A Multi-Stage Multi-Horizon Stochastic Equilibrium Model of Multi-Fuel Energy Markets
Working paper 2/2015
A Multi-Stage Multi-Horizon Stochastic Equilibrium Model of
Multi-Fuel Energy Markets
Zhonghua Sua,∗, Ruud Egginga , Daniel Huppmannc,d , Asgeir Tomasgarda,b
a
Norwegian University of Science and Technology, Department of Industrial Economics and Technology
Management, NO-7491 Trondheim, Norway
b
SINTEF Technology and Society, Applied Economics, Postboks 4760 Sluppen, 7465 Trondheim,
Norway cGerman Institute for Economic Research (DIW Berlin), Mohrenstraße 58, 10117 Berlin,
Germany dDepartment of Civil Engineering & Systems Institute, Johns Hopkins University, 3400 N.
Charles St., Baltimore, MD 21218, USA
CenSES working paper 2/2015
ISBN: 978-82-93198-15-4
Corresponding author. Tel: +47 73593656
Email address: [email protected] (Zhonghua Su)
Abstract
We develop a multi-horizon stochastic equilibrium model for analyzing energy markets, with particular attention to infrastructure development and renewable energy policies in perfect and imperfect
market structures. By decoupling short-term operational decisions feedback from long-term strategic investment decisions, the multi-horizon approach allows to consider long-term and short-term
uncertainties while maintaining favorable computational complexity. We illustrate for a representative stylized example how considering long and short-term uncertainty affects infrastructure
development and trade patterns and look into the Value of Stochastic Solution (VSS). Although
the VSS values are modest, the impact of addressing uncertainty is significant in infrastructure
sizing and timing decisions. We illustrate how in a perfect market much higher expansions occur in production, transmission and generation compared to an imperfect market. Independent of
market structure, more uncertainty generally favors gas production and gas-fired power generation,
with some small but insightful exceptions. As such, the model provides a two-fold contribution
to the literature, as it is the first stochastic multi-energy equilibrium model, as well as the first
multi-horizon equilibrium model.
Keywords: OR in energy, Stochastic programming, Multi-horizon scenario tree,
Generalized Nash equilibrium(GNE), Mixed complementarity problem (MCP)
Preprint submitted to Elsevier
July 3, 2015
1. Introduction
Global energy and climate policy drive the development of the energy system in a sustainable direction (Tester et al., 2005). The increased use of intermittent renewable energy
supply adds yet another uncertainty dimension to the environment wherein energy companies and regulators operate. Fast development of renewable energy technologies together
with rapid change in the use patterns of traditional fossil fuels makes it more important than
ever before to understand the interplay between different energy carriers and sources in the
energy system (Rhodes, 2007). Policy makers are responding to the challenges in different
ways, e.g., phasing out nuclear capacity (Rüdig, 2000), enhancing renewable energy supply,
proposing carbon capture and storage, and the EU Emissions Trading System (EU ETS).
The shale gas development in the USA during the past ten years (Jacoby et al., 2011) has
demonstrated the global impacts of local developments through the effect on international
oil and gas prices and consumption. In Europe, climate and energy policy is linked with
other political objectives like energy supply safety and developing new jobs.
In this paper we propose a modeling approach for analyzing the dynamic and complex
energy markets considering long-term and short-term uncertainties, with particular attention
for infrastructure development for fossil and renewable energy supply in perfect and imperfect
market structures. Such a model can analyze, for instance, the consequences of technology
development, policy development and demand trends for which the long term picture is
uncertain, while considering the impact of an increased share of intermittent renewable
energy which exposes the energy systems to much higher short-term variations in supply
than in the past. In addition to uncertain load, now also the energy generation is more and
more difficult to control. For many years policy makers and companies have relied on largescale models when making policy and investment decisions (Ventosa et al., 2005). Often such
models considered deterministic supply and demand. We investigate the situation where the
models include both short-term and long-term uncertainties. In addition we include the
possibility to examine strategic behavior through open-loop Cournot equilibria.
There is hardly any literature on stochastic energy market equilibrium models with a
multi-fuel energy system perspective. We give a brief overview of other relevant literature.
An important branch in the literature consists of optimization models focusing on energy
2
system investment decisions for one or several energy carriers. For instance, Sullivan (1988)
formulated oil and gas field planning using mixed integer programming. Alguacil et al. (2003)
modeled long-term electricity transmission expansion planning as mixed-integer, nonlinear
and non-convex problems. Hellemo et al. (2012) studied how to design natural gas infrastructure with an operational perspective by deterministic mixed-integer linear programming.
Optimization models for multi-fuel energy system often treat the energy sectors in detail,
while modeling the rest of the economy more simplistic and often exogenous given. An example is the TIMES model, which applies linear programming subject to various political and
technical constraints, to design the energy system at the lowest cost (Loulou et al., 2005). In
recent years, energy system models have also been linked to macroeconomic models in order
to better capture the interrelations between the energy system and the rest economy. For
instance, the Global Change Assessment Model (GCAM) includes interactions between the
energy, land use, agricultural, global economic and technology systems in a single framework
(Calvin et al., 2009). Golombek et al. (2013) analyzed the effects of the European market
liberalization using LIBEMOD MP, a multi-market equilibrium model. LIBEMOD MP is
a partial equilibrium model using functional forms commonly used in computational general equilibrium problems to represent energy goods production, trade and consumption in
Western European countries.
There is quite some literature on stochastic energy sector models, though there are very
few articles on stochastic energy system models. Goel and Grossmann (2004) took advantage
of stochastic programming to study the investment and operational planning of offshore gas
fields. They considered the uncertainties in gas reserves and applied traditional scenario
trees (Birge and Louveaux, 1997) to describe the information structure. Compared to deterministic models, their model could provide solutions with significantly higher expected net
present value (ENPV). Similar results have been shown by many authors (Prékopa et al.
1980; Wallace and Fleten 2003; Gupta and Grossmann 2014).
Garcia-Gonzalez et al. (2008) applied two-stage stochastic programming to cope with
the combined optimization of a wind generator and a pumped-storage facility, where market
price and wind generation were viewed stochastic. Hellemo et al. (2013), they analyzed
investments in natural gas fields and infrastructure by using multi-horizon scenario trees
3
to depict uncertainties. This is an early paper that includes operational details in a longterm planning model, recognizing that the short-term operations affect the design of the
system. PROMETHEUS is a stochastic world energy/technology model based on Monte
Carlo simulations, which produces joint empirical distributions of many future economic
variables in the world energy system (Fragkos et al., 2015).
The stochastic models mentioned above are all optimization models (and one simulation
model, (Fragkos et al., 2015)). One aspect such models do not cover is strategic behavior of
market players; such behavior will generally lead to market outcomes that do not maximize
overall social welfare. Specifically, equilibrium problems much better reflect the interactions
among all the energy market players wherein each one solves an optimization problem with
an own objective, possibly including shared constraints and the players problems being linked
through market-clearing conditions (Gabriel et al., 2012). If this is a one-level game, the
result is often a Generalized Nash Equilibrium (GNE) (Debreu, 1952). There are many
examples in the literature of such equilibrium models, modeled as mixed complementarity
problem (MCP). For instance, Bushnell (2003) implemented a mixed linear complementarity
model to study hydro-thermal electricity competition in the Western United States, where
the results verified that some firms act strategically in this market. Haftendorn and Holz
(2010) applied mixed complementarity models to study international steam coal trade. The
simulation results indicated that the real-world markets outcomes were close to what can be
expected in a perfectly competitive setting. The following are some examples of single-fuel
or single-commodity market equilibrium models. Genc et al. (2007) introduced a two-stage
game with probabilistic scenarios and analyzed capital investment the electric power sector
in Ontario assuming imperfect competition after a major market deregulation in the late
1990s. In Zhuang and Gabriel (2008), deregulated natural gas markets were formulated
as stochastic MCP, where first-stage and second-stage decisions were related to long-term
contract decisions and spot market activities respectively. Besides, Egging (2010, 2013)
developed a multi-stage stochastic MCP for the global natural gas market as well as an
algorithm decompose and solve such large-scale stochastic models.
In contrast to such one-level games, multi-level games have not gotten much attention in
the literature. Two reference are: Hobbs et al. (2000) analyzed the strategic behavior of a
4
dominant power generating firm in an electric power network by using a Stackelberg game
model, a two-level game formulated as a mathematical program with equilibrium constraints
(MPEC). In numerical experiments on a 30-busses network Hobbs et al. show optimal bidding behavior by a single generating firm in consideration of the other firm’s reactions.
Next, Murphy and Smeers (2005) compared Stackelberg equilibrium models with Cournot
equilibrium models for electricity generation capacity expansion in imperfectly electricity
markets. The Stackelberg model in their case has several leaders which results in an equilibrium problem with equilibrium constraints (EPEC), which is more difficult to handle both
computationally and when it comes to interpreting the equilibria that arise.
None of these approaches is capable to reflect and address both long term uncertainties in
for instance technology development, climate change, international trade trend, governmental
policy on CO2 emission, mix of energy sources, as well as the short term uncertainties, such
as in the production of renewable energy sources (e.g. solar, wind), seasonal demand and
other unexpected events (e.g., electricity blackout).
In this paper, we present a multi-stage multi-horizon stochastic equilibrium model of
multi-fuel energy markets, which is a one-level game.
Our work extends the model in Egging and Huppmann (2012) and Huppmann and Egging
(2014) who proposed a deterministic equilibrium model of multi-fuel energy markets to
study the overall effects of market power, fuel substitution and infrastructure constraints on
global energy markets. A main challenge for such a model is the computational complexity
regarding the growth of the scenario tree size. To address this, we customize the multihorizon scenario trees introduced by Kaut et al. (2014) for equilibrium models. To the best
of our knowledge this is the first stochastic multi-fuel energy system equilibrium model as
well as the first multi-horizon scenario tree equilibrium model.
The rest of this paper is organized as follows. In section 2, we explain modeling approach
background on the aspects of game theory, stochastic programming, as well as the multihorizon scenario trees, and present the optimization problem of each player and the resulting
mixed complementarity problem. In section 3, the market set-up and three model variants
are introduced. Section 4 analyzes the results specifically. Section 5 briefly concludes and
considers future work.
5
2. A multi-stage multi-horizon stochastic equilibrium model of multi-fuel energy
markets
In this section, we explain and present the multi-stage multi-horizon stochastic equilibrium approach. First, we discuss the model in the context of game theory and stochastic
programming to clarify the dynamics and information structure. Second, we illustrate the
multi-horizon scenario tree approach, accounting for uncertainties and decisions in different
time scales. Third, the optimization problem of each player is specified. Finally, we show the
Karush-Kuhn-Tucker (KKT) conditions of all optimization problems, which together with
the market clearing conditions constitute the MCP.
