Influence of feasibility constrains on t (1)

Electric Power Systems Research

  journalhomepage: www.elsevier.comlocateepsr

  Influence of feasibility constrains on the bidding strategy selection in a day-ahead electricity market session

  a ,

  Alberto Borghetti b ∗ b , Stefano Massucco , Federico Silvestro

  a Dept. of Electrical Engineering, University of Bologna, Viale risorgimento 2, 40136 Bologna, Italy b Dept. of Electrical Engineering, University of Genova, via all’Opera Pia 11a, 16145 Genova, Italy

  articleinfo

  abstract

  Article history:

  Large part of liberalized electricity markets, including the Italian one, features an auction mechanism,

  Received 18 December 2008

  called day-ahead energy market, which matches producers’ and buyers’ simple bids, consisting of energy

  Received in revised form 10 June 2009 Accepted 26 July 2009

  quantity and price pairs. The match is achieved by a merit-order economic dispatch procedure indepen-

  Available online 5 September 2009

  dently applied for each of the hours of the following day. Power plants operation should, however, take into account several technical constraints, such as maximum and minimum production bounds, ramp

  constraints and minimum up and downs times, as well as no-load and startup costs. The presence of

  Keywords:

  Electricity market

  these constraints forces to adjust the scheduling provided by the market in order to obtain a feasible

  Bidding strategies

  scheduling. The paper presents an analysis of the possibility and the limits of taking into account the

  Feasibility constrains

  power plants technical constraints in the bidding strategy selection procedure of generating companies

  Game theory

  (Gencos). The analysis is carried out by using a computer procedure based both on a simple static game-

  Unit commitment

  theory approach and on a cost-minimization unit-commitment algorithm. For illustrative purposes, we present the results obtained for a system with three Gencos, each owning several power plants, trying to model the bidding behaviour of every generator in the system. This approach, although complex from the computational point of view, allows an analysis of both price and quantity bidding strategies and appears to be applicable to markets having different rules and features.

  © 2009 Elsevier B.V. All rights reserved.

  1. Introduction

  ing costs and physical constraints, are obtained by the auctioneer’s optimization computer program; side payments to some genera-

  Large part of liberalized electricity markets, including the Ital-

  tors are therefore needed in order to cover all the costs declared in

  ian one, features an auction mechanism, called day-ahead energy

  their bids [1,3] . The presence of transmission constraints justifies

  market, which matches producers’ and buyers’ simple bids, consist-

  the wide interest in the development of approachesmethods able

  ing of energy quantity and price pairs. 1 The match is achieved by a

  to solve security constrain unit commitment (SCUC) problems (e.g.

  merit-order economic dispatch procedure independently applied

  for each of the hours of the following day. This type of public

  The paper focuses on the former market architecture with sim-

  day-ahead energy market is referred in the literature as a power

  ple bids, which does not use side payments. The rational operator is

  exchange (e.g. [1] ) or MinISO [2] in order to distinguish it from

  expected to present bids to this market with the purpose of maxi-

  more centralized market architectures defined as power pools or

  mizing its benefits. For the particular case of a generating company

  MaxIso. In the latter architecture, generators provide extensive data

  (Genco), the optimal bidding strategy is the one that allows the

  other than price–quantity pairs by means of complex bids, such as

  attainment of good profits and, at the same time, results in feasible

  startup costs, ramp rate limit, etc. With these extensive data, the

  schedules of its power plants, taking into account all their technical

  unit commitment (UC) and dispatch that maximize social welfare,

  characteristics and constraints. For this latter purpose, in general,

  taking into account all important aspects of generator’s operat-

  Gencos have at their disposal detailed software tools which can solve the so-called cost-based UC problem, i.e., they are able to cal- culate the optimal scheduling and power dispatching in order to feasibly satisfy an assigned load profile with the minimum variable

  ∗ Corresponding author. Tel.: +39 051 2093475; fax: +39 051 2093470.

  production costs (e.g. [5,6] ).

  E-mail addresses: alberto.borghettiunibo.it (A. Borghetti), stefano.massuccounige.it (S. Massucco), fsilvestroepsl.die.unige.it (F. Silvestro).

  Many optimization approaches have been applied to address

  1 The producers’ offers state the aim to sell a certain amount of energy at a given

  the optimal bidding strategy selection, e.g. [7–16] . In particular, for

  price or higher. The buyers’ offers state the aim to buy a certain amount of energy

  both the case of Gencos without market power and for the case

  at a given price or lower.

  of a single company with market power in the system – i.e., for

  0378-7796 – see front matter © 2009 Elsevier B.V. All rights reserved. doi: 10.1016j.epsr.2009.07.011

  A. Borghetti et al. Electric Power Systems Research 79 (2009) 1727–1737

  the case of a single company able to influence the market clearing

  the payoff matrix of the game represented in normal-form allows

  price (MCP) by means of its own bidding strategy – the optimal

  the direct treatment of nondifferentiable and nonconvex functions

  bidding strategy selection problem can be suitably transformed in

  (e.g. [35–40] ). Therefore, the proposed analysis is carried out by

  cost-minimization problems and, therefore, can be addressed by

  using a computer procedure coupling the use of a multiperiod and

  using a traditional cost-based UC algorithm (e.g. [15] ). If MCP is

  multiplayer payoff matrix – calculated by applying simple auction

  assumed as an exogenous (random) variable, each generating unit

  rules to the combinations of the operators’ pure strategies for all the

  can be considered separately [17] .

