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.     

Leader-Follower Equilibria for Electric Power and NO x Allowances Markets

This paper investigates the ability of the largest producer in an electricity market to manipulate both the electricity and emission allowances markets to its advantage. A Stackelberg game to analyze this situation is constructed in which the largest firm plays the role of the leader, while the medium-sized firms are treated as Cournot followers with price-taking fringes that behave competitively in both markets. Since there is no explicit representation of the best-reply function for each follower, this Stackelberg game is formulated as a large-scale mathematical program with equilibrium constraints. The best-reply functions are implicitly represented by a set of nonlinear complementarity conditions. Analysis of the computed solution for the Pennsylvania–New Jersey–Maryland electricity market shows that the leader can gain substantial profits by withholding allowances and driving up NO _x allowance costs for rival producers. The allowances price is higher than the corresponding price in the Nash–Cournot case, although the electricity prices are essentially the same.We are grateful to two anonymous referees for their insightful comments that helped us improve the paper. This work is partially supported by NSF grants CS 0080577 and 0224817, by USEPA STAR grant R82873101-0, and by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, Office of Science, U.S. Department of Energy, under Contract W-31-109-Eng-38. Any opinions or errors are the responsibility of the authors and not the sponsoring agencies.     

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.     

Integrated risk management for an electricity producer

Electricity suppliers face two sources of risk: uncertainty of spot prices and uncertainty of production costs. Uncertainty in selling the output is usually “solved” by signing forwards and the two sources of risk are dealt with separately. However, managing the integrated risk optimally is the direction we will suggest. We intend to analyse the problem a power producer is confronted to upon acting in a market where spot and forward agreements are available. Since forwards allow to sell production in advance at a given price but do not hedge against cost volatility, the total risk can be reduced by selling also in the spot market. The analysis is further detailed to encompass the spread option inherent in electricity production. We find a benchmark value for forwards to sign and the optimal spot/forward combinations. The analysis is carried out by accounting for market figures for input and output variables in the German market EEX.     

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.     

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.     

Pricing electricity derivatives within a Markov regime-switching model: a risk premium approach

In this paper we derive analytic formulas for electricity derivatives under assumption that electricity spot prices follow a 3-regime Markov regime-switching model with independent spikes and drops and periodic transition matrix. Since the classical derivatives pricing methodology cannot be used in the case of non-storable commodities, we employ the concept of the risk premium. The obtained theoretical results are then used for the European Energy Exchange data analysis. We calculate the risk premium in the case of the calibrated 3-regime MRS model. We find a time varying structure of the risk premium and an evidence for a negative risk premium (or positive forward premium), especially at short times before delivery. Finally, we use the obtained risk premium to calculate prices of European options written on spot, as well as, forward prices.     

Analysis on the forward market equilibrium model

We establish the existence results for the Allaz-Vila [B. Allaz, J.-L. Vila, Cournot competition, forward markets and efficiency, J. Econ. Theory 59 (1993) 1-16] forward market equilibrium model when the M producers have different linear cost functions. We also consider an example with three asymmetric producers. The computational results supplement the conclusion in that the forward trading would increase market efficiency.     

Open versus closed loop capacity equilibria in electricity markets under perfect and oligopolistic competition

We consider two game-theoretic models of the generation capacity expansion problem in liberalized electricity markets. The first is an open loop equilibrium model, where generation companies simultaneously choose capacities and quantities to maximize their individual profit. The second is a closed loop model, in which companies first choose capacities maximizing their profit anticipating the market equilibrium outcomes in the second stage. The latter problem is an equilibrium problem with equilibrium constraints. In both models, the intensity of competition among producers in the energy market is frequently represented using conjectural variations. Considering one load period, we show that for any choice of conjectural variations ranging from perfect competition to Cournot, the closed loop equilibrium coincides with the Cournot open loop equilibrium, thereby obtaining a ‘Kreps and Scheinkman’-like result and extending it to arbitrary strategic behavior. When expanding the model framework to multiple load periods, the closed loop equilibria for different conjectural variations can diverge from each other and from open loop equilibria. We also present and analyze alternative conjectured price response models with switching conjectures. Surprisingly, the rank ordering of the closed loop equilibria in terms of consumer surplus and market efficiency (as measured by total social welfare) is ambiguous. Thus, regulatory approaches that force marginal cost-based bidding in spot markets may diminish market efficiency and consumer welfare by dampening incentives for investment. We also show that the closed loop capacity yielded by a conjectured price response second stage competition can be less or equal to the closed loop Cournot capacity, and that the former capacity cannot exceed the latter when there are symmetric agents and two load periods.     

Analysis of relationship between forward and spot markets in oligopolies under demand and cost uncertainties

In this paper, we consider interaction between spot and forward trading under demand and cost uncertainties, deriving the equilibrium of the multi-player dynamic games. The stochastic programming and worst-case analysis models based on discrete scenarios are developed to analyze the impact of demand uncertainty and risk aversion on oligopoly (forward and spot) markets’ structure in terms of the forwards and spot pricing, traded quantities and production. A real case of the Iberian electricity market is studied to illustrate performance of the models. The numerical experiments show that cost uncertainty impacts on the strategic decisions more than demand uncertainty.     

Bidding in sequential electricity markets: The Nordic case

For electricity market participants trading in sequential markets with differences in price levels and risk exposure, it is relevant to analyze the potential of coordinated bidding. We consider a Nordic power producer who engages in the day-ahead spot market and the hour-ahead balancing market. In both markets, clearing prices and dispatched volumes are unknown at the time of bidding. However, in the balancing market, the market participant faces an additional risk of not being dispatched. Taking into account the sequential clearing of these markets and the gradual realization of market prices, we formulate the bidding problem as a multi-stage stochastic program. We investigate whether higher risk exposure may cause hesitation to bid into the balancing market. Furthermore, we quantify the gain from coordinated bidding, and by deriving bounds on this gain, assess the performance of alternative bidding strategies used in practice.     

