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STOCHASTIC POWER GENERATION UNIT
COMMITMENT IN ELECTRICITY MARKETS: A NOVEL
FORMULATION AND A COMPARISON OF SOLUTION
METHODS
SANTIAGO CERISOLA, ÁLVARO BAÍLLO, JOSÉ M. FERNÁNDEZ-LÓPEZ,
ANDRÉS RAMOS
Instituto de Investigación Tecnológica (IIT)
Escuela Técnica Superior de Ingeniería ICAI. Universidad Pontificia Comillas
Alberto Aguilera 23, 28015 Madrid, Spain, [email protected]
RALF GOLLMER
Institut für Mathematik University Duisburg-Essen
Subject classification:
Programming: Stochastic programming, integer programming, Lagrangean
relaxation, Benders decomposition.
Production/scheduling: unit commitment decisions under uncertainty.
Area of Review: Environment, Energy, and Natural Resources.
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In this paper we propose a stochastic unit commitment model for a power
generation company that takes part in an electricity spot market. The relevant feature of
this model is its detailed representation of the spot market during a whole week,
including seven day-ahead market sessions and the corresponding adjustment market
sessions. This representation takes into account the influence that the company’s
decisions exert on the market clearing price by means of a residual demand curve for
each market session. Uncertainty is introduced in the form of several possible spot
market outcomes for each day, which leads to a weekly scenario tree. The model also
represents in detail the operation of the company’s generation units, as usual in unit
commitment models.
The proposed unit commitment model leads to large-scale mixed linear-integer
problems that are hard to solve with current commercial optimizers. This suggests the
use of alternative solution methods. In this paper four solution approaches are tested
with a realistic numerical example in the context of the Spanish electricity spot market.
The first one is direct solution with a commercial optimizer, which illustrates the
mentioned limitations. The second method is a standard Lagrangean relaxation
algorithm. The third and fourth methods are two original variants of Benders
decomposition for multistage stochastic integer programs. We analyze the advantages
and disadvantages of these four methods and establish a comparison between them. The
results obtained suggest interesting conclusions and lines for future research.
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1 INTRODUCTION
Planning the operation of power generation units has traditionally been a field of
intense work for operations research academics and practitioners due to the complexity
of the mathematical programming models that arise in this context.
Some of these difficulties are caused by the particular features of electricity as a
commodity. For example, electric energy must be supplied at the same rate (power) at
which it is consumed, given that no efficient large-scale electric energy storage facilities
exist. Moreover, the demand for power depends on uncertain factors. Some of these
factors are relevant for short-term planning purposes (one-week time scope) such as
temperatures. Others have an impact in the long-term evolution of demand, such as the
GDP of the country of interest.
Power generation units are also a source of modeling difficulties such as the
following: i) Their operation is limited by a variety of constraints (e.g. minimum and
maximum power outputs, limited available energy, start-up and shut-down ramps,
maximum hourly load variations, minimum number of hours of continuous operation),
ii) their cost functions present non-convexities (e.g. no-load costs, start-up costs), iii)
their primary source of energy may be subject to some sort of uncertainty (e.g., the cost
of fossil fuels, the availability of hydro or wind energy), and iv) they suffer from
unforeseen outages.
Additional complications emerge from the presence of a transmission network
through which power flows according to Kirchhoff’s laws from generation units to
consumption sites.
In this paper we consider the problem of planning the operation of a diversified
portfolio of generation units with a one-week time scope. This problem, usually known
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as unit-commitment (UC) problem, consists of deciding the hourly power output of each
generation unit during the week of study, which implies choosing the generation units
that must be operating at each hour. This requires the use of binary variables, given that
fossil-fuel thermal units have start-up and no-load costs (due to the consumption of fuel
in order to reach and keep their working temperatures).
When the UC problem includes both thermal and hydro units it is sometimes
referred to as short-term hydrothermal coordination problem. Hydro units differ from
thermal units in a number of aspects: their variable costs are significantly smaller (they
are generally neglected), their available energy is limited (it depends on inflows), they
can store energy for future use (if they have an upstream reservoir) and they can even
sometimes accumulate additional reserves (by pumping water from downstream
reservoirs).
The UC problem has been an active field of research during decades (Sheblé
1994), (Sen and Kothari 1998). Two main lines of research have received particular
attention. On the one hand, researchers have struggled to develop new optimization
algorithms in order to accelerate the search for a solution of this NP-hard problem (see
(Padhy 2004) for a recent survey). On the other hand, authors have included new
modeling features into the basic UC model, typically enhancements of the technical
representation of the power system (Chattopadhyay and Momoh 1999), (Lai and
Baldick 1999), (Ma and Shahidehpour 1999), (Xi, Guan et al. 1999), (Gollmer, Nowak
et al. 2000), (Al-Agtash 2001), (Li and Shahidehpour 2003), (Lu and Shahidehpour
2004), (Yamin, Al-Agtash et al. 2004). In some cases, these improvements have been
oriented to the explicit consideration of uncertainty with respect to input data, in
particular with respect to the demand for power that has to be supplied (Takriti, Birge et
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al. 1996), (Bacaud, Lemacheral et al. 2001), (Nowak and Römisch 2001), (Shiina and
Birge 2004).
Until recent years, the purpose of UC models was to derive minimum-cost
generation schedules while meeting the hourly demand for power as well as additional
power reserve constraints in the power system of interest. This was consistent with the
regulatory framework, under which electric utilities had the obligation to serve specific
regions with tariffs that guaranteed the full recovery of their costs. In the last two
decades, however, the power industry in many countries has suffered a complex process
of reforms which has included the introduction of competition in the business of power
generation. Under this new regulatory context, cost-recovery is no longer guaranteed for
generation companies and the profitability of their units depends on their ability to
produce with lower costs than their rivals.
In general, competition is organized in the form of wholesale electricity markets
in which trading takes place with different time horizons. Electricity spot markets (in
which electricity is traded for immediate delivery) are particularly important, given the
non-storable nature of electricity. The spot price of electricity (as with other
commodities) is used as a reference for longer-term transactions.
In many cases, the bulk of the spot market transactions occur the day prior to
physical delivery, through what is known as a day-ahead market. These transactions are
usually based on the offers and bids submitted by generators or power purchasers
indicating the price at which they are willing to sell or buy different blocks of energy in
each time period (e.g. an hour). A price of electricity results for each time period (the
market clearing price) which is used for all transactions if a uniform-pricing scheme is
adopted (which is frequently the case).
