Download expansion planning in electricity markets. two different

14th PSCC, Sevilla, 24-28 June 2002
Session 43, Paper 4, Page 1
EXPANSION PLANNING IN ELECTRICITY MARKETS. TWO DIFFERENT
APPROACHES
Mariano Ventosa
Rafael Denis
Carlos Redondo
INSTITUTO DE INVESTIGACIÓN TECNOLÓGICA
Universidad Pontificia Comillas
Alberto Aguilera 23
28015 Madrid, SPAIN
[email protected]
Abstract – This paper presents two different
approaches to model expansion planning in electricity
markets under imperfect competitive conditions. Both
approaches consider a market in which firms compete in
quantity as in the Nash-Cournot game. However, the two
models differ from each other in how each firm makes its
expansion-planning decisions. In the first model, the key
assumption that permits its formulation as a Mixed Linear
Complementarity Problem (LCP) is that firms not only
decide their output in a Cournot manner but also their
new generating capacity. In contrast, in the second model,
there is a leader firm that anticipates the reaction of the
follower firms as in the Stackelberg game. The latter
model is formulated as a Mathematical Program with
Equilibrium Constraints (MPEC), wherein the leader firm
decides its optimal new capacity subject to a set of market
equilibrium constraints. Both models have been
implemented in GAMS. A simple example is also
presented to illustrate their comparison and application.
Keywords: Complementarity Problem, Nash-Cournot,
Mathematical Programming with Equilibrium Constraints,
Stackelberg.
1 INTRODUCTION
In deregulated power markets, electricity generation
becomes a liberalized activity in which both expansion
planning and operation scheduling no longer depend on
administrative and centralized procedures but rather on
decentralized decisions of the generation companies
whose goals are to maximize their own profit. Given
this competitive framework, electricity firms assume
much more risk as they become more responsible for
their own decisions. Therefore, generation firms need
suitable decision-support models that not only consider
technical operation constraints but also the interaction
of all market participants.
The appearance of wholesale electricity markets in
many countries has greatly stimulated the efforts of the
research community to develop models that consider
imperfect competition [1] since in many cases it is only
few firms that compete. Scott and Read [2] developed a
Cournot-based model with emphasis on hydro operation
using dual dynamic programming. An analytical
statement of this problem was presented in [3]. More
recently, Hobbs utilized the Linear Complementarity
Problem (LCP) to model spatial Cournot competition
among electricity producers [4] while Wei and Smeers
[5] addressed the same challenge using the Variational
Inequality (VI) approach. Although there is a large
number of papers on the subject of oligopoly pricing
and operation of electricity markets, there are few
publications regarding investment in a deregulated
context. In [6] and [7] two Cournot-based expansion
models are described. The first model computes the
market equilibrium using an iterative search procedure.
The second one calculates the equilibrium with a oneshoot procedure taking advantage of its Variational
Inequality (VI) formulation. In addition, another
approach can be found in the second article, labeled as
close loop Cournot model −equilibrium subject to
equilibrium constraints−, which is very difficult to
handle numerically.
This paper presents two different approaches to
model expansion planning1 in electricity markets under
imperfect competitive conditions. Both approaches
consider a market in which firms compete in quantity as
in the Nash-Cournot game. However, the two models
differ from each other in how firms make their
expansion-planning decisions.
In the first approach named in this paper as Cournotbased model, the key assumption that permits its
formulation as a Mixed Linear Complementarity
Problem (LCP) is that each firm not only decides its
output in a Cournot manner but also its new generating
capacity. Consequently, in this approach, the NashCournot equilibrium [8] defines a set of prices,
quantities and rated capacity of power plants that
simultaneously satisfies the first order optimality
conditions of all firms. The resulting LCP model can be
directly solved taking advantage of its complementarity
structure [9], whose particularities allow the use of
special
resolution
methodologies
incorporated
nowadays in commercial solvers.
In the second approach named in this paper as
Stackelberg-based model, there is a leader firm that
anticipates the reaction of the follower firms as in the
Stackelberg game [8]. This model is formulated as a
Mathematical Program with Equilibrium Constraints
(MPEC), wherein the leader firm decides its capacity
expansion subject to a complementarity problem that
defines the market equilibrium in prices and quantities.
1
This paper focuses on how firms behave when making investment
decisions. For this reason and for the sake of clarity, some important
topics of expansion planning have been ignored such as uncertainty,
system reliability or the binary nature of investing in a new power
plant.
14th PSCC, Sevilla, 24-28 June 2002
It is well known that MPEC models are difficult to
solve since there are no powerful methodologies to deal
with this kind of problems. Consequently, this model
has to be solved by general-purpose non-linear solvers.