2.1. The model in the context of game theory and stochastic programming
We assume that the market players make decisions for time periods simultaneously based
on symmetric common knowledge. Simultaneous games are sometimes called open-loop
games, since there is no feedback/observation of rivals’ decisions before each player makes
the decision. Since all decisions are made simultaneously, it is a static game. A dynamic
game is often called bi-level or multi-level game and would lead to a different situation where
the players observe other players’ actions and adjust their own behaviors based on that. This
type of strategies cannot be analyzed by our model.
Figure 1: The two-stage traditional scenario tree
Here it is important to realize that each player decides on his/her strategy for the whole
planning time horizon at time 0. In this one-level game, the strategy determines the ac6
tions/decisions contingent on the scenario. In other words, each player’s strategy is made
at time 0 for all possible scenario outcomes in the future. The fact of that time passes and
uncertainty is resolved is represented by a scenario tree. In Figure 1, this is exemplified by
a two-stage traditional scenario tree. Though each player’s strategy is made at time 0, the
implemented actions/decisions depends on which scenario is finally realized. Every time new
information is available to the decision maker, the pre-decided, scenario-dependent action
is implemented. Each point in time in the tree where more information is available and
a decision is implemented, is called a stage. Hence, we have a one-level game where each
player’s contingent action is depicted in a multi-stage scenario tree and each player’s decision
problem is a multi-stage stochastic program.
An alternative would be to model the decision process as a dynamic game. A two-level
game, for instance a Stackelberg game, can be formulated as a Mathematical Program with
Equilibrium Constraints (MPEC), where there are one leader and a set of followers. Further,
if the first level problem turns into a gaming situation among many players, the formulation
turns into an Equilibrium Problem with Equilibrium Constraints (EPEC). In both cases
both solving the resulting model and the interpretation of equilibria (which typically are not
unique) are more complex.
Our choice for a one-level game, where each player’s decision problem is formulated as a
stochastic program, results in a mixed complementarity problem (MCP). When the players
are following Cournot strategies, the resulting equilibrium is a Generalized Nash Equilibrium
(GNE) (Facchinei and Kanzow, 2007). The main underlying assumption in our game is
that the players make their strategies here and now and simultaneously under symmetric
information. While decisions are contingent on the realization of uncertainty, they do not
change as a consequence of observing other players’ actions.
2.2. The multi-horizon scenario tree approach
The multi-horizon scenario tree is an alternative formulation for the traditional scenario
tree. In a traditional scenario tree, the tree divides into branches corresponding to different
realizations of uncertainty. If there are many uncertain events or possible realizations, the
number of branches becomes very large, posing limits to the computationally tractability.
7
In multi-horizon scenario trees we decouple the representation of short-term and long-term
uncertainty. Hereby we limit drastically the number of branches to end up with a much
smaller, albeit it approximate, scenario tree. Figure 2 shows an example of a multi-horizon
scenario tree, with three long-term stages.
Figure 2: The multi-horizon scenario tree
The big red dots in Figure 2 represent the starting points of the main periods, termed
(strategic) investment nodes or long-term scenario nodes. The big red dots together with
the connecting red branches constitute the main tree, describing the long-term uncertainties/scenarios. In this main tree the time between uncertain events is typically one or several years. In long-term scenario nodes market players make investment decisions. Given
the multi-year lead times for major infrastructure construction projects, capacity expansion
decisions in one investment node will become available for use in its succeeding investment
nodes. Connected to each investment node, there is a green sub-tree, representing the shortterm uncertainties/scenarios. The green ovals represent short-term scenarios. The blue dots
in the ovals are representative short-term periods (for instance hours, days or seasons); in
these short-term nodes, market players make operational decisions such as production rates
or storage injection rates.
In the multi-horizon tree, the terminal leaves of the operational sub-trees are not linked to
succeeding investment and operational nodes. This decoupling of the short-term scenarios
from the long-term main tree is the key difference from traditional scenario trees. For
comparison, Figure 3, shows a traditional scenario tree where the operational nodes are all
connected to succeeding investment nodes to construct the scenario tree. Consequently, the
8
Figure 3: The traditional scenario tree
traditional approach creates a tree of much larger size to represent the same number of
uncertain events. Naturally, ignoring the connections of operational nodes with succeeding
nodes has limitations. For example, natural gas in storages carried over from one time period
to the next, hydro-power production levels affecting reservoir levels in future periods and
the depletion of oil and gas reserves, cannot be captured accurately in this approach. This
is the price for improving the computational tractability. How important this is must be
considered in each specific application. However, not only stochastic models but also many
large-scale deterministic models do suffer from those same limitations.
9
Table 1: Common notation
Notation
Sets:
a∈A
b∈B
c∈C
d∈D
e, f ∈ E
g∈G
n, k ∈ N
o∈O
r∈R
s∈S
u∈U
v∈V
w∈W
Parameters:
b(u)
dfw
duru
prbB
W
prw
Explanation
transmission arcs
short-term scenarios (c.f., Figure 9)
energy transformation technologies (e.g., electric power plants)
final demand sectors
energy carriers/fuels
environmental emission types
geographical nodes
storage technologies
geographical regions
suppliers of fuels and energy
short-term scenario nodes representing operational periods (or time slices).
(c.f., Figure 9)
storage loading/unloading cycles1
long-term scenario nodes representing stages
short-term scenario b that contains node u
discount factor
relative
duration of a short-term period (hour/day/season) (with
P
dur
B
u = 1, ∀b ∈ B)
u∈Ub
conditional probability of the realization of short-term scenario b (given
that the connected long-term scenario node is realized). This is used in
the emission permit auctioneer problem only.
probability of the realization of long-term scenario node w
2.3. Optimization problem of each player
In the model the following market players are represented: suppliers, transmission system
operator, transformation technology operators, storage operators, emission permit auctioneer, and final demand sectors. Each market player solves an own optimization problem (the
final demand sectors implicitly through their inverse demand curves.) We suppose that all
players have common knowledge of scenarios/future uncertainties. By solving this simultaneous game model, equilibria are reached contingent on scenarios. Suppliers, transmission
system operator, transformation technology operator and storage operators make long-term
investment decisions as well as short-term operational decisions. In contrast, the emission
permit auctioneer and the final-demand sectors make short-term/operational decisions only.
1
Loading/unloading cycles are a flexible mechanism to connect sub-periods within short-term scenarios.
E.g., pumped storage is typically used to overcome short-term fluctuations, hence a loading cycle would
connect sub-periods within a day. Underground natural gas storage is used to overcome seasonal fluctuations,
hence a loading cycle would include all sub-periods wherein injection or extraction can take place.
10
Suppliers can exert market power, depicted with a conjectural variation (CV) approach
(Boots et al., 2004). All other market agents in the model are price takers. In the following we extend the deterministic formulation in Huppmann and Egging (2014) to allow the
representation of uncertainty and multi-horizon stochastic scenario trees as discussed above.
This involves a two-fold contribution to the modeling literature, as we point out in a later
section. Generally, the changes concern variables to be scenario-node dependent instead of
natural-time dependent. Moreover, in the objective functions (except of the emission permit
auctioneer), long-term scenario nodes w are weighed by the probabilities prwW , and shortB
term scenario nodes u by the conditional probability prb(u)
. Table 1 shows common notation
used in the paper.
In the following we present the optimization problems of all market players. We discuss
the supplier formulation extensively, and keep the discussion for the other players succinct.
For more details refer to Huppmann and Egging (2014). The Karush-Kuhn-Tucker conditions
for all players are presented in Appendix A.
The supplier
Suppliers produce and sell fuels or energy carriers (e.g., biomass , electricity) using infrastructure services for transmission, transformation and storage. Each supplier maximizes his
expected net profit over all investment and operational nodes. The cost in each investment
node is weighed by the absolute probability prwW and discounted by dfw . Meanwhile, the
B
revenue and cost in each operational node are weighed by the conditional probability prb(u)
,
the relative duration duru , and also discounted. The optimization objective of supplier s is
shown in Eq.(1).
See Table 2 for notations used in the supplier problem. Supplier profit comes from selling
fuels to final demand sectors q D , subtracting cost of: production costP (·), transmission
q A , transformation q C , storage injection q O− and extraction q O+ , investment in production
capacity z P , and purchasing emission permits. In the summation term collecting the fuel
transmission fees for fuel e, term a ∈ A+
ne represents the subset of arcs ending at node n.
Similarly, in the summation term for storage cost, o ∈ OeE represent the subset of technologies
for storing fuel e. As in the deterministic model, the supplier owns produced energy carriers
11
Table 2: Notations relevant to the supplier
Notation
Subsets and
mappings:
a ∈ A+
ne
a ∈ A−
ne
e ∈ EcC−
(e, f ) ∈ EcC
f ∈ EcC+
nA+ (a)
nA− (a)
o ∈ OeE
u ∈ UbB
V
u ∈ Uvo
v ∈ VoO
v U (uo)
Ww→
Ww←
x(w)
Variables:
P
qwusne
A
qwusa
C
qwusnce
O−
qwusno
O+
qwusno
D
qwusnde
P
zwsne
P
αwusne
O
αwvsno
P
ζwsne
φwusne
Parameters:
P
avlwusne
capP
wsne
costP
wusne (·)
S
courwsnd
P
depww0 sne
emsP
wsneg
expP
wsne
P
golwsne
P
invwsne
P
linwsne
lossP
sne
qudP
wsne
Explanation
subset of arcs transmitting fuel e ending at node n
subset of arcs transmitting fuel e starting at node n
subset of input fuel(s) e of energy transformation technology c
input/output fuel mapping of energy transformation technology c
subset of output fuel(s) f of energy transformation technology c
end node of transmission arc a (singleton)
start node of transmission arc a (singleton)
subset of storage technologies storing fuel e
subset of short-term scenario nodes along a short-term scenario b
mapping among short-term scenario node, loading cycle and technology
subset of loading cycles with respect to storage technology o
mapping among loading cycle, short-term scenario node and technology (singleton)
subset of successor nodes of long-term scenario node w
subset of predecessor nodes of long-term scenario node w
long-term scenario x that contains node w
quantity produced
quantity transmitted through arc a
quantity put into energy transformation technology c
quantity to be injected into storage o
quantity to be extracted from storage o
quantity sold to final demand sector d
production capacity expansion
dual for available production capacity constraint
dual for injection/extraction balance constraint
dual for production capacity expansion limit constraint
dual for mass-balance constraint
availability factor of production capacity
existing capacity (before investment) of production
production cost function of supplier s for fuel e in geographical node n
Cournot market power parameter of supplier s in sector d of geographical node n
depreciation factor of production capacity expansion
emission of type g during production process of fuel e in node n by supplier s
limit of production capacity expansion
logarithmic term in the production cost function (golP ≥ 0)
unit cost of production capacity expansion
linear term in the production cost function (linP ≥ 0)
loss rate during production process of fuel e in node n
quadratic term in the production cost function (qudP ≥ 0)
throughout the whole value chain, after transmission, storage, but also transformation. In
the supplier objective, the value of courS determines to what extent suppliers exert market
power. Values range from 0 to 1; a value of 1 represents a Nash-Cournot supplier and a
12
value of 0 represents a perfectly competitive supplier.
