D. Irwin.

��


DISCUSSION

Laureano F. Escudero

Universidad Miguel Hernández de Elche, Spain

1 Introduction

The paper �Mathematical Programming and Electricity Markets� by

A.J. Conejo and F.J. Prieto presents in a uni�ed and elegant way the math-

ematical modeling of the main operational problems to be addressed by the

agents operating in an open electricity market. The agents are the Gen-

erating Companies (GENCOs), the Consumption Companies (CONCOs)

and the Energy Services Companies (ESCOs), among others. Usually, the

GENCOS sell energy to the CONCOs and the ESCOs through bilateral

contracts and the power pool in order to maximize their pro�t. The ESCOs

purchase energy from the GENCOs for selling it to the CONCOs as well.

In this note we will focus on the bilateral contract portfolio selection by

the GENCOs and ESCOs as a contribution to the timely approach presented

in the Conejo-Prieto paper. The aim of that paper on relation with the sub-

ject is presenting models for determining the pro�tability of a contract that

is proposed to a GENCO or an ESCO, given the already committed con-

tracts and the expected pro�t to obtain from the power pool along the time

horizon under consideration. The energy sales bilateral contract allocation

is one of the main decisions that a GENCO has to address in a deregulated

energy market together with the estimation of the water future value if any,

the operation decisions for the generation units in the new environment and

the power pool energy bidding, among others.

Both types of problems, namely, the single contract evaluation and the

contract portfolio selection (that considers simultaneously the evaluation

of a set of contracts) take into account the uncertainty associated with

the energy prices and the energy requirements from the already committed

contracts. Our approach also considers the price taker view. It can be

extended to the price maker view, although the modeling presents bigger

dimensions.

One important di�erence between both approaches is related to the

anticipativity character of the decisions about the power generation and the


energy selling and purchasing through the pool along the time horizon as it

is considered in the Conejo-Prieto approach. In our approach those variables

are not anticipated and, then, they are associated with given scenarios.

2 Problem description and modeling approach

Let the following additional notation to the Conejo-Prieto notation. (It

will follow it as much as possible.)

New sets and data parameters

, set of scenarios under consideration.

K, set of candidate selling bilateral contracts.

w(!), weight factor representing the likelihood that the modeler

associates with scenario !, for ! 2 .

lkt(!), energy requirement from contract k during period t under

the realization of scenario !, for k 2 K, ! 2 .

�kt(!), (average) value of market prices of the energy to be de-

livered from contract k during period t under scenario !,

for k 2 K, ! 2 .

�, feasible region for the set of candidate selling contracts to

be chosen.

�i(!), feasible operating region of generator i under scenario !,

for ! 2 .

Variables

pit(!), power output of generator i during period t under scenario

!, for ! 2 .

st(!), total power output of the GENCO that is allocated to the

pool during period t under scenario !, for ! 2 .

Æk 2 f0; 1g, variable such that its value is 1 if candidate contract k is

chosen and, otherwise, its value is zero, for k 2 K.

The Deterministic Equivalent Model (DEM) of the two-stage stochastic

version for the GENCO's contract selection problem can have the form


E!

nw(!)

Xt

Xj

�jt(!)ljt(!)o+ max

Æk8k2K;pit(!)8i;t;!;st(!)8t;w

E!

nw(!)

Xt

��t(!)st(!)�

Xi

cit(pit(!)) +Xk

�kt(!)lkt(!)Æk�o

subject to (A.1)Xi

pit(!) = st(!) +Xj

ljt(!) +Xk

lkt(!)Æk 8t; !

pit(!) 2 �i(!) 8i; t; !

Æk 2 � 8k

Model (A.1) maximizes the expected pro�t from the optimal allocation

of power generation between the power pool and the bilateral contracts

over all scenarios. The constraints are the operation restrictions on the

generators over all scenarios, the satisfaction of the existing contracts and

the constraints related to the feasibility of the new contracts.

Note that in the approach shown by model (A.1) the contract related

variables (i.e., the 0�1 variables) represent the decisions to be made in the

�rst-stage (by considering all given scenarios but without subordinating to

any of them), an approach so-called two-stage full-recourse. On the other

hand, the pool selling energy as well as the power generation during each

period are scenario dependent. The pool selling is in competence with the

(existing and new) energy selling contracts for utilizing the power generation

during the periods under each scenario.

Model (A.1) assumes that the GENCO`s power generation system is

only included by the thermal units. Along the lines presented in the Conejo-

Prieto paper it is easy to include hydropower generators in model (A.1) as

well, such that the water volume stored in the reservoirs links the time

period submodels. Escudero et al. (1996, 1999), among others, present con-

tinuous non-linear models to deal with hydro generators under uncertainty

in the water exogenous in�ow along a time horizon.

Another natural extension of model (A.1) is the bilateral contract portfo-

lio selection for an ESCO, where its activity includes purchasing and selling

energy in the power pool and committing bilateral contracts as a buyer and

as a seller as a well. Under the previous assumptions of a price taker agent,


the extension of model (A.1) can have the form

E!

nw(!)

Xt

Xj

�jt(!)ljt(!)o+ max

Æk8k2K;pit(!)8i;t;!;st(!)8t;w;�m8m2M ;bt(!)8t;!

E!

nw(!)

Xt

��t(!)[st(!)� bt(!)]�

Xi

cit(pit(!))+

+Xk

�kt(!)lkt(!)Æk �Xm

�mt(!)rmt(!)�m�o

subject to (A.2)Xi

pit(!) + bt(!) +Xm

rmt(!)�m =

= st(!) +Xj

ljt(!) +Xk

lkt(!)Æk 8t; !

pit(!) 2 �i(!) 8i; t; !

Æk 2 � 8k

�m 2 � 8m

Æk; �m 2 �� 8k;m

where the additional parameters and variables are as follows.

K, set of candidate selling bilateral contracts.

M , set of candidate purchasing bilateral contracts.

�mt(!), (average) value of market prices of the energy to be pur-

chased from contract m during period t under scenario !,

for ! 2 .

�, feasible region for the set of energy selling contracts to be

chosen.

�, feasible region for the set of energy purchasing contracts

to be chosen.

