Bilevel programming for price-based electricity auctions: a revenue-constrained case

Abstract

This paper describes the application of bilevel programming to a class of real-life problems in the field of electric power systems. Within the context of electricity markets, market-clearing procedures, i.e., auction models, are used by an independent entity to schedule generation offers and consumption bids as well as to determine market-clearing prices. This paper addresses a mathematically challenging type of auction, denoted as price-based market clearing, wherein, as a distinctive feature, market-clearing prices are explicitly incorporated in the formulation of the optimization process. This paper shows that bilevel programming provides a suitable modeling framework for price-based market clearing. Furthermore, based on practical modeling aspects, an equivalent single-level primal-dual transformation into a mixed-integer program can be implemented. Such transformation relies on the application of duality theory of linear programming. The bilevel programming framework for price-based market clearing is applied to a revenue-constrained auction model similar to those used in several European electricity markets. As a major contribution, bilinear terms associated with both generation revenue constraints and the duality-based transformation are equivalently converted into linear forms with no additional binary variables. Simulation results show the effective performance of the proposed approach and its superiority over current industry practice.

Appendices

Appendix A: Linearization of the product of a binary variable and a continuous variable

Given the product of a binary variable $u \in \{0,1\}$ and a continuous variable $\tau \in [\tau ^{min},\tau ^{max}]$, a linear equivalent can be found as follows (see Floudas 1995): (1) replace the product $u\tau$ by a new continuous variable $x$, and (2) introduce new inequalities $\tau ^{min}u \le x \le \tau ^{max}u$ and $\tau ^{min}(1-u) \le \tau -x \le \tau ^{max}(1-u)$. Thus, if $u$ is equal to 0, $x$ is also equal to 0 while $\tau$ is bounded by its upper and lower limits. Conversely, if $u$ is equal to 1, $x$ is set equal to $\tau$ and is bounded by the upper and lower limits for $\tau$.

Appendix B: Nomenclature

1.1 Indices

$b$ :

Supply offer block index.

$h$ :

Demand bid block index.

$i$ :

Generating unit index.

$j$ :

Consumer index.

$l$ :

Time period index.

$t$ :

Time period index.

1.2 Sets

$B_i$ :

Index set of energy blocks offered by unit $i$.

$H_j$ :

Index set of energy blocks bid by consumer $j$.

$I$ :

Index set of generating units.

$J$ :

Index set of consumers.

$T$ :

Index set of time periods.

$\Gamma$ :

Feasibility set of the vector of market-clearing prices.

1.3 Functions

$\mathcal {F}(\cdot )$ :

Objective function of the price-based market-clearing procedure.

$\mathcal {G}(\cdot )$ :

Constrained functions involving scheduling variables.

$\mathcal {H}(\cdot )$ :

Constrained functions relating both scheduling and dispatching variables and market-clearing prices.

$\mathcal {R}_i(\cdot )$ :

Minimum revenue function offered by unit $i$.

1.4 Constants

$DT_i$ :

Minimum down time of unit $i$.

$F_{i}$ :

Number of periods during which unit $i$ must be initially scheduled off due to its minimum down time constraint.

$FT_{i}$ :

Coefficient of the fixed term of the minimum revenue function offered by unit $i$.

$L_{i}$ :

Number of periods during which unit $i$ must be initially scheduled on due to its minimum up time constraint.

$n$ :

Dimension of vector $\mathbf {v}$.

$n_T$ :

Number of time periods.

$O^{d}_{hjt}$ :

Price of the $h$th energy block bid by consumer $j$ in period $t$.

$O^{g}_{bit}$ :

Price of the $b$th energy block offered by unit $i$ in period $t$.

$O_{it}^{su}$ :

Start-up offer price of unit $i$ in period $t$.

$P_{jt}^{d,max}$ :

Upper bound for the power consumption of consumer $j$ in period $t$.

$P_{jt}^{d,min}$ :

Lower bound for the power consumption of consumer $j$ in period $t$.

$P_{i0}^{g}$ :

Initial power output of unit $i$.

$P_{it}^{g,max}$ :

Upper bound for the power output of unit $i$ in period $t$.

$P_{it}^{g,min}$ :

Lower bound for the power output of unit $i$ in period $t$.

$RD_i$ :

Ramp-down rate of unit $i$.

$RU_i$ :

Ramp-up rate of unit $i$.

$S_i^0$ :

Number of periods during which unit $i$ has been scheduled off prior to the first period of the time span (end of period 0).

$SD_i$ :

Shutdown ramp rate of unit $i$.

$SU_i$ :

Start-up ramp rate of unit $i$.

$UT_i$ :

Minimum up time of unit $i$.

$UT_i^{0}$ :

Number of periods during which unit $i$ has been scheduled on prior to the first period of the time span (end of period 0).

$V_{i0}$ :

Initial on/off status of unit $i$.

$VT_{bit}$ :

Slope of block $b$ of the minimum revenue function offered by unit $i$ in period $t$.

$\gamma _{it}^{l,min}$ :

Lower bound for dual variable $\gamma _{it}^l$.