(
X
max
q P ,q A ,q C
w∈W,u∈U
q O− ,q O+ n∈N,e∈E
D P
q ,z
W
B
dfw duru prw
prb(u)
X
S
S
D
D
courwsnd
ΠD
(·)
+
(1
−
cour
)p
wunde
wsnd wunde qwusnde
d∈D
− costP
wusne (·) −
X
a∈A+
ne
−
X
X
A
pA
wua qwusa −
C
pC
wunce qwusnce
c∈C
O−
O+
O+
pO−
wuno qwusno + pwuno qwusno
o∈OeE
)
−
X
P
P
P
P
pG
wb(u)ng emswsneg qwusne − invwsne zwsne
(1)
g∈G
X
P
P
P
P
qwusne
≤ avlwusne
capP
+
dep
z
0
0
wsne
w wsne w sne
s.t.
P
(αwusne
)
(2)
if v ∈ VoO
(3)
(φwusne )
(4)
P
(ζwsne
)
(5)
←
w0 ∈Ww
X
X
O+
duru qwusno
=
V
u∈Uvo
P
(1 − lossP
sne )qwusne +
O−
duru (1 − lossO−
o )qwusno
V
u∈Uvo
X
+
o∈OeE
O+
qwusno
=
X
d∈D
X
C
transf C
wncf e qwusncf +
D
qwusnde
+
A
(1 − lossA
a )qwusa
a∈A+
ne
c∈C,f ∈EcC−
X
O
(αwvsno
)
X
C
qwusnce
+
X
a∈A−
ne
c∈C
A
qwusa
+
X
O−
qwusno
o∈OeE
P
zwsne
≤ expP
wsne
The supplier problem includes several constraints. In the production capacity constraint
(Eq.(2)), the production capacity capP including previous additions z P (depreciation by depP
) is weighted by the availability factor avlP .
2
The storage injection/extraction constraint,
Eq.(3) specifies that within a loading/unloading cycle the fuel quantities injected q O− (after
O+
injection loss lossO−
must balance. Mass balance Eq.(4) contains on
o ) and extracted q
the left hand side all supply sources and on the right hand side all sinks. For each fuel
e in geographical node n, the total loss-corrected production, transformation output, import and extraction from storage q O+ must be cover, hence be equal to, the total sales q D ,
transformation inputs, export, and storage injection q O− . Eq.(5) stipulates that produc2
For conventional fuels this will be a value close to, and at most 1, for renewables this value will be lower
dependent on the type.
13
tion capacity expansions z p should not exceed limit expP . Compared to the deterministic
model, this paper excludes fossil fuel reserve constraints. How to address such constraints
in a multi-horizon scenario approach will be discussed in future work. Finally, for reference
regarding the production cost functions (c.f., Golombek et al. (1995)), Eqs.(6)-(8) below are
the probabilistic specification of Eqs. (2a)-(2c) in Huppmann and Egging (2014).
The problem structure of the three infrastructure service providing players presented next
is very similar. The service providers collect fees from renting out capacities, while incurring
operational cost for the operation of the infrastructure, emission cost as well as capacity
investment expenses. Due to the price-taking nature assumed for all infrastructure service
providers, we can aggregate the capacity and operations for each infrastructure type by geographical node. Hence, there will be (at most) one transmission arc representing electric
power transmission between nodes A and B, there will be (at most) one transformation unit
generating coal-based electricity in node A, etc. As in the supplier problem above, in each
problem the investment cost is weighed by the long-term scenario node probabilities only,
whereas the operational and emission cost are additionally weighed by the short-term conditional probabilities. All losses due to transmission, transformation and storage operations
are accounted for in the supplier mass balance (Eq.(4)).
P
P
P
P
P
2
costP
wusne (·) = (linwsne + golwsne )qwusne + qudwsne (qwusne )
+
P
golwsne
P
cd
apwusne
−
P
qwusne
ln 1 −
P
qwusne
!
P
cd
apwusne
!
P
∂ costP
qwusne
wusne (·)
P
P
P
P
= linwsne + 2qudwsne qwusne − golwsne ln 1 −
P
P
∂ qwusne
cd
apwusne
"
!
#
P
P
∂ costP
qwusne
qwusne
wusne (·)
P
P
P
= golwsne avlwusne depw0 wnse ln 1 −
+
if w0 ∈ Ww←
P
P
P
∂zw
0 sne
cd
apwusne
cd
apwusne
X
P
P
P
P
P
where cd
apwusne = avlwusne capwsne +
depw0 wnse zw0 nse
(6)
(7)
(8)
←
w0 ∈Ww
The transmission system operator
Transmission arcs in the model are directional and, naturally, specific to energy carriers. If an oil pipeline and an electricity transmission cable both allow exports from two
geographical node A to B, they are included in the model as different arcs with their own
capacities.
The transmission system operator (TSO) maximizes the expected net present profit from
renting out arc capacity f A while subtracting costs of capacity investment, arc operation
14
Table 3: Notation relevant to the transmission system operator
Notation
Variables:
A
fwua
A
pwua
A
zwa
A
ζwa
A
τwua
Parameters:
capA
wa
depA
ww0 a
emsA
wag
expA
wa
A
invwa
lossA
a
trf A
wa
Explanation
quantity transmitted by the transmission system operator
market-clearing price of using transmission arc capacity
transmission arc capacity expansion
dual for transmission arc capacity expansion limit
dual for transmission arc capacity constraint
existing capacity of arc a
depreciation factor of arc capacity expansion
emission of type g during transmission process through arc a
limit of transmission arc capacity expansion
unit cost of arc capacity expansion
loss rate during transmission process through arc a
tariff of using transmission arc a
and emission (see Eq.(9)). See Table 3 for relevant notation.
max
f A ,z A
X
X
A
W
B
A
G
A
A
A A
dfw duru prw
prb(u)
(pA
−
trf
)f
−
p
ems
f
−
inv
z
wua
wa wua
wag wua
wa wa
wb(u)ng
w∈W,u∈U
(9)
g∈G
s.t.
A
fwua
≤ capA
wa +
X
A
depA
w0 wa zw0 a
A
(τwua
)
(10)
A
(ζwa
)
(11)
←
w0 ∈Ww
A
zwa
≤ expA
wa
The transformation technology operator
The transformation function in the model converts one or more fuels into one or more
other fuels. For example, a biomass-based power generation plant transforms biomass into
electric power. The objective of the transformation technology operator (TTO) is shown in
Eq.(12). See Table 4 for relevant notation.
Compared to the TSO, the TTO objective and capacity constraint Eq.(13) are very
similar, however there is an additional constraint, Eq.(14). This constraint regulates that at
least some minimum share of the aggregate output should be based on specific input fuel(s);
it can be used as a technical restriction or policy regulation, e.g., a bio-fuel mandate.
15
Table 4: Notation relevant to the transformation technology operator
Notation
Variables:
C
fwunce
C
pwunce
C
zwnc
C
βwunce
C
ζwnc
C
τwunc
Parameters:
capC
wnc
depC
ww0 nc
emsC
wceg
expC
wnc
C
invwnc
C
shrwnce
transf C
wncef
trf C
wnc
Explanation
quantity of fuel e put into energy transformation technology c
market-clearing price of using transformation technology capacity
transformation technology capacity expansion
dual for minimum input share of fuel e regarding transformation technology c
dual for transformation technology capacity expansion limit constraint
dual for energy transformation technology capacity constraint
existing capacity of transformation technology c (as measured in output fuel)
depreciation factor of transformation capacity expansion
emission of type g during transformation process of (input) fuel e
limit of transformation technology capacity expansion
unit cost of transformation technology expansion
minimum share of fuel e regarding energy transformation technology c
transformation rate of technology c from input fuel e to output fuel f in node n
tariff of using energy transformation technology c
"
max
f C ,z C
X
W
B
dfw duru prw
prb(u)
#
(pC
wunce
−
C
trf C
wnc )fwunce
−
w∈W
u∈U
e∈EcC−
X
C
C
pG
wb(u)ng emswceg fwunce
−
C
C
invwnc
zwnc
g∈G
(12)
X
s.t.
C
C
transf C
wncef fwunce ≤ capwnc +
X
C
depC
w0 wnc zw0 nc
C
(τwunc
)
(13)
C
(βwunce
)
(14)
C
(ζwnc
)
(15)
←
w0 ∈Ww
(e,f )∈EcC
C
shrwnce
X
C
transf C
wnce0 f fwunce0 ≤
(e0 ,f )∈EcC
X
C
transf C
wncef fwunce
f ∈EcC+
C
zwnc
≤ expC
wnc
The storage operator
Naturally, storage units are fuel-specific. Storage aids suppliers to transfer an amount
of fuel to other periods within the loading/unloading cycle. All costs, emissions and losses
are accounted for during injection. Limitations for total storage capacity, injection and
extraction capacity respectively are given by Eqs.(17)-(19). In the same order, Eqs.(20)-(22)
represent the expansion limitations. Table 5 contains the relevant notation.
16
Table 5: Notation relevant to the storage technology operator
Notation
Variables:
O−
fwuno
O+
fwuno
O−
pwuno
pO+
wuno
O
zwno
O−
zwno
O+
zwno
O−
ζwno
O+
ζwno
O
ζwno
κO−
wuno
κO+
wuno
O
τwvno
Parameters:
capO
wno
capO−
wno
capO+
wno
depO
ww0 no
depO−
ww0 no
depO+
ww0 no
emsO−
wog
expO
wno
expO−
wno
expO+
wno
O
invwno
O−
invwno
O+
invwno
lossO−
o
trf O−
wno
Explanation
gross quantity injected into storage o
gross quantity extracted from storage o
market-clearing price of injection into storage o
market-clearing price of extraction from storage o
yearly storage capacity expansion
injection capacity expansion
extraction capacity expansion
dual for injection capacity expansion limitation
dual for extraction capacity expansion limitation
dual for yearly storage capacity expansion limitation
dual for storage injection capacity constraint
dual for storage extraction capacity constraint
dual for storage capacity constraint in loading cycle v
existing capacity for fuel stored in storage technology o over one loading cycle
existing capacity for fuel injection into storage
existing capacity for fuel extraction rate from storage technology o
depreciation factor of yearly storage capacity expansion
depreciation factor of storage injection capacity expansion
depreciation factor of storage extraction capacity expansion
emission of type g (accounted at injection) during storage injection/extraction process
limit of yearly storage capacity expansion
limit of storage injection capacity expansion
limit of storage extraction capacity expansion
unit cost of yearly storage capacity expansion
unit cost of storage injection capacity expansion
unit cost of storage extraction capacity expansion
loss rate (accounted at injection) during storage injection/extraction process
tariff for using injection capacity into storage technology o
"
max
f O− ,f O+
z O ,z O− ,z O+
X
W
B
dfw duru prw
prb(u)
O−
O−
O+
O+
(pO−
wuno − trf wno )fwuno + pwuno fwuno
w∈W,u∈U
#
−
X
O− O−
pG
wb(u)ng emswog fwuno
−
g∈G
17
O
O
invwno
zwno
−
O− O−
invwno
zwno
−
O+ O+
invwno
zwno
(16)
s.t.