This paper is organized as follows. Section 2
presents an overview of the modeling assumptions
made. Section 3, discusses the mathematical structure of
both the models. Section 4 has to do with their
implementation. Section 5 describes a case study and
Section 6 provides the conclusions drawn from the
study. Finally, in order to make the paper easier to read,
the detailed mathematical formulation has been placed
in appendixes. Appendix A outlines the notation used
for the mathematical expressions. Appendix B states the
investment optimization program of each firm, and
Appendixes C and D show in detail the mathematical
formulation of both the proposed models.
2
MODELING ASSUMPTIONS
2.1 Demand and Generation System Representation
The expansion model considers a hiperanual scope
divided into different time segments: periods,
subperiods and load levels (group of hours with a
similar demand level). The periods coincide with years,
each subperiod groups several months while the
grouping of the peak, plateau, and off-peak hours makes
up the load levels.
In a market environment the modeling of the demand
must consider the reaction of the quantity demanded to
changes in the price. It is assumed in this paper that the
total demand at each load level is a linear function of
the price.
The model considers the particular characteristics of
each type of generation plant: thermal, hydro and
pumped plants.
Thermal generators are modeled considering their
rated power output and their quadratic fuel
consumption. Some relevant short-term limitations of
thermal units, such as ramp rates and minimum stable
output, are neglected in this long-term model. This
makes the model more tractable since these limitations
have low impact on expansion plans.
When modeling hydro plants, their rated power
output and reservoirs limited capacity is considered. For
each plant, hydro inflows and its initial and final
reservoir levels limit the hydraulic energy available.
Finally, the pumped-storage plants are modeled
considering their rated power output when generating,
their rated power consumption when pumping, their
maximum energy pumped capacity and their
performance in the pumping and generation cycle.
2.2 Short Term Market Assumptions
This subsection describes how firms make their
operational decisions −output of each firm−.
Since market mechanisms in a deregulated context
decide the actual operation of the generation units,
market equilibrium must be considered in order to
Session 43, Paper 4, Page 2
properly model the optimal energy scheduling of each
company.
As in [6] and [7], in both the models presented in this
paper, the well-known Cournot equilibrium has been
adopted for representing the short-term market behavior
since there typically are only few competitors in
electricity markets. In this oligopolistic model, firms
compete in quantity, i.e. each firm chooses an output
quantity to maximize its profit, and the price is derived
from the demand function. The Nash-Cournot market
equilibrium defines a set of outputs such that no firm,
taking its competitors’ output as given, wishes to
change its own output unilaterally.
A relevant assumption of this model is that firms’
operation decision-making occur simultaneously.
Therefore, modeling this type of market equilibrium
requires the simultaneous consideration of each firm’s
profit maximization problem. These optimization
problems are linked together through the market price
resulting from the interaction of all of them.
This scheme is shown in Figure 1, where z represents
the market profit −market revenues minus operating
costs− of each company e∈[1,...,E], x represents the
short-term decision variables −output of each unit− and
the set of constraints h and g represents the generation
limits. The electricity market, which is the link among
all optimization problems, is modeled by the demand
function that relates the supplied demand −total output
of all producers− to the electricity price.
Optimization Program
of Firm 1
Optimization Program
of Firm e
Optimization Program
of Firm E
maximize : z 1 ( x1 )
max imize : z e ( x e )
max imize : z E ( x E )
subject to : h1j = 0
g k1 ≤ 0
subject to : hej = 0
g ke ≤ 0
subject to : h Ej = 0
g kE ≤ 0
Price-m(x)=0
Electricity Market
Figure 1 Market equilibrium.
2.3 Two Different Investment Assumptions
In this paper, two different hypothesis regarding how
firms make investment decisions −new capacity
installed− in an imperfect competitive context are
considered: Cournot game and Stackelberg game.
Cournot-based model: the first model assumes that
firms make their capacity-expansion decisions as in the
Cournot model (assumption also employed in the shortterm market model described in 2.2). Thereby, each
firm chooses its new maximum capacity so that its own
profit is maximized. Formally, the Cournot investment
market equilibrium defines a set of capacities such that
no firm, taking its competitors’ capacities as given,
wishes to change its own capacity unilaterally.
Unlike the Stackelberg game, no firm has first mover
advantage since Cournot assumption implies that firms’
investment decision-making occur simultaneously.
14th PSCC, Sevilla, 24-28 June 2002
Session 43, Paper 4, Page 3
Stackelberg-based model: the second model assumes
that firms compete in capacity as in the Stackelberg
model. There is a leader firm that first decides its
optimal capacity. The follower firms then make their
optimal decisions knowing the capacity of the leader