  hours of the following day – with a cost-based UC algorithm, which

  Many methods have been proposed to solve the strategic bid-

  allows the refined calculation of the production costs for each of

  ding problem under the assumption of an exogenous MCP by

  the power plants of a specific Genco and to take into account all the

  using dynamic programming (e.g. [14] ), stochastic linear pro-

  relevant constraints.

  gramming [18] , mixed integer programming (MIP) (e.g. [13] ), and

  The analysis is carried out by choosing a preliminary limited

  population-based search methods (e.g. [19,20] ). Two-level opti-

  number of pure strategies for each Genco in order to limit the com-

  mization approaches are often applied in order to represent the

  putational effort. The discretization of pure bidding strategies may

  strategic interaction among suppliers (e.g. [21] ), also in hybrid

  be improved in a following refining stage [38,39] . The use of the

  markets where electrical energy and spinning reserve are simul-

  payoff matrix results in a procedure almost independent of the spe-

  taneously traded (e.g. [22] ) or in the presence of future contracts

  cific market rules, although we have implemented it with reference

  (e.g. [23] ) and bilateral contracts (e.g. [24] ). Also the influence of

  to a uniform-price day-ahead energy auction. The uniform price can

  extra objectives, such the minimization of supplier emission of pol-

  be differentiated by taking into account the network constraints

  lutants (e.g. [25] ), or the influence of unit reliability (e.g. [26] ) has

  between different zones of the system.

  been analyzed. The competition process can also be represented as

  The structure of the paper is the following. Section 2 describes

  a dynamic feedback system (e.g. as in [27] ).

  the scheme proposed for the analysis of the bidding decisions. Then,

  In order to explicitly represent individual market power, i.e.,

  the assumptions and the details of a simplified implementation into

  its ability to manipulate market price via its strategic bidding

  a computer code are presented in Section 3 , namely the main char-

  behaviour, Gencos bidding in an oligopolistic electricity market

  acteristics of the various strategies, the selection criteria of the most

  can be modelled as a supplier game. In particular, recent paper

  convenient bidding strategy based on the game theory and the inte-

  [28] thoroughly analyzes Nash equilibria (NE) and the conditions

  gration with a typical cost-based UC code. The integration with the

  for such equilibria to exist when Gencos game through their sup-

  UC code permits to analyze heuristics procedures conceived with

  ply functions. However, as mentioned in [28] , it is not rigorously

  the aim to select bidding strategies that are expected to result in fea-

  defined the link between market spot prices and onoff variables of

  sible schedules of the power plants. Section 4 presents the results of

  the UC problem, with fixed and startup costs, as well as with non-

  the analysis carried out for a system with three Gencos, each own-

  zero lower production bounds. Therefore, in game-theory based

  ing several power plants. The results show the bidding behaviour

  methods, UC is, in general, not included in the Gencos gaming

  of every generator in the system. This approach, although com-

  strategy, i.e., it is assumed to be known (e.g. [28,29] ). However,

  plex from the computational point of view, illustrates the effects

  it is a general belief that UC will remain an important support to

  of the considered power plant costs and constraints on the bidding

  the hourly bidding strategy builder tool, if accurate forecast of the

  strategy selection in a typical day-ahead electricity market session.

  Genco loads and hourly prices will be used as inputs for solving the

  Section 4 concludes the paper.

  UC problem (e.g. [2,30–32] ).

  As mentioned, in several electricity markets, day-ahead energy auctions adopt relative simple procedures in which they initially

  2. Structure of the procedure adopted for the analysis

  neglect transmission line capability constraints and network losses, as well as detailed power plant inter-temporal constraints, which

  As already mentioned, the procedure adopted for the proposed

  are accounted for by ex-post procedures, i.e., adjustment market

  analysis is based on the coordinated use of a normal-form game-

  sessions. As shown in [33] for the case of transmission constraints

  theory representation of the day-ahead electricity market and of

  and network losses, these simple auction procedures may result

  an algorithm for the solution of UC problems.

  in some loss of economic efficiency and cross-subsidies between

  The scheme of the procedure is shown in Fig. 1 . As described

  market participants with respect to the solution of a detailed opti-

  below, the procedure is based on the calculation of the payoff

  mization model.

  matrix, which represents the normal-form of the game for each

  In this paper, we do not consider improved energy auction

  of the periods t of the considered horizon T (e.g. the 24 h of the

  procedures that may be implemented in order to avoid these inef-

  following days), and, then, by a feasibility enforcement procedure

  ficiencies and we assume that the electricity market is based on

  that applies a cost-based UC program in order to update the payoff

  simple day-ahead energy auctions that clear the market at every

  matrix.

