Semi-Lagrangean approach for price discovery in markets with non-convexities

From standard economic theory, the market clearing price for a commodity is set where the demand and supply curves intersect. Convexity is a property that economic models require for a competitive equilibrium, which is efficient and well-behaved and provides equilibrium prices. However, some markets present non-convexities due to their cost structure or due to some operational constraints that need to be addressed. This is the case for electricity markets where the electricity producers incur costs for shutting down a generating unit and then bringing it back on. Non-convex cost structures can be a challenge for the price discovery process, since the supply and demand curves may not intersect, or if they intersect, the price found may not be high enough to cover the total cost of production. We apply a Semi-Lagrangean approach to find a price that can be applied in the electricity pool markets where a central system operator decides who produces and how much they should produce. By applying the model to an example from the literature, we found prices that are high enough to cover the producer’s total costs, and follows the optimal solution for achieving mining cost in production. The prices are an alternative solution to the price discovery problem in non-convexities economies; in addition, they provide nonnegative profits to all the generators without the use of side-payments or up-lifts, and closes the integrality gap.     

Evaluating Individual Market Power in Electricity Markets via Agent-Based Simulation

We use agent-based simulation in a coordination game to analyse the possibility of market power abuse in a competitive electricity market. The context of this was a real application to the England and Wales electricity market as part of a Competition Commission Inquiry into whether two particular generators could profitably influence wholesale prices. The research contributions of this paper are both in the areas of market power and market design policy issues for electricity markets, and in the methodological use of large industry-wide evolutionary simulation models.     

Asymmetric Supply Function Equilibria with Forward Contracts

We consider markets in which firms offer supply functions, rather than a quantity or price alone: the most important examples are wholesale electricity markets. The equilibria in such markets can be hard to characterize. In many cases, whole families of supply function equilibria occur so there are difficulties in determining which equilibrium will be chosen. In this paper, we consider supply function equilibria, when firms hold forward contracts, which is common in electricity markets. Under the assumption that contract positions have been fixed in advance, we characterize the families of supply function equilibria in a duopoly. The existence of forward contracts implies a tightening of the conditions for an equilibrium, and a greater likelihood that no equilibrium solution exists. In the case of three firms, there can be at most one supply function equilibrium, provided that the lowest demand be small enough.     

Neuro-fuzzy approach for short-term electricity price forecasting developed MATLAB-based software

Bid and offer competition is a main transaction approach in deregulated electricity markets. Locational marginal prices (LMP) resulting from bidding competition determine electricity prices at a node or in an area. The LMP exhibits important information for market participants to develop their bidding strategies. Moreover, LMP is also a vital indicator for a Security Coordinator to perform market redispatch for congestion management. This paper presents a method using modular feed forward neural networks (FFNN) and fuzzy inference system (FIS) for forecasting LMPs. FFNN is used to forecast the electricity prices in a short time horizon and FIS to forecast the prices of special days. FFNN system includes an autocorrelation method for selecting parameters and methods for data preprocessing and preparing historical data to train the artificial neural network (ANN). In this paper, the historical LMPs of Pennsylvania, New Jersey, and Maryland (PJM) market are used to test the proposed method. It is found that the proposed neuro-fuzzy method is capable of forecasting LMP values efficiently. In addition, MATLAB-based software is designed to test and use the proposed model in different markets and environments. This is an efficient tool to study and model power markets for price forecasting. It is included with a database management system, data classifier, input variable selection, FFNN and FIS configuration and report generator in custom formats.     

The forward dynamics in energy markets – infinite-dimensional modelling and simulation

In this paper an infinite-dimensional approach to model energy forward markets is introduced. Similar to the Heath–Jarrow–Morton framework in interest-rate modelling, a first-order hyperbolic stochastic partial differential equation models the dynamics of the forward price curves. These equations are analysed, and in particular regularity and no-arbitrage conditions in the general situation of stochastic partial differential equations driven by an infinite-dimensional martingale process are studied. Both arithmetic and geometric forward price dynamics are studied, as well as accounting for the delivery period of electricity forward contracts. A stable and convergent numerical approximation in the form of a finite element method for hyperbolic stochastic partial differential equations is introduced and applied to some examples with relevance to energy markets.     