firm.
This model can be seen as a two-stage game. In the
first stage, the leader firm decides its optimal capacity
anticipating the future reaction of the follower firms. In
the second stage, the leader firm and the follower firms
compete in the short-term market in a Cournot manner
(as it is described in 2.2). This model is similar to the
close loop Cournot model presented in [7]. They are
both different types of two-stage model.
The Stackelberg-based model seems to be more
realistic than Cournot-based one, since it reflects
sequential decision-making −first investment and then
operation−. Nevertheless, from a theoretical point of
view, the main difference between both assumptions is
that in the Stackelberg game the leader firm has first
mover advantage. That is, the leader firm is allowed to
make a new-capacity decision and the followers have to
adapt to the new situation. This theoretical gain is
illustrated in the case study (Section 5), wherein the
leader company (firm C) builds more capacity in the
Stackelberg-based model and obtains a slightly higher
profit than in the Cournot-based model.
these equations make the equilibrium solution
technically feasible.
The third one is formed by the complementary
slackness conditions associated to the inequality
constraints g. In fact, the Cournot-based model has the
structure of an MCP problem because the
complementary slackness conditions have the structure
of a Complementarity Problem. This set of equations
consists of the inequality constraints multiplied by their
corresponding dual variables µ equal to zero; next, the
inequality constraints themselves; and finally, the
explicit statement of dual variables µ as negative ones.
Grouping together all firms’ optimality conditions
−gradient of the Lagrangian function with respect to the
decision variables x, equality constraints h and the
complementary slackness conditions associated to g−
leads to a Mixed Complementarity Problem (MCP)
formulation [9].
3 MODELS STATEMENT
In this section, the mathematical structure of both the
models considered in this paper is stated.
Electricity Market
3.1 Cournot-based Model
When firms compete in capacity and quantity in a
Cournot manner, the investment and operation market
equilibrium problem (Figure 1) can be stated in terms of
an MCP scheme. This MCP structure is obtained by
means of setting the Karush-Kuhn-Tucker’s first order
optimality conditions associated to the maximization
programs −investment and operation− of each firm (see
Figure 2). Therefore, the solution of this non-linear
system of equations provides the prices, outputs and
new capacity installed that solve the investment and
operation Cournot market equilibrium.
In Figure 2, for each firm e, L represents the
Lagrangian function of the corresponding optimization
problem and λ and µ represent the dual variables
associated to the set of h and g constraints respectively.
The optimality conditions can be grouped together into
three sets of equations.
The first one cancels the gradient of the Lagrangian
function with respect to the decision −investment and
operation− variables x. These equations explain how
firms must use their resources, i.e. make their decisions,
to achieve their maximum profit.
The second set, the gradient of the Lagrangian
function with respect to the dual variables λ, coincides
with the equality constraints themselves h. Therefore,
Optimality Conditions
of Firm 1
Optimality Conditions
of Firm e
Optimality Conditions
of Firm E
∂L1
=0
∂x1
∂L1
1
∇λ L ( x,λ, µ ) = 1 = h1j = 0
∂λ j
µk1 ⋅ gk1 = 0 gk1 ≤ 0 µk1 ≤ 0
∂Le
=0
∂xe
e
∂L
∇λ Le ( x,λ, µ) = e = hej = 0
∂λ j
µke ⋅ gke = 0 gke ≤ 0 µke ≤ 0
∂LE
=0
∂x E
∂LE
E
∇λ L (x,λ, µ) = E = h Ej = 0
∂λ j
µkE ⋅ gkE = 0 gkE ≤ 0 µkE ≤ 0
∇xL1 ( x,λ, µ ) =
∇x Le (x,λ, µ ) =
∇x LE ( x,λ, µ) =
Price-m(x)=0
Figure 2 Cournot-based model as an MCP.
As a result of the modeling assumptions described in
Section 2, the Cournot-based model is a Linear
Complementarity Problem (LCP), which assures
solution existence and uniqueness in case that thermal
units’ marginal costs are strictly monotone increasing
[5]. Formally, an LCP is an MCP composed only by
linear equations, excluding the complementarity
slackness conditions [10].
Formulating the Cournot-based model as an LCP
problem makes it easy to handle numerically as there
exist specific algorithms devoted to Complementarity
Problems.
3.2 Stackelberg-based Model
In this two-stage game, the leader firm chooses its
optimal capacity in a first stage, as in the Stackelberg
game, and then in a second stage all the firms compete
in quantity as in the Cournot game. Formally, this twostage game can be stated as a bilevel optimization
program, i.e. an optimization problem subject to a group
of equilibrium constraints. Therefore, the Stackelbergbased model turns out to have a Mathematical Program
with Equilibrium Constraints (MPEC) structure [11]
(see Figure 3).
The upper-level is the investment optimization
problem of the leader firm, whose objective function is
to maximize its profit z leader . The investment decision
variables of the leader firm α are decided in this upper-
14th PSCC, Sevilla, 24-28 June 2002
level taking into account not only the constraints i(α)
but also the equilibrium constraints. Notice that i(α)
only regard investment decisions and operational
variables x −the outputs and prices− are settled by the
equilibrium constraints.
z 1eader ( x,α )
i (α ) ≤ 0 ← Investment Constraints

