Bilevel programming solution algorithms for optimal price-bidding of energy producers in multi-period day-ahead electricity markets with non-convexities

Abstract

We consider the problem of devising optimal price-offers (bids) for an energy producer participating in a multi-period day-ahead electricity market which exhibits non-convexities due to the discrete nature of the generation units’ commitments and quantities. The problem definition assumes perfect knowledge of the market’s technical characteristics, as well as of the bidding offers of the remaining producers. The problem is formulated as a bilevel optimization model with integer decision variables and linear constraint sets at both levels. The producer acts as the upper-level decision maker, aiming to find the optimal bidding offers that will maximize his individual profit upon clearing of the market, while an independent system operator acts as the lower-level decision maker, aiming to ensure satisfaction of the demand for energy at the minimum total bid-cost. Utilizing the theoretical properties of this problem, we develop both a heuristic as well as an exact algorithmic solution methodology for tackling it. More effective between the two naturally turns out to be the heuristic approach, which works iteratively, optimizing a single price-offer at each iteration, given that the remaining ones are kept fixed at their current values. We present experimental results demonstrating that it provides high quality solutions, while exhibiting reasonable computational requirements. We also demonstrate how the underlying theory can be utilized for the generation of valid inequalities to a suitable relaxation of the original formulation, in which the so-called bilevel feasibility of the obtained solution is not guaranteed. These inequalities are exploited within a cutting-plane framework by the exact solution approach for identifying the global optimum of the problem.

Introduction

We consider the problem of devising optimal price-offers (bids) for maximizing the individual profit of an energy producer who participates in a multi-period day-ahead electricity market. The problem definition assumes full knowledge of the market’s technical characteristics, as well as of the price-offers of all the remaining producers. The market is regulated by an independent system operator (ISO), who aims to satisfy the demand for energy at the lowest total bid-cost. Each participating producer is faced with the problem of maximizing his own profit, subject to the restriction that the system operator’s cost is minimized, i.e., a bilevel optimization problem. A key characteristic of this problem rendering its exact solution a tedious task is the presence of non-convexities stemming from the discreteness of the participating producers’ unit commitments and quantities.

We formulate a bilevel optimization model for this problem with bilinear objectives, integer decision variables and linear constraint sets at both levels, and we utilize integer parametric programming theory to develop specialized algorithmic methodologies for solving it. The first of them is an iterative heuristic approach, which successively optimizes the bidding offer of a single time period while keeping all the other ones fixed at their current values. This is accomplished through the comparative evaluation of the distinct lower-level optimal solutions identified by varying parametrically the single price-offer subject to optimization. We then elucidate how this methodology can be utilized to enable the generation of valid inequalities to a suitable relaxation of the original problem in which the so-called bilevel feasibility of the obtained solution is not guaranteed. In turn, these inequalities can be utilized within a cutting plane solution framework for identifying the exact optimum of the problem.

The remainder of this paper is structured as follows. Section 2 reviews the related literature, while Sect. 3 presents the detailed problem definition and the associated bilevel optimization model formulation. Section 4 develops the proposed heuristic solution approach, while Sect. 5 extends the underlying theory for the development of the exact solution methodology. Section 6 illustrates the application of the proposed methodologies on a small case study and presents extensive experimental results demonstrating their computational performance. Finally, Sect. 7 summarizes the research findings of this work and points to promising directions for future research.

Literature review

In this section, we present a literature review on bilevel optimization models in the context of electricity markets and on specialized solution algorithms for their treatment. Typically, such models are utilized for devising optimal price-offers of energy producers, or for clearing an energy market fairly and compensating the generation units for any losses. The most typical approach for solving these models is their reformulation as a single-level optimization model through the substitution of the lower-level problem by its first order KKT optimality conditions. Naturally, this approach is only applicable when the lower-level optimization problem is convex.

Bilevel optimization models in energy markets

In the context of energy markets, Barroso et al. (2006), Bakirtzis et al. (2007), and Ruiz and Conejo (2009) develop bilevel programming models whose lower-level optimization problem is convex; this enables the development of KKT-based methodologies for their solution. Similar bilevel models for addressing optimal strategic bidding problems energy producers are often faced with have also been proposed by Hobbs et al. (2000), and Li and Shahidehpour (2005).

Another technique that has been proposed for the treatment of bilevel optimization models in the context of electricity markets is the discretization of the upper-level decision maker’s strategy space. Such is the case with the works of Zhang et al. (2000), and Soleymani et al. (2008). Naturally, this does not necessarily lead to the global optimum of the problem, since it a-priori excludes certain solutions from the feasible space.

Many researchers have proposed the development of heuristic/metaheuristic solution approaches for addressing the strategic bidding bilevel optimization model. Typical examples are the papers by Ma et al. (2006), Bajpai and Singh (2008), Zhang et al. (2011), and Foroud et al. (2011). The bilevel optimization model proposed by Weber and Overbye (2002) is also treated with a solution approach that settles for local optima.

Other related bilevel optimization models are formulated in the works of Gross and Finlay (2000), and Fernández-Blanco et al. (2017). The authors of the former work study a framework for the analysis and formulation of bids in competitive electricity markets and develop a solution methodology exploiting a Lagrangian relaxation based approach. The authors of the latter work develop a nonlinear bilevel optimization model for the clearing of a day-ahead market under marginal pricing, which is reformulated as a single-level mixed integer linear program using linear programming duality and KKT optimality conditions.

A survey on optimization models for bidding in day-ahead electricity markets was recently published by Kwon and Frances (2012). Both deterministic as well as stochastic models are reviewed in this survey, as well as models that include unit commitment decisions. The classification scheme adopted by the authors examines the degree to which competition from other producers is directly incorporated into these models.

In an attempt to address the inherent stochasticity that the strategic bidding optimization problem exhibits, many authors, such as Badri et al. (2008), have developed stochastic models for its formulation. Vahidinasab and Jadid (2009) propose a multi-objective model that incorporates the suppliers’ emission of pollutants, utilize the ε-constraint reduced feasible region method in order to deal with the multiple objectives, and solve the single-level problem that results after the substitution of the lower-level problem by its first order optimality conditions with generic optimization software. Gabriel and Leuthold (2010) transform the problem into an MPEC first, reformulate it as a MILP using disjunctive constraints and linearization, and solve the resulting single-level model with generic optimization software, too.

The model that we study in the present work exhibits similarities to the ones that have been developed by Pereira et al. (2005) and Fampa et al. (2008). The lower-level of these two models, however, does not include binary variables for the formulation of the production units’ commitment. This allows the authors to transform the model into a single-level optimization problem through the replacement of the lower problem by its first order optimality conditions.

Solution methodologies for integer bilevel programming

The bilevel programming solution methodologies that we develop in the current work exploit a special attribute of the problem under consideration which is not always present in generic formulations, i.e., the fact that the leader’s decisions affect the cost of alternative lower-level solutions but not the lower-level feasible region. Most existing methodologies for bilevel programming do not depend on such special assumptions regarding the structure of the underlying problem. While this renders them robust and powerful, sometimes it comes at the price of necessitating substantial computational resources for their execution.

Moore and Bard (1990) develop one of the earliest branch and bound algorithms for mixed integer bilevel programming, highlighting the significant differences that the underlying theory exhibits as compared to that of integer single-level programming. Tsoukalas et al. (2009) develop a global optimization methodology for generalized semi-infinite, continuous minimax and bilevel optimization problems, which utilizes an auxiliary optimization problem for determining whether it is possible to attain a specific objective value. By performing a search on candidate objective values, the global optimum is identified.

Gümüs and Floudas (2005), and Domínguez and Pistikopoulos (2010) develop solution algorithms necessitating the substitution of all integer variables by expressions involving only binary variables. The former algorithm, which is an extension of the global optimization framework for continuous bilevel programming developed by Gümüs and Floudas (2001), handles several classes of bilevel programs, including one with purely integer decision variables at both levels. The latter algorithm, which is an extension of the global optimization approach developed by Pistikopoulos et al. (2003), is multi-parametric, capable of handling pure-integer and mixed integer bilevel programming problems. Multi-parametric is also the algorithm proposed by Faisca et al. (2007) for the solution of bilevel programs, in which the authors express the lower-level optimization problem parametrically using the decision variables of the upper-level problem.

Wiesemann et al. (2013) examine the computational complexity of pessimistic bilevel programming problems and study the conditions under which the existence of an optimal solution is guaranteed. For a special class of bilevel models in which the feasible set of the lower-level problem does not depend on the leader’s decisions, they also develop an iterative solution procedure which generates a sequence of finite-dimensional semi-infinite programming approximation problems. Kleniati and Adjiman (2015) extend the global optimization solution framework they had previously developed for continuous bilevel programming (Kleniati and Adjiman 2014a, b) to the case of mixed integer bilevel programming.

Many generic solution methodologies for integer bilevel programming utilize problem relaxations in which bilevel feasibility is suppressed, as is also the case with the current work. Typical examples are the works of DeNegre and Ralphs (2009), Caramia and Mari (2015), Fischetti et al. (2017), and Wang and Xu (2017). Whereas in our case the cuts for eliminating bilevel infeasible solutions utilize integer parametric programming theory holding true due to the independency of the lower-level feasible set from the upper-level decision variables, the cuts utilized by the above works are based on central properties holding true in more general problem formulations. More specifically, DeNegre and Ralphs (2009) extend the standard branch and cut solution methodology for integer single-level programming to the case of integer bilevel programming, utilizing cuts which are based on the general theory of eliminating non-integer solutions from convex hulls of non-integral polyhedrons. Caramia and Mari (2015) develop two solution algorithms for purely integer bilevel programming, the first of which utilizes suitable cuts based on valid bounds on the nonlinear expression of the optimal objective. The second algorithm is a branch and cut methodology, which, upon each branching decision, utilizes valid inequalities to eliminate large sets of bilevel infeasible solutions. Fischetti et al. (2017) develop another branch and cut solution methodology, which applies a family of cuts based on upper bound assertions of the lower-level optimal objective. Wang and Xu (2017) develop a branch and bound type solution algorithm termed watermelon algorithm, which utilizes disjunctive cuts based on the general optimality conditions that the optimal solution to the integer bilevel program must satisfy.