Behavioural simulations in spot electricity markets

We study the consistency of behavioural simulation methods used to model the operations of wholesale electricity markets. We include different supply and demand representations and propose the Experience-Weighted Attractions method (Camerer and Ho, 1999) to encompass several behavioural paradigms. We compare the results across assumptions and to standard economic theory predictions. The match is good under flat and upward-slopping supply bidding, and also for plausible demand elasticity assumptions. Learning is influenced by the number of bids per plant and the initial conditions. The simulations perform best under reinforcement learning, less well under best-response and especially poorly under fictitious play. The overall conclusion is that simulation assumptions are far from innocuous. We link their performance to underlying features, and identify those that are better suited to model liberalised electricity markets.     

Joint Modelling of Gas and Electricity Spot Prices

Abstract

The recent liberalization of electricity and gas markets has resulted in the growth of energy exchanges and modelling problems. In this article, we jointly model gas and electricity spot prices using a mean-reverting model that fits the correlation structures for the two commodities. The dynamics are based on Ornstein processes with parameterized diffusion coefficients. Moreover, using the empirical distributions of the spot prices, we derive a class of such parameterized diffusions that captures the most salient statistical properties: stationarity, spikes and heavy-tailed distributions. The associated calibration procedure is based on standard and efficient statistical tools. We calibrate the model on French market for electricity and on UK market for gas, and then we simulate some trajectories that reproduce well the observed prices behaviour. Finally, we illustrate the importance of the correlation structure and of the presence of spikes by measuring the risk on a power plant portfolio.     

Liquidity risks on power exchanges: a generalized Nash equilibrium model

The extreme volatility of electricity prices makes their financial derivatives important instruments for asset managers. Even if the volume of derivative contracts traded on Power Exchanges has been growing since the inception of the restructuring of the sector, electricity remains considerably less liquid than other commodity markets. This paper assesses the effect of limited liquidity in power exchanges using an equilibrium model where agents cannot hedge up to their desired level. Mathematically, the problem is formulated as a two stage stochastic Generalized Nash Equilibrium with possibly multiple equilibria. Computing a large panel of solutions, we show how the risk premium and players profits are affected by illiquidity. We also show that the illiquidity in the FTR market affects the trades in the electricity futures market.     

Some inequalities related to the analysis of electricity auctions

Most balancing markets of electric power are organized as uniform-price auctions. In 2001, the balancing market of England and Wales switched to a pay-as-bid auction with the intention of reducing wholesale electricity prices. Numerical simulations of an electricity auction model have indicated that this should lead to decreased average prices. In this work we prove two inequalities which give an analytic proof of this claim in the same model.     

Comparing efficiency across markets: An extension and critique of the methodology

The use of non-parametric frontier methods for the evaluation of product market efficiency in heterogeneous markets seems to have gained some popularity recently. However, the statistical properties of these frontier estimators have been largely ignored. The main point is that non-parametric frontier estimators are biased and that the degree of bias depends on specific sample properties, most importantly sample size and number of dimensions of the model. To investigate the effect of this bias on comparing market efficiency, this contribution estimates the efficiency for several datasets for two main product categories. Following (Zhang, Y., Bartels, R., 1998. The effect of sample size on the mean efficiency in DEA with an application to electricity distribution in Australia, Sweden and New Zealand. Journal of Productivity Analysis, 9(3), 187-204.), these results comprise re-estimates for the larger samples limiting their size to that of the smaller samples when the model dimensions for different samples are identical. Furthermore, sample sizes are adjusted to mitigate the eventual differences in dimensions in specification. This allows comparing market efficiency for different markets on a more equal footing, since it reduces the bias effect to a minimum making the comparison of market efficiency possible. However, the article also points out the critical limitations of this [Zhang, Y., Bartels, R., (1998). The effect of sample size on the mean efficiency in DEA with an application to electricity distribution in Australia, Sweden and New Zealand. Journal of Productivity Analysis 9 (3), 187–204] approach in certain respects. Apart from reporting these negative results, we also offer some suggestions for future work.     