  hour of the following day without consideration of inter-temporal constraints, i.e., it does not allow a bidder to explicitly specify

  2.1. Normal-form game-theory approach

  some technical constraints, such as ramp rate and minimum up and down times. Therefore, the differences between hourly energy

  A finite number of pure strategies are defined for each market

  programs and feasible generation schedules must be compensated

  participant or player, i.e., for each energy demand and generation

  by each Genco through its participation to the following adjust-

  bidder. For each combination of strategies and for each period,

  ment and balancing market sessions (as described for example in

  the market results (i.e., the MCP and the accepted bids) are deter-

  mined through a procedure that implements the market specific

  The present paper aims at investigating how a Genco bidding

  rules.

  strategy selection procedure may take into account the power

  As mentioned, we assume that the day-ahead market consists

  plant operational constraints and, also, try to guide market auc-

  of a uniform-price energy auction based on the participants’ bids.

  tion results toward almost feasible generation schedules in order

  Only the information provided by simple bids is taken into account

  to limit the costly participation to the adjustment market sessions.

  in the market-clearing mechanism. Simple bids is presented for

  Among the various game-theory based approaches proposed for

  each generating unit or consumption site and consists of a (not-

  the analysis of oligopolistic electricity markets, the one based on

  decreasing or not-increasing) staircase of energy quantity and price

  A. Borghetti et al. Electric Power Systems Research 79 (2009) 1727–1737

  Fig. 1. Structure of the adopted feasibility-constrained bidding selection procedure.

  pairs. Both the aggregated curves of all the generation and demand

  so that

  bids are computed and the market clears at the matching point of these two curves.

  ∀ d (2)

  Genco bids are expected to internalize all the operating costs,

  P g m ≤P g ≤P M g or P g =0

  ∀ g (3)

  including startup and shutdown costs. Therefore, also these costs appear to deserve to be taken into account in the bidding strategy selection procedure.

  Nz ⎡

  ⎛

  ⎞ ⎤

  −1

  Moreover, specific market rules can incorporate the possibility

  P =

  ⎣ ptdf ,z · ⎝

  P g −

  P d ⎠ ⎦ ≤P M

  that the market is split into several regional markets or zones due

  to the presence of tie-line constraints. The revenues for each Genco can be calculated with respect to the specific zonal prices relevant

  ∀ ∈ ˝l

  to the regions where the production units are located.

  The proposed strategy selection procedure of each Genco is

  ⎛

  ⎞

  based on the following optimization problem solved independently

  Nz

  ⎝

  for each of the hour t of the following day in order to obtain each

  P g −

  P d ⎠ =0

  element of the payoff matrix that represents the game in normal-

  ⎡

  where

  Nz

  Nh

  max ,P

  ⎣

  d,h (s i ) ·P d,h (s i )

  - Nz is the number of zones;

  - ˝, ˝l, ˝d z,i , ˝g z,i are the set of players, of network links

  ⎤

  between different zones, of demand sites and generation units in zone z that belongs to player i, respectively;

  Nh

  −

  g,h (s i ) ·P g,h (s i ) ⎦

  - Nh is the maximum number of price-quantity steps of the

  i ∈˝ g ∈ ˝g z,i h =1

  non-decreasing or non-increasing staircase that represents each generation or demand bid, respectively;

  A. Borghetti et al. Electric Power Systems Research 79 (2009) 1727–1737

  ,P d,h are the price and quantity pairs of each step h of demand

  called non-credible threats are disregarded. This is justified by the

  bid d, whilst g,h ,P g,h are the price and quantity pairs of each step

  assumption of participants’ rational behaviour, which prevents the

  h of generation bid g;

  adoption of the strategies against the profits maximization goal.

  -P m d ,P M d ,P g m ,P g M are the lower and upper limits of the consumption

  P d of demand site d and of the output P g of generation unit g,

  2.2. Criterion for the selection of the optimal bidding strategy

  respectively;

  -P is the power flow absolute value in network link and P M the

  After the calculation of the payoff matrix, for the solution of

  maximum limit;

  problem (6) , i.e., for the selection of optimal bidding strategy s i ∗ of

  - ptdf ,z is the , z element of the power transfer distribution factors

  each Genco i, the definition of a criterion is needed. Such a crite-

  (ptdf) matrix, i.e., the sensitivity of the power flow in network link

  rion should provide the profits maximization, taking into account

  to an injection at zone z (and equivalent extraction at zone Nz);

  that the competitors’ choice is not known and, therefore, taking

  - ␳ is the vector of the MCPs, one for each zone z, whilst P g and

  into account the associated risks. We have implemented a typical

  P d are the vectors of all the accepted demand and generation

  minimum-risk criterion that consists in the choice of the so-called

  quantities, i.e., the vector of all the selected P d and P g values, one

  maxmin strategy, i.e., the optimal strategy is the one that ensures

  for each demand site d and generation unit g, respectively

  the maximum profit at the end of the following day by assuming that the competitors will choose, in every period t, the combination

  The price and quantity pairs of each bid depend on the particular

  of strategies that results in the minimum profit level V- i,t (s i,t ) for

  bidding strategy s i ∈S i of each player i, being S i the relevant set of

  each strategy s i,t of the operator of interest (Genco i):

  strategies (also called strategy space).