X
O−
duru fwuno
≤ capO
wno +
X
O
depO
w0 wno zw0 no
O
(τwvno
)
if v ∈ VoO
(17)
O−
depO−
w0 wno zw0 no
(κO−
wuno )
(18)
O+
depO+
w0 wno zw0 no
(κO+
wuno )
(19)
O
zwno
≤ expO
wno
O
(ζwno
)
(20)
O−
zwno
≤ expO−
wno
O−
(ζwno
)
(21)
O+
zwno
≤ expO+
wno
O+
(ζwno
)
(22)
←
w0 ∈Ww
V
u∈Uvo
O−
fwuno
≤ capO−
wno +
X
←
w0 ∈Ww
O+
fwuno
≤ capO+
wno +
X
←
w0 ∈Ww
The emission permit auctioneer
The emission permit auctioneer allocates quota for environmental emissions (e.g., CO2 )
to other market players. The model can represent emission ceilings that count at the global,
regional or nodal level, i.e., Eqs.(24)-(26). Besides, emission taxes for the three levels are
considered too, enforcing that emission prices have strictly positive lower bounds even when
emission ceilings are not binding. See Table 6 for relevant notation. In contrast to the
formulation of the deterministic model where the dual variables are with respect to longterm periods, here the dual variables are with respect to short-term scenarios, accounting
for tax values being short-term scenario-dependent.
Table 6: Notation relevant to the emission auctioneer
Notation
Subsets:
n, k ∈ Nr
r ∈ Rn
Variables:
G
fwbng
G
pwbng
µglob
wbg
µreg
wbrg
µnod
wbng
Parameters:
quotaglob
wg
quotareg
wrg
quotanod
wng
taxglob
wbg
taxreg
wbrg
taxnod
wbng
Explanation
mapping from region to geographical node
mapping from geographical node to region
quantity of emission type g in node n
market-clearing price of environmental emission of type g
dual for global emission constraint of type g
dual for regional emission constraint of type g
dual for nodal emission constraint of type g
global emission quota
regional emission quota
nodal emission quota
tax for global emission of type g
tax for regional emission of type g
tax for nodal emission of type g
18
max
fG
X
W
dfw prw
prbB
X
reg
glob
nod
G
G
taxwbrg − taxwbng fwbng
pwbng − taxwbg −
(23)
r∈Rn
w∈W,b∈B
n∈N,g∈G
X
s.t.
G
fwbng
≤ quotaglob
wg
(µglob
wbg )
(24)
G
fwbng
≤ quotareg
wrg
(µreg
wbrg )
(25)
G
fwbng
≤ quotanod
wng
(µnod
wbng )
(26)
n∈N
X
n∈Nr
Final demand
Each fuel can be used by final demand sectors with some efficiency eff D to provide
services. The inverse demand curve of fuel e by sector d is Eq.(27). See Table 7 for relevant
notation.

!
X
D
D
 D
pD
wunde = eff wnde intwund − slpwund
D

eff D
wndf qwusndf
s∈S,f ∈E
!
−
euccD
wunde
−
D
euclwunde
X
D
qwusnde
−
s∈S
X
D
pG
wb(u)ng emswdeg
(27)
g∈G
h
D
D
In Eq.(27), the first part, i.e., eff D
wnde intwund − slpwund
P
s∈S,f ∈E
D
eff D
wndf qwusndf
i
, de-
picts the composite sector price index (energy service price weighted by the service effiD
ciency); the second part, i.e., −euccD
wunde − euclwunde
of a fuel; the third part, i.e., −
P
g∈G
P
s∈S
D
qwusnde
, depicts the end use cost
D
pG
wb(u)ng emswdeg , represents the emission cost of using a
fuel.
Moreover, consumption is subject to other constraints of, e.g., government regulation and
technology restrictions. Eq.(28) stipulates that at least a share shrL of total consumption
in final demand sector(s) d ∈ DlL of using fuel(s) e ∈ EblL is supplied by the subset of fuel(s)
e ∈ ElL . For example, in (EC, 2009), Sweden set the target that by 2020 renewable sources
(e ∈ ElL ) should account for 49% (shrL ) of total energy (e ∈ EblL ) consumption. Besides,
M
Eq.(29) regulates that at least a share shrM of total consumption of fuel(s) e ∈ Em
is realized
M
from using transformation technology(s) c ∈ Cm
. For instance, the EU proposed that by
M
2020 10% (shrM ) consumption of the transport fuels (e ∈ Em
) come from transformation
19
Table 7: Notation relevant to the demand
Notation
Subsets:
M
c ∈ Cm
d ∈ DlL
e ∈ ElL
bL
e∈E
l
M
e ∈ Em
Variables:
pD
wunde
L
βwunl
M
βwunm
Parameters:
eff D
wnde
emsD
wdeg
euccD
wunde
D
euclwunde
intD
wund
L
shrwnl
M
shrwnm
slpD
wund
ΠD
wunde (·)
Explanation
transformation technology(ies) satisfying transformation mix constraint m
demand sector(s) where fuel mix constraint l applies
fuel(s) accounting for at least a share of total consumption of involved fuel(s)
b L in fuel mix constraint l
e∈E
l
fuel(s) involved in fuel mix constraint l
fuel(s) satisfying transformation mix constraint m
final demand sector price
dual of fuel mix constraint l
dual of generation mix constraint m
efficiency of demand satisfaction in demand sector d at node n by fuel e
emission of type g during consumption process of fuel e in node n
constant end use cost parameter
linear end use cost parameter
intercept of inverse demand curve of demand sector d in node n
minimum share regarding sector fuel mix constraint l
minimum share regarding generation mix constraint m
slope of inverse demand curve of demand sector d in node n
inverse demand curve of demand sector d regarding fuel e
M
technologies (c ∈ Cm
) with renewables as inputs (EC, 2009), e.g., bio-diesel transformed
from oil seeds.
X
L
shrwnl
s∈S,d∈DlL
X
s∈S,d∈D
M
f ∈Em
D
eff D
wnde qwusnde
L
(βwunl
)
(28)
M
(βwunm
)
(29)
s∈S,d∈DlL
e∈ElL
blL
e∈E
M
shrwnm
X
D
eff D
wnde qwusnde ≤
D
qwusndf
≤
X
C
transf C
wncef qwusnce
s∈S,(e,f )∈EcC
M
c∈Cm
Market clearing
The optimization problems of individual market players are linked by the market clearing
conditions (MCC). Eq.(30) ensures market balance for transmission capacity between suppliers and the TSO. Eq.(31) ensures that transformation capacity offered by the TTO equals
what is used by the suppliers. Eqs.(32) and (33) balance the markets for storage injection
20
and extraction respectively. Finally, Eq.(34) ensures balance in the emission permit market.
X
A
A
qwusa
= fwua
(pA
wua )
(30)
(pC
wunce )
(31)
O−
O−
qwusno
= fwuno
(pO−
wuno )
(32)
O+
O+
qwusno
= fwuno
(pO+
wuno )
(33)
(pG
wbng )
(34)
s∈S
X
C
C
qwusnce
= fwunce
s∈S
X
s∈S
X
s∈S
X
duru
e∈E,u∈UbB
X
P
emsP
wsneg qwusne +
s∈S
X
D
emsD
wdeg qwusnde
s∈S,d∈D
!
+
X
a∈A+
ne
A
emsA
wag fwua
+
X
C
emsC
wceg fwunce
+
c∈C
X
O−
emsO−
wog fwuno
G
= fwbng
o∈O
2.4. Mixed complementarity problem
All optimization problems presented above are convex optimization problems with polyhedral feasible regions, therefore the Karush-Kuhn-Tucker (KKT) conditions are necessary
and sufficient for optimal solutions (Cottle et al., 1992). By combining the KKTs deduced
from all players’ optimization problems with the market clearing conditions we get a mixed
complementarity problem (MCP). This MCP is programmed in GAMS (Brook et al., 1988)
and can be parameterized to reflect different markets and case studies.
Stochastic mixed complementarity programs suffer from scalability problems (Egging,
2013). It takes many hours to solve a two-season, five-period, 28 region data set with the
deterministic version of the model (Huppmann and Egging, 2014). Obviously, incorporating
uncertainty increases the model size drastically, and we may expect that the model will
need very much time to solve, if at all. It is beyond the scope of this paper to develop a
decomposition approach which might reconcile the solution time. Instead, in the following
we introduce synthetic market cases to show and discuss the traits of the model as well as
general insights useful for policy development and investment decisions under uncertainty.
3
3
The dual horizon stochastic oligopoly data instance presented in the next section contains 41695 variables
(and equations; an MCP is a square system). The model solution time for this data instance is about ten
minutes on a computer with a CPU of 7.88 GB and 3.40 GHz, using PATH version 24.2.1 (Ferris and Munson,
2000).
21
3. Case study
In the following we introduce the market setup which we will consider with both a
perfectly competitive and a Cournot oligopoly market structure. We solve three variants,
distinguished by the level of uncertainty: deterministic (i.e., no uncertainty), long-term
uncertainty only, and both long-term and short-term uncertainty.
Table 8: General information of the market
Element
4 geographical nodes:
2 regions:
4 suppliers:
2 demand sectors:
3 types of fuel/energy-carrier:
2 types of transmission arc:
2 types of storage technology:
2 types of loading cycle:
2 types of transformation technology:
1 type of emission:
1 type of tax:
Comment
East, South, West, North.
Northeast, Southwest.
SA, SB , SC , SD .
industry, commercial.
electricity, natural gas, renewable.
electricity cable, natural gas pipeline.
natural gas storage, electricity storage.
yearly for gas, daily for electricity.
gas-to-power, renewable-to-power.
carbon dioxide.
regional carbon tax.