��, feasible region for the combined set of energy selling and

purchasing contracts to be chosen.

bt(!), continuous variable that represents the energy that the

ESCO purchases from the power pool during period t un-

der scenario ! , for ! 2 .


rmt(!), continuous variable that represents the energy that the

ESCO purchases from contract m during period t under

scenario ! , for ! 2 .

Æk 2 f0; 1g, variable that takes the value 1 if candidate selling contract

k is chosen and, otherwise, its value is zero, for k 2 K.

�m 2 f0; 1g, variable that takes the value 1 if candidate purchasing

contract m is chosen and, otherwise, its value is zero, for

m 2M .

So, model (A.2) determines the selling / purchasing contract policy (so-

called �rst-stage policy) that an ESCO must follow while restructuring a

contract portfolio. The restructuring is aimed to maximize the expected

pro�t from trading in the power pool and obtaining the income from the

energy selling contracts minus the cost of power generation and the cost

of energy purchasing contracts along the given time horizon over the set of

scenarios under consideration. The constraints are related to power gener-

ation conditions, the selling and purchasing contract stipulations and the

expected power demand to be satis�ed.

The scenario related variables are named second stage variables. See

that the Conejo-Prieto approach, so-called simple recourse, requires to an-

ticipate the power pool selling / purchasing decisions without knowing the

scenario to occur. However the approach given by (A.1)-(A.2) assumes

that the decision-maker does not need this type of anticipative decisions

and (s)he can subordinate it to the occurrence of the scenarios. This sec-

ond approach when appropriate results in a generation policy with greater

pro�t.

A natural extension of model (A.2) also includes the hydropower gener-

ation capability.

With some frequency both types of models (A.1) and (A.2) include

within the environment presented by the Conejo-Prieto paper some model-

ing structures for allowing further pro�t hedging. The new structures model

some types of �nancial contracts where the transactions are settled down

by di�erences.


3 Algorithmic approaches. Brief reference

The Benders (1962) Decomposition method can be applied to exploit

the so-called compact representation of the DEMs (A.1) and (A.2). The

�rst application of the method to the two-stage stochastic linear programs

is due to Van Slyke and Wets (1969). See also in Birge and Louveaux (1993)

among others some schemes for dealing with the integer version of the LP

models. Given the (presumably) large-scale instances of the models (A.1)

and (A.2), their decomposition in smaller models is a key for success.

On the other hand, we can also consider some other types of mathemati-

cal representations, speci�cally the so-called splitting variable representation

via scenario, since it is very amenable for some approaches to deal with 0�1

variables; it allows siblings of the set �� of 0 � 1 variables, see Alonso et

al. (2000) and below. Escudero et al. (1999) among others present detailed

algorithms for solving the LP version of this type of models, by using an

Augmented Lagrangian approach.

In spite of the good performance of the above approaches for the LP ver-

sion of the stochastic models, our previous experience with the 0�1 version

is not very promising when using Lagrangians. On the other hand, Benders

Decomposition schemes can have a better performance for smaller instances

than many instances of the models (A.1) and (A.2); see Caroe and Tind

(1998). The splitting variable representation of (A.1) and (A.2) replaces in

the models the variables Æ and � by the siblings Æ(!) and �(!), respectively,

and appending additionally the so-called non-anticipativity constraints

Æk(!)� Æk(!0) = 0 !; !0 2 n ! 6= !0; k 2 K;

�m(!)� �m(!0) = 0 !; !0 2 n ! 6= !0;m 2M:

(A.3)

For solving (A.1)-(A.2) we propose a two-stage version of a Branch-and-

Fix Coordination (BFC) algorithmic approach presented in Alonso et al.

(2000) for solving multi-stage 0 � 1 stochastic programs. A Branch-and-

Cut scheme can be used for the optimization of each scenario submodel

in (A.1)-(A.2), where the constraints (A.3) are utilized for coordinating

the selection of the branching nodes and the branching variables as well

as the variable �xing and node pruning. Caroe and Schultz (1998) use a

similar decomposition approach. However that approach focuses more on

using Lagrangian relaxation to obtain strong upper bounds and less on node

branching and variable �xing. In a di�erent context, see in Nürnberg and


Römisch (2000) a Lagrangian based 0�1 stochastic dynamic approach. See

also Takriti and Birge (2000).

References

Alonso A., Escudero L.F. and Ortuño M.T. (2000). BFC, a Branch-and-Fix

Coordination Algorithmic Framework for Solving Stochastic 0� 1 Pro-

grams. Trabajos de I+D I-2000-2, Centro de Investigación Operativa,

Universidad Miguel Hernández, Elche (Alicante), Spain.

Benders J.F. (1962). Partitioning Procedures for Solving Mixed Variables

Programming Problems. Numerische Mathematik 4, 238-252.

Birge J. and Louveaux F.V. (1997). Introduction to Stochastic Program-

ming. Springer.

Carøe C.C. and Schultz R. (1998). A Two-Stage Stochastic Program for

Unit Commitment under Uncertainty in a Hydro Power System. SC

98-13, Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB), Ger-

many.

Carøe C.C. and Tind J. (1998). L-Shaped Decomposition of Two-Stage

Stochastic Programs with Integer Recource. Mathematical Programming

83, 451-464.

Escudero L.F., de la Fuente J.L., García C. and Prieto F.J. (1996). Hy-

dropower Generation Management under Uncertainty via Scenario Anal-

ysis and Parallel Computation. IEEE Transactions on Power Systems

11, 683-690.

Escudero L.F., de la Fuente J.L., García C. and Prieto F.J. (1999). A par-

allel Computation Approach for Solving Multistage Stochastic Network

Problems. Annals of Operations Research 90, 131-160.

Nürnberg R. and Römisch W. (2000). A Two-Stage Planning Model for

Power Scheduling in a Hydro-Thermal System with Uncertainty. Preprint

2000-11, Institut für Mathematik, Humboldt Universität zu Berlin, Ger-

many.

Takriti S. and Birge J.R. (2000). Lagrangean Solution Techniques and

Bounds for Loosely Coupled Mixed-Integer Stochastic Programs. Oper-

ations Research 48, 91-98.


Van Slyke R. and Wets R. J-B. (1969). L-Shaped Linear Programs with

Applications to Optimal Control and Stochastic Programming. SIAM

Journal on Applied Mathematics 17, 638-663.

��

Steven A. Gabriel

University of Maryland, U.S.A.