  T

  We focus the analysis to the Gencos. Each element of the Genco payoff matrix, for each period t, is composed by a vector of values

  s ∗

  i = argmax

  V i (s i ,s ), one for each Genco i, being s ( ∈S ) the vector (and the

  relevant set) of the strategies corresponding to every player with

  where

  the exclusion of Genco i. Each value V i (s i ,s −i ) represents the profit obtained by Genco i during the considered period t, i.e., the differ-

  V- i,t (s i,t ) = min s V i,t (s i,t ,s

  −i,t ) ∀ s

  ence between the revenues and costs. The revenues are calculated by the summation of the all the products between price z and

  The literature on the subject often refers to another criterion

  selected output P g of each generating unit g owned by Genco i in

  based on Nash equilibria (NE) [41] , under the assumption that NEs

  every zone z. The costs are the summation of the operating costs

  may provide a coherent set of offers (e.g. [28,42] ). If all the play-

  associated to all the generating units owned by the Genco.

  ers choose to follow NEs, each Genco not only has the potential to

  Each Genco is considered as an intelligent agent that chooses

  obtain a satisfactory profit but also has nothing to gain by being the

  its bidding strategy in order to maximize its profits, under the

  only one to modify its own offer. Indeed, NE, if it exists, is a combi-

  nation of strategies s ∗ = {(s ∗ assumption that each Genco could estimate also the cost or bene- ∗ i ,s −i ) } so that each operator could not fit functions of the others market participants and their possible

  obtain any benefit by unilaterally deviating from it:

  strategies, but, obviously, does not know their final choice. The

  V i,t ∗ (s ∗ i,t ,s ∗ −i,t ) ≥V i,t (s i,t ,s ∗ −i,t )

  ∀ i, ∀ s i

  model is therefore conceived under the complete (but imperfect) knowledge assumption.

  This calculation may also allow inferring how a repeated game

  The problem solved by each Genco i to find the T-elements vector

  will be played, in the sense that if all players predict that a particular

  of the optimal strategies s ∗ i for all periods t could be represented as

  NE will occur, then no player has an incentive to play differently.

  an optimization problem

  2.3. Feasibility enforcement by using a cost-based UC program

  T

  s i ∗ = argmax s ,s

  V i,t (s i,t ,s

  −i,t )

  A Genco is expected to select the optimal strategy also taking

  into account that the load profile attributed by the market-clearing

  T Nz

  solution must be feasibly covered by its power plants. This is of

  = argmax s ,s

  importance even for the markets in which, as the Italian one, the

  i

  −i t =1 z =1 g ∈ ˝g

  clearing mechanism does not take into account all the power plants

  z,i

  constraints, in particular those that couple the decisions in dif- ferent periods, such as the typical minimum up and down time

  without violating any physical and operating constraint of the

  constraints, ramp constraints and the optimal use of an assigned

  generation units, being binary variable u g,t the commitment state

  water quantity for the reservoirs of the hydro power plants.

  during period t of each generation unit g of zone z owned by Genco

  For this purpose, the proposed procedure includes an itera-

  g i, c (p g,t ) the variable operating cost of generation unit g working

  tive process, in which, for each Genco, a feasible solution of the

  at production level p g,t during period t and g (u g,t ,u g,t −1 ) the tran-

  corresponding cost-based UC problem is calculated for the entire

  sition cost incurred at every change of the commitment state of

  load profile of the following day defined by the selected strate-

  generation unit g between two consecutive periods t − 1 and t.

  gies sequence and the payoff matrix is updated on the basis of the

  In order to calculate these optimal bidding strategies, the algo-

  production costs provided by the UC solutions.

  rithm carries out a preliminary elimination of the strategies that are

  For each Genco i, the cost-based UC problem may be written as

  dominated by the others, for each period t and Genco i. A strategy

  the problem of minimizing the sum of operating costs and transi-

  s d i is said to be dominated when it provides profits V d i,t lower than

  tion costs of committed units between consecutive periods

  those provided by every other strategy s i , for every combination of

  T

  Nz

  strategies s that could be chosen by the competitors:

  −i

  min

  u g,t c g (P g,t ) + g (u g,t ,u g,t −1 ) (11)

  V i,t d (s d i ,s −i )

  ∀ s i ∈S i :s i = s d i , ∀ s −i ∈S −i

  t =1 z =1 g ∈ ˝g z,i

  The elimination of the dominated strategies significantly

  so that to satisfy the Genco’s load profile D (T-dimensional vector

  reduces the computational efforts and it implies that the so-

  of load demands D z,t in each zone z and time period t) defined by

  A. Borghetti et al. Electric Power Systems Research 79 (2009) 1727–1737

  the payoff matrix elements associated to the last-selected optimal strategy s ∗ i , namely

  P g,t =D z,t

  i,t (s (V- ∗ i,t ))