3.1. The market setup
Table 8 summarizes the setup of the market. The market contains four geographical
nodes each with two demand sectors. On nodes North and East there is production of
natural gas as well as renewable (energy) by supplier S A and S B respectively. The other
two suppliers S C in South and S D in West produce renewable only. (Renewable could be
biomass, wind, etc.) Fuel-specific directional transmission arcs (see Figure 4) can be used
by all suppliers to transmit fuels (energy carriers) abroad. All nodes have transformation
capacity to allow electricity production from natural gas as well as renewable. Gas storage
exists in the regions without own production: West and South, whereas electricity storage
exists only in North. Within each year four periods are distinguished representing days and
nights in summer as well as winter. We have defined a loading/unloading cycle (Eq.(3)) of
one day for (pumped) electricity storage: what is injected (extracted) during a summer day
must be extracted (injected) during a summer night. (The same for winter day and night.)
For gas storage the loading cycle covers all periods within a year.
22
To switch between a perfectly competitive market (”Case-Com”) and a Cournot oligopoly
(”Case-Oli”) the ”Cournot parameter” courS in Eq.(1) is used. In Case-Com, courS = 0 for
all suppliers; whereas in Case-Oli, all courS = 1.
Figure 4: The geographical nodes connected
For each demand sector, the slope of the inverse demand function doesn’t change over
years, however the intercept increases 10% every ten years. In this paper, the unit of measurement for capacities is GWh/day. The initial gas production capacities are 32.64 in East
and 26.52 in North. The initial renewable production capacities are 22 in East, 60 in North,
40 in West, and 16 in South. The initial transformation (output) capacity of renewable-fired
power plant is 5 in every geographical node, and the transformation rate is 0.378. The
initial transformation (output) capacity of gas-fired power plant is 10 in every geographical
node, and the transformation rate is 0.352 in North and West, and 0.396 in East and South.
The initial transmission capacity is shown in Figure 4 (along arcs). The initial gas storage
capacity is 300 in West and 200 in South. The initial electricity storage capacity is 300 in
North.
3.2. The three model variants
To inspect the traits of our proposed model, we compare it with two other models. One
is a deterministic equilibrium model, the other a stochastic equilibrium model with only
23
long-term uncertainties. Both these models can be formed by adequately parametrizing
the multi-horizon model formulation presented above. In the three models, we assume the
discount factor dfw to be 10% per year for investment decisions.
Deterministic model
The first model used for contrasting results is a deterministic parametrization of our
model formulation (hereafter: ”DET”). DET is identical to the original deterministic model,
except that we have omitted the fossil fuel reserves constraints. Figure 5 illustrates the deterministic scenario tree represented as a multi-horizon scenario tree with two time resolutions.
(Since there is neither long-term nor short-term uncertainty, this is a degenerated multihorizon scenario tree.) There are four strategic/long-term stages, considered to represent
the period from 2010 to 2050 in ten-year steps. Figure 6 shows how the four representative
periods mentioned above (summer day, summer night, winter day and winter night) are
connected to long-term stage.
Figure 5: Scenario tree representation of the deterministic equilibrium model
Figure 6: Representative periods
Stochastic model with long-term uncertainty only
The second model used for comparing results is a four-stage stochastic equilibrium model
with long-term but without short-term uncertainty (hereafter ”STO-L”). We want to emphasize that this model by itself would have been a significant contribution as there are no
publications about stochastic multi-fuel equilibrium models in the current literature. Figure
24
Figure 7: Scenario tree of the stochastic equilibrium model with only long-term uncertainties
Note: K1 is the cost parameter adjustment relative to the average value.
7 shows the scenario tree represented in STO-L. In each (long-term) investment node, the
tree divides into two branches corresponding to different realizations of the uncertain events
in the next stage.
Specifically, the considered long-term uncertainty concerns the production cost of natural
gas (for all suppliers). In different branches of the tree, the quadratic term in the production
cost function (i.e., qudPwsne in Eq.(6)) is adjusted using parameter K1 to depict deviations in
parameter values in STO-L relative to the counterpart values used in DET. The adjustments
are assumed symmetrical, as illustrated in Figure 7. In the case studies we assume K1 = 0.4.
A dual-horizon stochastic equilibrium model
The stochastic equilibrium model with both long-term and short-term uncertainties (hereafter: ”STO-LS”) reflects the second contribution of this paper. To the best of our knowledge
the current literature does not present multi-horizon stochastic equilibrium models even for
single fuel markets. Figure 8 illustrates the four-stage dual-horizon scenario tree, used to
reflect uncertainties in long-term production cost of natural gas (as in STO-L), and addi25
Figure 8: Multi-horizon scenario tree of the stochastic equilibrium model with both long-term and short-term
uncertainties
Figure 9: Short-term uncertainty analysis
tionally short-term uncertainties in the availability of renewable production.
In DET and STO-L the availability of renewable energy production is assumed to be
50%. In STO-LS we use parameter K2 to depict the deviations in availability relative to
the counterpart values in DET. Figure 9 shows how in STO-LS the values used in DET
are adjusted in each operational node. For instance, in operational node u11 (in short-term
scenario b1) the availability in u11 is the value in DET multiplied by (1 + K2), etc. We
assume K2 = 0.4.
26
4. Results analysis
The investment decisions obtained from the cases and the expected values of the stochastic solutions allow us to analyze how long-term and short-term uncertainties affect the decisions of market players and the social welfare and profits connected to the solutions. The
three uncertainty variants DET, STO-L and STO-LS are solved assuming two different market structures, resulting in six cases total. As indicated above, the model is coded in GAMS
and we use the PATH solver (Ferris and Munson, 2000) to solve all problem instances. In the
following we discuss the competitive variants side by side, before moving to the oligopoly variants. Then, we analyze social welfare and the value of stochastic solution for each player. It
will become clear that although expected profits and social welfare do not vary much among
the scenarios, there are significant differences in investment decisions, as well as in volumes
produced, traded and consumed.
4.1. Perfectly competitive results (Case-Com)
First we compare each player’s investment specifically in production, transformation and
transmission capacities. Next, we analyze the total expected (average) investment with
respect to three different models.
Table 9: Investment in production capacity in the perfectly competitive markets
Supplier
S B (in EAST)
S B (in EAST)
S A (in NORTH)
Fuel
GAS
REN
GAS
Model
DET
STO-L
STO-LS
DET
STO-L
STO-LS
DET
STO-L
STO-LS
stage 1
w1
22.8
23.9
25.7
stage 2
w2 w3
4.5
4.7
38.3
39.3
41.0
w4
7.4
6.9
4.5
stage 3
w5 w6
w7
0.9
0.6
6.3
2.9
5.7
5.7
1.3
1.2
Average
22.8
27.8
29.3
4.5
3.9
0.7
38.3
42.5
44.2
Production capacity investment in Case-Com
Table 9 presents investment in production capacity for the three variants. (Note that
the lead time to bring capacity online is one stage, hence there is no investment in the last
27
stage, nodes w8 -w15 ). To facilitate the comparison, we replicate the deterministic values for
all scenario nodes they should be compared to. (As such, only value 4.5 for S B , REN, DET
is duplicated in this table).
Table 9 (right hand side) shows that (average) gas production capacity investment for
S B in DET, STO-L and STO-LS is 22.8, 27.8 and 29.3 respectively, and for S A 38.3, 42.5
and 44.2. Hence, gas production capacity investment increases when uncertainty increases
in both regions EAST and NORTH, regardless of the uncertainty type. In contrast, the
average renewable capacity investment in EAST is lower when increasing uncertainty, with
values 4.5, 3.9 and 0.7 respectively.
In all three variants the majority of natural gas production capacity investment occurs in
the first stage, whereas renewables expansions occur only in later stages. All gas production
expansions in the first stage are higher for higher uncertainty levels. In later stages no
additional expansions occur in DET, whereas they do occur in the stochastic models in
scenarios where the uncertain production cost turn out lower than average (w3 and w7 ). Here
we see that in STO-L the later stage expansions are higher than in STO-LS, however not
enough to compensate for the lower first-stage expansions. As such, the maximum invested
capacities (in w14 and w15 ) in STO-L of (23.9+7.4+0.9=) 32.2 in EAST and (39.3+5.7+1.3=)
46.3 in NORTH are lower than the corresponding maximum values of (25.7+6.9+0.6=) 33.2
and (41.0+5.7+1.2=) 47.9 in STO-LS. When considering the renewable production capacity
expansions in STO-L (in EAST), we see that when natural gas production cost turns out
high (w2 ) the addition of 4.7 is slightly higher than the 4.5 in DET. In contrast, when gas
production cost turn out low no renewable expansions occur. In the third stage, only when
natural gas production cost again turn out high (w4 ) an additional expansion of renewable
capacity occurs. As such, in half of the scenarios in STO-L the renewables capacity is higher
than in DET by the end of the time horizon, and in two of them significantly so. In STO-LS,
investment in renewable production capacity in EAST does not occur until the third stage,
and only in the scenario where gas production cost turn out much higher (w4 ). Still the
expansion of 2.9 is much lower than in the other model variants.
This shows how adding short-term uncertainty in renewables production favors investment in gas production capacity.
28
Table 10: Investment in transformation capacity in the perfectly competitive markets
Node
EAST
Fuel
GAS
EAST
NORTH
REN
REN
WEST
REN
Model
DET
STO-L
STO-LS
STO-L
DET
STO-L
STO-LS
DET
STO-L
stage 1
w1
7.6
7.7
10.0
stage 2
w2 w3
2.4 2.4
1.5 5.2
3.3
w4
0.2
1.2
6.3
6.3
6.3
2.6
2.6
stage 3
w5 w6
w7
0.4
0.2
Average
9.9
11.2
11.7
0.3
6.3
6.3
6.3
2.6
2.6
Transformation capacity investment in Case-Com
Table 10 shows transformation (i.e., power generation) capacity investment in the three
model variants. The expansions in gas-fired power generation are similar to the results
for gas production capacity: both adding long and short-term uncertainty leads to higher
expansions in the first stage, with values 7.6, 7.7, 10.0, as well as in expectation: 9.9, 11.2
and 11.7.
In the second stage in DET the investment in gas-fired power generation capacity in
EAST is 2.4, anticipating the further increase in electricity demand in future stages. In
STO-L there is rather high investment of 5.2 when gas production costs have turned out
low (w3 ), compared to only 1.5, when natural gas production costs have turned out high
(in node w2 ). In STO-LS the first stage expansion is 30% higher than in DET and STO-L.
In the second and third stage additional expansions happen in the low gas production cost
scenarios, to allow more gas-fired power to be delivered to the end-use sectors.