1 General comments

This paper provides an excellent overview of the ever-changing, com-

petitive electricity marketplace from an optimization point of view. This

perspective is particularly signi�cant in many markets across the world due

to the e�ects of deregulation and restructuring which have greatly changed

the industry.

The authors describe the various players in this new electricity mar-

ket such as generation companies (GENCOs), energy service companies

(ESCOs), market operators, etc., and give a concise de�nition of typical

optimization problems faced by each of these players. In most cases, the

resulting optimization problem is challenging to solve given the integer and

nonlinear aspects as well as the large size. They also consider various prob-

abilistic aspects of these optimization problems such as stochastic prices.

2 Equilibrium models

Another signi�cant area of research in modeling electrical power mar-

kets, not covered in this paper, involves incorporating game-theoretic ele-

ments from the Nash-Cournot perspective; see for example Hobbs (1999)

and Wei and Smeers (1999). This perspective attempts to model the com-

petitive aspects of market players suitable to an oligopoly and would build

on models discussed in this paper.


Typically these types of models result in some form of equilibrium prob-

lem, i.e, a variational inequality, nonlinear or linear complementarity prob-

lem. These classes of mathematical programs generalize optimization mod-

els and recently there has been extensive algorithmic research devoting to

e�ciently solving these challenging problems. See Ferris and Pang (1997)

for a discussion of engineering and economics applications of this class of

problems.

Some recent examples of related algorithms include: the B-di�erentiable

Newton methods (Pang (1990), Xiao and Harker (1994ab)), the NE/SQP

methods (Gabriel and Pang (1992), Gabriel and Pang (1994), Pang and

Gabriel (1993), Gabriel (1998a)), the path search approaches (Ralph (1994),

Dirkse and Ferris (1995)), the approaches based on the Fischer function Fis-

cher (1995), De Luca, Facchinei, and Kanzow (1996), Facchinei and Kanzow

(1997), and smoothing approaches such as Chen and Mangasarian (1996),

Chen and Harker (1997), as well as Gabriel (1998b).

3 Stochastic prices

Another important research area in electrical power markets concerns

modeling the stochastic nature of spot market prices. As discussed in this

paper, these prices enter into the optimization problems for several players

via the objective function. Since hourly prices can exhibit �uctuations in

multiples of 100 times the usual values, capturing the probabilistic nature

of these spikes is quite important. In e�ect, fortunes can be made or lost in

a matter of hours with these large �uctuations.

In summary, this paper provides an excellent summary of typical opti-

mization problems faced by current electrical power market players.

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��

Francisco D. Galiana

McGill University, U.S.A.

I would categorize the authors' contribution as one in portfolio man-

agement, albeit within a very complex and �noisy� environment, namely a

large inter-connected power system. The authors propose two investment

choices, namely physical bilateral contracts and the pool market, although

eventually, I presume one could also extend these ideas to include reserve

and regulation markets. The basic problem attacked by the authors is this:

To what degree should each competing agent participate in these two types

of markets? This answer is sought under the rational assumption that

the competing agents, GENCOS, ESCOS, DISCOS, CONCOS have as a

general goal to maximize their expected pro�t in the face of uncertainty

about prices and the behaviour of the competition. Thus, the authors have

proposed several systematic formulations to this di�cult problem under

varying assumptions, namely, from the perspective of the di�erent agents,

and assuming that the agents are or are not price-takers. The authors also

consider hydrothermal systems and time horizons of di�erent lengths. The

issue of market quotas is indeed a challenging research topic but it is a tool


that may provide a handle to deal with market power. Each of the sev-

eral problem variations is carefully and systematically de�ned as a Mixed

Integer Non-Linear Program.

In this paper, the authors address some of the toughest and most perti-

nent problems still to be solved in the area of electricity markets. I am not

sure that they have provided the �nal answer, time and future studies will

tell, but they have indeed proposed an approach that has a good chance of

success. What is more, the mere step of identifying these issues and formu-

lating possible systematic solutions is in itself an important contribution,

for it brings order where today there is considerable confusion, and it opens

the way to many interesting research projects.

��

Antonio Gómez Expósito and José L. Martínez Ramos

Universidad de Sevilla, Spain

This paper addresses the operational problems that new electricity mar-

ket partners are now facing. Essentially, most of them can be formulated as

optimization problems, the di�erences being in the objective function, the

constraints and time frames. Out of the scope of the paper, forecasting and

regression techniques constitute the other pillar of the new paradigm.

We share most of the point of views presented in this timely paper.

However, we would like to raise a few questions for discussion. In order to

better convey our ideas, we will �rst present a general optimization problem

from which the particular cases can be subsequently derived. Both the short

and medium-term time horizons will be separately discussed.

Consider a hypothetical market agent with self-production capability,

which is allowed to trade energy both through the pool and by signing pur-

chase and sale bilateral contracts. Using the notation of the paper whenever

possible, the mid-term decision problem can be stated as follows:

max E!

nXt

h�t(!) � s

0t �

Xi

cit(!; pit)

+Xj

�(s)

jt� ljt �

Xk

�(b)

kt� rkt

io


subject to s0t + dt +Xj

ljt =Xk

rkt +Xi

pit 8t

pit 2 �i 8i; t (D.1)

F(pit) 2 E(!)

�(s)

jt2 �(s)

8j; t

�(b)

kt2 �(b)

8k; t

ljt 2 L 8j; t

rkt 2 R 8k; t

This general model can be compared with that corresponding to the ESCO

in the paper (equation (3.1)) by setting dt = 0. The following comments

are in order:

� In the mid-term, uncertainty not only a�ects pool prices but also

production costs, as a consequence of fuel price volatility (this is very

clear nowadays).

� For an ESCO, st � bt in the paper is actually a single variable s0t, as

there are no means to separate both. In practice, however, decompos-

ing s0t into two positive variables may be a requirement of the problem

solver.

� Both purchase and sale contracts should be given a symmetrical treat-

ment in our opinion. In the contract design phase, both the unit price

and the energy bought and sold during the time period should be vari-

ables rather than constants subject to uncertainty. Of course, bounds

on these extra variables are required (nobody would pay a price much

higher than the one expected in the pool). If prices are input data,

then the optimum amount of energy traded through bilateral con-

tracts should still be determined, and viceversa. It is not completely

clear for us the aim and formulation of the contract selection phase,

which can be a byproduct of the design phase. If an ESCO is o�ered

a completely de�ned contract, then its decision should be based on

known values of ljt, �(s)

jt(or rkt, �

(b)

kt), as the possible deviations are

managed and charged by the market operator.