LP models for pricing diffuse nitrate discharge permits

Nitrate discharges from diffuse agricultural sources significantly contribute to groundwater and surface water pollution. Tradable permit programs have been proposed as a means of controlling nitrate emissions efficiently, but trading is complicated by the dispersed and delayed effects of the diffuse pollution. Hence, markets in nitrate discharge permits should be carefully designed to account for the underlying spatial and temporal interactions. Nitrate permit markets can be designed similar to the modern electricity markets which use LPs to find the equilibrium prices because the two trading problems have close analogy. In this paper, we propose alternative LP models to find efficient permit prices for year-ahead markets. The model structure varies depending on the catchment hydro-geology and long-term goals of the community. We show how the market price structures are driven by the constraint structure under different environmental conditions. We discuss the physical and economic conditions required to assure consistent prices, the modeling of essential and optional constraints in an LP, and the problem of balancing resource allocation over time among delayed-response discharge units. We then extend the LP model to balance resource allocation over time and to improve the market performance.     

Australian electricity market and price volatility

Australian Electricity Market has experienced high price volatility since the deregulation in early 1990s. In this exploratory and preliminary analysis of 2010 data from South Australian electricity market we identify and exhibit a number of phenomena which, arguably, contribute to (A) high cost of electricity supply to consumers and (B) volatility in spot prices. These phenomena include: (i) Distinct bidding patterns of some generators occurring in trading intervals corresponding to periods of low, medium and high spot prices, (ii) Low correlation between electricity demand and spot prices on days when spot price spikes are observed, (iii) Failure of the lottery model and associated Markowitz-type optimisation approaches to adequately explain the shifting structure of generators’ bids and (iv) Unexpectedly high contribution to the consumers costs and risks from the relatively small number of trading intervals where spot price spikes were observed.     

Calibration of a multifactor model for the forward markets of several commodities

We propose a model for the evolution of forward prices of several commodities, which is an extension of the factor forward model in [1, 2], to a market where multiple commodities are traded. We calibrate this model in a market where forward contracts on multiple commodities are present, using historical forward prices. First, we calibrate separately the four coefficients of each individual commodity, using an approach based on quadratic variation/covariation of forward prices. Then, with the same technique, we pass to the estimation of the mutual correlation among the Brownian motions driving the different commodities. This calibration is compared to a calibration method used by practitioners, which uses rolling time series and requires a modification of the model, but turns out to be more accurate in practice, especially with a low frequency of observed transaction. We present efficient methods to perform the calibration with both methods, as well as the calibration of the intercommodity correlation matrix. Then we calibrate our model to WTI, ICE Brent and ICE Gasoil forward prices. Finally we present a method for estimating spot volatility from forward parameters, with an application to the WTI spot volatility.     

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.     

A Non‐Gaussian Ornstein–Uhlenbeck Process for Electricity Spot Price Modeling and Derivatives Pricing

A mean‐reverting model is proposed for the spot price dynamics of electricity which includes seasonality of the prices and spikes. The dynamics is a sum of non‐Gaussian Ornstein–Uhlenbeck processes with jump processes giving the normal variations and spike behaviour of the prices. The amplitude and frequency of jumps may be seasonally dependent. The proposed dynamics ensures that spot prices are positive, and that the dynamics is simple enough to allow for analytical pricing of electricity forward and futures contracts. Electricity forward and futures contracts have the distinctive feature of delivery over a period rather than at a fixed point in time, which leads to quite complicated expressions when using the more traditional multiplicative models for spot price dynamics. In a simulation example it is demonstrated that the model seems to be sufficiently flexible to capture the observed dynamics of electricity spot prices. The pricing of European call and put options written on electricity forward contracts is also discussed.     

Models for capacity and electricity market design

The paper employs Operations Research methods for analysis of electricity and capacity markets. We provide two algorithms that determine the optimal capacity structure with account of fixed and variable costs. The first one relates to the case where there are several capacity types, and for each type the capacity constraint is not binding. The second algorithm is applicable when electricity is produced by standard small generators with the same capacity and different costs. Then we study two typical architectures of the market and examine their Nash equilibria. We consider a uniform price supply function auction in the electricity market. For pay-as-bid and uniform price versions of the capacity market design, we compare the equilibrium outcomes with the optimal capacity structure. The paper shows that the market equilibrium corresponds to the optimal capacity structure under conditions of pure competition, full rationality, and completely informed agents in the market. However, under more realistic assumptions, selection of the optimal structure is unlikely. Finally we provide the auction design that realizes such selection of capacities and does not require any additional information of each producer besides his own production costs. We establish sufficient conditions for perfect competition in the market.     

Strategic bidding in continuous electricity auctions: an application to the Spanish electricity market

In this paper we introduce an asymmetric model of continuous electricity auctions with limited production capacity and bounded supply functions. The strategic bidding is studied with this model by means of an electricity market game. We prove that for every electricity market game with continuous cost functions a mixed-strategy Nash equilibrium always exists. In particular, we focus on the behavior of producers in the Spanish electricity market. We consider a very simple form for the Spanish electricity market: an oligopoly consisting just of independent hydro-electric power production units in a single wet period. We show that a pure-strategy Nash equilibrium for the Spanish electricity market game always exists.     

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.