Turning the focus to renewable power generation. Every renewable-fired power plant
has an initial transformation (output) capacity of 5. We observe that most investments of
renewable-fired power plant take place in North and West whose initial renewable (maximum)
production capacities (60 and 40 respectively) are higher than East and South (23.64 and
26.52 respectively). It reflects that the transformation technology operator makes investment
decision accounting for the local production capacity of the primary energy. In NORTH and
WEST, all investments occur in the first stage. In North, the investment solutions from all
three models are the same (6.3). Note that the additional 6.3 (more precisely 6.34) together
29
with the initial capacity 5 (total 11.34) is just enough to transform all available renewable
produced in North (60 × 50% = 30, 30 × 0.378 = 11.34). By observing the results, the
renewable quantities put into renewable-fired power plant in North are 30 (in most of shortterm scenarios) after the first stage investments. In West, the investment solutions from
DET and STO-L can be explained in the same way. In contrast to in STO-L, in STO-LS
where renewable production uncertainty is considered, there is no investment of renewablefired power plant in West. An observation in line with our intuition is the investment in
STO-L of 1.2 in EAST in w4 when gas production cost have turned out much higher making
renewables more interesting. Additionally adding short-term uncertainty of the renewable
availability makes the renewable less interesting again: no expansion in STO-LS.
These results illustrate how expensive gas favors renewable power generation but that
the uncertainty of renewable production puts its downstream use to generate electricity at
a disadvantage compared to gas-fired power generation.
Table 11: Investment in transmission capacity in the perfectly competitive markets
Fuel
GAS
GAS
GAS
Outward
EAST
NORTH
NORTH
Inward
SOUTH
SOUTH
WEST
Model
DET
STO-L
STO-LS
DET
STO-L
STO-LS
DET
STO-L
STO-LS
stage 1
w1
34.7
34.2
34.4
16.7
16.6
14.8
29.9
30.7
30.3
stage 2
w2 w3
0.5
w4
stage 3
w5 w6
5.3
5.9
0.4
0.7
0.2
2.3
3.1
0.7
2.4
w7
1.2
2.1
1.2
1.1
Average
34.7
37.2
37.8
16.7
17.5
15.9
29.9
31.4
31.8
Transmission capacity investment in Case-Com
Table 11 shows the transmission capacity expansions. It turns out that only gas pipelines
are expanded, whereas electricity lines are not. The transmission expansions present a mixed
picture, as there is a less explicit relation between additional uncertainty and expansions.
On average arcs EAST to SOUTH and NORTH to WEST are expanded more when adding
more uncertainty, whereas North to SOUTH is expanded most in STO-L. However, we still
see that the expansions in later stages tend to follow the pattern: scenarios with lower gas
production cost result in higher expansions in later stages.
30
Expected total investment in Case-Com
Figure 10 shows the total expected investment for five types of capacities, summarizing
our findings. Across the board we observe that adding long-term uncertainty in natural
gas production cost (STO-L vs DET) results in higher (expected) investment in natural gas
production, transmission and transformation and lower (expected) investment in renewable
production. Additionally adding short-term uncertainty in renewable availability (STO-LS)
results in even more investments in natural gas production and transformation; but lower
investment in renewable production and transformation. Considering the unpredictability
of renewable energy availability, players would shift investment to more reliable natural gas.
Figure 10: The total expected investments in the perfectly competitive markets
4.2. Cournot oligopoly results (Case-Oli)
In this section, we present the results in a market structure wherein all suppliers are considered as Cournot-oligopoly suppliers. First, we compare each player’s investment specifically in production, transformation and transmission capacities. Next, we analyze the total
expected (average) investment with respect to three different models.
31
Table 12: Investment in production capacity in the Cournot oligopoly markets
Supplier
S B (in EAST)
S B (in EAST)
S A (in NORTH)
Fuel
GAS
REN
GAS
Model
DET
STO-L
STO-LS
STO-L
DET
STO-L
STO-LS
stage 1
w1
9.3
9.4
11.2
stage 2
w2 w3
w4
4.7
4.1
stage 3
w5 w6
w7
1.0
0.3
4.5
8.1
7.9
10.9
0.5
0.5
3.7
4.5
1.0
Average
9.3
12.0
13.3
1.1
8.6
10.0
13.2
Production capacity investment in Case-Oli
Table 12 presents investment in production capacity in the Cournot oligopoly markets for
the three variants. The added production capacities are much lower than the expansions in
the perfectly competitive markets (see Table 9), so as to deploy market power by (relatively)
decreasing the outputs (by lower capacities) in later stages. Table 12 (right hand side)
shows that (average) gas production capacity investment for S B in DET, STO-L and STOLS is 9.3, 12.0 and 13.3 respectively, and for S A 8.6, 10.0 and 13.2. Hence, gas production
capacity investment increases when uncertainty increases in both regions EAST and NORTH,
regardless of the uncertainty type.
In all three variants the majority of natural gas production capacity investment occurs
in the first stage. However, when gas production costs turn out low in STO-L and STOLS (e.g. w3 , w7 ), there are still gas expansions in later stages. In the second stage in
DET, the investment in gas production in NORTH is 0.5, anticipating the further increase
in gas demand in future stages. In STO-L and STO-LS, there is a high investment of 3.7
and 4.5 respectively, when gas production costs have turned out low (w3 ). In contrast, no
gas expansions occur when gas production cost turn out high (w2 ). Similarly as in the
competitive case, we see investment in STO-L (of 4.5) in EAST in w4 when gas production
cost have turned out much higher, making renewables more interesting. Additionally adding
short-term uncertainty of the renewable availability makes the renewable less interesting
again: no expansion in STO-LS.
32
Transformation capacity investment in Case-Oli
When considering the transformation (i.e., power generation) capacity investment in the
three model variants, it turns out that only renewable-fired power generation capacities
are expanded, whereas gas-fired power generation capacities are not. The expansions of
renewable-fired power generation occur in NORTH and WEST. As a matter of fact, the
expansions in the three model variants are the same as those in the perfectly competitive
markets (see Table 10). Note that in both markets, the (maximum) renewable production
capacities in NORTH (60) and WEST (40) are the same in all stages. As was explained for
the result in Case-Com, the expansions are just enough to satisfy the transformation demand
of the upstream industry.
Table 13: Investment in transmission capacity in the Cournot oligopoly markets
Fuel
GAS
GAS
GAS
GAS
ELE
ELE
Outward
EAST
EAST
NORTH
NORTH
NORTH
EAST
Inward
SOUTH
NORTH
SOUTH
WEST
EAST
NORTH
Model
DET
STO-L
STO-LS
DET
STO-L
STO-LS
DET
STO-L
STO-LS
DET
STO-L
STO-LS
DET
STO-L
STO-LS
STO-L
stage 1
w1
11.0
11.4
11.7
6.5
6.7
6.6
11.5
11.4
11.0
20.9
20.7
20.5
3.4
3.4
3.4
stage 2
w2 w3
0.3 0.3
1.6
0.9
2.5
3.1
0.1
0.6
0.5
0.5
0.9
1.0
0.1
2.7
2.7
0.6
0.9
1.0
w4
stage 3
w5 w6
w7
1.2
1.1
0.3
1.2
0.2
0.9
0.9
0.4
0.4
0.6
0.2
0.2
0.3
0.3
0.4
Average
11.3
12.5
12.8
6.5
8.3
8.4
11.5
11.9
11.6
21.0
22.1
20.0
4.1
4.2
4.2
0.1
Transmission capacity investment in Case-Oli
Table 13 shows the arc capacity expansions. Both gas pipelines and electricity lines
are expanded, which show a mixed relation between additional uncertainty and expansions.
Generally, in the stochastic models the expected investment is higher compared to deterministic model DET, mostly due to expansions in later stages. This shows how the recourse
decisions in the stochastic models provide flexibility to adjust investments accounting for the
33
realization of uncertainties. For example, in STO-L the expansion of the electricity transmission line from EAST to NORTH takes place in the third stage; and only in the scenario
where gas production cost has turned out much lower consistently (in both w3 and w7 ).
Figure 11: The total expected investments in the Cournot oligopoly markets
Expected total investment in Case-Oli
Figure 11 shows the total expected investment for five types of capacity, summarizing
our findings. Across the board we observe that adding long-term uncertainty in natural
gas production cost (STO-L vs DET) results in higher (expected) investment in natural gas
production and transmission and lower (expected) investment in renewable transformation.
The figure indicates the tendency for Cournot oligopoly suppliers to invest more in natural gas production capacity (thus to produce more in the next stage) accounting for the
possibility of decreased gas production cost. Additionally adding short-term uncertainty in
renewable availability (STO-LS) results in even more investment in natural gas production;
and naturally in lower investment in renewable production and transformation. Considering
similar results for perfectly competitive markets discussed above, we conclude that no matter
what the market structure is, accounting for the unpredictability of renewable energy favors
natural gas production and gas-fired power generation due to its more reliable availability.
34
4.3. Social welfare analysis
Expected social welfare (ESW) is the sum of all players’ expected benefits/surplus. We
calculate the ESW of the solutions obtained in the six situations analyzed above, by freezing
the investments decisions and evaluate the expected social welfare these solutions would
obtain in the STO-LS setting. (Naturally, this calculation results in the same value for
both STO-LS cases.) Using common terminology, for DET, this means we determine the
Expected Value of Expected Value Solution, the EEV (Birge and Louveaux, 1997).
In the following, ESW(•, ◦) represents the expected social welfare in the dual-horizon
stochastic framework by using the investment solution obtained from a specific model (• ∈
DET, STO-L or STO-LS) and market structure (◦ ∈ Case-Com,Case-Oli).
The expected social welfare of using the investment solution value from each model in the
perfectly competitive markets for the three cases ESW(DET,Case-Com), ESW(STO-L,CaseCom) and ESW(STO-LS,Case-Com) are 54124 , 54159 and 54178 respectively. Model STOLS provides (slightly) higher expected social welfare compared to the other models in the
perfectly competitive markets.
How much higher the value in the stochastic case is, is called the Value of Stochastic
Solution (VSS).
4
In fact, the recourse solution (obtained from STO-LS) will never be inferior to the other
two solutions in contributing to the expected social welfare (Birge and Louveaux, 1997).
The expected social welfare values in the Cournot oligopoly markets in ESW(DET,CaseOli), ESW(STO-L,Case-Oli) and ESW(STO-LS,Case-Oli) are 50553, 50626 and 50805. The
aggregate ESW increases when more uncertainty is considered, similar to what we observed
above for (for Case-Com). To the best of our knowledge, there is no theoretical proof that
in imperfect markets the aggregate VSS has to be nonnegative. Considering VSS under
strategic behavior, for instance Genc et al. (2007) has found that in a game-theoretic setting
the group of suppliers, and each separate one, may have a negative VSS (consumer surplus
4
More precise: to quantify how a single player’s stochastic solution performs better than the deterministic
solution in a stochastic optimization problem, the term value of stochastic solution (VSS) is used (Birge and
Louveaux, 1997). Since a perfectly competitive equilibrium can be obtained by a social welfare maximizing
model with a single agent, the meaning of VSS transfers quite naturally to the aggregate social welfare
obtained by a perfectly competitive market , but conceptually also to individual market agents.