Spatial dependencies of wind power and interrelations with spot price dynamics

Wind power has seen strong growth over the last decade and increasingly affects electricity spot prices. In particular, prices are more volatile due to the stochastic nature of wind, such that more generation of wind energy yields lower prices. Therefore, it is important to assess the value of wind power at different locations not only for an investor but for the electricity system as a whole. In this paper, we develop a stochastic simulation model that captures the full spatial dependence structure of wind power by using copulas, incorporated into a supply and demand based model for the electricity spot price. This model is calibrated with German data. We find that the specific location of a turbine – i.e., its spatial dependence with respect to the aggregated wind power in the system – is of high relevance for its value. Many of the locations analyzed show an upper tail dependence that adversely impacts the market value. Therefore, a model that assumes a linear dependence structure would systematically overestimate the market value of wind power in many cases. This effect becomes more important for increasing levels of wind power penetration and may render the large-scale integration into markets more difficult.     

Multi-layer model of correlated energy prices

In this article we develop an extension of the affine jump-diffusion modeling framework and use it to build an intuitive and tractable model of an energy price complex. The development is motivated by the need to model prices of electricity while capturing their dependence on the price of other energy commodities. Such a model is essential for valuing a range of typical derivatives traded in the electricity markets: cross-commodity spread options, cross-location spread options, fuel-switching powerplants, etc. We give an approximate pricing method for these derivatives together with precise error bound estimates.     

Numerical calculation of an asymmetric supply function equilibrium with capacity constraints

Producers submit offer curves to a procurement auction, e.g. an electricity auction, before uncertain demand has been realised. In the supply function equilibrium (SFE), every firm commits to the offer curve that maximises its expected profit, given the offer curves of competitors. The equilibrium is given by a system of differential equations. In practice, it has been very difficult to find valid SFE, i.e. non-decreasing solutions, from this system, especially for asymmetric producers. This paper shows that valid SFE can be calculated by means of a shooting algorithm that combines numerical integration with an optimisation procedure that searches for an end-condition. Multiple/parallel shooting is used for ill-conditioned cases.     

Tacit collusion games in pool-based electricity markets under transmission constraints

Harrington et al. (Math Program Ser B 104:407–435, 2005) introduced a general framework for modeling tacit collusion in which producing firms collectively maximize the Nash bargaining objective function, subject to incentive compatibility constraints. This work extends that collusion model to the setting of a competitive pool-based electricity market operated by an independent system operator. The extension has two features. First, the locationally distinct markets in which firms compete are connected by transmission lines. Capacity limits of the transmission lines, together with the laws of physics that guide the flow of electricity, may alter firms’ strategic behavior. Second, in addition to electricity power producers, other market participants, including system operators and power marketers, play important roles in a competitive electricity market. The new players are included in the model in order to better represent real-world markets, and this inclusion will impact power producers’ strategic behavior as well. The resulting model is a mathematical program with equilibrium constraints (MPEC). Properties of the specific MPEC are discussed and numerical examples illustrating the impacts of transmission congestion in a collusive game are presented.     

A computational consumer-driven market model: statistical properties and the underlying industry dynamics

We present an agent-based market model in which social emulation by consumers and the adaptation of producers to demand play a significant role. Our theoretical approach considers boundedly-rational agents, heterogeneity of agents and product characteristics, and the co-evolution of consumers’ desires and firms’ adaptation efforts. The model reproduces, and allows us to interpret, statistical regularities which have been observed in the evolution of industrial sectors, and that seem to be also significant in the case of discretionary consumption activities. Thus, we suggest new determinants and explanations (from the consumer-side) for these stylized facts, and we obtain new theoretical patterns which may be of help to better understand the dynamics of discretionary goods markets. This model and results may contribute to guide future research on the field of consumer market.     