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After the clearing of the day-ahead market, generation companies have the
obligation either to produce the energy they have sold or to purchase it in subsequent
last-minute adjustment market mechanisms (where they can also sell more). This
determines an hourly power profile that the company has to cover with its generating
portfolio.
Given that the company has the possibility of deciding up to a certain point the
amount of energy that it wants to sell in each hour, a number of authors have termed
this process as “self-scheduling” (García-González and Barquín 2000), (Conejo, Arroyo
et al. 2002), (de la Torre, Arroyo et al. 2002), (Yamin and Shahidehpour 2004), (Yamin
2005).
In this competitive framework, the objective of a generation company when
deciding its generation schedule for the following week is no longer the minimization of
its production costs, but rather the maximization of its profit (i.e. the difference of its
revenues in the wholesale electricity market and its costs). This has revitalized the
interest in UC models, leading to new developments both from a modeling and from an
algorithmic perspective (Hobbs, Rothkopf et al. 2001).
The most evident approach in order to evaluate the company’s profit due to a
certain generation schedule is to represent the electricity market by means of a series of
hourly electricity prices. However, in general, the production decisions of a generation
company, even if small, have an impact on the price of electricity.
The influence of a certain generation company on the price of electricity has been
modeled by a number of authors through a function that expresses the amount of energy
that the company can sell at each price, the residual-demand function (García-González,
Román et al. 1999), (García-González and Barquín 2000), (Baíllo, Ventosa et al. 2001),
(de la Torre, Arroyo Sánchez et al. 2002). The residual-demand function is a condensed
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representation of the rest of agents operating in the electricity market of interest. This
modeling enhancement has the cost of yielding a non-concave revenue function, which
can be modeled with binary variables but complicates the search for the maximum-
profit generation schedule. Other authors have proposed different condensed
representations of electricity spot markets (Nowak, Nürnberg et al. 2000), (Anderson
and Philpott 2003). An alternative is to explicitly represent other generation companies
(Pereira, Granville et al. 2005).
Uncertainty is again a relevant ingredient of these new UC models. The main
source of uncertainty is now the wholesale electricity market, due to the fact that the
decisions made by the rest of agents are not known in advance.
In some cases, authors represent the uncertain nature of the electricity market in
terms of scenarios of electricity prices (Takriti, Krasenbrink et al. 2000), (Tseng 2001),
(Valenzuela and Mazumdar 2003). As mentioned, this is only valid if the company of
interest is small with respect to the market. In contrast, other authors combine the
consideration of uncertainty and of the influence of the company on the price of
electricity (Anderson and Philpott 2002), (Baíllo, Ventosa et al. 2004), although their
purpose was deriving optimal offers for the electricity spot market, rather than
computing an optimal UC schedule.
Another recent subject of research is the management of the risk exposure of a
generation company making scheduling decisions under uncertainty in the context of a
wholesale electricity market (Conejo, Nogales et al. 2004), (Yamin and Shahidehpour
2004). For this purpose, uncertainty in the electricity market is represented by means of
price scenarios, assuming that the generation company is small with respect to the
market.
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Researchers have also considered the upstream fuel markets when making power
generation unit commitment decisions in electricity markets under uncertainty, (Takriti,
Supatgiat et al. 2001), (Takriti, Krasenbrink et al. 2000).
In this paper we propose a one-week stochastic UC model for a company owning
a diversified portfolio of generation units that sells its production in an electricity spot
market in which two daily market mechanisms are available: a day-ahead market and an
adjustment market. The objective is the maximization of the company’s profit, defined
as the difference between the revenues obtained by the company in each market
mechanism and its production costs. Revenues are given as the product of the price of
electricity in each hour and the company’s sales in that hour. We consider the
company’s influence on the price of electricity and represent this influence making use
of the concept of residual demand. This causes the revenues of the company to be
expressed as non-linear and non-convex functions of the company’s sales. We formulate
these non-convex functions using piecewise linear approximations and binary variables.
We explicitly take into account market uncertainty for UC decisions by means of
a scenario tree in which we represent each spot market scenario in terms of a series of
residual-demand curves. A multistage setting is considered where each stage
corresponds to one day. This multistage stochastic UC model including a spot market
representation with two market mechanisms and based on residual demand curves is an
original contribution (Baíllo 2002). This model paves the way toward the consideration
of market risk, which is one of the future lines of research we envisage.
The proposed model results in a large-scale mixed-integer mathematical
programming problem which requires a great computational effort to be solved. For a
small number of scenarios (e.g. four weekly scenarios), a commercial mixed-integer
programming (MIP) optimizer is the right choice. However, if a realistic representation
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of uncertainty is required, a specific decomposition algorithm is required. In this paper
we evaluate the performance of Lagrangean relaxation and two original variants of
Benders decomposition algorithm (Cerisola 2004), which is the second contribution of
this paper. We also present a comparison of these approaches and discuss their
advantages and shortcomings.
It is not the purpose of this paper to provide a state-of-the-art representation of the
power system (i.e. the generation units and the transmission network). We therefore use
a simplified representation of the generation units and do not consider the influence or
limitations imposed by the transmission network. As has been mentioned, there are
already a significant number of contributions from other authors in this field of
research.
The paper is organized as follows. In section 2 we describe the electricity spot
market that we consider for our UC model. This is relevant because each electricity spot
market has its own particular features. In section 3 we explain the mathematical
formulation of our stochastic UC model. Section 4 presents a realistic numerical
example based on the Spanish power system and discusses the results obtained from its
direct solution with a commercial optimizer. In section 5 we develop a Lagrangean
relaxation approach to solve the problem and evaluate its performance with the
numerical example. We also discuss the results we have observed when trying to obtain
an initial point for the Lagrangean relaxation algorithm. In section 6 and section 7 we
propose two similar variants of nested Benders decomposition algorithm for the
stochastic UC problem with complementary advantages and disadvantages. In section 8
we summarize the main conclusions of this research and propose future lines of
research.
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2 A DESCRIPTION OF THE ELECTRICITY SPOT MARKET
Agents participating in a wholesale electricity market require different market
mechanisms to perform their transactions in the way that best suits their business
strategy. The majority of wholesale electricity markets include a spot market in which
electricity is traded for immediate delivery and that is considered as a reference for the
rest of transactions.
Electricity spot markets typically include several market mechanisms. The
Spanish electricity spot market consists of a day-ahead market, a congestion
management procedure, an adjustment market, a reserves market and a balancing
mechanism. In this paper, we consider a simplified spot market consisting of a day-
ahead market and an adjustment market.