The application of some existing solution methodologies for bilevel programming depends strongly on the integrality of the upper and/or the lower-level variables. Such is the case with the works of Köppe et al. (2010), Xu and Wang (2014), and Lozano and Smith (2019). Köppe et al. (2010) develop a parametric programming solution methodology for bilevel programs with continuous upper-level and integer lower-level decision variables, which expresses the lower-level objective as a function of the upper-level decision variables. Xu and Wang (2014) develop a branch-and-bound algorithm for mixed integer linear bilevel programming, in which each branching decision is associated with several subproblems. Finally, Lozano and Smith (2019) present an exact solution algorithm for mixed integer bilevel programming, which utilizes a single-level value function reformulation to obtain lower and upper bounds on the optimal objective.

Mitsos (2010) and Yue et al. (2019) develop KKT-based solution methodologies for integer bilevel programming, which necessitate the introduction of dual variables as well as big-M formulations for the treatment of the associated complementary slackness constraints. The global optimization solution approach developed by Mitsos (2010), which is an extension of the deterministic algorithm for bilevel linear programming proposed by Mitsos et al. (2008), handles mixed integer nonlinear bilevel programming problems, utilizing fixed value and optimal value function reformulations in order to obtain lower and upper bounds on the optimal objective. Yue et al. (2019), on the other hand, develop a decomposition algorithm for mixed integer linear bilevel programming, which utilizes a master problem and suitable subproblems on a projection-based single-level problem reformulation. The algorithm necessitates the introduction of a number of constraints which grows exponentially with the number of lower-level integer variables. For this reason, the authors propose the adoption of a decomposition approach that employs column and constraint generation in order to overcome the associated computational difficulties. This algorithm, as well as the algorithm by Lozano and Smith (2019), are tested on a large collection of problems with variable and considerable size with very satisfactory results.

Problem definition and formulation

We consider a set of energy generation units participating in a multi-period day-ahead electricity market. Each corresponding producer must submit his energy price-offers (bids) for the planning horizon to an ISO, who is responsible for clearing the market and determining the unit commitments and energy dispatches that will satisfy the energy demand at the minimum total system bid-cost. After the optimal generation plan is determined, each participating producer is compensated in full for his startup cost, and is also paid a market clearing price for each MWh he contributes to the system, according to the clearing payment scheme in effect. This price may be the same for all producers under a uniform market clearing scheme, or the corresponding submitted price-offer under a pay-as-bid market clearing scheme. In the former case, the uniform clearing price is also known as system marginal price (smp), since it represents the marginal cost for energy, i.e., the additional cost that should be paid for increasing the demand by one MWh.

Each producer faces the problem of selecting his optimal price-offer for each time period of the planning horizon. Here, the term optimal pertains to the fact that the producer’s profit upon clearing of the market should be maximized. Assuming perfect information of the market’s technical characteristics, the corresponding profit maximization problem of an individual strategic producer is modeled as a bilevel optimization problem. In a recent chapter book, Kwon and Frances (2012) also hint towards a bilevel formulation for addressing this type of problems, providing proper justification for the approach that we adopt in the present work. A basic mathematical formulation for this problem preceded by the corresponding mathematical notation in the case of a pay-as-bid clearing scheme is presented below:

Sets:

I :

Production units, indexed by i (the index of the strategic producer is 1)

Parameters:

T :

Number of time periods of the planning horizon,

p i,t :

Price-offer of producer i for one unit of energy in time period t (i ∈ I: i > 1, t = 1,…,T),

C 1 :

Price cap for the price-offers of the strategic producer,

c 1 :

Unit variable production cost of the strategic producer,

s i :

Startup cost of unit i (i ∈ I),

m i :

Technical minimum of unit i (i ∈ I),

M i :

Technical maximum of unit i (i ∈ I),

d t :

Demand for energy in time period t (t = 1,…,T),

z i,0 :

Binary parameter denoting the status of unit i at the beginning of the planning horizon (i ∈ I)

Decision variables:

p 1,t :

Price-offer of the strategic producer for one unit of energy in time period t (t = 1,…,T),

q i,t :

Energy quantity of unit i in time period t (iI, t = 1,…,T),

z i,t :

Binary variable that takes the value 1 if the energy quantity of unit i in time period t is positive, and 0 otherwise (iI, t = 1,…,T),

y i,t :

Binary variable that takes the value 1 if unit i is switched on in time period t while being off in time period t − 1, and 0 otherwise (iI, t = 1,…,T)

$$\mathop {\text{Max}}\limits_{{p_{1,t} }} \, F_{1} = \sum\limits_{t = 1}^{T} {\left( {p_{1,t} - c_{1} } \right)q_{1,t} }$$
(1)
$${\text{s}} . {\text{t}} .\quad c_{1} \le p_{1,t} \le C_{1} ,\quad t = \, 1, \ldots ,T$$
(2)
$$p_{1,t} \in Z^{ + } ,\quad t = 1, \ldots ,T$$
(3)
$$\left( {y_{i,t} ,\,z_{i,t} ,\,q_{i,t} } \right) \in \arg \mathop {\hbox{min} }\limits_{{y_{i,t} ,z_{i,t} , \, q_{i,t} }} \, f = \sum\limits_{i \in I} {\sum\limits_{t = 1}^{T} {\left( {p_{i,t} q_{i,t} + s_{i} y_{i,t} } \right)} }$$
(4)
$${\text{s}} . {\text{t}} .\quad \sum\limits_{i \in I} {q_{i,t} } = d_{t} ,\quad t = 1, \ldots ,T$$
(5)
$$m_{i} z_{i,t} \le q_{i,t} \le M_{i} z_{i,t} ,\quad i \in I,\quad t = 1, \ldots ,T$$
(6)
$$y_{i,t} \ge z_{i,t} - z_{i,t - 1} ,\quad i \in I,\quad t = 1, \ldots ,T$$
(7)
$$y_{i,t} ,z_{i,t} \,\,{\text{binary}},\quad i \in I,\quad t = 1, \ldots ,T$$
(8)
$$q_{i,t} \in Z^{ + } ,\quad i \in I,\quad t = 1, \ldots ,T$$
(9)

The unit index 1 is used to specify the strategic producer whose profit is maximized in the upper-level objective F1 (1). This profit depends on his energy quantities, q1,t, which are lower-level decision variables, as well as on his price-offers, p1,t, which are upper-level decision variables. The start-up cost is not included in the upper-level objective, since producers are typically compensated in full for such costs. Constraint set (2) imposes a lower and an upper bound on the price-offers of the strategic producer. More specifically, typical market rules dictate that each price-offer must be at least equal to the associated unit’s variable production cost, and at most equal to a price cap set by the market regulator. Constraint set (3) additionally restricts these variables to non-negative integer values. The lower-level optimization problem is defined by (4)–(9). Expression (4) minimizes the ISO objective, f, i.e., the total system bid-cost for satisfying the demand for energy. Constraint set (5) is the market clearing constraint set ensuring energy balance (production equal to demand) in each time period. Constraint set (6) imposes the technical minima and maxima of the participating units. Constraint set (7) ensures correct values for the decision variables yi,t which are used to impose the start-up costs in the lower-level objective. More specifically, the difference (zi,t − zi,t−1) can take any of the values − 1, 0 and 1. Variable yi,t should take the value 1 in the latter case, and this is correctly imposed by this constraint. In the other two cases, both 0 and 1 are feasible for yi,t, but the value 0 is implicitly imposed by the fact that the coefficient of yi,t in the minimization objective is positive. Finally, constraints (8) and (9) impose integrality on the lower-level decision variables. Note that c1 and C1, as well as mi, Mi and dt for all i and t always take positive integer values in the above formulation, with 1 < mi < Mi.

At first glance, it might seem unrealistic for a particular producer to have full knowledge of his competitors’ bids. Note, however, that in a realistic environment, each participating producer might end-up solving a sequence of optimization models such as the above, using educated estimates of the other producers’ bids, based on historic data. This would enable the comparative evaluation of alternative self-bidding strategies based on the thorough examination of different scenarios and assumptions. In addition, the above formulation and the subsequent solution methodologies developed next could also be fruitfully utilized for identifying equilibrium points within an iterative game setting, in which each producer takes turn responding to the bids of the remaining producers that have been previously announced by solving his own profit maximization problem and announcing his own bids in return. As no generic solution methodologies are available for this class of problems, the development of specialized solution approaches could facilitate considerably such research pursuits.

The model formulation (1)–(9) fits in the general bilevel optimization modeling framework, which focuses on optimization problems involving two decision makers acting hierarchically. The first of them, called the leader, aims to choose the values of the decision variables that he controls, so as to optimize his own (upper-level) objective function. With known the upper-level decision variable values, a second decision maker called follower aims to choose the values of the decision variables that he, in turn, controls, so as to optimize his own (lower-level) objective function. In our case, the leader corresponds to the strategic producer, while the decision variables that he controls are the price-offers for energy. On the other hand, the follower corresponds to the ISO, while the decision variables that he controls are the unit commitments and the energy quantities of the participating producers.

A key characteristic of the above formulation is that the upper-level decision variables do not appear at the lower-level constraint set; thus, the follower’s feasible region is not influenced by the leader’s decisions. However, the comparative evaluation of alternative lower-level solutions is influenced by the leader’s decisions, since the upper-level decision variables appear in the follower’s objective. The term reaction set is used to denote the set of responses of the follower for a particular leader action, i.e., the set of optimal solutions to the lower-level problem for a particular set of upper-level decision variable values. Finally, the term inducible region (IR) is the set of every upper-level feasible solution, and corresponding lower-level optimal solution, i.e., the set over which the leader may optimize his objective. A solution that belongs to the IR, i.e., a solution for which the lower-level decision variables constitute a lower-level optimal solution for the associated upper-level decision variable values, is called bilevel feasible. In the problem formulation (1)–(9), the leader’s objective is bilinear since the generation unit quantities are determined at the lower level, while all the decision variables that he controls are integer. On the other hand, the lower-level problem is a linearly constrained integer program with a bilinear objective, since this objective becomes linear when the leader’s decisions are known.

