Optimal Pricing in Markets with Non-Convex Costs

We consider a market run by an operator who seeks to satisfy a given consumer demand for a commodity by purchasing the needed amount from a group of competing suppliers with non-convex cost functions. The operator knows the suppliers' cost functions and announces a price/payment function for each supplier, which determines the payment to that supplier for producing different quantities. Each supplier then makes an individual decision about how much to produce (and whether to participate at all), in order to maximize its own profit. The key question is how to design the price functions. This problem is relevant for many applications, including electricity markets. The main contribution of this paper is the introduction of a new pricing scheme, \name (\acr ) pricing, which is applicable to general non-convex costs, allows using general parametric price functions, and guarantees market clearing, revenue adequacy, and ecomonic efficiency while supporting comptitive euqilibrium. The name of this scheme stems from the fact that we directly impose all the equilibrium conditions as constraints in the optimization problem for finding the best allocations, as opposed to adjusting the prices later to make the allocations an equilibrium. While the optimization problem is, of course, non-convex, and non-convex problems are intractable in general, we present a tractable approximation algorithm for solving the proposed optimization problem. Our framework extends to the case of networked markets, which, to the best of our knowledge, has not been considered in previous work

Optimal Pricing in Markets with Non-Convex Costs

We consider a market run by an operator who seeks to satisfy a given consumer demand for a commodity by purchasing the needed amount from a group of competing suppliers with non-convex cost functions. The operator knows the suppliers' cost functions and announces a price/payment function for each supplier, which determines the payment to that supplier for producing different quantities. Each supplier then makes an individual decision about how much to produce (and whether to participate at all), in order to maximize its own profit. The key question is how to design the price functions. This problem is relevant for many applications, including electricity markets. The main contribution of this paper is the introduction of a new pricing scheme, \name (\acr ) pricing, which is applicable to general non-convex costs, allows using general parametric price functions, and guarantees market clearing, revenue adequacy, and ecomonic efficiency while supporting comptitive euqilibrium. The name of this scheme stems from the fact that we directly impose all the equilibrium conditions as constraints in the optimization problem for finding the best allocations, as opposed to adjusting the prices later to make the allocations an equilibrium. While the optimization problem is, of course, non-convex, and non-convex problems are intractable in general, we present a tractable approximation algorithm for solving the proposed optimization problem. Our framework extends to the case of networked markets, which, to the best of our knowledge, has not been considered in previous work

Unit commitment : tight formulation and pricing

In recent years, with increasing renewable supply variability, thermal power plants have started up and shut down more frequently. These discrete commitment decisions, optimized in the unit commitment (UC) problem, have an impact on system operations as well as generation expansion planning (GEP). The non-convex costs associated with the commitment decisions may also lead to generators' incentive to deviate from the optimal dispatch under locational marginal prices. In this dissertation, we first propose a convex relaxation of UC based on a primal formulation of the Lagrangian dual problem. This convex relaxation is used (i) to solve the convex hull pricing problem in polynomial time, providing prices with better incentives in non-convex electricity markets, and (ii) to construct a computationally efficient GEP model that represents operational flexibility limits. Next, we present a tight formulation for the commitment of combined-cycle units with representation of their transition ramping. Finally, we propose a pricing method that reduces out-of-market payments in multi-interval real-time markets.Electrical and Computer Engineerin

A compensation-based pricing scheme in marketswith non-convexities

A compensation-based pricing scheme is a market clearing mechanism that may be applied when a uniform, linear pricing scheme cannot support equilibrium allocations in the auction markets. We analyze extensions of our previously proposed pricing scheme [14] to include various possible representations of bids that reflect some non-convex costs and constraints. We conclude with a discussion on directions for future research.auction design, electricity market, non-convex bids, minimum profit condition, unit commitment constraints

Distributed Stochastic Market Clearing with High-Penetration Wind Power

Integrating renewable energy into the modern power grid requires risk-cognizant dispatch of resources to account for the stochastic availability of renewables. Toward this goal, day-ahead stochastic market clearing with high-penetration wind energy is pursued in this paper based on the DC optimal power flow (OPF). The objective is to minimize the social cost which consists of conventional generation costs, end-user disutility, as well as a risk measure of the system re-dispatching cost. Capitalizing on the conditional value-at-risk (CVaR), the novel model is able to mitigate the potentially high risk of the recourse actions to compensate wind forecast errors. The resulting convex optimization task is tackled via a distribution-free sample average based approximation to bypass the prohibitively complex high-dimensional integration. Furthermore, to cope with possibly large-scale dispatchable loads, a fast distributed solver is developed with guaranteed convergence using the alternating direction method of multipliers (ADMM). Numerical results tested on a modified benchmark system are reported to corroborate the merits of the novel framework and proposed approaches.Comment: To appear in IEEE Transactions on Power Systems; 12 pages and 9 figure