We assume that both the day-ahead market and the adjustment market are
constituted by twenty-four hourly auctions that take place one day in advance. These
auctions run based on simple sell offers and purchase bids. Participants are allowed to
operate as a portfolio with no limits on the number of offers or bids they wish to submit
and they are allowed to adopt different strategies for different hours. Hourly auctions
are mutually independent. Consequently, although the twenty-four hourly auctions that
form the day-ahead market are cleared at the same time, the result of one of these
auctions is based only on the offers and bids submitted by participants for that specific
hour.
The clearing process for a certain hourly auction evolves as follows. The set of
sell offers submitted for that hour yields an aggregate offer curve, whereas the buy bids
form an aggregate demand curve. The intersection of both curves determines the
volume of transactions *q that should take place.
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In this paper a marginal or uniform pricing scheme is adopted for all the hourly
auctions that constitute the spot market. The clearing price is determined by the
intersection of the aggregate sell offer and buy bid curves. In this manner, an offer is
rejected if and only if its price is greater than the market-clearing price. Similarly, a bid
is rejected if and only if its price is lower than the market-clearing price.
3 MODEL FORMULATION
3.1 Modeling the rest of agents
As mentioned, there are two possible approaches to model the behavior of the rest
of agents in the spot market. If the company of interest is unable to affect the spot price
with its decisions then the rest of the world can simply be modeled by means of the spot
price. On the contrary, if the company has a non-negligible influence on the spot price,
then a more complex approach has to be adopted. In this paper we use the concept of
residual demand to represent the rest of agents. Due to uncertainty with respect to
rivals’ decisions, some of the input data of our model are random parameters. We will
use bold characters to distinguish them.
In each hourly auction of the spot market, n , the amount of energy that a
generation company is able to sell, nq , depends on the clearing price, np . This is due to
the combined effect of the demand at that price, ( )n nD p , and the supply of the rest of
generation companies at that price, ( )restn nS p .
( ) ( ) ( ) ,restn n n n n n n n= − = ∀q D p S p R p , (1)
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where ( )n ⋅R is the residual demand faced by the company in auction n . To obtain
( )n nR p , the company only needs to know the demand, ( )n nD p , and the aggregate
offer, ( )n nS p , as it can obtain ( )restn nS p by subtracting its own offer.
( ) ( ) ( ) ,rest ownn n n n n nS n= − ∀S p S p p . (2)
Conversely, the clearing price can be expressed as a function of the company’s
sales. We refer to this function as inverse residual demand function,
( )1 ,n n n n−= ∀p R q . (3)
Given the inverse residual demand function of the company of interest, the
revenue of the company, nρ , can also be expressed as a function of its production,
( ) ( ) ,n n n n n n= ∀ρ q p q q . (4)
It should be noted that the revenue function is non convex in general.
If we assume that the aggregate offer and demand curves of each hourly auction
are published after its clearing, the company can easily obtain its residual demand and
its revenue function for each auction.
As can be seen in Figure 1, in a spot market in which offer curves are expressed as
stepwise functions (e.g. Spanish OMEL), inverse residual demand functions and
revenue functions have vertical segments that are not easy to represent in a
mathematical programming framework. To overcome this difficulty we suggest
approximating these functions by means of piecewise linear functions. The accuracy of
these approximations increases with the number of steps of the residual demand curve.
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0
10
20
30
40
50
60
0 2000 4000 6000 8000 10000 12000Energy (MWh)
Pric
e (€
/MW
h)
Residual-demand curve Piecewise linear approximation
Figure 1. Example of residual demand curve
In a spot market in which offer curves are defined as piecewise linear functions
(e.g. French PowerNext), inverse residual demand functions are naturally expressed as
piecewise linear functions whereas revenue functions turn out to be piecewise quadratic.
Needless to say, a piecewise quadratic function can be approximated as accurately as
desired by means of a piecewise linear function.
We therefore assume that both inverse residual demand functions and revenue
functions can be expressed as piecewise linear functions with J segments. Each
segment j is defined by its lower bound, jnq , and its upper bound, 1j n+q . We assign a
binary variable jnu to each segment j , such that 1jn =u if the company’s sales in hour
n are higher than jnq and 0jn =u in other case. We also define a continuous variable
jnv to represent the portion of segment j that is filled. Segment j can only be used if
segment 1j − is fulfilled.
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2np
jnp
1j np +
Jnp
1nq 2nq jnq 1j nq + Jnq
1jnδ
1nv 2nv jnv
Inverse residual demand function
1np
1j nρ +
2nρ
jnρ
1j nρ +
Jnρ
1nq 2nq jnq 1j nq + Jnq
1jnγ
1nv 2nv jnv
Revenue function
Figure 2. A piecewise linear representation of residual-demand and revenue functions.
Making use of the slopes of these piecewise linear functions, the following
expressions provide the clearing price and the company’s revenues for each level of
sales nq ,
1 ,n n jn jnj n= + ∀∑p p δ v , (5)
1 ,n n jn jnj n= + ∀∑ρ ρ γ v , (6)
,n jnj n= ∀∑q v , (7)
( ) ( )1 1 1 , ,j n j n jn jn jn j n jn j n+ + +− ≤ ≤ − ∀ ∀u q q v u q q , (8)
1 , ,jn j n j n−≤ ∀ ∀u u . (9)
3.2 Modeling the market clearing process
As previously described, each of the spot market mechanisms (day-ahead market
and adjustment market) is constituted by N hourly independent auctions, so that the
clearing process in a given hour n is based only on the offers and the bids submitted by
participants for that specific hour. As discussed in 3.1, while considering a generation
company with a non-negligible influence on the clearing price, the hourly auction n can
be modeled by means of the company’s sales, nq , and the residual demand curve faced
by the company, ( )n nR p . The clearing price, n*p is such that ( )n n n=*R p q .