In the case of a uniform clearing scheme, the upper-level objective of the above formulation is expressed as follows instead:

$$\mathop {\text{Max}}\limits_{{p_{1,t} }} \, F_{1} = \sum\limits_{t = 1}^{T} {\left( {\lambda_{t} - c_{1} } \right)q_{1,t} } ,$$
(10)

where λt is the smp of time period t (t = 1,…,T). In order to capture the true marginal cost of the associated time period, λt is defined as the dual variable of the corresponding energy balance constraint (5). This introduces considerable difficulties because the lower-level problem is an integer program and does not possess dual variables in the traditional sense. Several approaches have been proposed for computing λt, many of which suffer from inequities necessitating additional uplifts and side-payments in order to reach a market equilibrium that fairly clears the market (see Andrianesis et al. 2013a, b, for example); as a consequence, the relevant research is very active. In the current work, we focus on the algorithmic aspect of the problem, and we intentionally do not deal with such market design issues. Nevertheless, we note that one of the most common methodologies for computing the smp is to find the optimal solution of the integer linear problem (4)–(9) first, and then to solve the continuous problem that results after the integer variables are fixed to their optimal values in this solution. Based on marginal pricing theory (Schweppe et al. 1988), the energy commodities are paid at the shadow price of the market clearing constraint computed this way.

Under a uniform pricing scheme, the smp is not always unambiguously defined, due to the presence of the indivisibilities and the fact that alternative lower-level optimal solutions may lead to different smp definitions. In order to be able to eliminate these ambiguities, should they arise, the imposition of a conflict resolution set of rules is necessary and is typically in effect as a common practice in actual realistic markets. For the remainder of this work, we utilize the following widely adopted set of such rules, which unambiguously determine the smp in any possible case that can arise:

  • Rule 1 If at the optimal ISO solution there is a production unit whose energy dispatch in time period t is strictly between its technical minimum and its technical maximum, then the smp of this time period is equal to the price-offer of this unit.

  • Rule 2 If at the optimal ISO solution no production unit prescribed by Rule 1 exists in time period t and there is at least one production unit whose energy dispatch is equal to its technical minimum, then the smp of this time period is equal to the minimum price-bid of any unit producing at its technical minimum.

  • Rule 3 If at the optimal ISO solution all participating units produce at their technical maximum in time period t, then the smp of this time period is equal to the maximum price-bid of any unit producing at its technical maximum.

To see why the above set of rules handles any possible case that might arise, note that one can easily prove that if at the optimal solution to model (4)–(9) there are more than one units producing strictly between their technical minimum and technical maximum in some time period, then the price-bids of these units for this time period must be equal, and there is an optimal solution in which this is true for exactly one of them. Naturally, this unit is the optimal choice for satisfying the extra energy unit demand in this case, since the price-offer of this unit will never be greater than the price-offer of any unit producing at its technical minimum. The rationale behind Rule 2 is straightforward since the minimum price-bid unit producing at its technical minimum stems as the optimal choice for providing an extra MWh of energy when all participating units produce either at their technical minimum or at their technical maximum. Finally, the rationale behind Rule 3 becomes clear when one considers that if the energy demand is reduced by one MWh when all participating units produce at their technical maximum, then the optimal choice for accommodating the corresponding perturbation, i.e., for reducing the total energy supply by one unit, will be the maximum price-bid unit out of them. It becomes obvious from this discussion that Rules 2 and 3 exploit the right and left hand-side shadow price definitions, respectively, in order to overcome the obstacles raised by the integralities that the model formulation involves.

Note that the exact set of rules utilized for the smp definition is a market design issue; thus, the above rules may slightly vary from case to case. This, however, does not have any significant impact on the solution methodologies that we develop next for the treatment of the problem. In fact, these methodologies are independent of the set of rules in effect, in that they can be suitably modified to accommodate any such set. Another important market design issue is the choice of the upper-level objective function. For reasons of completeness, all the proposed methodologies that we develop next are suitably adjusted both for the case of a pay-as-bid market clearing scheme in which expression (1) is adopted as the upper-level objective, as well as for the case of a uniform price market clearing scheme in which expression (10) is adopted instead.

Even under the assumption that the feasible region of problem (1)–(9) is non-empty and compact, an optimal solution may not exist. This is a well-known pitfall in bilevel programming (Bard 1998) that may occur when the optimal solution of the lower problem is not unique. The basic theory of bilevel optimization (Candler and Norton 1977) prohibits the cooperation between the upper and the lower-level decision makers. Thus, it is not possible for the upper-level decision maker to force the lower-level decision maker to choose a particular lower-level optimal solution in the case of multiple optima, which, in turn, implies that the strategic producer may not always be able to attain his maximum profit.

Most approaches that have been proposed for circumventing this difficulty modify slightly the problem definition and the associated model formulation. A highly popular one called optimistic (pessimistic) approach (see Loridan and Morgan 1996), suggests the selection of the most (least) favorable solution to the upper-level decision maker in case of multiple lower-level optima. This implies that there is some way for the upper-level decision maker to convince the lower-level decision maker to choose a particular lower-level optimal solution. In the particular application under study, the unit with the lowest variable production cost is typically favored in order to resolve such conflicts, mainly because such units push towards lower total system costs. This motivates the units to reduce their costs and become more competitive. In the present work, we adopt the optimistic approach for the resolution of such conflicts, because it guarantees the existence of an optimal solution under reasonable regularity assumptions (Dempe 2002). It should be clarified that this choice regarding the proposed approach for dealing with multiple lower-level optima does not affect crucially the applicability of the proposed methodology; it only affects which actual solution will be identified as optimal.

In our case, the feasible region of the lower-level problem is nonconvex due to the integrality of the decision variables yi,t, zi,t and qi,t. If this were not the case, however, then this set would be convex, since all constraints are expressed with linear functions. The existence of an optimal solution to the lower-level problem is ensured due to the fact that the feasible set is closed and compact, while all decision variables have finite bounds. Under reasonable assumptions such as that of the optimistic approach adoption, the global solution of the bilevel problem is also ensured. In fact, the integrality of the decision variables ensures that the cardinality of the feasible set of both the lower as well as the upper-level problem is finite.

In view of the absence of generic solution methodologies for integer bilevel programming, in the next section we develop specialized solution methodologies for handling the optimization model under consideration. In doing so, we exploit the theoretical properties of this model, and in particular, the key fact that the upper-level decision variables appear at the lower-level objective, but not in any of the lower-level constraints. First, we develop a heuristic solution methodology, which, despite its relatively low computational requirements, appears to provide high quality solutions. This significant advantage makes this methodology suitable for the treatment of large realistic problems. For reasons of completeness, we also develop an exact solution methodology for the treatment of the problem. While the considerably larger computational requirements of this methodology limit its applicability on small sized problems only, the associated framework that we develop opens up new interesting research directions towards the development of efficient exact algorithmic methodologies for the solution of the problem. In addition, we also elaborate on the conditions under which the applicability of the proposed methodologies remains valid on more complex model extensions that may involve additional problem characteristics.

Heuristic solution methodologies

The single period case

Consider the special case of Problem (1)–(9), in which the planning horizon consists of a single time period. In this case, each producer must submit a single price-bid to the ISO, while index t is naturally suppressed as redundant. Kozanidis et al. (2013) have developed an exact solution algorithm for this problem, which utilizes the fact that the optimal ISO objective is a non-decreasing, piecewise linear concave function of the strategic producer’s price-bid. For reasons of completeness, we repeat this key result and its proof next.

Proposition 1

For the single period version of the problem, the optimal ISO objective is non-decreasing, piecewise-linear and concave as a function of the strategic producer’s bid.

Proof

The fact that it is non-decreasing is trivial, since increasing the value of the strategic producer’s bid does not alter the feasible region of the problem, but only increases the total system bid-cost of the solutions in which unit 1 participates. The fact that it is also piecewise-linear and concave is due to Noltemeier (1970).□

The heuristic solution methodology that we propose for the treatment of the multi-period problem version utilizes the above solution methodology for the single period version of the problem; therefore, we first revisit its main steps in reasonable detail. Note that this methodology needs to be properly adjusted to account for the integrality of variables p1,t in our case, since these variables are treated as continuous in the work by Kozanidis et al. (2013). This adjustment, however, is trivial as shown next.

Suppose that we solve the lower-level problem for the two extreme values of the price-offer subject to optimization, i.e., for the unit production cost and for the price cap. The fact that f*(p1) is non-decreasing implies that f*(c1) ≤ f*(C1). The fact that f*(p1) is concave implies that the line connecting the points (c1, f*(c1)) and (C1, f*(C1)) provides a lower bound on f*(p1) for any value of p1 in the interval [c1, C1] (see Fig. 1). Finally, the fact that f*(p1) is piecewise linear implies that there are distinct lower-level optimal solutions, depending on the exact value of the strategic producer’s price-offer; moreover, there exists one such solution for each linear segment comprising f*(p1).

Fig. 1
figure1

Lower bound on f*(p1) determined by f*(c1) and f*(C1)

If the unit commitments and energy quantities at the optimal solution for p1 = c1 coincide with those at the optimal solution for p1 = C1, then this dispatch is also optimal for any feasible value of p1, and q *1 remains constant in the entire interval [c1, C1]. Consider now the case in which the optimal ISO solution for p1 = c1 is different than the one for p1 = C1. In this case, we first need to investigate whether additional lower-level optimal solutions besides these two exist, for some different integer value of p1 strictly between c1 and C1. Note that when p1 is increased above c1, the slope of f*(p1) remains constant and equal to q *1 (c1) for as far as this optimal solution remains unchanged. Similarly, when p1 is decreased below C1, the slope of f*(p1) remains constant and equal to q *1 (C1) for as far as this optimal solution remains unchanged. Consider now the value of p1, say u, for which these two lines intersect (see Fig. 2), even if this value is not integer. If f*(u) is lower than the value of these two lines at the intersection point, W, then additional lower-level optimal solutions exist. Otherwise, the search terminates, since the concavity of f* implies that no additional ISO optimal solutions exist. The algorithm continues similarly, aiming to identify all the lower-level optimal solutions that can result by varying p1.