  ∀ t, ∀ z

  g ∈ ˝g z,i

  if maxmin criterion (8) is adopted, or

  P g,t =D z,t (V i,t ∗ )

  ∀ t, ∀ z

  g ∈ ˝g z,i

  if the NEs defined by (10) are followed (as illustrated by Fig. 1 ). The UC solution should not violate physical and operating constraints Fig. 2. Example of payoff matrix for the case of three market participants. of the generation units, both of thermal and hydro ones, such as

  the minimum and maximum output limits, the minimum up and

  reference to an offer which will be called “at the marginal costs”,

  down times, the ramp constraints and reservoir storage constraints

  calculated for each step as the ratio between the operating cost vari-

  (e.g. [5] ). As the UC calculation is carried out independently for each

  ation and the corresponding output variation. We consider offer

  Genco and each zone, by assuming the corresponding load profile

  curves composed by a number of equal power steps with the first

  as defined by the payoff matrix, the UC solution also satisfies the

  steps grouped in order to fulfil the minimum power output of the

  network constraints between zones.

  station.

  For each hour of the following day the payoff matrix is indeed

  2.4. Construction of the bidding offer curves

  a multi-dimensional array, obtained by the market-clearing solu- tion that, at each hour, matches the Gencos step-wise bids with the

  The last step of the procedure is the translation of the selected

  forecasted demand level. For each hour, the array dimensionality is

  most convenient strategies in bidding offer curves for each hour and

  equal to the number of Gencos and the bound on each dimension is

  generating unit or group of units. Such a procedure should take into

  equal to the number of the strategies assumed for the correspond-

  account all the specific electricity market administrative rules and

  ing Genco. Each element of the matrix is given by the vector of

  regulations.

  the expected profits for all the Gencos for a particular combination

  The UC solution provides both a more refined calculation of the

  of pure strategy. The market-clearing calculation is carried out for

  payoff matrix elements and also a feasible scheduling with respect

  each strategy combination and for each period. Fig. 2 illustrates the

  to inter-temporal constraints. As we assume that these constraints

  structure of the payoff matrix for the particular case of three Gen-

  are not explicitly enforced in the electricity market auctions, it

  cos, in which Genco 1 operates with 3 strategies, Genco 2 with 4

  appears useful that, in every period, each Genco may differenti-

  strategies and Genco 3 with 2 strategies.

  ate the offers of the power plants that the bidding procedure has selected as not in operation from the offers relevant to the power

  3.2. Implementation of two strategy selection criteria

  plants that are expected to be dispatched, in order to limit the costly participation to the adjustment market sessions. For such

  As mentioned before, two bidding selection criteria have been

  a purpose, a specific heuristic procedure is here introduced, able

  implemented in order to select strategy s i,t ∗ by each Genco i in each

  to reduce the probability the onoff commitment of some units

  hour t: namely, the maxmin criterion and the NE-based one. Both

  selected by the bidding procedure will be changed by the market

  criteria use the calculated payoff matrix.

  auction results in several periods.

  Fig. 3 illustrates the procedure adopted for the implementation

  The following Section 3 describes the assumptions and the

  of the maxmin criterion, for the case of the 4 strategies of Genco 2 of

  details of the computer procedure developed for the proposed anal-

  Fig. 2 . First of all, the minimum profit values V- i,t (s i,t ) are calculated,

  ysis of the influence of feasibility constrains on the bidding strategy

  for each strategy s i,t of the considered Genco i and for every hour

  selection.

  t, as specified by (9). On the basis of these minimum profit values,

  a forward dynamic algorithm is implemented over the entire 24-

  3. Details of the implemented procedure

  h optimization horizon, in order solve problem (8), i.e., to find the final maximum profit. By using the forward dynamic algorithm, not

  The various blocks illustrated in Fig. 1 have been implemented

  only the summation of variable operating costs c g (p g,t ), associated

  in Matlab scripts. In the implemented procedure, we do not con-

  to each generation unit g owned by the considered Genco working

  sider network constraints and we consider only Gencos as market

  at production level p g,t during period t, but also the summation of

  participants (i.e., the level of the demand is fixed and known for

  all transition costs g (u g,t ,u g,t −1 ), incurred at every change of the

  each hour of the following day). Their objective function is given

  commitment state u g,t of a generation unit between two consecu-

  by (6) , i.e., the maximization of expected profits (expected revenues

  tive periods t − 1 and t, could be efficiently taken into account as

  minus the estimated operating costs) in the day-head market with-

  requested by (6) .

  out cooperation with other market participants and without taking into consideration the existence of other electricity market sessions (e.g. balancing and spinning reserve sessions).

  3.1. Payoff matrix calculation Each producer offers a step-wise offer curve for each of his

  generating units depending on the selected bidding strategy, con- sidered to be the same for all the units of the same zone. The list of implemented strategies refers to both price strategies and pro-

  Fig. 3. Maxmin criterion: example of the scheme of the forward dynamic program

  duction reduction strategies. The strategies are formulated with

  implemented for market participant 2 of Fig. 2 .