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3.3 Modeling the operation of the generation units
In this paper we approximate the production costs of thermal unit t in each hour
n as a linear function,
( ) , ,t t t t t t t t t tn n n no k f k t nβ α= + + ∀ ∀c q u q , (10)
where to are unit t ’s variable O&M costs, in €/MWh, tnq is the net output of unit t , in
MW, tf is the fuel cost, in €/Tcal, tβ is the independent term of the heat rate function,
in Tcal/h, { }0,1∈tnu is the commitment state (on/off) for unit t in hour n , tα is the
linear term of the heat rate function, in Tcal/MWh, and tk is the self-consumption
coefficient of unit t , in p.u. Thermal units also have a gross maximum capacity t
q , in
MW, a gross minimum stable output tq , in MW, as well as up- and down-ramp-rate
limits, tl , tl , in MW/h,
, ,tt t t t t t
n n nq k q k t n≤ ≤ ∀ ∀u q u , (11)
1 , ,tt t t
n nl l t n−− ≤ − ≤ ∀ ∀q q . (12)
Furthermore, the commitment state of every thermal unit in each hour, tnu ,
depends on the commitment state of that thermal unit in the hour before, 1tn−u , and on
the starting and stopping decisions tny and t
nz respectively, for that hour,
1 , ,t t t tn n n n t n−= + − ∀ ∀u u y z . (13)
Thermal units are also subject to special requirements with respect to their
commitment dynamics. For example, a thermal unit t is frequently required to produce
a minimum number of hours, tN , before it stops. Similarly, once it stops, it must
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remain offline a minimum number of hours, tN , before it can produce again. We
formulate these minimum up-time and minimum down-time requirements as follows:
1 1, , , 1, , 1t t ttn n n t n Nν ν+ − − ≤ ∀ ∀ = −…u + u u , (14)
1 1, , , 1, , 1t t tn n n tt n Nν ν+ − − ≥ ∀ ∀ = −…u + u u , (15)
Our model manages hydro reserves in an aggregate manner by integrating hydro
plants located in the same river basin into an equivalent hydro unit, h . The detail of the
hydro networks can be considered in a subsequent decision stage in order to derive a
more precise hydro schedule. We consider a constant energy/flow ratio for each
equivalent hydro unit and express hydro reserves in terms of stored energy, in MWh.
Equivalent units can also operate in pumping mode. The state of each equivalent
reservoir h is evaluated as follows,
1 , ,h h h h h h h h
n n n n n nk i h nη−= − + − + ∀ ∀w w q s b , (16)
where hnw is the energy stored by unit h at the end of hour n in MWh, h
nq is its net
output in hour n in MW, hk is its self consumption coefficient, in p.u., hni are the net
inflows it receives in hour n , in MWh, hns is the energy spilt in MWh, h
nb is the energy
pumped in hour n in MWh and hη is the performance of the pump-turbine cycle, in
p.u.
Each unit has gross maximum generation and pumping capacities, h
q , hb , both in
MW. Its reservoir has a maximum and a minimum operating level, hw , hw , in MWh,
0 , ,hh h
n k q h n≤ ≤ ∀ ∀q , (17)
0 , ,hh
n b h n≤ ≤ ∀ ∀b , (18)
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0 , ,hn h n≤ ∀ ∀s , (19)
, ,hh h
nw w h n≤ ≤ ∀ ∀w . (20)
In this paper we assume that unit h has a certain amount of energy, 0hw , available
for the planning horizon. A medium-term hydrothermal model can determine this
energy.
3.4 The energy balance equation
If the generation company sells for hour n an amount of energy 1nq through the
day-ahead market and an amount of energy 2nq through the adjustment market, the
company must produce this energy with its generation units. To that end we formulate
the following energy balance equation:
1 2 , ,t h hn n n n nt h t n⎡ ⎤+ = + − ∀ ∀⎣ ⎦∑ ∑q q q q b . (21)
It is important to notice that each hourly auction of both market mechanisms is
represented in our model as described in sections 3.2 and 3.3.
3.5 Modeling the problem’s objective function
In the new deregulated framework the objective of a risk-neutral generation
company operating in a spot market such as the one we consider in this paper is the
maximization of its expected profit. In the particular context of this paper the
company’s profit, P , is a random variable that can be calculated as follows:
( ) ( ) ( )1 1 2 2 t tn n n n n nn t
⎡ ⎤= + −⎣ ⎦∑ ∑P ρ q ρ q c q . (22)
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3.6 Compact formulation
In this section we summarize the formulation that we propose for the stochastic
UC problem. We formulate some of the constraints in a compact manner to simplify the
forthcoming description of the decomposition algorithms that we have used to solve this
problem.
( ) ( ) ( )1 2
1 1 2 2, ,
, , ,,
maxn n
t t t tn n n n
h hn n
t tn n n n n nn t
⎡ ⎤= + −⎣ ⎦∑ ∑q q
q u y zq b
P ρ q ρ q c q , Objective function,
1 1,n nQ n∈ ∀q , Day-ahead market constraints,
2 2,n nQ n∈ ∀q , Adjustment market constraints,
{ }, , , ,t t t t tn n n n Q t∈ ∀q u y z , Thermal unit constraints,
{ }, , ,h h h hn n n Q h∈ ∀q b w , Hydro unit constraints,
1 2 , ,t h hn n n n nt h t n⎡ ⎤+ = + − ∀ ∀⎣ ⎦∑ ∑q q q q b , Energy balance equation.
4 A NUMERICAL EXAMPLE
4.1 Description of the numerical example
We address the weekly UC problem of a fictitious power generation company
owning a number of randomly chosen generation units present in the Spanish Power
System (see Table 1 for a description of the generation portfolio).
TOTAL Nuclear Hydro Pumping Fuel Gas Coal
3.834 0.16 1.521 0.1 0.157 0.157 1.739 [GW]
Percentage 4.17 39.67 2.61 4.09 4.09 45.36 [%] Table 1. Generating power capacity considered in our example.
The generation company must decide the optimal UC schedule for its
hydrothermal portfolio of generators under uncertainty of the behavior of the rest of
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agents in the spot market. In other words, the generation company does not know the
residual demand that it will be facing in each of the hourly auctions of the seven day-
ahead and adjustment market sessions that will take place in the week of study.
Therefore it must decide its UC based on historic information about the behavior of the
rest of agents. We assume this historic information to be readily available.
Based on this historic information we assume that the generation company
constructs a multistage scenario tree for the week of study such as the one shown in
Figure 3.
Sat Sun Mon Tue Wen Thu Fri Figure 3. Multistage scenario tree.
Each node of the scenario tree corresponds to one spot market session, including a
day-ahead market session and an adjustment market session. This implies that each
node comprises twenty-four residual demand functions for the day-ahead market,
twenty-four residual demand functions for the adjustment market and a twenty-four
hour schedule for the generation portfolio of the company including UC decisions.