� The general model (D.1) considers additional energy constraints F(pit)

linking some or all time periods (e.g., take-or-pay contracts).


The GENCO case is obtained by setting s0t = st, dt = 0 and rkt = 0 in

(D.1). For a CONCO, s0t = �bt and ljt = 0 (see the comments below on

the utility function ut(dt)).

In the short term, bilateral contracts constitute constant terms in the

objective function. Hence, the general model (D.1) becomes:

max E!

nXt

h�t(!) � s

0t �

Xi

cit(pit)io

subject to s0t + dt +Xj

ljt =Xk

rkt +Xi

pit 8t (D.2)

pit 2 �i 8i; t

F(pit) 2 E

Again, when dt = 0, some di�erences between (D.2) and (3.2) can be

noticed:

� Pool prices are uncertain even the day before, particularly in poorly

competitive markets. Hence, what is maximized is still the expected

pro�t.

� An ESCO may not be allowed to sell energy to the pool, in which case

s0t = �bt.

� It seems that a single generator is considered in (3.2).

� It is not clear for us why the term dt appears in (3.2) but not in (3.1).

If it is a forecasted value corresponding to non-eligible customers it

should be dt(!). If it is constant it could either be included inP

jljt

or explicitly appear as dt.

For a GENCO, s0t = st, dt = 0 and rkt = 0 in (D.2). On the other

hand, the pure price-taker and price-maker models constitute extreme the-

oretical cases, as any GENCO with two or more generators may be able,

to some extent, to alter market prices. This should be modeled through

the dependence of prices on the energy sold by the GENCO, �t(!; st), or,

equivalently, on the market quota, qt.

As in the medium-term case, the optimization problem of a CONCO in

the short term is obtained from (D.2) with s0t = �bt and ljt = 0. In our


opinion, obtaining the consumer utility function ut(dt) is not and easy task.

An alternative would be to impose bounds on dt

dt � dt � dt

and on the total energy

"d �Xt

dt � "t

re�ecting the fact that the CONCO is willing to shift energy consumption

from peak to valley hours.

Finally, regarding the market clearing procedures, the meaning and via-

bility of single-round auctions is quite clear. However, a multi-round auction

procedure seems to be rather dependent on the initial values of prices and

on the way prices are modi�ed to balance the load. Convergence problems

such as oscillatory behavior, as stated by the authors, but also slow con-

vergence and even divergence problems might appear. Authors' comments

on the viability of energy markets based on multi-round auctions would be

welcomed.

��

Narcís Nabona

Universitat Politècnica de Catalunya, Spain

1 Introduction

The paper proposes models that could be applied by several agents in

the electricity markets to maximize their pro�ts when the market structure

is based on a generation auction system. This is a timely subject and the

paper presents an interesting and uni�ed approach to the di�erent models,

for which the authors should be commended.

2 General remarks

In order to fully appreciate scenario models, details as to how many

scenarios are to be used and how they are generated when there are sev-

eral sources of uncertainty should be given. The important question is not


whether a solution to the resulting optimization challenge can be obtained,

but whether the solution obtained is of any worth when the problem is

solved with a limited number of scenarios.

It is clear that the available modeling languages considerably ease prob-

lem de�nition and replications in scenario based models. It is also clear

that, even for a limited number of scenarios and using several simplifying

assumptions, the models de�ned yield a large optimization problem. With

the available optimization software or through specially developed codes,

an optimal solution to these problems can be obtained.

However, the fact that a particular model could be used to obtain rea-

sonable results when employing a huge number of scenarios is no guarantee

that the solution obtained will be valid when run with a limited number

of scenarios. This is an important practical question regarding the models

proposed, and is not addressed by the authors.

The proposed models ignore the long-term-short-term hierarchy. Long-

term optimization can provide energies to be generated by thermal units

over the whole short-term period to be considered, and target volumes for

reservoirs. The length of short-term optimizations is then a week at most,

and has constraints to impose the amount of energy to be generated by a

unit or group of units over the week, and �nal volumes in reservoirs are

data. The resulting short-term problem is complicated and big enough.

The authors consider several months or a full year with hourly periods

which can easily lead to problems with scenarios in the million variable

range or over.

Variable pit and others are said to be energies from model (3.1) onwards,

but they are declared as power in models (2.1) through (2.4). Is the duration

of periods assumed to be always uniform for a given model?

Details or bibliography dealing with the issue of how do the authors

propose to transform a MINLP into a MILP by adding binary variables

would be welcome.

3 Model (2.1) for a GENCO with little market power

It is not clear how long period t may be, but since variables are powers,

t should refer to periods of no more than two or three hours. However, for


�t(w) and ljt(w) to have meaning, a very large number of scenarios should

be required, specially if contracts are monthly or yearly arrangements and

hundreds (thousands) of successive periods are to be considered. It may be

assumed that the model presented also considers start-up and shut-down

of units (governed by parameters included in the operating region �i of

each unit); thus, under simplifying assumptions, one would get a very large

MILP.

It is not clear from the context whether the requirements ljt(w) (data)

from contracts satisfy a condition such as:Xt

ljt(w) � Æt = ~lj8j;8w

where Æt is the duration of periods (assumed to be all of equal length) and

e ~lj the energy to be supplied through contract j.

3.1 The case with hydrogeneration

With hydrogeneration, the model presented is more complicated for

three reasons: the uncertainty in water in�ows, which makes the num-

ber of necessary scenarios an order of magnitude higher; the nonlinearities

required by the hydrogeneration function; and the need for a function of

expected future pro�ts for each reservoir.

According to this discussors experience, it is not advisable to give values

to water stored in reservoirs at the end of the last period because water

volumes are very sensitive to these values, as optimal �nal volumes of stored

water change greatly when water value is changed; thus small estimation

errors in water value functions will lead to wide oscillations in results.

In order to obtain solutions with such a model, many simplifying as-

sumptions will have to be made while many more scenarios than before

should be taken into account. Therefore, poorer results than without hy-

drogeneration are to be expected.