Fig. 2
figure2

Upper bound on f *(p1) determined by two distinct ISO optimal solutions

Once the above procedure has been completed, the energy dispatch of the strategic producer in each p1-interval associated with a distinct ISO optimal solution is known. The optimal value of p1 is then identified by comparing the maximum profit the strategic producer can attain in each of these intervals. Under a uniform pricing scheme, two distinct cases need to be considered for computing this profit, depending on whether he is marginal or not. If he is marginal, this profit is naturally maximized when p1 becomes equal to the associated right endpoint; otherwise, the smp and subsequently this profit, too, remain constant for any integer p1-value in the associated interval. Under a pay-as-bid clearing scheme, on the other hand, the optimal value of p1 in each of these intervals is equal to the associated right endpoint, unless the strategic producer’s dispatch in that period is equal to 0, in which case any feasible p1-value is optimal. After the strategic producer’s maximum profit in each distinct ISO optimal solution is identified, the maximum out of them is naturally chosen as the optimal one. More details on this solution methodology for the single-period version of the problem are provided in Kozanidis et al. (2013).

The multi-period case

Switching our attention to the multi-period version of the problem, note that, given the price-offers of the participating producers, the lower-level problem remains an integer linear program. Therefore, its optimal objective is still a non-decreasing piecewise linear concave function of any single price-bid of the strategic producer. Thus, the optimal value of any single price-bid for the current values of all the remaining ones can be computed with the same exact procedure that was used in the single-period case. This involves again identifying the distinct lower-level optimal solutions that result by varying parametrically the price-offer subject to optimization, and selecting the corresponding value that results in the maximum profit.

Starting from an initial set of feasible price-bids, the proposed heuristic solution methodology works iteratively, optimizing at each iteration the price-bid of a single time period given that the remaining ones are kept fixed at their current values. Under a uniform pricing scheme, the optimal value of a price-offer within a particular p1-interval, say [at, bt], is not unique, unless the strategic producer is marginal. In order to break such ties, the heuristic proceeds by choosing as optimal p1-value one of the two corresponding endpoints. If the previous value of p1,t is equal to one of the two endpoints, then the algorithm just alternates this choice by selecting as optimal p1-value the other endpoint, in order to explore additional neighborhood directions for possible objective improvement. Otherwise, it selects as optimal p1-value the left endpoint, which corresponds to the lowest cost, in order to increase the price-offer competitiveness of the strategic producer. The procedure terminates as soon as a full cycle in which the profit of the strategic producer remains unchanged is encountered. This design avoids the execution of meaningless cycles, which, for example, can come about when the optimal values of some price-offers are changing constantly, even though the associated ISO optimal solution and the corresponding strategic producer profit remain unaltered.

It is easy to verify that the profit of the strategic producer cannot worsen in any two consecutive iterations of the heuristic. This holds true, simply because keeping the value of any of his bids the same as before ensures that this profit will remain unchanged, too. The rationale behind the above criterion for algorithmic termination is to keep going as long as there is an improvement in this profit, and to stop as soon as a full cycle in which no such improvement has been observed is encountered. The only case in which this would not lead to termination after a finite number of iterations is if a strictly positive objective improvement could be perpetually maintained. This, however, cannot happen due to the integrality of the decision variables and the consequent finite cardinality of the problem’s feasible set.

Under a pay-as-bid clearing scheme, things are more straightforward. Assuming a strictly positive energy dispatch for the strategic producer in the associated interval, the optimal p1-value is unique and equal to this interval’s right endpoint. If the strategic producer’s dispatch is equal to 0, on the other hand, then the algorithm selects as optimal p1-value the left endpoint, unless this value coincides to the previous p1-value in which case the right endpoint is selected instead. The same conditions used for algorithmic termination in the case of uniform pricing are also applied in the case of a pay-as-bid clearing scheme. Repeating this iterative procedure several times with various initial sets of price-bids (seeds) provides many alternative solutions, the best of which is naturally the one the algorithm returns upon termination.

As far as the issue of addressing more complex models is concerned, note that the above procedure is straightforward and can be applied to similar strategic bidding optimization problems fitting this modeling framework, under the assumption that the lower-level problem remains linear. This implies that the model formulation can be extended to incorporate additional restrictions that may be present in different applications even if the formulation of these restrictions necessitates the introduction of integer decision variables, as long as the modeling of the associated constraints remains linear.

The heuristic solution algorithm

In this subsection, we present the proposed heuristic solution methodology for the multi-period version of the problem in a step-by-step basis, using pseudocode for the reader’s convenience.

figurea

Exact solution methodologies

Motivation

The integer parametric programming theory utilized in the development of the proposed heuristic solution methodology can also be utilized within the context of a cutting-plane solution methodology for finding the exact optimum of the problem. More specifically, it can be suitably modified to enable the generation of valid inequalities for excluding solutions identified by a suitable relaxation of the original problem which do not qualify for global optimality. Note that, in bilevel programming, the theory for obtaining bound information on the optimal objective through suitable relaxations exhibits significant differences with that of typical single-level optimization problems (Bard 1998).

We consider the relaxation of the bilevel optimization model (1)–(9) in which the restriction that the follower’s response must belong to the reaction set is suppressed, i.e., we relax the requirement that the set of unit commitments and energy quantities constitutes an optimal ISO solution in conjunction with the corresponding set of strategic producer price-offers. This requirement, formally termed as bilevel feasibility, is a key prerequisite for global optimality. The following is a well-known result in the context of bilevel optimization, which is utilized in the development of the proposed exact solution algorithm for the treatment of the problem:

Proposition 2

The optimal objective value to the problem that results after bilevel feasibility is relaxed from the original formulation is a valid upper bound on the optimal objective of the original problem.

Proof

The proof is trivial, since relaxing bilevel feasibility enlarges the feasible set of the upper-level problem through the inclusion of those bilevel infeasible solutions which are feasible with respect to the remaining problem constraints, without excluding any other feasible solution. □

The result of Proposition 2 has been utilized in many previous works, such as those by Caramia and Mari (2015), Fischetti et al. (2017), Wang and Xu (2017), and Lozano and Smith (2019), among others. Consider the optimal solution to the problem that results after bilevel feasibility is relaxed in the original formulation. If this solution happens to be bilevel feasible, then, naturally, it is also the exact optimal solution of the original problem. If not, then in order to pursue the search for the optimal solution, one needs to exclude this solution from further consideration. We show next how the integer parametric programming theory exploited in the development of the proposed heuristic solution approach can be suitably modified in order to accomplish this, too.

The price-offers of the strategic producer appear as objective coefficients of his energy quantity variables in the ISO optimization problem. For a particular set of values of these price-offers, it is trivial to solve the lower-level problem and identify its optimal solution. Based on fundamental integer parametric programming theory, this solution remains optimal for a sufficiently small simultaneous perturbation of some of the decision variables’ objective coefficients. More specifically, Geoffrion and Nauss (1977) have showed that when the objective coefficients of a minimization integer program are linearly perturbed through a single scalar parameter, then its optimal objective is piecewise-linear, continuous, and concave on its finite domain as a function of this parameter. In our case, by solely perturbing the strategic producer’s price-offers, this property allows us to identify interval ranges, such that, when each of these bids lies in its corresponding interval, the lower-level optimal solution remains unchanged. After identifying these interval ranges, we use typical integer programming modeling techniques to generate a valid inequality imposing the truly optimal lower-level solution. Besides excluding the previously identified bilevel infeasible solution from further consideration, this procedure also enforces the truly optimal lower-level solution for large value-combinations of the price-offers. The specifics of this procedure are explained next.

For t = 1,…,T, assume that p b1,t is the strategic producer’s price-offer for time period t in the identified bilevel infeasible solution. It is trivial to find the truly optimal ISO solution for (p b1,1 , p b1,2 ,…, p b1,T ). We seek maximum θ1 and θ2 values, such that this solution remains optimal when each price-offer p1,t belongs to the interval [p b1,t  − θ1, p b1,t  + θ2]. The left endpoint of this interval is identified by finding the maximum θ1 value for which the slope of the function f* remains unchanged when, starting from their initial values, these price-offers are simultaneously decreased by θ1. Similarly, the right endpoint of this interval is identified by finding the maximum θ2 value for which the slope of the function f* remains unchanged when, starting from their initial values, these price-offers are simultaneously increased by θ2. Of course, only integer values are of interest in each of these intervals. Once the maximum θ1 and θ2 values have been identified, the following crucial result justifies the validity of the proposed valid inequality:

Proposition 3

The optimal ISO objective for (p b1,1 , p b1,2 ,…, p b1,T ) remains unchanged when each price-offer p1,t belongs to the interval [p b1,t  − θ1, p b1,t  + θ2].

Proof

Consider the ISO optimal objective for (p b1,1 , p b1,2 ,…, p b1,T ), say f *1 , and assume that the ISO optimal objective is also equal to f *1 when each price-offer p1,t is equal to [p b1,t  − θ1], as well as when each price-offer p1,t is equal to [p b1,t  + θ2]. If there exists some combination of p1,t-values with each p1,t belonging to [p b1,t  − θ1, p b1,t  + θ2], such that the optimal ISO solution, say f *2 , is different than f *1 , then this is a contradiction, since it directly negates the concavity and monotonicity of objective f* as a function of the scalar parameter that linear perturbs it. Therefore, the fact that the optimal ISO objective is the same when each price-offer p1,t is equal to [p b1,t  − θ1], as well as when it is equal to [p b1,t  + θ2], implies that it will also be the same for any value-combination of the price-offers within these intervals. □

After incorporating the ensuing valid inequality into the model formulation, the relaxed problem is solved again and its next optimal solution is identified. The procedure continues similarly, eventually terminating as soon as the first bilevel feasible solution is encountered, which naturally constitutes the problem’s exact optimum. An elucidation of the valid inequalities generation procedure is presented next in sufficient detail, in order to ensure material completeness and comprehension.

