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.     

On the existence of supply function equilibria

Supply function equilibria are used in the analysis of divisible good auctions with a large number of identical objects to be sold or bought. An important example occurs in wholesale electricity markets. Despite the substantial literature on supply function equilibria the existence of a pure strategy Nash equilibria for a uniform price auction in asymmetric cases has not been established in a general setting. In this paper we prove the existence of a supply function equilibrium for a duopoly with asymmetric firms having convex non-decreasing marginal costs, with decreasing concave demand subject to an additive demand shock, provided that the second derivative of the demand function is small enough and not increasing. The proof is constructive and also gives insight into the structure of the equilibrium solutions.     

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.     

Incentive based energy market design

Current energy market designs and pricing schemes fail to give investors the appropriate market signals. In particular, energy prices are not high enough to attract investors to build new or maintain existing power capacity. In this paper we propose a method to compute second-best Pareto optimal equilibrium prices for any market exhibiting non-convexities and, based on this result, an energy market design able to restore the correct energy price signals for supply investors.     

Monotonicity and Market Equilibrium

Economic theory relates prices to quantities via ” market curves.” Typically, such curves are monotone, hence they admit functional representations. The latter invoke linear pricing of quantities so as to obtain market values. Specifically, if higher prices call forward greater supply, a convex function, bounded below by market values, represents the resulting supply curve. Likewise, if demand decreases at higher prices, a concave function, bounded above by market values, represents the attending demand curve. In short, grantedmonotonicity, market curves are described by bivariate functions, either convex or concave, appropriately bounded by linear valuations of quantities. The bounding supply (demand) function generates ask (resp. bid)valuations. Exchange and trade, as modelled here, are driven by valuation differentials, called bid-ask spreads. These disappear, and market equilibrium prevails, if all ”inverse market curves” intersect in a common price. A main issue is whether and how market agents, by themselves, may reach such equilibrium. The paper provides positive and constructive answers. As vehicle it contends with bilateral transactions.     

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.     

Two-settlement electricity markets with price caps and Cournot generation firms

We compare two alternative mechanisms for capping prices in two-settlement electricity markets. With sufficient lead time, forward market prices are implicitly capped by competitive pressure of potential entry that will occur when forward prices rise above some backstop price. Another more direct approach is to cap spot prices through a regulatory intervention. In this paper we explore the implications of these two alternative mechanisms in a two-settlement Cournot equilibrium framework. We formulate the market equilibrium as a stochastic equilibrium problem with equilibrium constraints (EPEC) capturing congestion effects, probabilistic contingencies and horizontal market power. As an illustrative test case, we use the 53-bus Belgian electricity network with representative generator costs but hypothetical demand and ownership structure. Compared to a price-uncapped two-settlement system, a forward cap increases firms’ incentives for forward contracting, whereas a spot cap reduces such incentives. Moreover, in both cases, more forward contracts are committed as the generation resource ownership structure becomes more diversified.     

Pricing with markups in industries with increasing marginal costs

We study a game that models a market in which heterogeneous producers of perfect substitutes make pricing decisions in a first stage, followed by consumers that select a producer that sells at lowest price. As opposed to Cournot or Bertrand competition, producers select prices using a supply function that maps prices to production levels. Solutions of this type of models are normally referred to as supply function equilibria. We consider a market where producers’ convex costs functions are proportional to each other, depending on the efficiency of each particular producer. We provide necessary and sufficient conditions for the existence of an equilibrium that uses simple supply functions that replicate the cost structure. We then specialize the model to monomial cost functions with exponent \(q>0\) , which allows us to reinterpret the simple supply functions as a markup applied to the production cost. We prove that an equilibrium for the markups exists if and only if the number of producers in the market is strictly larger than \(1+q\) , and if an equilibrium exists, it is unique. The main result for monomials is that the equilibrium nearly minimizes the total production cost when the market is competitive. The result holds because when there is enough competition, markups are bounded, thus preventing prices to be significantly distorted from costs. Focusing on the case of linear unit-cost functions on the production quantities, we characterize the equilibrium accurately and refine the previous result to establish an almost tight bound on the worst-case inefficiency of equilibria. Finally, we derive explicitly the producers’ best response for series-parallel networks with linear unit-cost functions, extending our previous result to more general topologies. We prove that a unique equilibrium exists if and only if the network that captures the market structure is 3-edge-connected. For non-series-parallel markets, we provide an example that does not admit an equilibrium on markups.     