∂Le
e
∀e
∇ x L ( x,λ , µ ) = e = 0
∂x

∂Le
Equilibrium

→ ∇λ Le ( x,λ , µ ) = e = hej = 0 ∀e
Constraints
∂λ j

µke ⋅ gke = 0 gke ≤ 0 µke ≤ 0
∀e


maximize :
subject to :
Figure 3 Stackelberg-based model as an MPEC.
The lower level is stated by the first order optimality
conditions of each firm and represents the Cournot
short-term market equilibrium. As in the previous
model, these constraints have the structure of an MCP
due to the so-called complementary slackness condition.
However, unlike the Cournot-based model, now x only
represents the operation decision variables and h and g
only limit such operation variables.
In general, MPEC models are non-convex and
difficult to solve. As a result of this non-convexity, it is
very complicated to derive the conditions that assure
solution existence or uniqueness. In addition, there are
no powerful solvers to deal with this type of models.
4 IMPLEMENTATION
Both models presented in this article were
implemented in GAMS version 2.50. In the case of the
Cournot-based model, two specific commercial solvers
are available for such complementarity problems,
MILES and PATH [12]. Both solvers are based on a
generalization of the classic Newton method, in which
each subproblem is solved as a linear complementarity
problem using an extension of the Lemke´s algorithm.
When solving our model, PATH showed a better
performance as a result of its algorithmic enhancements.
In the case of the Stackelberg-based model, the
optimization problem has an MPEC structure. Unlike
MCP, there is no specific commercial solver for dealing
with MPEC. Consequently, these models have to be
solved by means of general-purpose Non-Linear
Programming (NLP) solvers such as MINOS or
CONOPT2. Solving our model, CONOPT2 is more
efficient2 than MINOS due to its non-linear equilibrium
constraints. Even though the solution of the MCP model
2
Usually, CONOPT2 is better suited for models with very nonlinear
constraints than MINOS. On the other hand, if the model has few
nonlinearities outside the objective function then MINOS would probably be more convenient.
Session 43, Paper 4, Page 4
is used as starting point of the MPEC model in order to
accelerate and ensure the convergence of CONOPT2,
the MPEC solving time is much longer than the MCP
one.
5 CASE STUDY
Complementing the theoretical comparison established in previous sections, this case study allows a
practical and computational comparison between both
the models. Focusing on the results, there are considerable similarities between both approaches; however,
from a computational point of view, the MCP approach
is significantly easier to handle numerically.
5.1 Test System Description
The scope of this sample case has been split into 11
periods (years) with 4 subperiods each (3 months) and 3
load levels (peak, plateau, and off-peak hours) with the
same duration. Table 1 describes the initial situation in
which the three firms considered are. In this case study,
firm C is the only one allowed to build new capacity.
Type
Nuclear
Thermal
Hydro
Total
Units Power (MW)
4
4000
8
3500
3
2600
15
10100
Firm A
25%
24.3%
77%
38.1%
Firm B
50%
50%
11.5%
40.1%
Firm C
25%
25.7%
11.5%
21.8%
Table 1 Generation capacities by firm
The slope of the demand curve is 4.82 €/MWh/GW.
The system met a maximum peak load of 8322 MW and
a yearly energy demand of 55488 GWh in the first year
of the study. It is assumed that demand increases at a
4% rate and so the maximum peak of demand in the
entire scope of the study is 10087 MW. The hydro
energy available every year is 1446 GWh. The only
technology considered convenient when building new
plants is CCGT. The cost of building new capacity is
482 €/kW. The discount rate utilized for the calculation
of the present value is 5%. Depreciation time is ten
years since building in the first period is not allowed.
5.2 Expansion Results
Some of the results obtained with both models are
shown in Table 2. These results are consistent with
theoretical predictions, since firm C builds more
capacity and makes a higher profit when behaving as
the leader firm of the Stackelberg model (MPEC).
Variable
MCP
MPEC
New capacity in the first year (GW), firm C 1.208 ↓ 1.375 ↑
Average prices in the first year (€/MWh)
33.55 ↑ 33.37 ↓
Profit Firm A (k€)
5065.8 ↑ 5040.7 ↓
Profit Firm B (k€)
8683.1 ↑ 8637.2 ↓
Profit Firm C (k€)
6028.8 ↓ 6032.2 ↑
Market Share Firm A (%)
24.7 ↑
24.5 ↓
Market Share Firm B (%)
41.6 ↑
41.2 ↓
Market Share Firm C (%)
33.7 ↓
34.3 ↑
Table 2 Main results of the case study.
14th PSCC, Sevilla, 24-28 June 2002
It is seen that firm C builds more capacity in the case
of the Stackelberg model. Since there is more capacity
in the system, average prices diminish slightly.
However, even though firm C has to deal with the cost
of higher investments and the slight reduction in prices,
its increase in market share provides it bigger profits, as
predicted through theory. Its competitors’ decrease in
share and prices leads them to lower profits. Still,
differences between models are minor ones.
Regarding the operation, the new plants run fully in
peaks. The rest of the time becomes the technology that
sets the marginal price.
5.3 Computational Issues
The size of the MCP −Cournot-based model− in this
case study is 3023 equations and 3057 continuous
variables. The size of the MPEC −Stackelberg-based
model− is 7634 equations and 7404 variables, and
GAMS identifies up to 10659 non-linear entries. A 1.5
GHz PC using the PATH solver spends 0.9 seconds in
the first case and 186 seconds for the MPEC when
using CONOPT2. Therefore, in this case study MCP is
much more tractable than MPEC from a computational
viewpoint.
6 CONCLUSION
This paper presents two different approaches for
addressing the expansion-planning problem in
deregulated electricity markets. Both approaches
consider the Cournot model for the short-term
competition although they differ in how firms decide
their optimal capacity. The first approach, where firms
compete in capacity in a Cournot manner, has an MCP
structure. The second one, where a leader firm decides
its capacity as in the Stackelberg model, has an MPEC
structure.
Both models have been developed and implemented
in GAMS. The application to a sample case clearly
shows that there are minor differences in most market
figures. However, the study also reveals that, from a
computational point of view, MCP solvers are far more
powerful than the methods available for computing
MPEC models. Therefore, the MCP approach should be
the considered when conducting real systems studies.
Finally, in order to improve both expansion models,
further work must be done so as to take account of