Valid inequalities’ generation

The valid inequality that we want to introduce in order to exclude a bilevel infeasible solution must impose the restriction that if each of the strategic producer’s bids, p1,t, belongs to the interval it has been associated with, then a particular unit commitment and energy quantity distribution must comprise the corresponding ISO optimal solution. Assume that the particular interval in question for price offer p1,t is denoted by [at, bt]. If both at ≠ c1 and bt ≠ C1, then the generation of the cut necessitates the introduction of two binary variables, say W tl and W tr , denoting whether p1,t is greater or equal to at and less or equal to bt, respectively. Mathematically, this is expressed through the following four constraints:

$$p_{1,t} \le \, \left( {C_{1} - a_{t} + \, 1} \right)W_{l}^{t} + \, \left( {a_{t} - \, 1} \right)$$
(11)
$$p_{1,t} \ge \, \left( {a_{t} - c_{1} } \right)W_{l}^{t} + c_{1}$$
(12)
$$p_{1,t} \le \, \left( {C_{1} - b_{t} } \right)\left( {1 \, - W_{r}^{t} } \right) \, + b_{t}$$
(13)
$$p_{1,t } \ge c_{1} + \, \left( {b_{t} + 1 - c_{1} } \right)\left( {1 \, - W_{r}^{t} } \right)$$
(14)

Constraints (11) and (12) impose the restriction pertaining to the left endpoint of the interval, i.e., W tl  = 1 if and only if p1,t  ≥ at. More specifically, if W tl  = 0 then p1,t ≤ at − 1 from constraint (11), while constraint (12) becomes redundant. On the other hand, if W tl  = 1 then p1,t  ≥ at from constraint (12), while constraint (11) becomes redundant. Similarly, constraints (13) and (14) impose the restriction pertaining to the right endpoint of the interval, i.e., W tr  = 1 if and only if p1,t ≤ bt. More specifically, if W tr  = 1 then p1,t ≤ bt from constraint (13), while constraint (14) becomes redundant. On the other hand, if W tr  = 0 then p1,t  ≥ bt + 1 from constraint (14), while constraint (13) becomes redundant. Of course, no corresponding binary variable needs to be introduced if the corresponding endpoint coincides with c1 or C1, respectively.

After the required binary variables have been properly defined for all price-offers, the imposition of a particular energy quantity, say Q, for production unit i in period t is accomplished by introducing the following two constraints:

$$q_{i,t} \le Q + (M_{i} - Q)\sum\limits_{t} {(2 - W_{l}^{t} - W_{r}^{t} )}$$
(15)
$$q_{i,t} \ge Q - Q\sum\limits_{t} {(2 - W_{l}^{t} - W_{r}^{t} )}$$
(16)

If W tl  = W tr  = 1 for t = 1,…,T, then the two summations in constraints (15) and (16) are eliminated and qi,t is set equal to Q. If at least one of these auxiliary variables is equal to 0, which implies that the corresponding price-offer does not belong to its associated interval, then both these constraints become redundant. Using such a pair of constraints for the energy quantity of each energy producer, we can impose a specific ISO optimal solution, thus eliminating a bilevel infeasible solution. Note that for units which are constrained to 0-quantity in the associated solution, the above two constraints can be replaced by the following equivalent constraint that directly fixes the status of unit i in period t:

$$z_{i,t} \le \sum\limits_{t} {(2 - W_{l}^{t} - W_{r}^{t} )} .$$
(17)

Relaxing bilevel feasibility

The most typical approach for relaxing bilevel feasibility in general bilevel optimization problems is the suppression of the lower-level objective; this transforms the problem into a single-level optimization model. This technique has been utilized in many previous works, such as those by Caramia and Mari (2015), Fischetti et al. (2017), Wang and Xu (2017), and Lozano and Smith (2019), among others. In the case of a pay-as-bid clearing scheme, this can be accomplished straightforwardly by removing the ISO objective from the original model formulation. Note that the objective of this relaxed single-level optimization model is quadratic, since it involves the product of two decision variables both treated at the same level. In the case of uniform pricing, on the other hand, an explicit representation of the system marginal price is not present in the problem formulation and needs to be incorporated. In order to accomplish this, we introduce extra constraints enforcing the correct smp definition for each time period of the planning horizon, according to the actual set of rules in effect. The exact procedure for doing this is illustrated in the appendix.

The exact solution algorithm

Having elucidated the various actions the proposed exact solution algorithm involves, we are now in a position to present it in a step-by-step basis using pseudocode for the reader’s convenience.

figureb

Computational results

We have implemented the proposed solution methodologies using C/C++ source code. In this section, we illustrate their application on a small case study, and we present extensive experimental results evaluating their relative computational performance. All tests were performed on a 6-Core @ 3.5 GHz 64-bit AMD Processor with 8 GB system memory, while the commercial optimization software LINGO 13.0 (2011) was internally utilized for the solution of the encountered optimization models.

A small case study

For illustration purposes, we consider first a small case study with 3 production units and a 4-period planning horizon. The technical characteristics and startup costs of the generation units, as well as the price-offers and the demand for energy in each time period are shown in Table 1. The technical minima and maxima are given in MW, the startup costs in €, the price-offers for energy in €/MWh, and the energy demand in MWh. The unit variable production cost of the strategic producer (generation unit 1) is 50 €/MWh, while the price cap is 100 €/MWh. We assume that all units are OFF at the beginning of the planning horizon, i.e., that zi,0 = 0 for i = 1,…,3.

Table 1 Case study data

First, we apply the proposed heuristic solution approach under a pay-as-bid clearing scheme. Initially, we set the price-offer of the strategic producer in time period t (t = 1,…,4) equal to the minimum price-offer of any other producer in the same time period, i.e., 57, 58, 58 and 62, respectively. In the first iteration, the algorithm optimizes the value of p1,1, while keeping the values of p1,2, p1,3 and p1,4 fixed. Table 2 presents the ISO optimal solution when p1,1 is set equal to 50, in which the strategic producer is rewarded with the maximum possible dispatch in each time period for his low price-offers. His total profit upon clearing of the market is equal to 500(50–50) + 500(58–50) + 500(58–50) + 500(62–50) = 14,000, while the optimal ISO objective is equal to 206,400. Note that the strategic producer’s profit in the first period of the planning horizon is equal to 0, despite the fact that his corresponding energy dispatch is equal to his technical maximum.

Table 2 Optimal ISO solution when p1,1 is set equal to 50

Table 3 presents the ISO optimal solution when p1,1 is set equal to 100. In this case, the strategic producer does not participate in the market in the first time period, as a result of his particularly high price-offer; his total profit upon clearing of the market is equal to 0(100–50) + 500(58–50) + 500(58–50) + 500(62–50) = 14,000, same as before, while the optimal ISO objective is equal to 213,040.

Table 3 Optimal ISO solution when p1,1 is set equal to 100

The parametric analysis outlined in Sect. 4.1 identifies the 3 distinct lower-level optimal solutions and corresponding p1,1 interval values depicted in Table 4. In the first two of these solutions, the optimal value of p1,1 is equal to the associated interval’s right end-point. In the third solution, on the other hand, any p1,1 value in the associated interval is optimal, since the strategic producer’s optimal dispatch in the first period is equal to 0. As shown in this table, the maximum profit that the strategic producer can attain is equal to 19,880, realized for p1,1 = 64. The corresponding ISO optimal objective is equal to 212,840. The subsequent solutions visited by the algorithm in the next iterations are presented in Table 5.

Table 4 ISO optimal solutions for p1,2 = 58, p1,3 = 58, p1,4 = 62
Table 5 Solutions visited by the heuristic algorithm in the case study under a pay-as-bid clearing scheme

As shown in Table 5, the algorithm identifies the 4 price-offers (64, 60, 65, 62) upon termination, each of which is optimal for the current values of the other three. This is realized at the end of the 7th iteration, which flags the completion of a full cycle (4 iterations) in which the values of the 4 price-offers and the corresponding strategic producer profit remain unchanged. The strategic producer profit of the solution returned by the algorithm is equal to 21,530, while the optimal ISO objective is equal to 216,090.

We also applied the heuristic algorithm using two different sets of initial price-offer values, i.e. (64, 60, 65, 67), which corresponds to selecting the maximum price-offer of any producer in each time period, and (64, 58, 65, 62), which corresponds to some random selection from the other producers’ price offers in the same time period. The collective results comparing the three corresponding final solutions are presented in Table 6. All three solutions are pretty close in terms of the optimal ISO objective, with the two solutions obtained with the first and the third set of price-offers coinciding and providing the same strategic producer profit, which is larger than that of the second one.

Table 6 Heuristic algorithm results for three different sets of initial price-offers (pay-as-bid pricing)

Table 7 presents results similar as those of Table 6 for the case that a uniform pricing clearing scheme is adopted. The final solution that the algorithm returns for the second set of initial price-offers is the same with the one returned under the pay-as-bid clearing scheme. On the other hand, the first and the third solution coincide and qualify as the best, with an associated ISO optimal objective equal to 210,090 and corresponding strategic producer profit equal to 21,530. Note that, for this particular example, the profit of the strategic producer in the solution that the algorithm returned for each of the three sets of initial price-offers is the same under both clearing schemes.