Market Price Calculations in Restructured Electricity Markets

In the traditional organisation of the power market, the generation Unit Commitment and Dispatch problem was solved as a cost minimisation problem. After deregulation of the electricity sector, the problem must be solved as a profit maximising problem. It is necessary to find feasible market prices. This is difficult, because simple marginal cost based prices not always cover startup and operation-independent costs, with the result that the generator would choose not to run with such prices. In this paper a market structure is proposed with a central market operator computing the market equilibrium for both energy and reserves, based on generator offers and consumer bids. It is shown that it is possible to find feasible market prices. Using a simple test system, it is shown that demand elasticity can have a profound impact on prices and generator revenues and profits during peaking hours.     

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.     

Average shadow price and equilibrium price: A case study of tradable pollution permit markets

Shadow prices indicate implicit values of limited resources at the margin and provide important information in decision making for resource management. In continuous economic models, shadow prices associated with demand-supply balance constraints represent consumers’ willingness to pay and producers’ marginal cost, hence they correspond to market equilibrium prices. As well known, however, marginal analysis fails in the case of discrete optimization, such as mixed integer programming. An alternative concept has been introduced in the literature to measure the value of an extra unit of a limited resource in such cases. This concept is based on average rather than marginal values, thus called the average shadow price, and interpreted in the same way as conventional shadow prices. Whether average shadow prices in a discrete economic model can serve as market equilibrium prices has not been addressed in the related literature. The present paper addresses this issue in an empirical setting. Using a tradable pollution permit market as an example, where firms’ YES/NO type technology adoption decisions are represented by binary variables, we show that the average shadow price of tradable permits can be interpreted as the equilibrium price only when certain conditions related to the cost structure and emission levels hold. On the other hand, we show that an iterative procedure based on individual firms’ cost minimizing behavior presents a better approach for finding a price that can eliminate or reduce the gap between demand and supply of permits in the market.     

An optimization-based conjectured supply function equilibrium model for network constrained electricity markets

This paper proposes a model to compute nodal prices in oligopolistic markets. The model generalizes a previous model aimed at solving the single-bus problem by applying an optimization procedure. Both models can be classified as conjectured supply function models. The conjectured supply functions are assumed to be linear with constant slopes. The conjectured price responses (price sensitivity as seen for each generating unit), however, are assumed to be dependent on the system line's status (congested or not congested). The consideration of such a dependence is one of the main contributions of this paper. Market equilibrium is defined in this framework. A procedure based on solving an optimization problem is proposed. It only requires convexity of cost functions. Existence of equilibrium, however, is not guaranteed in this multi-nodal situation and an iterative search is required to find it if it exists. A two-area multi-period case study is analysed. The model reaches equilibrium for some cases, mainly depending on the number of periods considered and on the value of conjectured supply function slopes. Some oscillation patterns are observed that can be interpreted as quasi-equilibria. This methodology can be applied to the study of the future Iberian electricity market.     

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.     

A Lagrangian decomposition approach for the pump scheduling problem in water networks

Dynamic pricing has become a common form of electricity tariff, where the price of electricity varies in real time based on the realized electricity supply and demand. Hence, optimizing industrial operations to benefit from periods with low electricity prices is vital to maximizing the benefits of dynamic pricing. In the case of water networks, energy consumed by pumping is a substantial cost for water utilities, and optimizing pump schedules to accommodate for the changing price of energy while ensuring a continuous supply of water is essential. In this paper, a Mixed-Integer Non-linear Programming (MINLP) formulation of the optimal pump scheduling problem is presented. Due to the non-linearities, the typical size of water networks, and the discretization of the planning horizon, the problem is not solvable within reasonable time using standard optimization software. We present a Lagrangian decomposition approach that exploits the structure of the problem leading to smaller problems that are solved independently. The Lagrangian decomposition is coupled with a simulation-based, improved limited discrepancy search algorithm that is capable of finding high quality feasible solutions. The proposed approach finds solutions with guaranteed upper and lower bounds. These solutions are compared to those found by a mixed-integer linear programming approach, which uses a piecewise-linearization of the non-linear constraints to find a global optimal solution of the relaxation. Numerical testing is conducted on two real water networks and the results illustrate the significant costs savings due to optimizing pump schedules.     