uncertainties such as demand growth, regulatory
changes and so forth.
7 ACKNOWLEDGEMENTS
This work has been developed under a research
project funded by Iberdrola, S.A. and with the support
of a grant from the Comisión Interministerial de Ciencia
y Tecnología.
Session 43, Paper 4, Page 5
REFERENCES
[1] E. Kahn, “Numerical techniques for analyzing market
power in electricity”. The Electricity Journal. pp. 34-43.
July 1998.
[2] T. J. Scott, and E.G. Read, “Modelling Hydro Reservoir
Operation in a Deregulated Electricity Market”.
International Transactions in Operational Research, Vol.
3 pp 243-253, 1996.
[3] J. Bushnell, “Water and Power: Hydroelectric Resources
in the Era of Competition in the Western US”, POWER
Conference on Electricity Restructuring, University of
California, Energy Institute. 1998.
[4] B. F. Hobbs. “LCP Models of Nash-Cournot Competition
in Bilateral and POOLCO-Based Power Markets”.
Proceedings, IEEE Winter Power Meeting, NY City, Feb.
1999.
[5] W. Jing-Yuan, Y. Smeers. “Spatial Oligopolistic
Electricity Models with Cournot Generators and
Regulated Transmission Prices”. Operations Research,
Vol. 47, No. 1, January-February, pp. 102-11, 1999.
[6] Chuang, A.S., Wu, F., Varaiya, P., “A Game-Theoretic
Model for Generation Expansion Planning: Problem
Formulation and Numerical Comparisons”. Transactions
on Power Systems, Vol. 16, NO 4, pp. 885-891, Nov.
2001.
[7] Murphy, F., Smeers, Y., “Capacity expansion in
imperfectly competitive restructured electricity market”.
Submitted to Econometrica, 2001.
[8] X. Vives, “Oligopoly Pricing: old ideas and new tools”.
MIT Press. 1999.
[9] M. Rivier, M. Ventosa, A. Ramos, “A Generation
Operation Planning Model in Deregulated Electricity
Markets based on the Complementarity Problem”, M.
Ferris, O. Mangasarian, J. Pang, Chapter 13 of
Applications and algorithms of complementarity. Kluwer
Academic Publishers, 2001.
[10] R.W. Cottle, J.S. Pang, and R.E. Stone. “The Linear
Complementarity Problem”. Academic Press, Boston,
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[11] Luo, Z., Pang, J. and Ralph, D. “Mathematical Programs
with Equilibrium Constraints”. Cambridge University
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[12] S.C. Billups, S. P. Dirkse, and M. C. Ferris. “A
comparison of large scale mixed complementarity
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Applications, 7:3-25, 1997.
APPENDIX A: NOTATION
In this appendix all symbols used in this paper are
identified and classified, according to their use, into
indices, parameters, variables and dual variables.
Index Description
Index Description
b Pumped-storage plants
p Periods
e Firms
s Subperiods
e* Leader firm
h Hydro units
n Load levels
Table 3 Indices.
t Thermal plants
t* New thermal plants
14th PSCC, Sevilla, 24-28 June 2002
Session 43, Paper 4, Page 6
Parameter Description
Ah , p , s
bb , bb
Variable Associated constraint
Hydro inflows for hydro plant h in subperiod s of
period p [TWh]
µ bR, p , s Scheduling of pumped-storage plant b in subperiod
Maximum and minimum capacity of pumped-storage
plant b when pumping [GW]
µ bR, p , s Maximum hydro energy reserve of pumped-storage
s of period p [k€/TWh]
plant b in subperiod s of period p [k€/TWh]
µ tTA*, p Increasing maximum rated capacity [k€/GW]
cit * Capacity investment cost [k€/GW]
d ' p, s,n
d p0 , s , n
Dp, s,n
Constant slope of the demand function in load level n in
subperiod s of period p [(k€/TWh)/GW]
Duration of load level n in subperiod s of period p [kh]
hh , hh
hb , hb
Maximum and minimum capacity of pumped-storage
plant b [GW]
Heat rate (linear [kTcal/TWh] and quadratic
[kTcal/(GW2·kh)] terms) of thermal plant t
o 't * , o ''t * Heat rate (linear [kTcal/TWh] and quadratic
[kTcal/(GW2·kh)] terms) of thermal plant t*
pt , pt
Maximum and minimum rated capacity of thermal plant
t [GW]
p t * , p t * Maximum and minimum rated capacity of thermal plant
t* [GW]
Rb
Rh , Rh
tp
vt , vt *
Maximum hydro energy reserve of pumped-storage
plant b [TWh]
Maximum and minimum hydro energy reserve of hydro
plant h [TWh]
Discount rate [p.u.]
Fuel cost of thermal plant t and t* [k€/kTcal]
ρb Performance of pumped-storage plant b [p.u.]
Table 4 Parameters.
Variable Description
bb , p , s , n
Power consumption by pumped-storage plant b in load
level n in subperiod s of period p [GW]
hb , p , s , n
Power generation by pumped-storage plant b in l load
level n in subperiod s of period p [GW]
hh , p , s , n
Power generation by pumped-hydro plant h in load
level n in subperiod s of period p [GW]
I t *, p
New capacity installed in central t* in period p [GW]
pt , p , s , n
Power generation by thermal plant t in load level n in
subperiod s of period p [GW]
pt *, p , s , n
Rh , p , s
Power generation by thermal plant t*in load level n in
subperiod s of period p [GW]
Hydro energy reserve of hydro plant h at the beginning
of subperiod s of period p [TWh]
Table 5 Decision variables.
Variable Description
g e, p , s , n
Total power generation sold in the market of firm e in
load level n, subperiod s, of period p [GW]
π p, s,n
System marginal price in load level n, subperiod s, of
period p [k€/TWh]
Table 6 Auxiliary variables.
Variable Associated constraint
µ
µ hR, p , s ; µ hR, p , s Maximum and minimum hydro energy reserve of
Electricity demand at price zero in load level n in
subperiod s of period p [(k€/TWh)/GW]
Maximum and minimum capacity of pumped-hydro
plant h [GW]
o 't , o ''t
µ tI*, p , s , n Maximum generation of new plants [k€/GW]
R
h, p, s
Hydro scheduling of hydro plant h in subperiod s
of period p [k€/TWh]
hydro plant h in subperiod s of period p [k€/TWh]
µ
p
t , p, s,n
;µ
p
t , p, s,n
Maximum and minimum rated capacity of thermal
plant t in subperiod s of period p [k€/GW]
p
µ t *, p , s , n Minimum rated capacity of new thermal plant t* in
subperiod s of period p [k€/GW]
µ
I
t *, p
Minimum installed capacity of new thermal plant t*
in period p [k€/GW]
µ hh, p , s , n ; µ hh, p , s , n Maximum and minimum capacity of hydro plant h
in subperiod s of period p [k€/GW]
µ
h
b, p, s,n
;µ
h
b, p, s, n
Maximum and minimum capacity of pumpedstorage plant b in subperiod s of period p [k€/GW]
Table 7 Dual variables.
APPENDIX B: INVESTMENT OPTIMIZATION
PROBLEM
This appendix states in detail the optimization
problem that defines the generation companies’
behavior (Figure 1). The goal of each firm is to
maximize its own profit subject to the set of constraints
that limits its long-term operation decisions.
B.1 Objective Function
The objective of each generation company is to
maximize its profit −market revenues minus operating
costs and investment costs− for the entire scope.
Maximize :