We are therefore assuming that uncertainty unfolds from one spot market session
to the following. In this particular numerical example we assume that there are two
possible market situations for each day. We also assume that the rest of agents will
behave in a consistent manner from Tuesday to Friday. This yields the eight-scenario
tree depicted in Figure 3.
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Building such a scenario tree is complicated and requires specific techniques
(Pflug 2001). However, we are specifically concerned with the solution of the resulting
multistage stochastic problem and a detailed description of how the scenario tree should
be constructed exceeds the purpose of this paper.
The residual-demand curves used in this example have been obtained from the
offers and bids submitted to the Spanish spot market between July 11th 2004 and July
23rd 2004 (OMEL).
4.2 Direct problem solution
We have implemented the multistage stochastic UC model in GAMS (Brooke,
Kendrick et al. 2004). The characteristics of the numerical example are shown in Table
2. As can be seen, it is a large-scale mixed linear-integer programming problem. We use
binary variables to model UC decisions and to represent non-concave revenue functions.
Rows Columns NonZero Integer Number of elements 186973 142817 75121 42916
Table 2. Size of the numerical example.
We have solved this problem with CPLEX 9.0 in a PC P-IV 2.8 GHz 512 MB.
The best feasible solution that we have obtained after two hours of computation is
5.401754 M€, with an upper bound of 5.403861 M€, which implies a tolerance of
0.039 %.
Figure 4 shows the prices that result from the UC schedule proposed by the
model. It is interesting to see that day-ahead market prices are similar to adjustment
market prices and that the model can be used to do arbitrage between both market
mechanisms.
Figure 5 depicts the hourly power output that the model suggests. It is clear that
the model recommends selling more when prices are expected to be higher. During this
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week, the model indicates that the company should participate in the adjustment market
to sell more power.
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Day-ahead market price
Adjustment market price
Figure 4. Day-ahead market prices and adjustment market prices, in €/MWh.
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Power production
Adjustment market sales
Day-ahead market sales
Figure 5. Day-ahead market sales, adjustment market sales and power production, in MWh.
Figure 6 represents the dynamic evolution of the commitment status of a certain
thermal unit as uncertainty with respect to the electricity spot market unfolds. Similarly,
Figure 7 shows the evolution of the reserves of one of the hydro equivalent units (we
force the model to use all hydro reserves, irrespective of the market scenario, which is a
simplification). Both figures illustrate the advantages of using a stochastic UC model.
Sat Sun Mon Tue Wen Thu Fri
Unit onUnit off
Figure 6. Commitment status for a particular generation unit.
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1 4 7 10 13 16 19 22 Figure 7. Evolution of the hydro reserves for a particular hydro unit.
Directly solving multistage stochastic UC problems with CPLEX has limitations
regarding the size of the problems that can be solved. This mainly depends on the
number of generation units that constitute the company’s portfolio and the number of
market scenarios that have to be considered. Due to this we have explored the
possibility of using alternative decomposition solution procedures.
5 LAGRANGEAN RELAXATION
5.1 Lagrangean relaxation of the stochastic UC problem
A large-scale problem with complicating constraints is particularly amenable for a
dual decomposition solution strategy. One possible approach is Lagrangean relaxation
(LR) (Geoffrion 1974).
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Traditionally, LR has been used to derive the solution of UC problems in two
different ways. The first one is known as component decomposition (Römisch and
Schultz 2001) and consists of relaxing the demand and reserve constraints, which
establish a link between the different generation units. The independent schedules thus
obtained do not satisfy the demand and reserve constraints but are usually very close to
the primal optimal solution (Merlin and Sandrin 1983), (Zhuang and Galiana 1988),
(Yan, Luh et al. 1993), (Takriti and Birge 2000). The second approach consists of
deriving an extended formulation of the stochastic problem that includes copies of the
decision variables for each scenario and explicitly incorporates the formulation of the
non-anticipativity constraints. The LR of these constraints leads to a Lagrangean
subproblem that is separable into individual subproblems, each one corresponding to
one scenario. This technique is commonly known as scenario decomposition (Nowak
and Römisch 2001).
We have applied LR to our stochastic UC problem by relaxing the energy balance
equation, which can be seen as a component decomposition approach. The Lagrangean
dual problem that results from this relaxation is formulated as follows:
( ) ( ) ( )( )
{ }{ }
1 2
1 1 2 2
1 2, ,, , ,
,
1 1
2 2
max
,min,
, , , ,
, , ,
n nt t t tn n n n
h hn n
n
t tn n n n n nt
n t h hn n n n n nt h
n n
n nt t t t tn n n n
h h h hn n n
Q n
Q n
Q t
Q h
⎧ ⎫⎡ ⎤+ −⎪ ⎪⎢ ⎥=⎪ ⎪⎢ ⎥⎡ ⎤− + − + −⎪ ⎪⎢ ⎥⎣ ⎦⎣ ⎦⎪ ⎪⎪ ⎪⎪ ⎪∈ ∀⎨ ⎬⎪ ⎪∈ ∀⎪ ⎪⎪ ⎪∈ ∀⎪ ⎪⎪ ⎪
∈ ∀⎪ ⎪⎩ ⎭
∑∑
∑ ∑q qq u y zq b
ρ q ρ q c qP
q q q q b
q
q
q u y z
q b w
λ
λ
Due to the relaxation of the energy balance equation, the inner maximization
problem naturally decomposes into a number of smaller subproblems. The first type of
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subproblem decides the offering strategy for the n -th hourly auction of the day-ahead
market, given the current value of the Lagrange multipliers:
( )1
1 1 1
1 1
maxn
n n n n
n nQ
−
∈
qρ q q
q
λ
The second type of subproblem selects the offering strategy for the n -th auction
of the adjustment market, given the value of the Lagrange multipliers:
( )2
2 2 2
2 2
maxn
n n n n
n nQ
−
∈
qρ q q
q
λ
The third type of subproblem optimizes the schedule of each thermal unit t , given
the value of the Lagrange multipliers:
( )
{ }, , ,max
, , ,
t t t tn n n n
t t tn n n nn
t t t t tn n n n Q
−
∈
∑q u y z
q c q
q u y z
λ
Finally, the fourth type of subproblem optimizes the schedule of each hydro unit
h , given the value of the Lagrange multipliers:
{ }
, ,max
, ,
h h hn n n
h hn n nn
h h h hn n n Q
⎡ ⎤−⎣ ⎦
∈
∑q b w
q b
q b w
λ
A straightforward economic interpretation can be suggested for the Lagrange
multiplier nλ . It can be understood as the marginal cost associated to a local variation
of the total quantity sold by the company in hour n . It can also be interpreted as the
marginal revenue obtained by the company due to the total energy produced with its
generation units in hour n .