4 Models (2.3) and (2.4) for a price taker and for a price

maker GENCO

These models are sound because they can be applied to a limited amount

of periods (for as long as the predicted prices �t or price-quota functions

�t(qt) can be predicted with su�cient accuracy). However, in cases of lim-

ited fuel supplies (or take-or-pay fuel contracts) or when there is hydrogen-

eration, problems should have a much greater scope and it could be assumed

that uncertainties and scenarios would be required as with models (2.1) and

(2.2).

5 Model (3.1) for bilateral contracts for ESCO's

As with model (2.1) the energies ljt, sold by the ESCO, appear to be

quite loose. Should there be a constraint of type:Xt

ljt = ~lj 8j

where ~lj is the energy to be supplied through contract j, or is it that ljtshould rather be a data ljt(w) de�ned for every realization of uncertainty

w, as occurs in the term to be added to the objective function?

6 Model (4.2) for a CONCO (from the pool perspective)

Model (4.2) seems to suggest that, since bt has no cost, there is no

need to produce any energy pt to have a dt as large as the consumer utility

functions ut(dt) may ask for, which is nonsensical.

7 Single-period auction problem (5.1)

The terminology employed in problem (5.1) regarding production is mis-

leading, as it does not show that m refers to a production unit, called pro-

ducer in the paper (generally having a minimum power output pm> 0),

which is decomposed into several successive bid stretches, called generation

blocks in the paper, with increasing cost.


8 Final remarks

The authors express the opinion that the �competitive market� is a

more appropriate structure to supply reliable and cheap electric energy to

consumers than a centralized public service. This discussor sees this state-

ment as being of a political nature, since instances of collusion between

agents and unreliable supply may occur and have occurred in this type of

�competitive markets�. Moreover, there are types of �competitive� market

structure other than that addressed in this paper.

In single-period and multi-period auction problems (sections 5.1 and

5.2), the objective function is termed net social welfare. This sounds too

bombastic. (Note that an increase in a stock exchange index, which is of

the same nature as the objective function in (5.1), is not net social welfare.)

��

Gerald B. Sheblé

Iowa State University, U.S.A.

This is a well-documented survey of various market structures and the

methods to solve such markets. It does not give a complete survey nor are

all of the assumptions stated. If the authors had included all variations, this

would be an interesting textbook. The authors concentrate on the concept

of a POOL where all of the data is manipulated by a central authority.

While this approach is a smooth transition from the regulated environment,

many economic studies have shown that there are many missing price signals

for this approach. This reviewer does note that many POOLS have been

established. The main di�erence between the electric markets and other

commodity and �nancial markets is the repeated play as noted in the paper.

As the price is updated, and not the quantity, these are called Walrasian

markets. I have found that all markets require multiple rounds for proper

price discovery as found in all commodity and �nancial markets. The price

has to change as each new event, each new contract, and each new piece of

information is revealed.

I do not understand the need for CONCOs as de�ned in this paper.

Consumers always maximize their use of desired quantities and available


budget. At least I try to do so, personally. Thus, CONCOs would be

served by ESCOs. Larger corporate consumers were previously covered in

the literature as ESCOs, as they would maximize their pro�t by playing the

market with the consumption divisions (CONCOs) as their only customers.

I wish that the authors would elaborate on this issue to clear the issue for

me, as I do not understand the distinction made by the authors to the

distinctions made in the literature.

I think that the authors mislead in the paragraph where they state

that the �one possible alternative is to summarize the expected behavior

of the energy market through weekly or monthly averages�. An approach

based on expected or planned values does work for other commodities. This

alternative approach works for all of the �nancial and commodity markets as

well documented in the �nancial literature. These markets are not perfectly

competitive. These techniques work well even if there is market power (an

easier case to solve). These techniques work well even if there is a price

leader and a price follower(s). It is also noted that these techniques work

primarily because the markets are decentralized. I would most strongly note

that liquid markets are not near perfect competition. Perfect competition

implies that �xed costs are not recovered. Such markets are ill liquid as

each company �les for bankruptcy until the market moves from the perfect

competition state.

The authors also do not list the complexity of the markets for spot

contracts, forward contracts, future contracts, and options on all of the

previous. The authors state that these markets are simply the �averages

over these time periods�. Review of the various texts available on �nancial

calculus applied to electric markets show that this is an oversimpli�cation.

Scenarios are one such approach to market analysis. There are other ap-

proaches based on real option valuation approaches that are just as accu-

rate and easier to perform when the information is available. The authors

should also note that the electric future markets in the United States have

collapsed. These markets are not used, as the underlying spot markets are

not su�ciently liquid to value the longer-term markets.

The authors state that single period auctions ignore the time dependent

constraints. I strongly disagree with these comments. The time depen-

dent constraints are the responsibility of those bidding. Just as economic

dispatch assumed that unit commitment had been optimized, the spot mar-

ket assumes that the forward contracts have been optimized within some


acceptable error tolerance. All other commodity markets function in this

manner. The responsibility for operating equipment properly is left in the

hands of the equipment owner. I think that it is very important for this to

be understood when discussing such auctions. As I do believe in markets, I

do believe in decentralized solutions. Decomposition techniques work very

well in operations research and in markets simulated by humans or adaptive

agents. While economists want the producers to bid at marginal costs, no

one in �nance or business would want to hear of this. Bids are always based

on the value of the commodity to the end user, with any intermediary value

added charges included. The same comments apply to the multiple period

presentation.

I also wish that the authors would consider the contract size when stat-

ing that Walrasian markets may show oscillatory behavior. Markets decide

on the quantity for each contract such that the number of contracts traded

leads to non-oscillatory behavior. As all of the commodity and �nancial

markets are Walrasian as de�ned by these authors, it is interesting that

these markets do not show oscillatory behavior. The authors should in-

clude the market friction caused by transaction fees and by brokerage fees.

This reviewer also sees very similar problems between the fossil fueled

thermal power plant and the hydro power plant. The hydro power plant has

been treated as stochastic based on the probability of water unavailability.