π

p , s , n ⋅ g e, p , s , n


2

 (1)
D
t
v
o
p
o
''
p
⋅
−
⋅
⋅
+
⋅
(
)
∑∑∑
p, s,n
p
t
t
t , p, s,n
t , p,n
t , p,s,n
 ∑

p
s
n
t ∈e


2
 − ∑ t p ⋅ vt * ( ocn ⋅ pt *, p , s , n + o ''t *, p , n ⋅ pt *, p , s , n ) 
 t *∈e

− ∑∑ cit * ⋅ t p ⋅ ( I p − I p −1 )
∀e
t *∈e p >1
B.2 Investment Constraints
The first set of built-capacity constraints (2) states
that total capacity in any period is to be greater or equal
than capacity in the previous.
The second set of constraints (3) limits the power
generated by the new plants in a given period. It is to be
less than their actual rated capacity. This constraint is
analogous to the upper bound in generation output,
which affects conventional thermal plants. However, in
the case of new plants this output cap is a decision
variable.
I p −1 − I p ≤ 0
⊥ µtTA
*, p
pt *, p , s , n − I p ≤ 0 ⊥ µtI*, p , s , n
∀p, t* ∈ e,e
(2)
∀p, s, n, t* ∈ e, e
(3)
14th PSCC, Sevilla, 24-28 June 2002
Session 43, Paper 4, Page 7
B.3 Operating Constraints
The operating constraints are those that regard
resources planning across multiple load levels such a
hydro energy scheduling, operation of pumped-hydro
units as well as those that represent upper and lower
limits of each generation unit.
Hydro scheduling. The energy available during each
subperiod is limited by the hydro inflows and the initial
and final reservoir levels of the subperiod. The initial
level of the first subperiod and the final level of the last
subperiod are fixed data for the optimization problem.
∑D
p, s,n
⋅ hh , p , s , n − Rh , p , s + Rh , p , s +1 − Ah , p , s ≤ 0 ⊥ µ hR, p , s
n
∀p, s, h ∈ e, e
(4)
Pumped-storage constraints. The first constraint
establishes a balance between pumped and generated
energy (5) while the second limits the maximum
pumped energy within each subperiod (6).
∑D
n
p, s,n
⋅ ( hb , p , s , n − ρb ⋅ bb , p , s , n ) ≤ 0 ⊥ µ
R
b, p, s
C.1 Optimality Conditions
Building the Lagrangian function of each firm’s
optimization problem3 Le and canceling its gradient
with respect to the decision variables leads to the
following set of equations:
∂Le
= D p , s , n ⋅ t p ⋅ vt ⋅ ( o 't + 2 ⋅ pt , p , s , n ⋅ ot '')
∂pt , p , s , n
+ D p , s , n ⋅ ( −π p , s , n + t p ⋅ d ' p , s , n ⋅ g e , p , s , n )
(
∀p, s, b ∈ e, e
(6)
∂Le
= D p , s , n ⋅ t p ⋅ vt * ⋅ ( o 't * + 2 ⋅ pt *, p , s , n ⋅ ot * '')
∂pt *, p , s , n
+ D p , s , n ⋅ ( −π p , s , n + t p ⋅ d ' p , s , n ⋅ g e , p , s , n )
n
Variable bounds. The vast majority of the variables
involved in the previous formulation are subject to the
following bounds:
R h ≤ Rh , p , s ≤ R h
⊥ µ hR, p , s ; µ hR, p , s
p
(
− µ
I
t *, p , s , n
−µ
p
t *, p , s , n
)=0
∂Le
= D p , s , n ⋅ ( −π p , s , n + t p ⋅ d ' p , s , n ⋅ g e , p , s , n )
∂hh , p , s , n
(7)
p t ≤ pt , p , s , n ≤ p t ⊥ µ t , p , s , n ; µ tp, p , s , n
∀ p, s, n, t ∈ e, e
(8)
h h ≤ hh , p , s , n ≤ h h
⊥ µ hh, p , s , n ; µ hh, p , s , n
∀ p , s , n , h ∈ e, e
(9)
bb ≤ bb , p , s , n ≤ bb
⊥ µ bb, p , s , n ; µ bb, p , s , n
∀ p , s , n, b ∈ e , e
(10)
hb ≤ hb , p , s , n ≤ hb
⊥µ
h
b, p, s, n
∀ p , s , n, b ∈ e , e
(11)
∂Le
= D p , s , n ⋅ ( −π p , s , n + t p ⋅ d ' p , s , n ⋅ g e , p , s , n )
∂hb , p , s , n
p t * ≤ pt *, p , s , n
⊥µ
p
t *, p , s , n
∀ p, s, n, t* ∈ e, e
(12)
− Dp, s, n ⋅ µ
I t * ≤ I t *, p
⊥ µ tI*, p
∀ p, t* ∈ e, e
(13)
;µ
h
b, p , s , n
B.4 Auxiliary constraints
For the sake of the clarity, the following equations
have been stated. However, in the implementation, the
total power generation and the price have been
substituted by these expressions:
Total power generation of each firm. The total power
