## Modeling and computing two-settlement oligopolistic equilibrium in a congested electricity network (2008)

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Venue: | Operations Research |

Citations: | 11 - 2 self |

### BibTeX

@ARTICLE{Yao08modelingand,

author = {Jian Yao and Ilan Adler and Shmuel S. Oren},

title = {Modeling and computing two-settlement oligopolistic equilibrium in a congested electricity network},

journal = {Operations Research},

year = {2008},

pages = {3447}

}

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### Abstract

A model of two-settlement electricity markets is introduced, which accounts for flow congestion, demand uncertainty, system contingencies and market power. We formulate the subgame perfect Nash equilibrium for this model as an equilibrium problem with equilibrium constraints (EPEC), in which each firm solves a mathematical program with equilibrium constraints (MPEC). The model assumes linear demand functions, quadratic generation cost functions and a lossless DC network, resulting in equilibrium constraints as a parametric linear complementarity problem (LCP). We introduce an iterative procedure for solving this EPEC through repeated application of an MPEC algorithm. This MPEC algorithm is based on solving quadratic programming sub-problems and on parametric LCP pivoting. Numerical examples demonstrate the effectiveness of the MPEC and EPEC algorithms and the tractability of the model for realistic size power systems. 1

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Citation Context ...ets as a two-period Nash-Cournot game: the forward market (period zero) and the spot market (period one) and we characterize the equilibrium of this game as a sub-game perfect Nash equilibrium (SPNE, =-=[10]-=-). In period zero, rational firms enter into forward contracts, forming rational expectations regarding the forward commitments of their rivals and the period-one equilibrium outcomes. Period-one is a... |

626 | The Linear Complementarity Problem
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Citation Context ...ar demand functions, quadratic generation cost functions and a lossless DC network, resulting in the preceding equilibrium constraints in the form of a parametric linear complementarity problem (LCP, =-=[6]-=-). When applied to realistic size systems, this EPEC model presents a computational challenge. Therefore, the second goal of this paper is to study the computational aspect of our EPEC model and, by e... |

202 |
Mathematical programs with equilibrium constraints
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Citation Context ...[37, 38]. As before, the model is formulated as an equilibrium problem with equilibrium constraints (EPEC) where each generation firm solves a mathematical problem with equilibrium constraints (MPEC, =-=[23]-=-) parameterized by the other firms’ forward commitments. The model assumes linear demand functions, quadratic generation cost functions and a lossless DC network, resulting in the preceding equilibriu... |

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Citation Context ...ould not exceed the thermal limits K c l of the transmission lines in both directions. The transmission network is modeled in terms of lossless DC (i.e. linear) approximation of Kirchhoff’s laws (see =-=[4]-=-). Specifically, flows on lines can be calculated using power transfer distribution factor (PTDF) D c l,i which specifies the proportion of flow on a line l ∈ L resulting from an injection of one-unit... |

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Citation Context ...rizontal market power is a two-settlement approach where forward contracts and real-time balancing transactions 1are settled at different prices. Both theoretical analysis and empirical evidences in =-=[1, 2, 8, 14, 26, 30]-=- have suggested that forward contracting decreases the incentives of sellers to manipulate spot market prices since, under two settlements, the volume of trading that can be affected by spot prices is... |

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Citation Context ...rizontal market power is a two-settlement approach where forward contracts and real-time balancing transactions 1are settled at different prices. Both theoretical analysis and empirical evidences in =-=[1, 2, 8, 14, 26, 30]-=- have suggested that forward contracting decreases the incentives of sellers to manipulate spot market prices since, under two settlements, the volume of trading that can be affected by spot prices is... |

87 | The Competitive Effects of Transmission Capacity in a Deregulated Electricity Industry
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Citation Context ...− ∑ i∈N ≥ 0, D c l,i rc i ≥ 0, l ∈ L l ∈ L ) = 0, i ∈ N The equilibrium problem among the above MPECs represents a “generalized Nash game” (see [16]), and it may have zero or multiple equilibria (see =-=[3]-=-). On the other hand, even if some pure-strategy equilibrium is found, it can be degenerate, that is, firms will find it optimal to barely congest some transmission lines so as to avoid congestion cha... |

82 |
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Citation Context ...rizontal market power is a two-settlement approach where forward contracts and real-time balancing transactions 1are settled at different prices. Both theoretical analysis and empirical evidences in =-=[1, 2, 8, 14, 26, 30]-=- have suggested that forward contracting decreases the incentives of sellers to manipulate spot market prices since, under two settlements, the volume of trading that can be affected by spot prices is... |

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Citation Context ...other hand, even if some pure-strategy equilibrium is found, it can be degenerate, that is, firms will find it optimal to barely congest some transmission lines so as to avoid congestion charges (see =-=[28]-=-). Moreover, this formulation would lead to a two-settlement model with three levels of decision, which makes an equilibrium solution for the two-settlement market computationally intractable. The sec... |