Table 7 Heuristic approach results for three different sets of initial price-offers (uniform pricing)

Next, we apply the exact solution algorithm under a pay-as-bid clearing scheme first. Solving the optimization problem that results after the lower-level objective is suppressed, we get the following price-offers and energy quantities, respectively, for the strategic producer: p1,t = (100, 100, 100, 100), q1,t = (500, 500, 500, 500). It is easy to confirm that this solution is not bilevel feasible, since solving the ISO’s problem for p1,1 = p1,2 = p1,3 = p1,4 = 100, we get an ISO optimal solution in which q1,t = (0, 0, 0, 0). Using integer parametric programming theory we find that the maximum simultaneous decrease on the 4 price bids of the strategic producer for which this solution remains unchanged is equal to 35. This implies that when p1,1 ∈ [65, 100] and p1,2 ∈ [65, 100] and p1,3 ∈ [65, 100] and p1,4 ∈ [65, 100] then q1,1 = q1,2 = q1,3 = q1,4 = 0 and the profit of the strategic producer is equal to 0. To express this restriction mathematically, we add binary variables k1,t for t = 1,…,4, such that k1,t is equal to 1 if p1,t  ≥ 65, and 0 otherwise. This is expressed as follows mathematically: k1,t  ≥ (p1,t − 64)/36 and k1,t ≤ (p1,t − 50)/15. Then, the cut in question is expressed by adding the following inequalities: z1,t ≤ (4 − k1,1 − k1,2 − k1,3 − k1,4) for t = 1,…,4. This excludes the previous bilevel infeasible solution from further consideration. Continuing adding similar cuts for each bilevel infeasible solution identified, the algorithm eventually reaches the exact optimal solution, which is p *1,t  = (64, 60, 58, 69), q *1,t  = (420, 470, 500, 380), with f * = 216,440 and F *1  = 21,800.

Under a uniform pricing clearing scheme, the algorithm identifies that the exact optimal solution is p *1,t  = (64, 60, 50, 70), q *1,t  = (420, 470, 500, 380), with smp *t  = (64, 60, 58, 70), f * = 212,820 and F *1  = 22,180. Thus, although being pretty close, the two optimal solutions under the two clearing schemes are not identical. In the uniform pricing case, the optimal ISO cost is lower, while the strategic producer’s optimal profit is slightly higher. For this small case study, the best solution identified by the heuristic algorithm in the case of a pay-as-bid clearing scheme approximates the truly optimal one with a percentage difference of 1.2%, while the corresponding approximate difference in the case of a uniform pricing clearing scheme is equal to 2.9%. Of course, the quality of the solutions returned by the heuristic algorithm can potentially be improved through further execution attempts with additional initial solutions. For each of the two clearing schemes, Table 8 presents the total execution time, the total number of times (runs) the ISO optimization problem was solved, and the total number of valid inequalities (cuts) added. Note that the latter two figures do not coincide, due to the fact that the identification of a bilevel infeasible solution sometimes leads to the update of an existing cut instead of the introduction of a new one.

Table 8 Execution time, number of runs and number of cuts on the case study for the exact solution algorithm

In order to provide further insight to the interested reader, we also illustrate the application of the branch and bound algorithm by Moore and Bard (1990) on the same case study under a pay-as-bid pricing scheme, after imposing the same limit on the size of the branch and bound tree for avoiding excessive run time. After performing the necessary initialization, the algorithm finds in Step 1 the optimal solution to the problem that results after both bilevel feasibility and integrality are relaxed from model (1)–(9). A valid upper bound on the optimal objective equal to 100,000 is computed this way, realized when the price-offers and the energy quantities of the strategic producer assume their maximum values in all time periods. Next, the algorithm computes the optimal solution to the same problem after bilevel feasibility is reinstated, which comprises a linear bilevel optimization model. The identified solution violates the integrality constraints so the algorithm proceeds by branching on one of the fractional decision variables. We adhere to the authors’ suggestion to branch on the largest fractional variable, but we always give priority to the unit commitment variables of the strategic producer, since the decision on the optimal unit commitment must precede that of the optimal energy quantities. Consequently, the first variable we branch on is variable z1,1.

The algorithm proceeds similarly by solving the problem that results after bilevel feasibility is relaxed, reinstating bilevel feasibility, solving again, and branching on a fractional decision variable. The optimal solution remains unchanged, so the lower-level objective is re-introduced and a new optimal solution is obtained. After several such branching decisions, we reach an integral solution in which p *1,t =(64, 66, 65, 67). The algorithm proceeds by solving the lower-level problem for these upper-level variable values, in order to obtain a feasible solution and update the incumbent. Unfortunately, the corresponding solution is of little use, since for these price-offers the strategic producer remains inactive in all time periods and therefore makes no profit.

The main characteristic in the subsequent iterations of the algorithm is that many feasible solutions for which the strategic producer makes no profit at all are visited before identifying one in which his actual profit is positive. Therefore, the lower bound on his optimal objective remains equal to 0 for quite a few iterations. After considerable branching and backtracking, the algorithm manages to improve this lower bound by identifying bilevel feasible solutions with positive profit for the strategic producer. Eventually, the optimal solution with profit equal to 21,800 is identified, for which p *1,t =(64, 60, 58, 69). Even then, however, the algorithm does not terminate, since the upper bound on the optimal objective provided by the high point solutions (solutions to the single-level problem resulting after the lower-level objective is suppressed) is substantially higher. In an attempt to examine these solutions through further branching, the algorithm reaches the maximum size on the branch and bound tree and terminates, returning the above as the best incumbent solution which also happens to be the optimal one.

The above discussion reveals that, due to the fact that it relaxes both bilevel feasibility and integrality, the algorithm of Moore and Bard (1990) necessitates substantial branching on the decision variables which inevitably has to be repeated several times for the general integer variables qi,t. An additional obstacle is introduced by the fact that branching is necessitated even when the relaxed solution is integral, since such a solution cannot in general be fathomed as in the case of single-level integer programming. In particular, excessive branching on decision variables qi,t is necessitated, as determined by the fact that these general integer variables have many alternative feasible values. The authors develop several variations of the algorithm for reducing its computational burden, but these render its nature heuristic, thus negating the guarantee that the final solution will be exactly optimal. On the other hand, the integrality constraints are always satisfied in every solution considered by our proposed algorithm, due to the fact that only bilevel feasibility is suppressed in the problem relaxation involved. Thus, by working with the original integer problem version, the computational burden of branching is avoided, while the major computational effort in our case is mainly devoted for the generation of the numerous cuts that need to be introduced for eliminating the bilevel infeasible solutions encountered.

In the small case study studied above, the fact that c1 = 50 ≤ pi,t for every i > 1 and t enables the strategic producer to provide the lowest price-offer in each time period. This is a rather uncommon assumption that, in general, does not hold true in practice. As it turns out, actually, it can be exploited by the strategic producer for gaining more profits. Assume, for example, that, even though c1=50, the price-offers of the strategic producer in each time period are constrained to be at least equal to 60. In this case, the exact optimal solution of the problem under a pay-as-bid clearing scheme is p *1,t = (61, 65, 63, 66), q *1,t = (420, 200, 330, 380), with f *= 216,690 and F *1 = 17,990. Thus, tightening the lower bounds on the strategic producer’s price-offers leads to an approximate 17.5% decrease of his individual profit. In the case of uniform pricing, the corresponding percentage decrease in profit is almost 7%, since the optimal solution is p *1,t = (63, 60, 60, 69), q *1,t = (420, 470, 330, 380), with smp *t = (63, 60, 60, 69), f *= 216,680 and F *1 = 20,680. Note that the strategic producer is marginal in every time period in this case.

The above analysis confirms not only that the interval ranges constraining the price-offers of the strategic producer affect strongly the maximum profit that he can attain, but also that he can often exploit the flexibility of being able to provide lower price-offers than his competitor generation units to his own advantage. Even so, we emphasize that this does not affect in any way the validity of the proposed solution methodology, whose application remains unchanged, independently of whether this assumption holds true or not.

Randomly generated problems

In this subsection, we test the performance of the proposed solution algorithms on randomly generated problems. For the heuristic algorithm, we generated random problem instances with the following sizes expressed as A × B where A = number of generation units and B = number of time periods: 3 × 4, 4 × 4, 5 × 24, 6 × 24, 7 × 24, 8 × 24, 9 × 24, and 10 × 24. The emphasis on the number 24 for B is mainly due to the fact that realistic day-ahead electricity markets are typically solved over a planning horizon consisted of 24 hourly time periods. Each of the remaining units, besides the one pertaining to the strategic producer, was assigned the technical characteristics (technical minimum/maximum, start-up cost and price bids) of a factual unit participating in the Greek electricity market, according to the data provided by Andrianesis et al. (2013b). The particular unit with which the association was made was selected randomly. The data pertaining to the strategic producer’s unit were then generated as follows: The technical minimum (m1) was an integer selected randomly in the interval \(\mathop {[\hbox{min} m_{i} }\limits_{i \, > 1} ,\mathop {\hbox{max} m_{i} }\limits_{i \, > 1} ]\), the technical maximum (M1) was set equal to m1 + range, where range was an integer selected randomly in the interval \(\mathop {[\hbox{min} }\limits_{i \, > 1} (M_{i} - m_{i} ),\mathop {\hbox{max} }\limits_{i \, > 1} (M_{i} - m_{i} )]\), the start-up cost (s1) was an integer selected randomly in the interval \(\mathop {[\hbox{min} s_{i} }\limits_{i \, > 1} ,\mathop {\hbox{max} s_{i} }\limits_{i \, > 1} ]\), the unit variable production cost (c1) was set equal to 0.9 \(\mathop {\hbox{min} }\limits_{i \, > 1, \, t} p_{it}\) rounded to the nearest integer, and the price-cap (C1) was set equal to 1.1 \(\mathop {\hbox{max} }\limits_{i \, > 1, \, t} p_{it}\), rounded to the nearest integer. Finally, the demand for energy in each time period t was an integer distributed uniformly in the interval \(\left[ {\sum\nolimits_{i} {(M_{i} - m_{i} ) + \mathop {\hbox{min} }\limits_{i} m_{i} } ,\sum\nolimits_{i > 1} {M_{i} + m_{1} } } \right]\).