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5.2 Solution of the numerical example
We have solved the Lagrangean dual problem of the numerical example using a
standard method based on the outer approximation of the dual function. We have
implemented this LR algorithm in GAMS in order to solve the subproblems and the
master dual problem with CPLEX. The LR algorithm includes enhancements such as
the dynamic update of the dual feasibility region to avoid the typical oscillatory
behavior (Jiménez and Conejo 1999). If we initialize the Lagrange multipliers to zero
and set a minimum tolerance of 0.01 %, we converge in 98 iterations to a value of the
dual function of 5.412936 M€, whereas the corresponding value of the outer
approximation is 5.412388 M€ (see Figure 8). Each iteration requires 10 s of CPU time
in a PC P-IV 2.8 GHz 512 MB, which means that about 15 minutes are required to
obtain this solution.
5.2
5.25
5.3
5.35
5.4
5.45
5.5
5.55
5.6
1 11 21 31 41 51 61 71 81 91Iterations
Obj
ectiv
e fu
nctio
n (M
€
Dual function Outer approximation of the dual function
Figure 8. Evolution of the LR algorithm.
As usual with LR algorithms, the dual solution thus obtained does not satisfy the
relaxed constraints. However, a feasible solution can be easily obtained either by fixing
the decisions taken for the spot market and optimizing the generation schedule or by
fixing the generation schedule and optimizing the decisions taken for the spot market.
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The best result is obtained by fixing the generation schedule, with solution
5.313693 M€, which is 1.6 % worse than the solution provided by CPLEX. Table 1
summarizes these results.
CPLEX LR solution best possible dual lower bound dual solution primal solution
5.401754 5.403861 5.412388 5.412936 5.313693 Table 3. Summary of the results obtained with the LR and comparison with those obtained with CPLEX.
5.3 Computing a warm start for the LR algorithm
One question that arises is whether we can save computational effort by selecting
better initial values for the Lagrange multipliers instead of making a cold start. One
possible form of computing an initial Lagrangean solution is to solve the original
problem with the integrality requirements relaxed and use the resulting values for the
dual variables of the energy balance equation as the initial values for the Lagrange
multipliers.
We have therefore solved with CPLEX the numerical example with the integrality
requirements relaxed and obtained 5.420223 M€. We have then used the values of the
Lagrange multipliers to evaluate the dual function and have obtained 5.412394 M€,
which is between the bounds of the solution provided by the LR algorithm. This initial
point is as close to the optimum of the dual problem as the solution obtained after 98
iterations of the LR algorithm.
A possible reason for this is that it may happen that the feasible region for the
stochastic UC problem satisfies the integrality property (Geoffrion 1974) which, in
principle, is not easy to prove. A necessary condition for this property to hold is that the
solution of the problem without the relaxed constraint but including the integrality
requirements is identical to the solution of the problem without the relaxed constraint
and with the integrality requirements relaxed. In this particular numerical example this
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property does not hold because these two solutions are 22.041479 M€ and
22.049839 M€, respectively. An interesting fact is that when the minimum up- and
down-time requirements are not considered, both solutions are the same
(22.061646 M€).
In any case, given that UC problems are solved with LR algorithms on a weekly
or even daily basis, it seems a good idea to use this initial point, given that it can save
significant computational effort.
6 ALGORITHM 1
6.1 Description of the algorithm
(Benders 1962) suggests and algorithm for problems with complicating variables
that iterates between a master problem (which includes the complicating variables) and
a linear subproblem (which comprises the rest of the problem). In (Van Slyke and Wets
1969) Benders algorithm is suggested as an approach to tackle the L-shaped two-stage
linear programs that typically appear in optimal control theory and stochastic
programming (hence the name “L-shaped method”).
The two-stage L-shaped method (Benders algorithm) is immediately extended to
multistage situations via nested decomposition and to stochastic situations with the use
of the multicut or the monocut version of the method (Birge and Louveaux 1988). This
extension is sometimes referred to as nested Benders decomposition.
In a multistage stochastic situation, the algorithm traverses the corresponding
scenario tree forth and back until convergence is reached. In this context, a variety of
partitioning strategies can be adopted. A natural choice is to solve one problem for each
node of the scenario tree. However, if it is possible to solve the problem for a single
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scenario, then a good idea is to start by solving the most restrictive scenario and then
proceed with the rest of the tree by solving the remaining subtrees (Figure 9). This
reduces the need for feasibility cuts and speeds up convergence.
Sat Sun Mon Tue Wen Thu Fri Figure 9. Proposed partitioning approach.
As commented before, the L-shaped method was originally conceived to address
two-stage problems with a linear second stage. In a multistage setting this implies that
all the subproblems must be linear programs. The presence of subproblems with
integrality requirements (we refer to these subproblems as ISPs) significantly
compromises the application of the the L-shaped method, given that the recourse
functions turn out to be non-convex (in the case of our maximization problem they are
non-concave). If the cuts generated in the backward passes are obtained from the
solution of the ISPs, Benders algorithm, in general, will not converge toward the
optimal solution. The reason is that the cuts generated in this fashion are tangent to the
non-concave recourse functions and may eliminate the optimal solution.
Due to this, when applying Benders decomposition to the solution of UC
problems authors typically adopt a partitioning approach in which they restrict the
presence of integrality requirements (e.g. binary variables) to the master problem (Ma
and Shahidepour 1998).
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However, if the cuts generated in the backward passes are obtained from the
solution of the subproblems with the integrality requirements relaxed (RSPs), the
optimal solution will surely not be eliminated because these cuts are tangent to the
recourse functions of the RSPs (we refer to these as relaxed recourse functions). These
recourse functions are concave and, in this maximization problem, run above the
original non-concave recourse functions of the ISPs.
According to these ideas, we suggest using the following variant of the nested
Benders decomposition to the solution of a multistage problem that includes integrality
requirements in the subproblems: In forward passes the ISPs are solved, whereas in
backward passes the RSPs are solved in order to generate new cuts, as shown in Figure
10 (Cerisola 2004).
Master problem
ISP 1
ISP 2
ISP N–1…
RSP N–1
ISP N RSP N
RSP 1
RSP 2…
Status of thermal unitsRemaining hydro reserves
Status of thermal unitsRemaining hydro reserves
Status of thermal unitsRemaining hydro reserves
Status of thermal unitsRemaining hydro reserves
Benders cuts
Benders cuts
Benders cuts
Benders cuts
Figure 10. Algorithm 1.