35
results were not analyzed.)
As we also observe in the following, even if the (aggregate) expected social welfare is
higher when using a stochastic model, this is no guarantee that individual VSS values are
nonnegative.
Specifically, the results show for each player’s VSS as a percentage. For the infrastructure
operators (TSO, TTO) the picture is mixed.
Table 14 shows that in none of the cases analyzed the suppliers are better off in the
model that considers uncertainty. This was also observed in, e.g., Genc et al. (2007), who
hypothesized that a prisoner’s dilemma type of effect can be responsible for this in the
imperfect market situation. The TSO benefits in the competitive case in STO-LS compared
to DET, but has lower profit, albeit very modestly, in the oligopoly case. In contrast, the
TTO profit decreases in the competitive case, and increases in the oligopoly case, and by
much higher relative VSS values. Making up the balance, the consumers benefit and have a
positive VSS independent of market structure and geographical location (Tables 15 and 16).
Table 14: The value of stochastic solution for each supplier
Supplier
SA
SB
SD
SC
in Case-Com
-3.58%
-4.00%
-3.81%
-0.49%
in Case-Oli
-1.98%
-0.24%
-3.27%
-0.59%
Table 15: The value of stochastic solution for consumer in each sector
Consumer
industry sector in East
commercial sector in East
industry sector in North
commercial sector in North
industry sector in West
commercial sector in West
industry sector in South
commercial sector in South
36
in Case-Com
1.37%
1.40%
2.52%
2.75%
6.33%
6.02%
2.71%
2.56%
in Case-Oli
1.20%
1.20%
2.48%
2.71%
6.33%
6.24%
2.98%
2.97%
Table 16: The value of stochastic solution for groups of players
all suppliers
all consumers
the TSO
the TTO
all players
in Case-Com
-3.41%
2.49%
1.09%
-4.38%
0.10%
in Case-Oli
-1.37%
2.56%
-0.03%
12.51%
0.50%
5. Conclusion and outlook for future work
We have presented a multi-horizon stochastic equilibrium model for analyzing energy
markets. This model provides a two-fold contribution to the literature, as it is 1. the
first stochastic equilibrium model with multiple energy carriers and fuels, and 2. the first
multi-horizon stochastic equilibrium model. By decoupling short-term operational decisions
feedback from long-term strategic investment decisions, our approach allows consideration of
both long-term and short-term uncertainty while maintaining reasonable computation time.
In a stylized but representative four-stage example considering long-term gas production cost
uncertainty and short-term uncertainty in renewable generation, we find significant impact of
addressing uncertainty in both infrastructure sizing and timing decisions, although the Values
of Stochastic Solution found are modest. As expected, in a perfectly competitive market
investment levels in production, transmission and generation capacities are much higher than
in an imperfect market. Generally, independent of market structure, the more uncertainty
is considered, the higher the expected investments are, not so much in the first but mostly
due to investments in later stages. Also independent of market structure, more uncertainty
generally favors gas production and gas-fired power generation, with some small but insightful
exceptions. Naturally, when only natural gas production cost uncertainty is considered, in
scenarios where gas production cost turn out high, renewable power generation is stimulated.
But when renewable production uncertainty is considered additionally, renewable electricity
generation loses out to gas-fired power generation. In contrast, considering natural gas
production cost uncertainty by itself, may still increase investment in gas production in
some nodes and transmission capacities in parts of the network.
Future work may concern efficient ways to incorporate restrictions that naturally couple
short-term uncertainty and operational decisions to future (long-term) stages, such as fossil
37
fuel reserves or hydropower production and reservoir levels. An alternate line of research is
the development of decomposition algorithms to allow solving larger than stylized examples
with realistic data.
References
Natalia Alguacil, Alexis L Motto, and Antonio J Conejo. Transmission expansion planning: a mixed-integer
LP approach. IEEE Transactions on Power Systems, 18(3):1070–1077, 2003.
John Birge and Frano̧ois Louveaux. Introduction to Stochastic Programming. Springer, New York, 1997.
Maroeska G. Boots, Fieke A.M. Rijkers, and Benjamin F. Hobbs. Trading in the downstream European gas
market: A successive oligopoly approach. The Energy Journal, 25(3):73–102, 2004.
Anthony Brook, David Kendrick, and Alexander Meeraus. Gams, a user’s guide. ACM Signum Newsletter,
23(3-4):10–11, 1988.
James Bushnell. A mixed complementarity model of hydrothermal electricity competition in the Western
United States. Operations Research, 51(1):80–93, 2003.
Katherine Calvin, James Edmonds, Ben Bond-Lamberty, Leon Clarke, Son H. Kim, Page Kyle, Steven J.
Smith, Allison Thomson, and Marshall Wise. 2.6: Limiting climate change to 450 ppm CO2 equivalent
in the 21st century. Energy Economics, 31(Suppl 2):107–120, 2009.
Richard W. Cottle, Jong-Shi Pang, and Richard E. Stone. The Linear Complementarity Problem. Society
for Industrial & Applied Mathematics, 1992.
Gerard Debreu. A social equilibrium existence theorem. Proceedings of the National Academy of Sciences
of the United States of America, 38(10):886, 1952.
EC. Directive 2009/28/EC of the European Parliament and of the Council of 23 April 2009 on the promotion of the use of energy from renewable sources and amending and subsequently repealing Directives
2001/77/EC and 2003/30/EC. Official Journal of the European Union, 140:47, 2009.
Ruud Egging. Multi-Period Natural Gas Market Modeling Applications, Stochastic Extensions and Solution
Approaches. PhD thesis, University of Maryland, 2010.
Ruud Egging. Benders decomposition for multi-stage stochastic mixed complementarity problems – applied
to a global natural gas market model. European Journal of Operational Research, 226(2):341–353, 2013.
Ruud Egging and Daniel Huppmann. Investigating a CO2 tax and a nuclear phase out with a multi-fuel
market equilibrium model. In IEEE Proceedings of the 9th Conference on the European Energy Market,
Florence, 2012.
Francisco Facchinei and Christian Kanzow. Generalized Nash equilibrium problems. 4OR, 5(3):173–210,
2007.
Michael C. Ferris and Todd S. Munson. Complementarity problems in GAMS and the PATH solver. Journal
of Economic Dynamics and Control, 24(2):165–188, 2000.
Panagiotis Fragkos, Nikos Kouvaritakis, and Pantelis Capros. Incorporating uncertainty into world energy
modelling: the PROMETHEUS model. Environmental Modeling and Assessment, pages 1–21, 2015.
38
Steven A. Gabriel, Antonio J. Conejo, J. David Fuller, Benjamin F. Hobbs, and Carlos Ruiz. Complementarity Modeling in Energy Markets. International Series in Operations Research & Management Science.
Springer, 2012.
Javier Garcia-Gonzalez, RM Ruiz de la Muela, L Matres Santos, and A Mateo González. Stochastic joint
optimization of wind generation and pumped-storage units in an electricity market. IEEE Transactions
on Power Systems, 23(2):460–468, 2008.
Talat S. Genc, Stanley S. Reynolds, and Suvrajeet Sen. Dynamic oligopolistic games under uncertainty: A
stochastic programming approach. Journal of Economic Dynamics and Control, 31(1):55–80, 2007.
Vikas Goel and Ignacio E. Grossmann. A stochastic programming approach to planning of offshore gas
field developments under uncertainty in reserves. Computers and Chemical Engineering, 28(8):1409–1429,
2004.
Rolf Golombek, Eystein Gjelsvik, and Knut Einar Rosendahl. Effects of liberalizing the natural gas markets
in western europe. The Energy Journal, 0:85–111, 1995.
Rolf Golombek, Kjell Arne Brekke, and Sverre A.C. Kittelsen. Is electricity more important than natural
gas? partial liberalizations of the western european energy markets. Economic Modelling, 35:99–111,
2013.
Vijay Gupta and Ignacio E Grossmann. Multistage stochastic programming approach for offshore oilfield
infrastructure planning under production sharing agreements and endogenous uncertainties. Journal of
Petroleum Science and Engineering, 124:180–197, 2014.
Clemens Haftendorn and Franziska Holz. Modeling and analysis of the international steam coal trade. The
Energy Journal, 31(4):201–225, 2010.
Lars Hellemo, Kjetil Midthun, Asgeir Tomasgard, and Adrian Werner. Natural gas infrastructure design
with an operational perspective. Energy Procedia, 26:67–73, 2012.
Lars Hellemo, Kjetil Midthun, Asgeir Tomasgard, and Adrian T. Werner. Multi-stage stochastic programming for natural gas infrastructure design with a production perspective. Stochastic Programming :
Applications in Finance, Energy, Planning and Logistics World Scientific Series in Finance, 4:259–288,
2013.
Benjamin F. Hobbs, Carolyn B. Metzler, and Jong-Shi Pang. Strategic gaming analysis for electric power
systems: An MPEC approach. IEEE Transactions on Power Systems, 15(2):638–645, 2000.
Daniel Huppmann and Ruud Egging. Market power, fuel substitution and infrastructure - a large-scale
equilibrium model of global energy markets. Energy, 75:483–500, 2014.
Henry D. Jacoby, Francis M. O’Sullivan, and Sergey Paltsev. The influence of shale gas on US energy and
environmental policy. Technical report, MIT Joint Program on the Science and Policy of Global Change,
2011.
Michal Kaut, Kjetil T. Midthun, Adrian S. Werner, Asgeir Tomasgard, Lars Hellemo, and Marte Fodstad.
Multi-horizon stochastic programming. Computational Management Science, 11(1-2):179–193, 2014.
Richard Loulou, Uwe Remme, Amit Kanudia, Antti Lehtila, and Gary Goldstein. Documentation for the
TIMES model, part II. Energy Technology Systems Analysis Programme, 2005.
Frederic H. Murphy and Yves Smeers. Generation capacity expansion in imperfectly competitive restructured
electricity markets. Operations Research, 53(4):646–661, 2005.
39
András Prékopa, Sándor Ganczer, István Deák, and Károly Patyi. The stabil stochastic programming model
and its experimental application to the electrical energy sector of the Hungarian economy. Stochastic
programming, pages 369–385, 1980.
Richard Rhodes. Energy Transitions: A Curious History. Center for International Security and Cooperation,
Stanford University, 2007.