  A. Borghetti et al. Electric Power Systems Research 79 (2009) 1727–1737

  The implemented computer tool also determines the NEs by

  procedures in order to obtain a feasible solution of the primal prob-

  means a complete enumeration algorithm. As we consider only

  lem from the solution of the dual. Various other techniques for the

  pure strategies, in some periods more than one NE may be obtained,

  solution of the dual problem have been proposed in the literature

  whilst, in others, none is found. In case of multiple NEs, the program

  on the subject, such as Bundle methods (e.g. [47,48] ). The quality

  discards those that are dominated by others, i.e., those that are char-

  of the obtained UC solution is indicated estimated by the duality

  acterized by inferior profit values V i,t ∗ for all the participants. Being

  gap value, i.e., by the difference between the optimal value of the

  K the set of multiple NEs that fulfil condition (10) , equilibrium h is

  objective function of the primal problem and the solution of the

  discarded if there exists, at the same hour t, another NE k able to

  Lagrangian dual [5] .

  assure better profits for each Genco i

  As illustrated in Fig. 1 , the UC results are used to update the relevant values of the payoff matrix. The optimal strategy selection

  V ∗,h (s i,t ∗,h t )

  ∀ i, ∀ k ∈ K, k = h

  criterion is then applied again to the updated payoff matrix and the iterative process is repeated if the resulting optimal strategy differs

  Moreover, we have often found (see Section 4 ) similar profits

  from the one selected before the application of the payoff matrix

  and market prices corresponding to different non-dominated NEs

  refinement by using the UC algorithm. We have not encountered

  relevant to the same hour. For this reason, the computer tool also

  convergence problems in this iterative process for the examined

  calculates, for each hour, the average value of the profits of each

  cases and the feasibility-constrained optimal sequence of strategies

  producer and the average value of the market prices relevant to all

  is usually found after few iterations.

  the calculated non-dominated NEs.

  3.4. Bidding offer construction

  3.3. The adopted cost-based UC program

  As illustrated in Fig. 1 , each Genco bidding selection procedure

  As already mentioned, our intention is that the proposed bidding strategy tool will also guide the market to a feasible dispatch-

  ends with the bidding offer construction, usually based on refined procedures able to meet all the specific electricity market admin-

  ing of the power plants, starting from the deeper knowledge that each Genco is able to obtain relevant to costs and con-

  istrative rules and regulations. As justified in Section 2.3 , we here examine the effects of a simple heuristic procedure that differenti-

  straints of his own generation units with respect to the data used in the payoff matrix calculation. This justifies the use of a UC

  ates the offers of the power plants that the bidding procedure has selected as not in operation from the offers of the power plants that

  computer program in order to refine the solution. The UC pro- gram provides a self-dispatching solution for each Genco and

  are expected to be dispatched. The latter offers are chosen on the

  for each zone, taking into account both the most convenient

  basis of the selected optimal strategy; the former ones are increased in order to reduce the probability that those power plants will be

  strategies defined by the game-theory based model and the most detailed information available on the operating costs and con-

  forced to operate by the market results. The considered heuristic is therefore not a part of the market-clearing mechanism but is

  straints.

  Several approaches have been proposed for the solution of the

  assumed to be introduced in the Genco bidding offer construction in order to limit the costly participation to the adjustment market

  UC problem. For an exhaustive overview we refer the reader to the recent survey [43] . Refined UC codes have been presented in the

  sessions.

  literature and are commercially available, which allow a detailed description of the various units, taking into account also the pres-

  4. Simulation results

  ence of hydro stations. The optimal scheduling of hydrostations should take into account various peculiar constraints that may

  4.1. Test system

  require the development of specific optimization procedures, as shown, for example in [44,45] .

  The considered test system is composed by three Gencos with

  The main points of the proposed analysis appears to be ade-

  only thermal units: the first (Genco 1) has 20 units (corresponding

  quately illustrated also by using a simple UC code that takes into

  to 54.7 of the system capacity), the second (Genco 2) has 10 units

  account only the presence of thermal units. The implemented

  (27.3 of the total capacity), and the third (Genco 3) has 4 units

  UC program minimizes the variable operating costs, described by

  (18 of the total capacity).

  quadratic functions, and the startup and shutdown costs of the

  The power plant characteristics, the parameter values of the

  Genco’s units in order to satisfy the load profile, over the 24-h

  variable-cost function, assumed to be quadratic, and the values

  horizon of the following day, that has been defined by the current

  of the startup costs have been adapted from [49] . Table 1 shows

  optimal strategy selected by applying one of the two implemented

  the values of the parameters of all the considered power plants,

  selection criteria. The main operating constraints and the physical

  where a, b, and c are the coefficients of the quadratic cost function,

  characteristics of the power generation system, usually considered

  and the minimum up and down times T are considered equal for

  for the problem of interest (e.g. [5] ), are enforced in the UC calcu-

  all.

  lation.