In each iteration, an upper bound is given by the solution of the master problem
and a lower bound is determined by the evaluation of the objective function of the
complete problem for the current solution. Observe that as long as the method to
approximate the recourse functions does not produce exact convexifications, the lower
bound may never reach the upper bound. This algorithm, which we will refer to as
Algorithm 1, finishes when the difference of the primal values obtained in two
consecutive iterations is lower than a specified tolerance.
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This approach does not eliminate the optimal solution and provides a feasible
solution, but it does not guarantee a minimum quality for the solution. The reason is that
the relaxed recourse functions may not approximate the original recourse functions
accurately enough. Nevertheless it seems a good idea to test this solution approach with
the stochastic UC problem formulated in this paper.
6.2 Solution of the numerical example
We have solved the numerical example using Algorithm 1 with the partitioning
approach shown in Figure 9. The constraints that establish a link between the
subproblems are those that represent the dynamics of the operation of generation units,
(13), (14), (15), and (16). The dual variables corresponding to these constraints are the
ones used to obtain Benders cuts.
The solution obtained with Algorithm 1 is 5.401882 M€ with an upper bound
provided by the master problem of 5.411203 M€ (0.17 %). Ten iterations were required,
as shown in Table 4. Each forward pass consumed about 150 s, while each backward
pass took about 8 s (Table 5 shows the details of one iteration). The ten iterations
involved about 1500 s of CPU time.
Iteration Solution Master problem(upper bound)
1 5.059272 1.356502 2 5.349088 5.782115 3 5.397756 5.445906 4 5.389815 5.4324 5 5.386587 5.413691 6 5.389619 5.411896 7 5.401837 5.411277 8 5.401876 5.411208 9 5.401882 5.411203 10 5.401882 5.411203 Table 4. Evolution of Algorithm 1.
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Pass Subtree Profit (M€) Time (s) Equations Variables Forward 1 5.411208 56.3 33230 25410
2 2.486954 30.8 28627 21761 3 1.320484 6 24018 18260 4 1.318075 5.2 24018 18260 5 0.677441 4.1 19284 14785 6 0.676719 4.2 19284 14785 7 0.626711 10.1 19284 14785 8 0.625106 8.3 19284 14785
Backward 8 0.626359 0.8 19284 14785 7 0.62797 0.7 19284 14785 6 0.677632 0.7 19284 14785 5 0.678313 0.7 19284 14785 4 1.318863 1.2 24019 18260 3 1.321161 1.7 24019 18260 2 2.490514 1.3 28629 21761
Table 5. Results corresponding to the 8th iteration of Algorithm 1.
The solution obtained with Algorithm 1 is better than the one obtained with
CPLEX. However, the upper bound given by Algorithm 1 does not provide an accurate
measure of the quality of this solution. The upper bound returned by CPLEX, 5.403861,
shows that the quality of the solution obtained with Algorithm 1 is much better than
what the algorithm suggests (Table 6).
CPLEX Algorithm 1 solution best possible solution upper bound
5.401754 5.403861 5.401882 5.411203 Table 6. Summary of the results obtained with Algorithm 1.
In conclusion, although the Benders cuts that are obtained with Algorithm 1 seem
to guide the master problem toward a reasonable solution, they do not provide an
accurate approximation for the non-concave recourse function and therefore do not
yield a good measure of the quality of this solution.
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7 ALGORITHM 2
7.1 Description of the algorithm
The generalized Benders’ decomposition (GBD) algorithm (Geoffrion 1972)
(Holmberg 1994) consists of iterating between the master problem and the LR of the
subproblem, where the relaxed equations are those that connect both stages. The
solution of the subproblems via LR is intended to handle non-linear subproblems. The
algorithm proceeds as the traditional Benders algorithm does. The extension of this
algorithm to a nested situation is immediate.
GBD suggests a refinement of Algorithm 1. The idea is that in backward passes,
we first solve the RSPs. Then we take the dual variables corresponding to the coupling
constraints and we use the corresponding Lagrange multipliers to evaluate the objective
function of the Lagrangean dual of the ISPs. The value of the objective function of the
Lagrangean dual of the ISPs for those multipliers is lower than the solution obtained for
the RSPs. This means that a Benders cut constructed with those multipliers and with the
objective function of the Lagrangean dual of the ISPs approximate more accurately the
recourse function of the ISPs. This improvement has the computational cost of solving a
MIP subproblem instead of a LP subproblem.
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Master problem
ISP 1
ISP 2
LR ISP 2
RSP 2
LR ISP 1
RSP 1
Status ofthermal
unitsRemaining
hydroreserves
Benders cuts
Lagrangemultiplers
Benders cuts
ISP N–1
…
LR ISP N
ISP N RSP N
LR ISP N–1
Benders cuts
RSP N–1
…
Status ofthermal
unitsRemaining
hydroreserves
Status ofthermal
unitsRemaining
hydroreserves
Status ofthermal
unitsRemaining
hydroreserves
Lagrangemultiplers
Lagrangemultiplers
Benders cuts
Lagrangemultiplers
Figure 11. Algorithm 2.
It is important to emphasize that in this case the Lagrangean dual of each ISP is
obtained by relaxing the constraints that establish a link with its ancestor. This
application of LR is quite different from the one suggested in section 5.
It is also important to notice that this algorithm is less computationally expensive
that GBD. In this algorithm we just evaluate the Lagrangean dual function of each ISP
for a given value of the Lagrange multipliers. In contrast, in GBD the Lagrangean dual
problem of each ISP is solved for the current proposal of the corresponding ancestor.
As with Algorithm 1, this algorithm, which we will refer to as Algorithm 2,
finishes when the difference of the primal values obtained in two consecutive iterations
is lower than a specified tolerance.
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7.2 Solution of the numerical example
We have solved the numerical example with Algorithm 2 and have obtained the
same solution as with Algorithm 1. The difference lies in the upper bound provided by
the solution of the master problem, which is more accurate in this case.
As with Algorithm 1, ten iterations were required to obtain the solution (Table 7).
Again, each forward pass consumed about 150 s. Backward passes now took about 15 s
(Table 8). The ten iterations involved about 1500 s of CPU time, which is quite similar
to the CPU time consumed by Algorithm 1.