75 |
Convergence properties of regularization schemes for mathematical programs with complementarity constraints
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Citation Context ... by the other MPECs’ decision variables (see [29] for more discussions on related topics). One solution approach is to derive the optimality conditions for the regularization scheme of the MPECs (see =-=[9, 32, 33]-=-), then either solve the nonlinear complementarity conditions of the EPEC as a whole [19, 34] or iteratively solve the nonlinear complementarity conditions of individual MPECs [19, 34]. The second app... |

66 |
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Citation Context ...In the spot market, each firm g ∈ G determines the outputs from its units at Ng. A variety of modeling approaches have been proposed to simulate generation firms’ decision making (see, for 7example, =-=[18, 27, 35, 36]-=-). One modeling consideration regarding the suppliers’ strategic behaviors in these models is whether or not they game the congestion prices set by the ISO. Following [18] and [27] we classify spot ma... |

65 | A smoothing method for mathematical programs with equilibrium constraints
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Citation Context ...s the penalty interior point algorithm (PIPA) and the piecewise sequential quadratic programming (PSQP, see also [21]) algorithm. More recent advances in MPEC algorithms can be found, for example, in =-=[5, 7, 11, 12, 13, 20, 31]-=-. Fukushima, Luo and Pang [11] present a sequential quadratic programming approach, through a reformulation of the complementarity condition as a system of semismooth equations by means of Fischer-Bur... |

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Citation Context ...In the spot market, each firm g ∈ G determines the outputs from its units at Ng. A variety of modeling approaches have been proposed to simulate generation firms’ decision making (see, for 7example, =-=[18, 27, 35, 36]-=-). One modeling consideration regarding the suppliers’ strategic behaviors in these models is whether or not they game the congestion prices set by the ISO. Following [18] and [27] we classify spot ma... |

44 |
A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints
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Citation Context ...s the penalty interior point algorithm (PIPA) and the piecewise sequential quadratic programming (PSQP, see also [21]) algorithm. More recent advances in MPEC algorithms can be found, for example, in =-=[5, 7, 11, 12, 13, 20, 31]-=-. Fukushima, Luo and Pang [11] present a sequential quadratic programming approach, through a reformulation of the complementarity condition as a system of semismooth equations by means of Fischer-Bur... |

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Citation Context ... l∈L (λ c l− Dc l,i − λc l+ Dc l,i D c l,i rc i + K c l − ∑ i∈N ≥ 0, D c l,i rc i ≥ 0, l ∈ L l ∈ L ) = 0, i ∈ N The equilibrium problem among the above MPECs represents a “generalized Nash game” (see =-=[16]-=-), and it may have zero or multiple equilibria (see [3]). On the other hand, even if some pure-strategy equilibrium is found, it can be degenerate, that is, firms will find it optimal to barely conges... |

31 |
Mathematical programs with complementarity constraints
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Citation Context ... by the other MPECs’ decision variables (see [29] for more discussions on related topics). One solution approach is to derive the optimality conditions for the regularization scheme of the MPECs (see =-=[9, 32, 33]-=-), then either solve the nonlinear complementarity conditions of the EPEC as a whole [19, 34] or iteratively solve the nonlinear complementarity conditions of individual MPECs [19, 34]. The second app... |

30 | Convergence of a penalty method for mathematical programming with complementarity constraints
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Citation Context ...s the penalty interior point algorithm (PIPA) and the piecewise sequential quadratic programming (PSQP, see also [21]) algorithm. More recent advances in MPEC algorithms can be found, for example, in =-=[5, 7, 11, 12, 13, 20, 31]-=-. Fukushima, Luo and Pang [11] present a sequential quadratic programming approach, through a reformulation of the complementarity condition as a system of semismooth equations by means of Fischer-Bur... |

30 | Exact penalization and stationarity conditions of mathematical programs with equilibrium constraints - Luo, Pang, et al. - 1996 |

27 | Some properties of regularization and penalization schemes for MPECs
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24 | QPECgen, a MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constaints
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Citation Context ...second-order optimality conditions; it also describes some iterative algorithms, such as the penalty interior point algorithm (PIPA) and the piecewise sequential quadratic programming (PSQP, see also =-=[21]-=-) algorithm. More recent advances in MPEC algorithms can be found, for example, in [5, 7, 11, 12, 13, 20, 31]. Fukushima, Luo and Pang [11] present a sequential quadratic programming approach, through... |

22 |
A smoothing method for a mathematical program with P -matrix linear complementarity constraints
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Long-term Contracts and Imperfectly Competitive Spot Markets: A Study of the U.K. Electricity Industry
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Citation Context ...rizontal market power is a two-settlement approach where forward contracts and real-time balancing transactions 1 are settled at different prices. Both theoretical analysis and empirical evidences in =-=[1, 2, 8, 14, 26, 30]-=- have suggested that forward contracting decreases the incentives of sellers to manipulate spot market prices since, under two settlements, the volume of trading that can be affected by spot prices is... |