  Each producer offers a step-wise offer curve for each of his

  The adopted UC solution algorithm is based on the Lagrangian

  generating units. We consider offer curves composed by 8 equal

  relaxation of the load balancing constraint. The Lagrangian relax-

  power steps and the first steps are grouped in order to fulfil the

  ation approach is often preferred due to its ability to include more

  minimum power output of the station. As already mentioned,

  detailed system representation than would be possible with other

  the strategies are formulated with reference to an offer at the

  techniques (e.g. [46] ). As known [5] , the Lagrangian relaxation

  marginal costs. As an example, Fig. 4 shows, for the case of the

  technique allows to decouple the problem into a minimization sub-

  first power plant of Genco 1, the quadratic function of the variable

  problem for each generating unit. Then the solution is guided by

  costs and the corresponding step-wise bidding offer at marginal

  the dual maximization problem. Each of the minimization sub-

  costs. We have chosen for each Genco five different strategies:

  problem is solved by means of a dynamic programming algorithm.

  bidding at price values 40 and 25 higher than marginal costs

  The solution of the dual problem is carried out through an itera-

  (strategy 1 and strategy 2, respectively), bidding at marginal costs

  tive procedure where the Lagrangian multipliers are updated by

  (strategy 3), bidding at lower power capacity, eliminating the

  using the so-called sub-gradient method with adequate heuristic

  lowest step above the minimum power output for each power

  A. Borghetti et al. Electric Power Systems Research 79 (2009) 1727–1737

  Table 1

  Data of the power plants of the three Gencos in the considered test system (m.u. indicates a generic monetary unit).

  Genco no.

  Unit no.

  Min output P m (MW)

  Max output P M (MW)

  Quadratic cost function coefficients

  Startup costs Minimum up and (m.u.)10

  down times T (h)

  a (m.u.h)

  b (m.u.MWh)

  c (10 −2 ) (m.u.MW 2 h)

  plants larger than 500 MW (strategy 4), and, finally, bidding at

  4.2. Bid strategy selected for the base test system

  price values 50 lower than marginal costs (strategy 5). There- fore, Gencos can choose both price and quantity strategies. For

  The adopted procedure is applied to the test system by consid-

  example, strategy 3 is typical of price-taker participant, whilst

  ering both the maxmin criterion and NEs.

  strategy 4 may be adopted by dominant players to drive market price.

  4.2.1. Maxmin criterion

  A twenty-four 1-h horizon is assumed, with a predefined typical

  The strategies selected by the three Gencos by adopting the

  demand profile characterized by a first peak at 12 a.m. and a second

  maxmin criterion are listed in Table 3 . The table presents both the

  peak at 8 p.m., as reported in Table 2 .

  results obtained before and after the application of the UC-based feasibility enforcement.

  The solution of bidding strategy selection procedure obtained before the application of the UC-based feasibility enforcement corresponds to a sequence of power plants startups and shut- downs with various violations of the minimum up and down times constraints. The list of units whose scheduling violates these con-

  Table 2

  Demand profile.

  Hour

  Demand (MW)

  Hour Demand (MW)

  Fig. 4. Example of the bidding offer at marginal costs and quadratic cost function

  of the first power plant belonging to Genco 1.

  A. Borghetti et al. Electric Power Systems Research 79 (2009) 1727–1737

  Table 3

  Strategies selected through the maxmin criterion.

  Before the application of the UC-based feasibility enforcement

  After the UC-based feasibility enforcement

  List of the units that violate min. up and down time constraints before the application of the UC procedure (maxmin criterion).

  Units that violate the constraints Min. up time

  Min. down time

  straints is reported in Table 4 . For the considered case, the UC-based feasibility enforcement converges and solves all the constraints vio- lations of Table 2 after 3 iterations when it is applied to Genco 1 and Genco 2 (with a reduction of the minimum expected profit equal to

  4.4 and 2.7, respectively) and after 6 iterations when it is applied to Genco 3 (with a profit reduction equal to 8.7).

  The feasibility enforcement does not change the strategies of Genco 1 and change the strategy of Genco 2 only in low load peri- ods (periods 1–6 and 22). The strategies of Genco 3 are the most affected.

  4.2.2. NEs Table 5 shows the strategies corresponding for each period to the calculated single dominant NE, when it exists, without the application of the UC-based feasibility enforcement.

  For the case of NE calculations, the results of periods 1 and 4 are not shown in Table 5 : in fact, in period 4 there is no equilibrium,

  Fig. 5. Results obtained by adopting the strategies selected by the maxmin criterion

  whilst in period 1 there are two equilibriums for the two triples of

  and by the NEs: (a) market prices and load demand and (b) profits for the generation

  strategies {1,1,3} and {1,2,1} (for the three Gencos respectively),

  companies.

  with market prices equal to 24.3 m.u. and 23.6 m.u.

  Table 5 shows that, as expected, the equilibrium is obtained

  4.3. Market results

  for strategies characterized by offers with high prices, because of the absence of a direct participation from the buyers. Only for the

  The adoption of the strategies provided by the proposed bid-

  smallest Genco (namely, Genco 3) the NE includes strategy 3 (at the

  ding selection procedure according to the maxmin criterion and

  marginal costs) in two low load periods (2 and 24).

  of those obtained by adopting the strategies relevant to the cal-

  Table 5

  Strategies corresponding to the NEs.