Iteration Solution Master problem(upper bound)
1 5.059272 1.356502 2 5.349088 5.771254 3 5.397756 5.440117 4 5.389815 5.426414 5 5.386576 5.406862 6 5.399464 5.405477 7 5.401837 5.404777 8 5.401866 5.404703 9 5.401882 5.404626 10 5.401882 5.404626 Table 7. Evolution of Algorithm 2.
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Pass Subtree Profit (M€) Time (s) Equations Variables Forward 1 5.404703 67.9 33230 25410
2 2.484833 47.8 28627 21761 3 1.319637 6.4 24018 18260 4 1.31725 4.6 24018 18260 5 0.677661 3.2 19284 14785 6 0.676946 3.8 19284 14785 7 0.626849 6.6 19284 14785 8 0.625348 5.8 19284 14785
Backward 8 0.626603 0.7 19284 14785 8 0.625361 0.7 19284 14785 7 0.62811 0.7 19284 14785 7 0.626856 0.7 19284 14785 6 0.677858 0.7 19284 14785 6 0.676976 0.7 19284 14785 5 0.678531 0.7 19284 14785 5 0.6777 0.7 19284 14785 4 1.318021 1 24019 18260 4 1.317251 1 24019 18260 3 1.320324 1 24019 18260 3 1.319641 1 24019 18260 2 2.488403 1.5 28629 21761 2 2.487503 1.5 28629 21761
Table 8. Results corresponding to the 8th iteration of Algorithm 2.
The advantage of Algorithm 2 with respect to Algorithm 1 is that it provides a
more accurate measure of the quality of the solution obtained. The disadvantages are
that the solution obtained is not necessarily better and that it may have a higher
computational cost, although this effect is negligible in the numerical example that we
have solved in this paper.
Table 9 summarizes the results obtained with Algorithm 2 and establishes a
comparison with those obtained with CPLEX and Algorithm 1.
CPLEX Algorithm 1 Algorithm 2 solution best possible solution upper bound solution upper bound
5.401754 5.403861 5.401882 5.411203 5.401882 5.404626 Table 9. Summary of the results obtained with Algorithm 2.
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8 COMPARISON OF SOLUTION METHODS
We have used four different methods to solve the numerical example that we have
proposed. In this section we summarize the advantages and disadvantages of each
solution method and suggest the circumstances under which each method is the right
choice.
Direct solution with a commercial optimizer has the important advantage of being
easy to implement in a modeling platform such as GAMS. In fact, even if the problem
of interest cannot be directly solved with a commercial optimizer, it is very
recommendable to start by implementing the model in a modeling platform, in order to
have a benchmark when validating other alternative solution methods. Another
advantage is that, when dealing with problems with sizes near to current computing
limitations, it is still possible to obtain a feasible solution with an accurate upper bound
that gives a good measure of the quality of this solution. The obvious disadvantage of
this method is that it has limitations regarding the size of the problems that can be
solved.
LR implies certain implementation work, which is a drawback with respect to
direct solution, although its implementation is quite straightforward. An advantage is
that it provides approximate solutions for problems that cannot be directly solved with a
commercial optimizer. Additionally, it is a well-known algorithm for which a variety of
enhancements have been proposed and that requires reasonable computation times
(about 15 minutes for the numerical example of this paper). Its implementation also
permits quickly obtaining an initial Lagrangean solution that, according to our
experience, may be very close to the actual Lagrangean dual optimum. A disadvantage
is that the primal solution obtained is not feasible for the original problem. However,
the possibility of deriving a feasible solution is at hand, by simply fixing part of the
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results (e.g. the generation commitment schedule) and determining the optimum value
for the rest of variables.
Algorithm 1 is harder to implement than LR. It requires decomposing the
constraints that establish a link between different subproblems and approximating the
recourse functions by Benders cuts. The advantages are that it provides a feasible
solution in every iteration and that it provides a final solution that is much better than
the one obtained with LR with similar computation times. A disadvantage is that it does
not provide a good measure of the quality of the solution obtained (the solution may be
much better than what the algorithm suggests).
Algorithm 2 is even harder to implement than Algorithm 1 because it requires
formulating the Lagrangean function for each subproblem. Apart from this, it has the
same advantages than Algorithm 1 with similar computation times. Additionally, it
provides a good measure of the quality of the solution obtained.
Table 10 summarizes the previous analysis of advantages and disadvantages and
permits a comparison of the four solution methods.
Direct solution LR Algorithm 1 Algorithm 2
Advantages
• Easy to implement. • Benchmark for other
methods. • Solutions are feasible.
• Not difficult to implement. • Well-known method. • Quick initial solution. • Reasonable CPU time.
• Feasible solutions. • Reasonable CPU time.
• Feasible solutions. • Reasonable CPU time. • Accurate measure of
the solution quality.
Disadvantages • Size limitations. • Computational cost
for large problems.
• Infeasible solutions. • Leads to poor feasible
solutions.
• Hard to implement. • Inaccurate measure of
the solution quality.
• Very hard to implement.
Table 10. Summary of advantages and disadvantages for the four solution methods.
9 CONCLUSIONS AND FUTURE RESEARCH
In this paper we have proposed a stochastic UC model for a generation company
that takes part in an electricity spot market. The original feature of this UC model is its
detailed representation of the spot market of interest, including the influence of the
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company’s decisions on the price of electricity and uncertainty with respect to the
behavior of the rest of agents. The operation of the generation units is also represented
in detail, as in other traditional UC models.
This stochastic UC model results in large-scale mixed linear-integer problems
when applied to realistic numerical examples. In this paper we have evaluated four
alternative solution methods: direct solution with a commercial optimizer, Lagrangean
relaxation, and two original extensions of the nested Benders decomposition. We have
also performed a detailed analysis of advantages and disadvantages and have suggested
the circumstances under which it would be reasonable to use each method.
This paper also suggests future lines of research. We have shown that a quick
Lagrangean solution can be obtained with a low computational cost and that this
solution is remarkably close to the solution of the Lagrangean dual problem. This result
can be guaranteed if the UC problem satisfies the integrality property. We have checked
that this is not the case for the numerical example proposed in this paper, but have also
indicated that it might hold for other cases in which certain constraints (minimum up-
time and down-time requirements) are not present. We consider this to be an interesting
future line of research.
From a different perspective, the good performance of the two extensions of
Benders algorithm is also noteworthy. Although this good performance cannot be
guaranteed a priori, our experience suggests that it should be expected for UC problems.
We also think that these algorithms may prove useful for other operation planning
models in the context of power generation.
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