generation of each firm represents the effective output
that is sold on the market.
g e, p , s , n = ∑ pc , p , s , n + ∑ pt *, p , s , n + ∑ hh , p , s , n + ∑ ( hb , p , s , n − bb , p , s , n )
c∈e
t *∈e
h∈e
b∈e
∀e, p, s, n
(14)
Price equation. The price is represented as a linear
function of the total purchased power in the market.


π p , s , n = t p ⋅ d ' p , s , n ⋅  d p0 , s , n − ∑ ge , p , s , n  ∀ p, s, n

e

(15)
APPENDIX C: COURNOT-BASED MODEL
In this appendix, the first order KKT optimality
conditions of each firm’s investment problem
(
)
− D p , s , n ⋅ µ hR, p , s − µ hh, p , s , n − µ hh, p , s , n = 0
(
R
b, p, s
+µ
R
b, p, s
) − (µ
h
b, p, s, n
−µ
h
b, p,s ,n
)=0
∂Le
= D p , s , n ⋅ (π p , s , n − t p ⋅ d ' p , s , n ⋅ g e , p , s , n )
∂bb , p , s , n
(
)
+ D p , s , n ⋅ ρb ⋅ µ bR, p , s − µbb, p , s , n − µ bb, p , s , n = 0
(
)
∂Le
= µ hR, p , s − µ hR, p , s −1 − µ hR, p , s − µ hR, p , s = 0
∂Rh, p , s
∂Le
= ( t p − t p +1 ) ⋅ cit * + ∑ µ tI*, p , s , n
∂I t *, p
s,n
TA
I
+ µ tTA
*, p − µ t *, p +1 + µ t *, p = 0
∀p, n, s, t* ∈ e, e
(17)
∀ p, s, h ∈ e, e
p
∀p, n, s, t ∈ e, e
(16)
)
− µ tp, p , s , n − µ t , p , s , n = 0
(5)
∀p, s, b ∈ e, e
∑ Dp, s, n ⋅ hb, p, s , n ≤ Rb ⊥ µbR, p, s
formulated in Appendix B is derived. These equations
make up the Cournot-based model which has the
structure of a Mixed Complementarity Problem (Figure
2). Since there are no equality constraints, the MCP
consists of two sets of equations. The equations that set
the gradient with respect to the decision variables equal
to zero are stated in C.1 and the equations that define
slackness conditions are listed in C.2.
∀p, n, s, h ∈ e, e
(18)
∀p, n, s, b ∈ e, e
(19)
∀p, n, s, b ∈ e, e
(20)
∀p, s > 1, h ∈ e, e
(21)
∀p, t* ∈ e, e
(22)
C.2 Complementary Slackness Conditions
As previously established in Section 2, in order to
complete the set of non-linear equations that define the
optimization problem of each company, the following
three sets of equations must be added to (16÷22). First
3
In this paper has been considered the equivalent minimization problem of each firm.
14th PSCC, Sevilla, 24-28 June 2002
Session 43, Paper 4, Page 8
of all, the inequality constraints (2÷13) multiplied by
their corresponding dual variables µ; next, explicit
statement of dual variables µ as negative ones; and
finally, the inequality constraints themselves.
C.3 Meaning of the Optimality Conditions
The optimality conditions provide useful information
about the role of each type of power plant not only in
firms’ optimal energy scheduling policy [9] but also in
firms’ optimal investment policy.
Focusing on firms’ optimal investment policy,
equation (17) says that when an enlarged-capacity plant
is generating at its maximum capacity, the dual variable
µ tI*, p , s , n is equal to firm’s marginal revenue minus
plant’s marginal cost. This is to say, the profit marginal
increase of having an additional MW available. This
dual variable could be named as capacity-availability
marginal profit.
− µ tI*, p , s , n = D p , s , n ⋅ (π p , s , n − t p ⋅ d ' p , s , n ⋅ g e, p , s , n )
− D p , s , n ⋅ t p ⋅ vt * ⋅ ( o 't * + 2 ⋅ pt *, p , s , n ⋅ ot * '')
(23)
In addition, it can be seen in equation (22) that
building new capacity is strongly subject to significant
time-coupling since its depreciation will certainly
involve the following periods. Therefore, it is easier to
comprehend the meaning of those optimality equations
avoiding this coupling.
Adding together equation (22) of a certain period p’
with the ones of the following periods:
 ∂Le