  Genco owner of the power plant that sets the price in the various periods for the two criteria.

  Period

  Maxmin

  NE

  A. Borghetti et al. Electric Power Systems Research 79 (2009) 1727–1737

  Table 7

  Marginal units that set the price in the various periods for the two criteria.

  List of the units that violate minimum up and down time constraints after the market-clearing simulation.

  Units that violate the constraints Min. up time

  Min. down time

  Maxmin criterion

  List of the units that violate min. up and down time constraints after the market- clearing simulation with the application of the heuristic procedure.

  Heuristic parameters

  Units that violate the constraints Min. up time

  Min. down time

  Maxmin criterion

  Genco 1

  ˛ 1 = 1.1, ˇ 1 = 0.9

  ˛ 2 = 1.1, ˇ 2 = 0.9

  ˛ 1 = 1.1, ˇ 1 = 0.9

  ˛ 2 = 1.1, ˇ 2 = 0.9

  Fig. 6. Results obtained by adopting the strategies selected by the maxmin criterion

  culated NEs produces, as expected, different market solutions. To

  and by the NEs, modified by the heuristic procedure: (a) market prices and load

  compare them, we here present the results obtained by the sim-

  demand and (b) profits for the generation companies.

  ulation of the day-ahead market by concurrently adopting, for the considered three Gencos, the strategies of Table 3 and we repeat the

  of putting them in operation during the day. Anyhow, the total

  simulation by adopting the strategies of Table 5 . In every period in

  profits for the three companies obtained from the market by using

  which there is not a unique NE, all the Gencos bid at the marginal

  the maxmin-selected strategies are larger than the minimum ones

  costs (strategy 3).

  evaluated in the bidding selection procedure: namely 49.6, 165.7

  Fig. 5 shows the comparison between the market clearing results

  and 211.3 larger for Gencos 1, Genco 2 and Genco 3, respec-

  obtained, both for the case of maxmin and NE selection criterion.

  tively.

  Fig. 5 (a) shows the given load profile and the market prices, whilst

  Table 6 reports, for each period, the Genco which owns the so-

  Fig. 5 (b) shows the corresponding profits of the companies.

  called marginal unit, i.e., the power plant which establishes the

  As expected, higher market prices result by adopting the strate-

  market price. The list of the marginal units is shown in Table 7 . It

  gies relevant to NEs, with respect to those obtained by using the

  can be noticed that Genco 1 results the owner of the marginal units

  strategies selected by maxmin criterion, which is based on the

  in 15 periods when maxmin-selected strategies are adopted and in

  concept of adversity to risk, above all in low load hours.

  22 periods when those relevant to NEs are considered.

  Fig. 5 (b) shows that, in the first three periods, the profits

  As already mentioned, the market-clearing procedure is

  obtained through the maxmin-selected strategies are particularly

  assumed not to take into account several power plants constraints.

  low (even slightly negative for Genco 1 and Genco 2). This is due to

  Table 8 shows the violations of the minimum up and down time

  the fact that minimum up and down constrains are not considered

  constraints that results both when maxmin and NE selected strate-

  binding at period 0 and startup costs are not assigned to period

  gies are used.

  1. Therefore, in order to save the startup costs, it results conve-

  Table 8 shows that, although the strategies selected according to

  nient to have some large plants in service already since period 1

  the maxmin criterion have been modified by the UC-based feasibil-

  at low load, thus supporting their high production costs, instead

  ity enforcement procedure, there are several constraints violations.

  Table 10

  Genco owner of the power plant that sets the price in the various periods for the two criteria, after the application of the heuristic procedure with the ˛ and ˇ factors values of Table 9 .

  Periods

  Maxmin

  NE

  A. Borghetti et al. Electric Power Systems Research 79 (2009) 1727–1737

  Table 11

  Marginal unit that sets the price in the various periods for the two criteria, after the application of the heuristic procedure with the ˛ and ˇ factors values of Table 9 .

  This is due to the fact that by concurrently adopting the maxmin

  ment of a feasible power scheduling, is needed as a decision support

  strategies for all the Gencos, each one of these sees competitors’

  tool for Gencos in oligopolistic markets.

  behaviours different from those conjectured by applying the pes- simistic minimum profit condition.

  However, a heuristic procedure can be implemented in order to

  Acknowledgements

  limit the violations of Table 8 , by using the feasible UC calculated by the proposed bidding selection procedure. Such a procedure dif-

  The authors would like to warmly thank Prof. C.A. Nucci for his

  ferentiates, in every period, the offers of the power plants that are

  helpful comments and R. Ferrante for his collaboration in perform-

  set not in operation in the feasible UC from the offers of the power

  ing the calculations. This work was supported in part by the Italian

  plants that are expected to be dispatched in the same feasible UC.

  Ministry of University and Scientific Research and in part by the

  The aim of the heuristic is to try to force the market to follow the

  University of Bologna under Project Decisopelet 2006.

  feasible scheduling that was devised by the UC-based feasibility enforcement.

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