The same is true for fossil plants when the fuel may not be available in

time. The regulated utility had a large amount of storage for each plant,

typically three to four months. The GENCO experience is to use just in

time inventory techniques to maximize pro�t. As the obligation to serve is

gone, the valued risk of not generating is decreased tremendously. Thus, the

expense of three months coal supply is not warranted. It should be noted

that the new Coal futures contract o�ered by the New York Mercantile

Exchange (NYMEX) was introduced to handle this new uncertainty. It

should be noted that the delivery method for the commodity was very hard

to solve. It was eventually decided that barge shipment of coal would be

used to guarantee the delivery of coal. Once again the availability of water

is a key issue to the production of energy as insu�cient water for navigation

has halted shipments in the past.

A GENCO will always schedule the amount of generation based on the

market price. If a uniform price is used, as I think the authors assume, then

the result is as stated. If a discriminatory price is used, then GENCOs bid


price and quantity to maximize pro�t and to recover all costs. I would prefer

to state that once a production schedule has been found by and estimated

market forecast, then bids could be generated that would maximize pro�t

taking possibilities.

I think that the authors should measure the size of a GENCO by the

percentage capability of their generators and not the number of generators.

We have investigated many measures for market power and have found that

all presently de�ned measures are misleading.

I wish that the authors had given a diagram of the various markets, as

I would call the exchange of bilateral contracts a market just as the pool

is considered a market. Most commodity markets are a combination of

bilateral contracts directly between the parties and a commodity market

for contract between the parties with credit risk shared with the exchange

participants.

I also would point to the Electric Power Research Institute Auction Sim-

ulator available for free at the following web site (http://www.ee.iastate.edu/

�sheble/download.html). This is a Walrasian auction for the single hour

market using adaptive agents (extremely simple ones at that) where oscil-

lation does occur but only when an agent is �red or hired.

I think that the authors have made a tremendous contribution to overview

so much detail in such a short paper. However, there are a few key con-

cepts that have not been mentioned in the electric energy literature. As

these concepts are too often overlooked, it is hard for one research group

(electrical engineers) to thoroughly state the assumptions of another group

(economists) when such are not normally stated. I have often enjoyed the

discussion one can start by mixing an economist with a �nancial analyst

and start asking questions when the assumptions and the jargon are not

identical. Each research group has grown a tower of jargon almost as a

means of protecting information from all others.

��


Rejoinder by Antonio J. Conejo and Francisco J. Prieto

We truly appreciate the relevant comments provided by the discussants

that have clari�ed, completed and enhanced our original work.

We would like to start by stressing that our paper provides a review of

some mathematical programming problems arising in electricity markets;

an exhaustive survey was not pursued. It is intended for an operations

research community not particularly familiar with such markets. Further-

more, in writing the paper we pursued a pedagogical approach emphasizing

clarity versus compactness. We did not particularly intend a computation-

ally oriented (and computationally e�cient) formulation.

Speci�c answers to discussants follow.

Prof. Escudero:

Prof. Escudero discussion clari�es the de�nitions of (i) the single con-

tract evaluation and (ii) the contract portfolio selection problems. The for-

mulations that he provides for these problems (in the cases of the GENCO

and the ESCO) are clearly more general and versatile, although more com-

plex, than the ones provided in the paper. We are grateful for these formu-

lations, that extend and improve our contribution.

Prof. Gabriel:

Prof. Gabriel has complemented our work including relevant references

on two research areas related to the one analyzed in our paper: equilibrium

models and price forecasting. We are grateful for the information provided,

which clari�es, complements and enhances the paper.

Prof. Galiana:

We appreciate the comments of Prof. Galiana. As he states, we believe

that the proposed framework can be extended to reserve and regulation


markets. We also agree with his appreciation that clear modeling and ap-

propriate solution frameworks are needed to bring order to electricity market

studies.

Prof. Gómez Expósito and Prof. Martínez Ramos:

The issues raised by Prof. Gómez Expósito and Prof. Martínez Ramos

are considered in detail below.

1. Uncertainty modeling. As the discussants indicate, to obtain a com-

prehensive medium-term model formulation, several other sources of

uncertainty would have to be taken into account. In particular, it

would be necessary to consider: (i) fuel cost uncertainty (as pointed

out by the discussants), (ii) demand uncertainty, or (iii) thermal plant

unavailability. Also, we did not explicitly model (iv) water in�ow un-

certainty for hydroelectric generation. In conclusion, we only modeled

price uncertainty to emphasize this source of uncertainty arising in

electricity markets.

2. Separation of selling and buying variables. As the discussants point

out, in many cases both variables could be merged into a single one.

However, in other cases, particularly for medium-term models, the use

of two di�erent variables might be useful. For example, the average

price at which an ESCO sells energy to the market should be higher

than that at which it would buy it, and in this case it would make

sense to separate purchase and sales variables.

3. Contract design. The problem of optimizing the design parameters

for a contract is of a di�erent nature from the problems we have

considered (optimal usage of the contracts). In particular, the design

of a contract would in general be the result of a negotiation between

the parts to attain an acceptable compromise in their objectives. This

would naturally lead to a game theoretic formulation, or to using an

equilibrium theory setting for the problem, such as the one discussed

in Gabriel et al. (2001), for example.

If these considerations are ignored in the formulation of the problem,

the results may be of limited usefulness. For example, if contract


prices are optimized by (only) one of the agents, the optimal result

should be that selling prices would equal in�nity, while buying prices

would equal zero.

The only alternative we can think of along the lines suggested by the

discussants is to introduce constraints on these design variables that

would try to capture the (expected) strategic behavior of the other

parts in the contract. Nevertheless, we think there may be better

approaches to handling this problem, such as those mentioned above,

and that these approaches lie beyond the scope of the present paper.

4. Inter-temporal constraints. We model single-unit inter-temporal re-

strictions through constraints of the type pit 2 �i. If multiple-unit

inter-temporal restrictions have to be modeled, the following type of

constraints should be included: pit 2 � where represents the set of

units i tied together through inter-temporal restrictions (e.g. regional

emission caps).

5. Next-day price uncertainty. We have not included next-day price un-

certainty to keep short-term operation problems deterministic as it

is customary. Recall that, rigorously speaking, the unit commitment

problem is not deterministic due to load uncertainty; however, it is

routinely considered a deterministic problem. Nevertheless, next-day

price uncertainty should be taken into account, but perhaps it would

be better to include it when designing a bidding strategy (which is

a problem not addressed in our paper), rather than when computing

optimal bidding quantities.