Wolfgang Rüdig. Phasing out nuclear energy in Germany. German Politics, 9(3):43–80, 2000.
J. Sullivan. The application of mathematical programming methods to oil and gas field development planning.
Mathematical Programming, 42:189–200, 1988.
Jefferson W. Tester, Elisabeth M. Drake, Michael J. Driscoll, Michael W. Golay, and William A. Peters.
Sustainable energy: choosing among options. MIT press, 2005.
Mariano Ventosa, Alvaro Baıllo, Andrés Ramos, and Michel Rivier. Electricity market modeling trends.
Energy Policy, 33(7):897–913, 2005.
Stein W. Wallace and Stein-Erik Fleten. Stochastic programming models in energy. In A. Ruszczynski and
A. Shapiro, editors, Stochastic Programming, volume 10:637-677 of Handbooks in Operations Research and
Management Science. Elsevier, 2003.
Jifang Zhuang and Steven A. Gabriel. A complementarity model for solving stochastic natural gas market
equilibria. Energy Economics, 30(1):113–147, 2008.
40
Appendix A. The Karush-Kuhn-Tucker conditions

W
B 
dfw duru prw
prb(u)
∂

costP
wusne (·)
P
∂ qwusne
+
X
P

pG
wb(u)ng emswsneg
g∈G
P
+αwusne
− (1 − lossP
sne )φwusne ≥ 0
⊥
W
B
A
dfw duru prw
pA
prb(u)
wua + φwusnA− (a)e − (1 − lossa )φwusnA+ (a)e ≥ 0 ⊥
X
W
B
dfw duru prw
prb(u)
pC
transf C
wunce + φwusne −
wncef φwusnf
P
qwusne
≥0
(A.1)
A
qwusa
≥0
(A.2)
f ∈EcC+
X
M
βwunm
transf C
wncef ≥ 0
⊥
C
qwusnce
≥0
(A.3)
W
B
O−
O
dfw duru prw
prb(u)
pO−
wuno − duru (1 − losso )αwv U (uo)sno + φwusne ≥ 0
⊥
O−
qwusno
≥0
(A.4)
W
B
O
dfw duru prw
prb(u)
pO+
wuno + duru αwv U (uo)sno − φwusne ≥ 0
(
X
D
W
B
D
D
dfw duru prw prb(u)
− intD
+
slp
eff
q
eff D
0
wund
wund
wndf wuns df
wnde
⊥
O+
qwusno
≥0
(A.5)
−
m∈M,f ∈EcC+
M
if c∈Cm
s0 ∈S,f ∈E
D
+euccD
wunde + euclwunde
X
D
qwus
0 nde +
s0 ∈S
S
D
+courwsnd
eff D
wnde slpwund
X
X
D
pG
wb(u)ng emswdeg
g∈G
D
D
D
eff D
wndf qwusndf + euclwunde qwusnde
)
f ∈E

L
L
X  (shrwnl
− 1)eff D
wnde βwunl
+φwusne +

shrL eff D β L
l∈L
if d∈DlL
wnl
wnde wunl
+
if e ∈ ElL


bL , e ∈
if e ∈ E
/ ElL 
l
X
M
M
shrwnm
βwunm
≥0
D
⊥ qwusnde
≥0
(A.6)
m∈M
M
if e∈Em
P
X P
W
B ∂ costw0 usne (·)
P
P
− avlw0 usne depww0 sne αw0 usne
dfw0 duru prw0 prb(u)
P
∂ zwsne
0
→
w ∈Ww
u∈U
P
avlwusne
W
P
P
+dfw prw
invwsne
+ ζwsne
≥0 ⊥
X
P
P
capP
depP
wsne +
w0 wsne zw0 sne − qwusne ≥ 0 ⊥
P
zwsne
≥0
(A.7)
P
αwusne
≥0
(A.8)
←
w0 ∈Ww
X
h
i
O−
O+
duru (1 − lossO−
o )qwusno − qwusno = 0 ⊥
O
αwvsno
(free)
V
u∈Uvo
if v ∈ VoO
41
(A.9)
P
(1 − lossP
sne )qwusne −
X
−
C
transf C
wncf e qwusncf
c∈C,f ∈EcC−
d∈D
X
X
D
qwusnde
+
X
C
qwusnce
+
A
qwusa
a∈A−
ne
a∈A+
ne
c∈C
X
A
(1 − lossA
a )qwusa −
X
+
O+
O−
=0 ⊥
qwusno
− qwusno
φwusne (free)
(A.10)
o∈OeE
X
L
−shrwnl
X
−
pA
wua
X
D
qwusndf
+
+
(A.11)
D
eff D
wnde qwusnde ≥ 0
⊥
L
βwunl
≥0
(A.12)
C
transf C
wncef qwusnce ≥ 0
⊥
M
βwunm
≥0
(A.13)
s∈S,(e,f )∈EcC
M
c∈Cm
s∈S,d∈D
M
f ∈Em
W
B
dfw duru prw
prb(u)
P
ζwsne
≥0
s∈S,d∈DlL
e∈ElL
b
M
−shrwnm
⊥
X
D
eff D
wnde qwusnde +
s∈S,d∈DlL
e∈ElL
P
expP
wsne − zwsne ≥ 0
trf A
wa
+
X
A
pG
wb(u)nA− (a)g emswag
A
+ τwua
≥0
A
⊥ fwua
≥0
(A.14)
A
A
depA
ww0 a τw0 ua + ζwa ≥ 0
A
⊥ zwa
≥0
(A.15)
A
A
depA
w0 wa zw0 a − fwua ≥ 0
A
⊥ τwua
≥0
(A.16)
A
⊥ ζwa
≥0
(A.17)
C
⊥ fwunce
≥0
(A.18)
C
⊥ zwnc
≥0
(A.19)
C
transf C
wncef fwunce ≥ 0
C
⊥ τwunc
≥0
(A.20)
C
transf C
wncef fwunce ≥ 0
C
⊥ βwunce
≥0
(A.21)
C
⊥ ζwnc
≥0
(A.22)
g∈G
X
W
A
dfw prw
invwa
−
→ ,u∈U
w0 ∈Ww
X
capA
wa +
←
w0 ∈Ww
A
expA
wa − zwa ≥ 0
X
W
B
C
G
C
dfw duru prw
prb(u)
− pC
+
trf
+
p
ems
wunce
wnc
wceg
wb(u)ng
g∈G
+
X
X
C
C
transfwncef
τwunc
+
C
C
C
shrwnce
0 transfwncef βwunce0
(e0 ,f )∈EcC
f ∈EcC+
X
−
C
C
transfwncef
βwunce
≥0
f ∈EcC+
W
C
dfw prw
invwnc
−
X
C
C
depC
ww0 nc τw0 unc + ζwnc ≥ 0
→ ,u∈U
w0 ∈Ww
X
capC
wnc +
←
w0 ∈Ww
C
−shrwnce
X
(e0 ,f )∈EcC
X
C
depC
w0 wnc zw0 nc −
(e,f )∈EcC
C
transf C
wnce0 f fwunce0 +
X
f ∈EcC+
C
expC
wnc − zwnc ≥ 0
42
W
B
dfw duru prw
prb(u)
−
pO−
wuno
+
trf O−
wno
+
X
O−
pG
wb(u)ng emswog
g∈G
O
O−
+duru τwv
U (uo)no + κwuno ≥ 0
O−
⊥ fwuno
≥0
(A.23)
W
B
O+
pO+
−dfw duru prw
prb(u)
wuno + κwuno ≥ 0
X
W
O
O
O
dfw prw
invwno
−
depO
ww0 no τw0 vno + ζwno ≥ 0
O+
⊥ fwuno
≥0
(A.24)
O
⊥ zwno
≥0
(A.25)
0
→
w ∈Ww
v∈VoO
X
W
O−
dfw prw
invwno
−
w
0
O−
O−
depO−
ww0 no κw0 uno + ζwno ≥ 0
⊥
O−
zwno
≥0
(A.26)
O+
O+
depO+
ww0 no κw0 uno + ζwno ≥ 0
⊥
O+
zwno
≥0
(A.27)
⊥
O
τwvno
≥0
→
∈Ww
u∈U
W
O+
dfw prw
invwno
X
−
→
w0 ∈Ww
u∈U
capO
wno +
X
O
depO
w0 wno zw0 no −
←
w0 ∈Ww
X
O−
duru fwuno
≥0
V
u∈Uvo
if v ∈ VoO
capO−
wno +
X
O−
O−
depO−
w0 wno zw0 no − fwuno ≥ 0
(A.28)
⊥
κO−
wuno ≥ 0
(A.29)
O+
fwuno
≥0
⊥
κO+
wuno ≥ 0
(A.30)
O
−zwno
+ expO
wno ≥ 0
⊥
O
ζwno
≥0
(A.31)
O−
−zwno
+ expO−
wno ≥ 0
⊥
O−
ζwno
≥0
(A.32)
O+
−zwno
+ expO+
wno ≥ 0
X
glob
reg
W
nod
dfw prw
prbB − pG
+
tax
+
tax
+
tax
wbng
wbng
wbg
wbrg
⊥
O+
ζwno
≥0
(A.33)
⊥
G
fwbng
≥0
(A.34)
G
fwbng
≥0
⊥
µglob
wbg ≥ 0
(A.35)
G
fwbng
≥0
⊥
µreg
wbrg ≥ 0
(A.36)
⊥ µnod
wbng ≥ 0
(A.37)
←
w0 ∈Ww
capO+
wno +
X
O+
depO+
w0 wno zw0 no −
←
w0 ∈Ww
r∈Rn
X
+µglob
wbg +
nod
µreg
wbrg + µwbng ≥ 0
r∈Rn
quotaglob
wg −
X
n∈N
quotareg
wrg −
X
n∈Nr
G
quotanod
wng − fwbng ≥ 0
43
A
fwua
−
X
A
qwusa
=0 ⊥
pA
wua (free)
(A.38)
pC
wunce (free)
(A.39)
O−
qwusno
=0 ⊥
pO−
wuno (free)
(A.40)
O+
qwusno
=0 ⊥
pO+
wuno (free)
(A.41)
pG
wbng (free)
(A.42)
s∈S
C
fwunce
−
X
C
qwusnce
=0 ⊥
s∈S
O−
fwuno
−
X
s∈S
O+
fwuno
−
X
s∈S
X
G
fwbng
−
duru
e∈E,u∈UbB
X
X
P
emsP
wsneg qwusne +
s∈S
D
emsD
wdeg qwusnde
s∈S,d∈D
!
+
X
a∈A+
ne
A
emsA
wag fwua
+
X
c∈C
C
emsC
wceg fwunce
+
X
o∈O
44
O−
emsO−
wog fwuno
=0 ⊥