= 0  → t p ' ⋅ cit * = − ∑ µ tI*, p , s , n


p ≥ p ', s , n  ∂I t *, p
p ≥ p ', s , n

∑
(24)
Equation (24) reflects that firm’s maximum-profit is
achieved when investment cost is equal to the sum of
the capacity-availability marginal profit from p’ to last
p. That is to say, each firm should build new capacity
up to the point where capacity investment cost −left
hand side− reaches the cumulative future profit of such
investment −right hand side−.
APPENDIX D: STACKELBERG-BASED MODEL
This appendix states the mathematical formulation
that defines the Stackelberg-based model described in
3.2. In this two-stage model, the equivalent optimization
program has the structure of an MPEC problem (Figure
3): profit maximization of the leader firm subject to
investment constraints and to the market equilibrium
constraints.
D.1 First stage. Investment Problem of the Leader Firm
In the first stage, the leader firm e* decides its
capacity considering that prices and productions are
provided by the market equilibrium, which is the second
stage.
Objective function of leader firm. The leader firm
seeks to maximize its profit −market revenues minus
operating costs and investment costs− for the entire
scope of the model.
Maximize :


π

p , s , n ⋅ g e, p , s , n


 (25)
D p , s , n ⋅  − ∑ t p ⋅ vt ( ot ⋅ pt , p , s , n + o ''t , p , n ⋅ pt2, p , s , n )
∑∑∑
 t∈e*

p
s
n


2
''
t
v
o
p
o
p
−
⋅
⋅
+
⋅
 ∑ p t * ( cn t *, p , s , n
t *, p , n
t *, p , s , n ) 
 t *∈e*

− ∑ ∑ cit * ⋅ t p ⋅ ( I p − I p −1 )
t *∈e* p >1
Investment constraints of the leader firm. In the
Stackelberg model, the investment constraints are those
that only regard the investment variables of the leader
firm. In particular, non-decreasing capacity of new
plants (26) and lower bound of new capacity (27) are
considered.
I p −1 − I p ≤ 0
⊥ µtTA
*, p
∀p, t* ∈ e *
(26)
I t * ≤ I t *, p
⊥ µ tI*, p
∀p, t* ∈ e *
(27)
Note that now, equation (3) is not an investment
constraint of the leader firm because it regards the
power generation of a new thermal plant, and power
output is to be considered as a second-stage variable in
this model.
D.2 Second stage. Market Equilibrium Constraints
As explained in Section 3, the short-term market
equilibrium −within the Stackelberg-based model− can
be stated as a MCP similar to the Cournot-based model.
However, in this equilibrium, which is the second
decision stage, the investment decision variables of the
leader firm are assumed fixed.
Like the Cournot-based model, there are no equality
constraints. Therefore, this MCP consists of two sets of
equations: optimality conditions and complementarity
slackness conditions.
Optimality conditions. This set of equations is
obtained by canceling the gradient of the Lagrangian
function of each firm’s optimization problem with
respect to the operational decision variables. Therefore,
these equations are a subset of the Cournot-based model
optimality conditions (see C.1): (16), (17), (18), (19),
(20) and (21). Needless to say that the gradient with
respect to the new capacity (22) is not to be considered
since the investment decisions of the leader firm are
variables of the first stage; therefore, they are assumed
fixed in these market equilibrium constraints.
Complementary slackness conditions. This group of
equations consists of three sets of equations: the
inequality constraints regarding operational decisions
(3÷12) multiplied by their corresponding dual variables
µ; next, explicit statement of dual variables µ as
negative ones; and finally, the inequality constraints
themselves. Obviously, the investment constraints of the
leader firm (26) and (27) are not to be considered since
they are already included in the first stage of the
Stackelberg-based model.