Electricity price modeling and asset valuation: a multi-fuel structural approach

We introduce a new and highly tractable structural model for spot and derivative prices in electricity markets. Using a stochastic model of the bid stack, we translate the demand for power and the prices of generating fuels into electricity spot prices. The stack structure allows for a range of generator efficiencies per fuel type and for the possibility of future changes in the merit order of the fuels. The derived spot price process captures important stylized facts of historical electricity prices, including both spikes and the complex dependence upon its underlying supply and demand drivers. Furthermore, under mild and commonly used assumptions on the distributions of the input factors, we obtain closed-form formulae for electricity forward contracts and for spark and dark spread options. As merit order dynamics and fuel forward prices are embedded into the model, we capture a much richer and more realistic dependence structure than can be achieved by classical reduced-form models. We illustrate these advantages by comparing with Margrabe’s formula and a simple cointegration model, and highlight important implications for the valuation of power plants.     

A two stage stochastic equilibrium model for electricity markets with two way contracts

This paper investigates generators’ strategic behaviors in contract signing in the forward market and power transaction in the electricity spot market. A stochastic equilibrium program with equilibrium constraints (SEPEC) model is proposed to characterize the interaction of generators’ competition in the two markets. The model is an extension of a similar model proposed by Gans et al. (Aust J Manage 23:83–96, 1998) for a duopoly market to an oligopoly market. The main results of the paper concern the structure of a Nash–Cournot equilibrium in the forward-spot market: first, we develop a result on the existence and uniqueness of the equilibrium in the spot market for every demand scenario. Then, we show the monotonicity and convexity of each generator’s dispatch quantity in the spot equilibrium by taking it as a function of the forward contracts. Finally, we establish some sufficient conditions for the existence of a local and global Nash equilibrium in the forward-spot markets. Numerical experiments are carried out to illustrate how the proposed SEPEC model can be used to analyze interactions of the markets.     

电力市场实用稳定性的判定条件

首次把实用稳定性的理论用于电力市场稳定性的研究中.结合Alvarado提出的电力市场动态模型,利用微分代数方程与特征值技术,从理论上研究电力市场的实用稳定性,并且给出了判断电力市场实用稳定、一致实用稳定和实用渐近稳定性的充分条件.利用这些实用稳定性条件,对于Alvarado给出的数值算例,可方便地利用初始数据判断电力市场模型的实用稳定性,并通过实例提供了假设模型中某个参数在电力市场变化中起主要作用,如何控制电力市场模型实用稳定性的方法.最后利用Maple软件包给出了参数变化引起模型实用稳定和不稳定的图形演示.     

Optimal bidding of a virtual power plant on the Spanish day-ahead and intraday market for electricity

We develop a multi-stage stochastic programming approach to optimize the bidding strategy of a virtual power plant (VPP) operating on the Spanish spot market for electricity. The VPP markets electricity produced in the wind parks it manages on the day-ahead market and on six staggered auction-based intraday markets. Uncertainty enters the problem via stochastic electricity prices as well as uncertain wind energy production. We set up the problem of bidding for one day of operation as a Markov decision process (MDP) that is solved using a variant of the stochastic dual dynamic programming algorithm. We conduct an extensive out-of-sample comparison demonstrating that the optimal policy obtained by the stochastic program clearly outperforms deterministic planning, a pure day-ahead strategy, a benchmark that only uses the day-ahead market and the first intraday market, as well as a proprietary stochastic programming approach developed in the industry. Furthermore, we study the effect of risk aversion as modeled by the nested Conditional Value-at-Risk as well as the impact of changes in various problem parameters.     

Equilibrium prices supported by dual price functions in markets with non-convexities

The issue of finding market clearing prices in markets with non-convexities has had a renewed interest due to the deregulation of the electricity sector. In the day-ahead electricity market, equilibrium prices are calculated based on bids from generators and consumers. In most of the existing markets, several generation technologies are present, some of which have considerable non-convexities, such as capacity limitations and large start-up costs. In this paper we present equilibrium prices composed of a commodity price and an uplift charge. The prices are based on the generation of a separating valid inequality that supports the optimal resource allocation. In the case when the sub-problem generated as the integer variables are held fixed to their optimal values possess the integrality property, the generated prices are also supported by non-linear price functions that are the basis for integer programming duality.