6. ESCO, GENCO and CONCO roles. To ensure the proper functioning

of the market we have assumed no vertical integration, i.e. we have

assumed that an ESCO is essentially an energy service company with

minimal self-production facilities and self-consumption requirements.

The same assumption applies to GENCOs (minimal self-consumption

requirements) and CONCOs (minimal self-production facilities).

7. Consistency of equations (3.1) and (3.2). Customer demand in equa-

tion (3.1) is included as part of the selling contract termP

jljt. As

previously stated, demand uncertainty has not been explicitly mod-

eled.

8. Utility function modeling. Utility functions are hard to derive for


certain types of CONCOs but not for others, e.g. an aluminum pro-

duction plant. The indirect way to model the utility function proposed

by the discussants is a simpli�cation that might be of interest in some

instances.

9. Single-round versus multiple-round market clearing procedures. A

single-round market clearing procedure emphasizes simplicity, trans-

parency and may guarantee no information manipulation. However,

it generally results in low economic e�ciency and cross-subsidies. A

multiple-round market clearing procedure emphasizes the achievement

of economic e�ciency (maximum social welfare) and tends to avoid

cross-subsidies. However, a more computationally-involved procedure

is required and some may interpret this as lack of transparency and

simplicity. We do believe that a multi-round procedure allows agents

to correct their respective decisions on-line with the valuable feed-back

information of previous rounds. A hybrid procedure with a prede�ned

number of rounds could probably be an excellent alternative. On the

other hand, multi-round auctions allow taking into account, in a sim-

ple manner, the transmission network (including both congestion and

losses), see Galiana et al. (2001) (cited in the paper), making unnec-

essary ex post congestion management and loss allocation procedures.

Prof. Nabona:

The issues raised by Prof. Nabona are considered below.

1. Appropriate subset of scenarios. The main purpose of our paper is

to review some mathematical programming problems arising in elec-

tricity markets, and to provide a description of their main character-

istics. We did not attempt a careful description of speci�c solution

techniques. We agree with the discussant that the selection of a lim-

ited set of representative scenarios is critical to obtain an accurate

solution at a reasonable computational cost.

2. Long-term and short-term hierarchy. In electricity markets the long-

term/ short-term hierarchy materialized in the e�ect of actual contract


decisions (which are mainly long-term) on short-term pool related de-

cisions. It should be noted that this (long-term/short-term) time cou-

pling is of a di�erent nature than the coupling in centralized electric

energy systems. We do recognize this coupling in the paper but do

not translate it into short-term problems for the sake of simplicity.

3. Power versus energy. In short-term models, both energy and power

can be used in an interchangeable manner, provided that the time

step is �xed, as it is the case in our short-term models. Energy is the

relevant quantity in long-term models. For simplicity, we have also

assumed a constant time step in these models. However, as indicated

by the discussant, power and energy should be di�erentiated if time

steps of di�erent size are considered.

4. Bibliography on transforming MINLP models on MILP models. We

believe that it is safer and more e�cient to use problem-speci�c fea-

tures to achieve that transformation, rather than relying on general

procedures. Basic ideas are reported in Bradley et al. (1977) and

Castillo et al. (2001); speci�c ideas related to electric power are re-

ported in Arroyo and Conejo (2000) (cited in the paper). General

methods for the solution of MINLP models are discussed in Floudas

(1995).

5. In the formulation of problem (2.1) the values of ljt are assumed to

be data supplied by the decision maker. The constraint suggested

by Prof. Nabona is very reasonable (either as an equality or as a set

of inequalities), but in this context it should be imposed in the data

generation phase, rather than explicitly in the problem.

6. Hydrogeneration. In the paper, water value functions are actually

used as an alternative to specify reservoir �nal levels. In this regard,

Prof. Nabona raises an interesting point. We believe that the sen-

sitivity of the solutions to water values mentioned in his comments

is in many cases a result of using linear approximations to these wa-

ter values. These linear functions give rise to extreme responses in

the solutions (either nearly empty or nearly full reservoirs) for small

changes in the functions. This undesirable behavior is signi�cantly

less likely to appear if the water value functions are constructed from

a reasonably large number of linear segments. As a consequence, it


is important to generate a su�ciently detailed approximation to the

water value functions using dynamic dual programming.

7. Problems (2.3) and (2.4). We agree with the comments of the discus-

sant.

8. In Problem (3.1), the values of ljt are also assumed to be data supplied

by the decision maker. The suggested constraint should be imposed

in the data generation phase. It would be very reasonable to assume

a dependence of ljt on the realization of the uncertainty, ljt(!), as

suggested by Prof. Nabona, and as indicated in the formulation of

Problem (2.1).

9. Problem (4.2). There is a typographical error in the de�nition of the

objective function of problem (4.2). It certainly should include the

cost of buying energy from the pool. Problem (4.2) objective function

should read

maxpt;8t;bt;8t

Xt

(ut(dt)� ct(pt)� �tbt)

The justi�cation of this expression should be apparent from the for-

mulation of problem (6).

10. Problem (5.1). In the formulation of problem (5.1), we stress its actual

structure. We have tried to avoid electric-power related issues in order

to broaden the intended audience of the paper. Other approaches

might lead to formulations that may not be as clear as a comprehensive

one.

11. Competitive markets. It was not our intention in the paper to take a

position either for or against the statement that �a competitive market

is a more appropriate structure to supply reliable and cheap electric

energy to consumers than a centralized public service�. We simply

state that the above paradigm has been, and is, behind the moves from

centralized operation approaches to market oriented ones in many

electric energy systems all over the world.

12. Net social welfare. As stated in standard microeconomic theory text-

books (e.g. Mas-Colell et al. (1999), the objective functions of problem

(5.1) and (5.2) represent net social welfare values if (as stated in the

paper) producers/consumers do bid at their marginal costs/utilities.

This net social welfare is the sum of the consumer surplus and the


producer surplus. A di�erent problem is the consideration of undesir-

able behaviors in oligopolistic markets and their e�ects on consumer

payments.

Prof. Sheblé:

The issues raised by Prof. Sheblé are considered below.

1. We wish to thank Prof. Sheblé for raising many interesting issues from

an economic and �nancial point of view.