Supply Function Equilibrium in Electricity Spot Markets with Contracts and Price Caps

In electricity wholesale markets, generators often sign long term contracts with purchasers of power in order to hedge risks. In this paper, we consider a market where demand is uncertain, but can be represented as a function of price together with a random shock. Each generator offers a smooth supply function into the market and wishes to maximize his expected profit, allowing for his contract position. We investigate supply function equilibria in this setting, using a model introduced by Anderson and Philpott. We study first the existence of a unique monotonically increasing supply curve that maximizes the objective function under the constraint of limited generation capacity and a price cap, and discuss the influence of the generators contract on the optimal supply curve. We then investigate the existence of a symmetric Nash supply function equilibrium, where we do not have to assume that the demand is a concave function of price. Finally, we identify the Nash supply function equilibrium which gives rise to the generators maximal expected profit. This work was supported by the Australian Research Council Grant RMG1965The authors would like to thank three anonymous referees for their helpful comments     

Marginal cost pricing and revenue reconciliation for electricity

This paper demonstrates how mathematical programming can be used to calculate efficient prices of electricity when demand fluctuates over time and where production comes from different types of plants. It is shown that prices have to be set according to marginal costs and differentiated between peak and off-peak load.Furthermore, the paper concerns the use of reconciliation methods for those cases where producers' surplus will be far above what might be considered as a fair rate of return. Here it is shown that it is extremely important to let the energy charges follow the marginal costs, while any modification of the efficient price should be directed to changes in fixed charges.     

Bayesian estimation of nonlinear equilibrium models with random coefficients

In this paper, we develop a conditional likelihood based approach for estimating the equilibrium price and shares in markets with differentiated products and oligopoly supply. We model market demand using a discrete choice model with random coefficients and random utility. For most applications, the likelihood function of equilibrium prices and shares is intractable and cannot be directly analyzed. To overcome this, we develop a Markov Chain Monte Carlo simulation strategy to estimate parameters and distributions. To illustrate our methodology, we generate a dataset of prices and quantities simulated from a differentiated goods oligopoly across a number of markets. We apply our methodology to this dataset to demonstrate its attractive features as well as its accuracy and validity. Copyright © 2014 John Wiley & Sons, Ltd.     

可再生能源配额制与碳排放权交易并行实施的政策效果

中国将同时实施可再生能源配额制与碳排放权交易,并以售电商为配额制的考核主体。本文建立两级供应链模型分析两者相互作用的政策效果。研究结果表明,引入碳排放权交易将导致火电价格上涨与可再生能源电力价格下降,有利于实现两者在同一价格水平竞争。然而,碳排放权交易亦将造成零售电价上涨,电力需求减少,且变化幅度与碳排放权价格和减排成本成正比。若配额比例与售电商差异性提高,在不影响环境成本内化于火电批发价格的同时,可降低售电商的转嫁成本。从而减小了可再生能源电力价格降幅与零售电价涨幅,并平抑碳排放权价格与减排成本上涨造成的电价与需求波动。     

Optimal procurement strategies for online spot markets

Spot markets have emerged for a broad range of commodities, and companies have started to use them in addition to their traditional, long-term procurement contracts (forward contracts). In comparison to forward contracts, spot markets offer products at essentially negligible lead time, but typically command a higher expected price for this added flexibility while also exhibiting substantial price uncertainty. In our research, we analyze the resulting procurement challenge and quantify the benefits of using spot markets from a supply chain perspective. We develop and solve mathematical models that determine the optimal order quantity to purchase via forward contracts and the optimal quantity to purchase via spot markets. We analyze the most general situation where commodities can be both bought and sold via a spot market and derive closed-form results for this case. We compare the obtained results to the reference scenario of pure contract sourcing and we include results for situations where the use of spot markets is restricted to either buying or selling only. Our approaches can be used by decision makers to determine optimal procurement strategies based on key parameters such as, demand and spot price volatilities, correlation between demand and spot prices, and risk aversion. The results of our analysis demonstrate that significant profit improvements can be achieved if a moderate fraction of the commodity demand is procured via spot markets. The results also show that companies who use spot markets can offer a higher expected service level, but that they might experience a higher variability in profits than companies who do not use spot markets. We illustrate our analytical results with numerical examples throughout the paper.