The heuristic algorithm was applied three times on each problem instance, each time with a different set of initial price-offers for the strategic producer. In the first case, the price-offer of each time period t was set equal to the minimum price-offer of any other producer in the same time period, in the second case it was set equal to the maximum price-offer of any other producer in the same time period, and finally, in the third case, it was set equal to some of the other bids in the same time period, with the selection being made randomly. As the total number of distinct problem instances solved for each problem size was equal to 20, the total number of times the heuristic was applied on each problem size was equal to 3 × 20 = 60. For each clearing scheme, the following table presents the average and maximum computational times for each problem size.

As the above results demonstrate, the computational times of the heuristic algorithm are quite reasonable, enabling the solution of problems whose size approaches that of realistic problems encountered in practical environments of the Greek electricity market. The variance of the computational times appears to be significant but not excessive. This is acceptable, considering that the optimization problem under consideration is highly non-convex and combinatorial. An interesting observation that can be made based on the results of Table 9 regards the fact that the computational times increase much faster with problem size in the case of a pay-as-bid clearing scheme than in the case of a uniform clearing scheme. This can be possibly explained by the fact that under a pay-as-bid clearing scheme there is a much larger set of alternative solutions that the algorithm must comparatively evaluate, due to the fact that the strategic producer’s profit is dependent on his exact price-offer even in those time periods in which he is not marginal.

Table 9 Computational times (in seconds) of the heuristic algorithm on random problem instances

For the needs of the present paper, we applied the exact solution algorithm on problem instances with sizes 3 × 4 and 4 × 4, which were the same as those in the case of the heuristic algorithm. For each of the two clearing schemes, Table 10 presents the average and maximum computational times, the average and maximum number of runs, and the average and maximum number of cuts utilized. These results demonstrate the excessive computational requirements of the algorithm. In particular, the computational times are considerably large, partially due to the significant number of runs and cuts that these two relatively small problem sizes necessitated. The computational resource requirements of the algorithm are undoubtedly substantial. Consequently, it can be applied on particularly small sized problems only, whereas its application on realistic problem instances seems implausible at the moment. The computational requirements appear much higher under a uniform clearing scheme, the most reasonable explanation for this being the significantly more complicated model formulation due to the necessity for the explicit system marginal price representation.

Table 10 Computational results for the application of the exact algorithm on random problem instances

Table 11 presents results regarding the quality of the solutions returned by the heuristic solution algorithm. More specifically, for each clearing scheme and each of the two problem sizes, 3x4 and 4x4, this table presents the average and maximum percentage difference between the strategic producer’s profit in the solution provided by the heuristic, and that in the optimal solution identified by the exact solution algorithm. These results demonstrate that the heuristic algorithm provides high quality solutions at least for these two particular problem sizes. The maximum percentage difference in the strategic producer’s profit is less than 3.5% in the worst case under both clearing schemes. Additionally, there does not seem to be significant difference in the heuristic algorithm’s effectiveness between the two clearing schemes. These results seem quite promising, leaving open the possibility that the heuristic algorithm may be capable of providing high quality solutions for problems of realistic size, too.

Table 11 Results regarding the quality of the solutions provided by the heuristic algorithm

In view of the crucial observation regarding the strategic producer’s potential of increasing his profits under the assumption of being in a position to provide lower price-offers than his competitors, and due to the fact that this assumption holds true in the experiments whose results are presented above, we have repeated these experiments after changing the lower bound on his price-offers from 0.9 \(\mathop {\hbox{min} }\nolimits_{i \, > 1, \, t} p_{it}\) to 1.1 \(\mathop {\hbox{min} }\nolimits_{i \, > 1, \, t} p_{it}\) rounded to the nearest integer (his unit variable production cost was kept unchanged). Table 12 presents the average and maximum percentage difference in his optimal profit under both clearing schemes over 20 problem instances for each problem size. These results confirm the key role that the interval ranges on the strategic producer’s price-offers play on the maximum profit he can attain, since the maximum percentage profit decrease is approximately equal to 12% in some cases.

Table 12 Percentage difference in optimal strategic producer profit under different lower bounds on pi,t

Summary and future work

In the present paper, we addressed a mixed integer bilevel optimization problem which can be utilized to devise optimal bidding strategies for energy producers participating in multi-period day-ahead electricity markets. We utilized the theoretical properties of this problem to develop a heuristic solution algorithm, which works in successive iterations by focusing on single time periods. Next, we demonstrated how the related theoretical groundwork can be modified to enable the generation of valid inequalities to a suitable relaxation of the problem in which the bilevel feasibility of the obtained solution is not guaranteed. The significance of these inequalities lies in that they can be embedded within a cutting plane algorithmic procedure for identifying the exact optimal solution of the problem. We implemented this solution methodology, and we illustrated its applicability as well as that of the heuristic solution algorithm on a small numerical example.

We concluded with experimental results demonstrating the computational capabilities of the proposed solution algorithms, as well as evaluating their relative performance. These results demonstrate that the computational requirements of the heuristic solution algorithm are very moderate, enabling the solution of problems whose size approaches that of realistic ones in reasonable times. The exact solution algorithm, on the other hand, exhibits significantly higher computational requirements, which prohibit its application on realistic problem instances at the moment.

The present research work can be extended in many suitable ways. As far as the model formulation is concerned, the future incorporation of additional problem characteristics that are present in realistic applications, such as minimum uptimes and downtimes, ramp-up/ramp-down constraints, and step-wise price-offers appears quite promising. Such a task will increase the model formulation complexity, making the problem more challenging and, at the same time, more realistic. In particular, although the minimum uptimes/downtimes and the ramp-up/ramp-down constraints increase the complexity of the optimization model, modifying the proposed solution methodologies to incorporate them can be carried out rather straightforwardly. Due to the fact that they are both typically modeled with linear constraints involving the existing decision variables zi,t and qi,t, respectively, the optimization model remains mixed-integer bilinear after their inclusion, which implies that the validity of the integer parametric programming property utilized in both methodologies is retained. Of course, the question of how much the computational performance of the two methodologies or the quality of the proposed heuristic will be affected by the inclusion of these problem characteristics remains open and stems as an interesting direction for future research.

Another interesting direction for future research is the further refinement of the proposed heuristic algorithm’s involved decisions, in order to expedite its computational performance. The question of how initial solutions (seeds) can be wisely selected for enhancing solution quality remains open, too. As far as the exact solution algorithm is concerned, its practical application seems to necessitate further algorithmic enhancements. In that direction, the question of whether exact optimality conditions for the lower-level problem can be devised and incorporated directly in the existing formulation in order to guide and expedite the search for the exact optimal solution appears quite promising. The fulfilment of this task stems as a possible way to overcome the significant computational obstacles that the current implementation exhibits.

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Appendix: Relaxing bilevel feasibility in the case of uniform pricing

Appendix: Relaxing bilevel feasibility in the case of uniform pricing

The procedure we adopt for relaxing bilevel feasibility in the case of uniform pricing necessitates the introduction of two binary variables wi,t and vi,t for each generation unit i and time period t. Variable wi,t takes the value 1 if and only if qi,t > mi, and 0 otherwise, while variable vi,t takes the value 1 if and only if qi,t < Mi, and 0 otherwise. We can modify accordingly constraints (11)–(14) to ensure that wi,t and vi,t correctly depict these two conditions as follows:

$$q_{i,t} \le \, \left( {M_{i} - m_{i} } \right)w_{i,t} + m_{i}$$
(18)
$$q_{i,t} \ge \, \left( {m_{i} + 1} \right)w_{i,t}$$
(19)
$$q_{i,t} \le M_{i} - v_{i,t}$$
(20)
$$q_{i,t} \ge M_{i} \left( {1 \, - v_{i,t} } \right)$$
(21)

When wi,t = 1, constraint (19) sets qi,t strictly greater than mi, while constraint (18) becomes redundant. On the other hand, when wi,t = 0, constraint (18) sets qi,t less or equal to mi, while constraint (19) becomes redundant. Similarly, when vi,t = 0, constraint (21) sets qi,t equal to Mi, while constraint (20) is redundant. When vi,t = 1, on the other hand, constraint (20) sets qi,t strictly less than Mi, while constraint (21) becomes redundant. After the introduction of these constraints, Rule 1 for time period t is enforced by adding the following two constraints for each unit i, in which K is a sufficiently large number:

$$\lambda_{t} \le p_{i,t} + (2 - w_{i,t} - v_{i,t} )K$$
(22)
$$\lambda_{t} \ge p_{i,t} - (2 - w_{i,t} - v_{i,t} )K$$
(23)

If mi < qi,t < Mi, then wi,t = vi,t = 1, so these two constraints set the smp of time period t equal to the price-offer of unit i; in any other case, they are both redundant. In order to express Rule 2, we additionally introduce a binary variable uj,t for each unit j > 1 and time period t, which takes the value 1 if and only if p1,t < pj,t, and 0 otherwise. Correct values for variables uj,t are ensured through the introduction of the following two constraints which are analogous to constraints (13) and (14):

$$p_{1,t} + \, 1 \, \le \, \left( {C_{1} - p_{j,t} + \, 1} \right)\left( {1 \, - u_{j,t} } \right) \, + p_{j,t}$$
(24)
$$p_{1,t } \ge c_{1} + \, \left( {p_{j,t} - c_{1} } \right)\left( {1 \, - u_{j,t} } \right)$$
(25)

For each unit j > 1 and time period t, we also introduce a binary variable bj,t which takes the value 1 if and only if both qj,t = mj (i.e., zj,t = 1 and wj,t = 0) and pj,t ≤ p1,t (i.e., uj,t = 0) hold, and 0 otherwise, as well as a binary variable gj,t, which takes the value 1 if and only if both q1,t = m1 (i.e., z1,t = 1 and w1,t = 0) and pj,t > p1,t (i.e., uj,t = 1) hold, and 0 otherwise. Correct values for variables bj,t and gj,t are ensured through the following eight constraints:

$$b_{j,t} \ge z_{j,t} - w_{j,t} - u_{j,t}$$
(26)
$$b_{j,t} \le z_{j,t}$$
(27)
$$b_{j,t} \le \, 1 \, - w_{j,t}$$
(28)
$$b_{j,t} \le \, 1 \, - u_{j,t}$$
(29)
$$g_{j,t} \ge z_{1,t} + u_{j,t} - w_{1,t} - \, 1$$
(30)
$$g_{j,t} \le z_{1,t}$$
(31)
$$g_{j,t} \le u_{j,t}$$
(32)
$$g_{j,t} \le \, 1 \, - w_{1,t}$$
(33)