2. CONCOs and ESCOs. An ESCO buys energy through bilateral con-

tracts and the pool, and sells it to customers. Its target is to maximize

its pro�ts, de�ned as the di�erence between electric energy selling

revenues and electric energy buying expenditures. No constraints are

imposed on power supply to customers other than those stated in

their respective contracts. The main business activity of an ESCO is

electricity.

A CONCO buys energy through bilateral contracts and the pool for

its own consumption. Its target is to maximize its pro�ts de�ned as

the di�erence between revenues from selling the products it produces

(its utility) and its production expenditures (simpli�ed in the paper

as the cost of buying electric energy), among them the cost of buying

electric energy. The CONCO production process may impose speci�c

constraints on the consumption of electric energy. Electricity is not

the main business activity of a CONCO.

Nevertheless, a CONCO can be considered an ESCO with a single

client: itself. Although the mathematical description of the problems

faced by both, ESCOs and CONCOs, are similar, we found conceptu-

ally useful the distinction, in a similar way as economists di�erentiate

among retailers and customers.

In practice, and in the electric pool market of mainland Spain, quite a

few large corporate consumers have became CONCOs, while divisions

of formerly vertically integrated utilities have became ESCOs.


3. Weekly or monthly averages. Due to the di�erent time scales under

consideration, there is a need to summarize information in the mod-

els. Unfortunately, market power is not uniform over time (at certain

hours a company may have a larger capacity to a�ect prices than at

other hours). Our aim with this remark was just to indicate that the

generation of planning information based on the use of these averages

might provide misleading information, unless some precautions were

taken, as the inherent nonlinear nature of market power might be

dampened through the use of (linear) averages.

We of course agree with Prof. Sheblé that perfect competition would

not happen in practice. Nevertheless, the behavior of some markets

may present some of the theoretical properties associated with this

model (absence of market power, for example). We have used the

label �perfect competition� mostly as a proxy for the properties of

interest in these cases.

4. Forward contracts, future contracts and options. As we indicate in

the introduction of the paper, and to avoid excessive complication in

the presentation, the models we describe are only appropriate for con-

sidering physical contracts. They might be extended to some simple

classes of �nancial (spot or forward) contracts on electricity. But as

indicated by Prof. Sheblé, the approach presented here would not be

appropriate for more complex kinds of �nancial contracts (future con-

tracts, options, etc.). As mentioned in his remarks, other tools would

be required to handle them.

5. Time dependence constraints. Before bidding, a producer should self-

schedule taking into account all its production constraints; then, it

bids with the goal that its optimal schedule (as determined by itself)

is accepted. This may happen or not; if it does, all its inter-temporal

constraints are satis�ed and its production is feasible; if it does not,

some of its inter-temporal constrains may be violated and its produc-

tion is infeasible. In this last case, additional mechanisms have to

be arranged (e.g. shorter time-horizon markets) to allow the producer

to achieve a feasible solution. This two- or multiple-step (multiple-

market) procedure may result in lower economic e�ciency.

On the other hand, a multi-period auction allows producers (does

not force them) to declare inter-temporal constrains, so that the auc-

tion results are feasible for all producers declaring inter-temporal con-


straints, no matter if they have achieved their respective production

targets or not.

It should be noted that a single-step (single-market) solution results

in equal or lower market clearing prices than a multi-step (multiple-

market) solution, even though the �rst step of the multi-step solution

may result in lower prices (at the cost of infeasibility) than the single-

step one.

6. Contract size in Walrasian markets. We agree with Prof. Sheblé that

the number of contracts would have an impact on the behavior of

the market. On the other hand, our remarks on possible oscillatory

behavior of prices were related to the procedure to determine a set

of market prices, rather than on the actual behavior of these market

prices over di�erent time periods. In some cases, it is possible that no

market equilibrium exists (in a theoretical sense), and an algorithm

that attempts to compute this equilibrium will present the behavior

we describe. This does not imply that actual market prices must be

equilibrium prices.

We agree with Prof. Sheblé's comments on (i) hydro versus thermal genera-

tion regarding fuel stocks, (ii) GENCO scheduling based on market prices,

and (iii) measures of market power.

We wish to thank again Professors Escudero, Gabriel, Galiana, Gómez

Expósito, Martí nez Ramos, Nabona and Sheblé for their insightful contri-

butions, that have enhanced the paper and broadened its scope.

References

Bradley S.P. Hax A.C. and Magnanti T.L. (1977). Applied Mathematical

Programming, Addison-Wesley Publishing Company.

Castillo E., Conejo A., Pedregal P., García R. and Alguacil N. (2002). Build-

ing and Solving Mathematical Programming Models in Engineering and

Science. John Wiley and Sons. To be published.

Gabriel S.A., Kydes A.S. and Whitman P. (2001). The National Energy

Modeling System: A Large-Scale Energy-Economic Equilibrium Model.

Operations Research 49, 14-25.


Mas-Colell A., Whinston M.D., Green J.R. (1995). Microeconomic Theory.

Oxford University Press.

Floudas C.A. (1995). Nonlinear and Mixed-Integer Optimization. Funda-

mentals and Applications. Oxford University Press.

Top

Volume 9, Number 1

June 2001

CONTENTS Page

A. J. Conejo and F. J. Prieto. Mathematical Program-

ming and Electricity Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

L.F. Escudero (comment) . . . . . . . . . . . . . . . . . . . . . . . . . . 23

S.A. Gabriel (comment) . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

F.D. Galiana (comment) . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

A. Gómez Expósito and J.L. Martínez Ramos

(comment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

N. Nabona (comment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

G.B. Sheblé (comment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

A. J. Conejo and F. J. Prieto (rejoinder) . . . . . . . . 45

G. Bergantiños and E. Sánchez. Weighted Shapley

Values for Games in Generalized Characteristic Function

Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

H. Norde and F. Patrone. A Potential Approach for

Ordinal Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

C. E. Escobar-Toledo. Industrial Petrochemical Pro-

duction Planning and Expansion: A Multi-Objective Linear

Programming Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

P. Borm and B. van der Genugten. On a Relative

Measure of Skill for Games with Chance Elements . . . . . . . . 91

L. Cánovas, M. Landete and A. Marín. Extreme

Points of Discrete Location Polyhedra . . . . . . . . . . . . . . . . . . . . 115

Documents

Mathematical programming and electricity markets