16 | Multi-Settlement Systems for Electricity Markets: Zonal Aggregation under Network Uncertainty and
- Oren
(Show Context)
Citation Context ...ype of game among them. Allaz and Vila [2] show that, as the number of forward trading periods increases, producers lose their ability to raise energy prices above their marginal cost. Kamat and Oren =-=[22]-=- analyze twosettlement markets over two- and three-node networks, and extend the results in [1, 2] to a system with uncertain transmission capacities in the spot market. Recent work in [37] and [38] f... |

14 |
Mathematical Programs with Complementarity Constraints and Game Theory Models in Electricity Markets
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Citation Context ...olution approach is to derive the optimality conditions for the regularization scheme of the MPECs (see [9, 32, 33]), then either solve the nonlinear complementarity conditions of the EPEC as a whole =-=[19, 34]-=- or iteratively solve the nonlinear complementarity conditions of individual MPECs [19, 34]. The second approach which we will follow in this research is to iteratively solve MPECs using MPEC-based al... |

11 |
Nash-Cournot Equilibria
- Hobbs, Pang
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Citation Context ...two-settlement market computationally intractable. The second approach assumes that the firms do not fully anticipate the impact of their production decisions on congestion charges (see, for example, =-=[25]-=-) which can be interpreted as a “bounded rationality” assumption. In this approach, the ISO is a Nash player that moves simultaneously with the generation firms. The firms determine their supply quant... |

7 |
Equilibrium Problems with Equilibrium Constraints: Stationarities, Algorithms, and Applications
- Su
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Citation Context ...olution approach is to derive the optimality conditions for the regularization scheme of the MPECs (see [9, 32, 33]), then either solve the nonlinear complementarity conditions of the EPEC as a whole =-=[19, 34]-=- or iteratively solve the nonlinear complementarity conditions of individual MPECs [19, 34]. The second approach which we will follow in this research is to iteratively solve MPECs using MPEC-based al... |

6 |
Network-constrained models of liberalized electricity markets: the devil is in the details. Energy Ecomnomics
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Citation Context ...In the spot market, each firm g ∈ G determines the outputs from its units at Ng. A variety of modeling approaches have been proposed to simulate generation firms’ decision making (see, for 7example, =-=[18, 27, 35, 36]-=-). One modeling consideration regarding the suppliers’ strategic behaviors in these models is whether or not they game the congestion prices set by the ISO. Following [18] and [27] we classify spot ma... |

4 |
Computing Cournot Equlibria in Two-settlement Electricity Markets with Transmission Contraints
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(Show Context)
Citation Context ...t and Oren [22] analyze twosettlement markets over two- and three-node networks, and extend the results in [1, 2] to a system with uncertain transmission capacities in the spot market. Recent work in =-=[37]-=- and [38] further extends the above results to more realistic multi-node and multi-zone networks. Yao, Oren and Adler [37] consider flow constraints, system contingencies and demand uncertainties in t... |

3 |
A smoothing method for a mathematilcal program with P-matrix linear complementarity constraints
- Chen, Fukushima
- 2004
(Show Context)
Citation Context |

3 |
2004 Electrical Engineering: Principles and Applications (3rd edition
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Citation Context ...3 and 31-52 are prone to congestion in this example. The method for calculating the state-dependent PTDF matrices from the network data can be found in standard electrical engineering textbooks (e.g. =-=[15]-=-) and will be omitted here due to space limitation. We assume six independent contingency states in the spot market. The first three states correspond to demand uncertainty, while all generation units... |

3 | Multi-Settlement Systems for Electricity Markets: Zonal Aggregation under Network Uncertainty and Market Power
- Kamat, Oren
- 2004
(Show Context)
Citation Context ...ype of game among them. Allaz and Vila [2] show that, as the number of forward trading periods increases, producers lose their ability to raise energy prices above their marginal cost. Kamat and Oren =-=[22]-=- analyze twosettlement markets over two- and three-node networks, and extend the results in [1, 2] to a system with uncertain transmission capacities in the spot market. Recent work in [37] and [38] f... |

3 | Cournot Equilibrium in Price-capped TwoSettlement Electricity Markets
- Yao, Willems, et al.
- 2005
(Show Context)
Citation Context ...n [22] analyze twosettlement markets over two- and three-node networks, and extend the results in [1, 2] to a system with uncertain transmission capacities in the spot market. Recent work in [37] and =-=[38]-=- further extends the above results to more realistic multi-node and multi-zone networks. Yao, Oren and Adler [37] consider flow constraints, system contingencies and demand uncertainties in the spot m... |

3 |
A Hybrid Bertrand-Cournot Model of Electricity Markets with Multiple Subnetworks. Working paper
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- 2006
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Citation Context ...an uncongested regime, have been eloquently demonstrated in [3]. Such discontinuities become intractable in a meshed system with multiple nodes. We partially address the above issue in a sequal paper =-=[39]-=- through a hybrid approach that requires apriority identification of “systematically congested” links (e.g. path 15 in California or 10 the link between France and the UK across the English Channel) w... |