Rule 2 for time period t is then expressed through the introduction of the following set of constraints. Constraints (34)–(35) are introduced only once as they pertain to unit 1, while constraints (36)–(37) are introduced once for each unit i > 1.

$$\lambda_{t} \le p_{1,t} + K(1 - z_{1,t} + w_{1,t} ) + \sum\limits_{j > 1 \, } {Kb_{j,t} } + \sum\limits_{j > 1 \, } {K(w_{j,t} + v_{j,t} - 1)}$$
(34)
$$\lambda_{t} \ge p_{1,t} - K(1 - z_{1,t} + w_{1,t} ) - \sum\limits_{j > 1} {Kb_{j,t} } - \sum\limits_{j > 1 \, } {K(w_{j,t} + v_{j,t} - 1)}$$
(35)
$$\lambda_{t} \le p_{i,t} + K(1 - z_{i,t} + w_{i,t} ) + \sum\limits_{{j > 1,j \ne i: \, p_{j,t} < p_{i,t} }} {K(z_{j,t} - w_{j,t} )} + Kg_{i,t} + \sum\limits_{j \ne i \, } {K(w_{j,t} + v_{j,t} - 1)}$$
(36)
$$\lambda_{t} \ge p_{i,t} - K(1 - z_{i,t} + w_{i,t} ) - \sum\limits_{{j > 1,j \ne i: \, p_{j,t} < p_{i,t} }} {K(z_{j,t} - w_{j,t} ) - Kg_{i,t} } - \sum\limits_{j \ne i \, } {K(w_{j,t} + v_{j,t} - 1)}$$
(37)

Constraint (36) sets λt less than or equal to pi,t plus a summation of non-negative terms each of which involves a multiplication with the sufficiently large number K. Similarly, constraint (37) sets λt greater than or equal to pi,t minus the summation of the same exact terms. Thus, if all these terms are equal to 0, then λt is set equal to pi,t by these two constraints; otherwise, they are both redundant. The term (1 − zi,t + wi,t) is equal to 0 if unit i produces at its technical minimum in time period t; otherwise, it is equal to 1. The term (zj,t − wj,t) is equal to 1 if unit j produces at its technical minimum in time period t; otherwise, it is equal to 0. The term gi,t is equal to 1 if unit 1 has smaller price-offer than unit i and produces at its technical minimum in time period t; otherwise, it is equal to 0. Finally, the term (wj,t + vj,t − 1) is equal to 1 if the output of unit j is strictly between its technical minimum and its technical maximum in time period t; otherwise, it is equal to 0. Thus, these two constraints set λt equal to the price-offer of unit i in time period t if unit i produces at its technical minimum, and in addition there is no other unit with smaller price-offer producing at its technical minimum, and no other unit producing strictly between its technical minimum and its technical maximum in the same time period; in any other case, these constraints are both redundant. Constraints (34)–(35) are similar to constraints (36)–(37), pertaining to the unit of the strategic producer. One can easily verify that these constraints determine the smp value correctly even when more than one units are simultaneously marginal according to Rule 2, both in the case that one of them is the strategic unit as well as in the case that it is not.

In order to express Rule 3, we additionally introduce a binary variable aj,t for each unit j > 1 and time period t, which takes the value 1 if and only if both qj,t = Mj (i.e., vj,t = 0) and pj,t > p1,t (i.e., uj,t = 1) hold, and 0 otherwise, as well as a binary variable hj,t, which takes the value 1 if and only if both q1,t = M1 (i.e., v1,t = 0) and pj,t ≤ p1,t (i.e., uj,t = 0) hold, and 0 otherwise. Correct values for variables aj,t and hj,t are ensured through the following six constraints:

$$a_{j,t} \ge u_{j,t} - v_{j,t}$$
(38)
$$a_{j,t} \le u_{j,t}$$
(39)
$$a_{j,t} \le \, 1 \, - v_{j,t}$$
(40)
$$h_{j,t} \ge \, 1 \, - v_{1,t} - u_{j,t}$$
(41)
$$h_{j,t} \le \, 1 \, - v_{1,t}$$
(42)
$$h_{j,t} \le \, 1 \, - u_{j,t}$$
(43)

Rule 3 for time period t is then expressed through the introduction of the following set of constraints. Constraints (44)–(45) are introduced only once as they pertain to unit 1, while constraints (46)–(47) are introduced once for each unit i > 1.

$$\lambda_{t} \le p_{1,t} + Kv_{1,t} + \sum\limits_{j > 1 \, } {K(z_{j,t} + v_{j,t} - 1)} + \sum\limits_{j > 1 \, } K \alpha_{j,t}$$
(44)
$$\lambda_{t} \ge p_{1,t} - Kv_{1,t} - \sum\limits_{j > 1 \, } {K(z_{j,t} + v_{j,t} - 1)} - \sum\limits_{j > 1 \, } K a_{j,t}$$
(45)
$$\lambda_{t} \le p_{i,t} + Kv_{i,t} + \sum\limits_{j \ne i \, } {K(z_{j,t} + v_{j,t} - 1)} + \sum\limits_{{j > 1,j \ne i,p_{j,t} > p_{i,t} \, }} K (1 - v_{j,t} ) + Kh_{i,t}$$
(46)
$$\lambda_{t} \ge p_{i,t} - Kv_{i,t} - \sum\limits_{j \ne i \, } {K(z_{j,t} + v_{j,t} - 1)} - \sum\limits_{{j > 1,j \ne i,p_{j,t} > p_{i,t} \, }} K (1 - v_{j,t} ) - Kh_{i,t}$$
(47)

Constraint (46) sets λt less than or equal to pi,t plus a summation of non-negative terms each of which involves a multiplication with the sufficiently large number K. Similarly, constraint (47) sets λt greater than or equal to pi,t minus the summation of the same exact terms. Thus, if all these terms are equal to 0, then λt is set equal to pi,t by these two constraints; otherwise, they are both redundant. The term vi,t is equal to 0 if unit i produces at its technical maximum in time period t; otherwise, it is equal to 1. The term (zj,t + vj,t − 1) is equal to 1 if the energy quantity of unit j in time period t is positive but strictly smaller than its technical maximum; otherwise, it is equal to 0. The term (1 − vj,t) is equal to 1 if unit j produces at its technical maximum in time period t; otherwise, it is equal to 0. Finally, the term hi,t is equal to 1 if unit 1 has greater or equal price-offer than unit i and produces at its technical maximum in time period t; otherwise, it is equal to 0. Thus, these two constraints set λt equal to the price-offer of unit i in time period t if unit i produces at its technical maximum, and in addition there is no other unit whose energy dispatch is positive but strictly smaller than its technical maximum, and no other unit with larger price-offer (for j > 1, j ≠ i) or larger or equal price-offer (for j = 1) producing at its technical maximum; in any other case, they are redundant. Constraints (44)–(45) are similar to constraints (46)–(47), pertaining to the unit of the strategic producer. One can easily verify in this case, too, that these constraints determine the smp value correctly even when more than one units are simultaneously marginal according to Rule 3, both in the case that one of them is the strategic unit as well as in the case that it is not.

With the introduction of the above constraints, the complete model for relaxing bilevel feasibility in the case of a uniform pricing clearing scheme is formulated as follows:

$$\mathop {\text{Max}}\limits_{{p_{1,t} }} \, F_{1} = \sum\limits_{t = 1}^{T} {\left( {\lambda_{t} - c_{1} } \right)q_{1,t} }$$
$${\text{s}} . {\text{t}} .\quad c_{1} \le p_{1,t} \le C_{1},\quad t = 1, \ldots ,T$$
$$\sum\limits_{i \in I} {q_{i,t} } = d_{t} \,, \, t = 1, \ldots ,T$$
$$m_{i,t} z_{i,t} \le q_{i,t} \le M_{i,t} z_{i,t} ,\quad i \in I,\quad t = 1, \ldots ,T$$
$$y_{i,t} \ge z_{i,t} - z_{i,t - 1} ,\quad i \in I,\quad t = 1, \ldots ,T$$
$${\text{constraints}}\,\,\, ( 1 8 )- (23),\quad i \in I,\quad \, t = 1, \ldots ,T$$
$${\text{constraints}}\quad (24){-}(33),\,\,(36){-}(43),\,\,(46) - (47),\quad i \in I,\quad \, i > 1,\quad \, t = 1, \ldots ,T$$
$${\text{constraints}}\quad ( 3 4 ){-}(35),\,\,(44){-}(45),\quad t = 1, \ldots ,T$$
$$p_{1,t} \in Z^{ + } ,q_{i,t} \in Z^{ + } \quad ,i \in I, \quad t = 1, \ldots ,T$$
$$y_{i,t} ,z_{i,t} ,w_{i,t} ,v_{i,t} \quad {\text{binary}},\quad i \in I,\quad t = 1, \ldots ,T$$
$$u_{i,t} ,b_{i,t} ,g_{i,t} ,a_{i,t} ,h_{i,t} \,\,\,{\text{binary}},\quad i \in I,\quad \, i > 1,\quad t = 1, \ldots ,T.$$

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Kostarelou, E., Kozanidis, G. Bilevel programming solution algorithms for optimal price-bidding of energy producers in multi-period day-ahead electricity markets with non-convexities. Optim Eng 22, 449–484 (2021). https://doi.org/10.1007/s11081-020-09521-y

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Keywords

  • Day-ahead electricity markets
  • Energy producers
  • Optimal bidding offers
  • Bilevel optimization
  • Integer parametric programming
  • Exact and heuristic solution algorithms