$\gamma _{it}^{u,min}$ :

Lower bound for dual variable $\gamma _{it}^u$.

$\delta _{hjt}^{d,max}$ :

Upper bound for the consumption in block $h$ bid by consumer $j$ in period $t$.

$\delta _{bit}^{g,max}$ :

Upper bound for the production in block $b$ offered by unit $i$ in period $t$.

$\epsilon _{it}^{min}$ :

Lower bound for dual variable $\epsilon _{it}$.

$\xi _{it}^{min}$ :

Lower bound for dual variable $\xi _{it}$.

$\sigma _{it}^{min}$ :

Lower bound for dual variable $\sigma _{it}$.

1.5 Variables

$a_{it}$ :

Variable equal to the product $v_{it}\gamma _{it}^l$.

$d_{it}$ :

Variable equal to the product $v_{it}\gamma _{it}^u$.

$e_{it}$ :

Variable equal to the product $v_{it}\xi _{it}$.

$k_{it}$ :

Variable equal to the product $v_{it-1}\xi _{it}$.

$m_{it}$ :

Variable equal to the product $v_{it}\epsilon _{it}$.

$o_{it}$ :

Variable equal to the product $v_{it+1}\epsilon _{it}$.

$p_{jt}^{d}$ :

Power consumption of consumer $j$ in period $t$.

$p_{it}^{g}$ :

Power output of unit $i$ in period $t$.

$q_{it}$ :

Variable equal to the product $v_{it}\sigma _{it}$.

$R_i$ :

Total revenue earned by unit $i$.

$R_i^{min}$ :

Minimum revenue unit $i$ is willing to earn.

$r_{it}$ :

Variable equal to the product $v_{it-1}\sigma _{it}$.

$s_{it}$ :

Start-up offer cost of unit $i$ in period $t$.

$v_{it}$ :

Variable that is equal to $1$ if unit $i$ is scheduled on in period $t$, being $0$ otherwise.

$w_{i}$ :

Variable that is equal to $0$ if unit $i$ is scheduled off along the time span, being $1$ otherwise.

$\alpha _{it}$ :

Dual variable related to the definition of $p_{it}^g$ as the sum of the generation levels awarded to the generation offer blocks.

$\beta _{bit}^l$ :

Dual variable related to the constraint imposing the lower bound for $\delta _{bit}^g$.

$\beta _{bit}^u$ :

Dual variable related to the constraint imposing the upper bound for $\delta _{bit}^g$.

$\gamma _{it}^l$ :

Dual variable related to the constraint imposing the lower bound for $p_{it}^{g}$.

$\gamma _{it}^u$ :

Dual variable related to the constraint imposing the upper bound for $p_{it}^{g}$.

$\Delta R_i$ :

Revenue margin of unit $i$.

$\delta _{hjt}^d$ :

Consumption level awarded to block $h$ bid by consumer $j$ in period $t$.

$\delta _{bit}^g$ :

Generation level awarded to block $b$ offered by unit $i$ in period $t$.

$\epsilon _{it}$ :

Dual variable related to the shutdown ramp rate constraint of unit $i$ in period $t$.

$\lambda _{t}$ :

Market-clearing price in period $t$.

$\mu _{jt}^l$ :

Dual variable related to the constraint imposing the lower bound for $p_{jt}^{d}$.

$\mu _{jt}^u$ :

Dual variable related to the constraint imposing the upper bound for $p_{jt}^{d}$.

$\xi _{it}$ :

Dual variable related to the ramp-up and start-up ramp rate constraint of unit $i$ in period $t$.

$\pi _{jt}$ :

Dual variable related to the definition of $p_{jt}^d$ as the sum of the consumption levels awarded to the demand bid blocks.

$\rho _{hjt}^l$ :

Dual variable related to the constraint imposing the lower bound for $\delta _{hjt}^d$.

$\rho _{hjt}^u$ :

Dual variable related to the constraint imposing the upper bound for $\delta _{hjt}^d$.

$\sigma _{it}$ :

Dual variable related to the ramp-down rate constraint of unit $i$ in period $t$.

1.6 Matrices and vectors

$\mathbf {A}$ :

Coefficient matrix associated with power balance constraints.

$\mathbf {B},\mathbf {C}$ :

Coefficient matrices associated with dispatching constraints other than power balance equations.

$\mathbf {b_1}$ :

Vector of the right-hand side coefficients of power balance constraints.

$\mathbf {b_2}$ :

Vector of the right-hand side coefficients of dispatching constraints other than power balance equations.

$\mathbf {c}$ :

Coefficient vector of the objective function of the lower-level primal problem.

$\mathbf {v}$ :

Vector of binary scheduling variables.

$\mathbf {y}$ :

Vector of continuous scheduling variables.

$\mathbf {z}$ :

Vector of dispatching variables.

$\varvec{\eta }$ :

Vector of dual variables associated with dispatching constraints other than power balance equations.

$\varvec{\lambda }$ :

Vector of market-clearing prices.