Abstract: We consider the problem of reducing the carbon emissions of a set of firms over a finite horizon. A regulator dynamically allocates emission allowances to each firm. Firms face idiosyncratic as well as common economic shocks on emissions, and have linear quadratic abatement costs. Firms can trade allowances to minimise total expected costs, from abatement and trading plus a quadratic terminal penalty. Using variational methods, we exhibit in closed-form the market equilibrium in function of regulator's dynamic allocation. We then solve the Stackelberg game between the regulator and the firms. Again, we obtain a closed-form expression of the dynamic allocation policies that allow a desired expected emission reduction. Optimal policies are not unique but share common properties. Surprisingly, all optimal policies induce a constant abatement effort and a constant price of allowances. Dynamic allocations outperform static ones because of adjustment costs and uncertainty, in particular given the presence of common shocks. Our results are robust to some extensions, like risk aversion of firms or different penalty functions. Based on joint work with S. Biagini.
Abstracts & Slides
Optimal dynamic regulation of carbon emissions market
Michel De Lara
Large-Scale Microgrids Optimal Management by Mixing Stochastic Dynamic Programming with Spatial Decomposition
Microgrids are recognized as a relevant tool to absorb decentralized renewable energies in the energy mix. However, the sequential handling of multiple stochastic productions and demands, and of storage, make their management a delicate issue. We add another layer of complexity by considering microgrids where different buildings stand at the nodes of a network and are connected by the arcs; some buildings host local production and storage capabilities, and can exchange with others their energy surplus. We formulate the problem as a multistage stochastic optimization problem, corresponding to the minimization of the expected temporal sum of operational costs, while satisfying the energy demand at each node, for all time. The resulting mathematical problem has a large-scale nature, exhibiting both spatial and temporal couplings. However, the problem displays a network structure that makes it amenable to a mix of spatial decomposition-coordination with temporal decomposition methods. We conduct numerical simulations on microgrids of different sizes and topologies, with up to 48 nodes and 64 state variables. Decomposition methods are faster and provide more efficient policies than a state-of-the-art Stochastic Dual Dynamic Programming algorithm. Moreover, they scale almost linearly with the state dimension, making them a promising tool to address more complex microgrid optimal management problems (joint work with Pierre Carpentier, Jean-Philippe Chancelier and François Pacaud).
Risk- and Variance-Aware Pricing in Wholesale Electricity Markets
Abstract: This presentation will describe a method to internalize the variability (variance) of renewable generation resources in wholesale electricity pricing procedures. Using a rich body of literature on chance-constrained and distributionally robust optimization in the context of wholesale electricity markets, this presentation seeks to interpret this variability using VaR and CVaR risk measures, which are common in portfolio optimization tasks. Using these risk measures and conic duality theory, we derive and analyze energy and balancing reserve prices that internalize the risk of system limit violations and the variance of system state variables. Finally, we outline an approach to roll out risk-hedging financial instruments (e.g. Arrow-Debreu Securities) to enable risk trading in wholesale electricity markets.
Thirty years of electricity markets
Abstract: This talk will look back at electricity markets since the 1990s. What challenges did they face then, and how successfully have they overcome them? How will markets need to change for a low-carbon future, and how can modelling help? What are the limits of modelling, and how can we present our findings to neither over- nor under-sell their usefulness?
Designing Risk-free Service for Renewable Wind and Solar Resources
Speakers: Aparna Gupta and Sai Palepu, Lally School of Management, RPI.
Abstract: Renewable generation is inherently stochastic. We develop a risk mitigation strategy by applying principles of securitization to the stochastic generation of wind and solar resources. We demonstrate the design of a risk-free tranche for the renewable assets as a risk-free service the renewable resource can offer in the day-ahead market. The risk-free tranche needs to be comparably risk-free relative to a benchmark for the power markets, against which we evaluate the tranche’s risk-reward performance. Analyzing the risk profiles of renewable power generation on different days of a year, we identify the critical determinants of the risk-free tranche and develop a valuation framework to determine the renewable resources’ bidding strategy in the day-ahead market. Analogous to a risk-free instrument in the financial markets, we create a risk-free benchmark for power markets in terms of a combined-cycle natural gas generator. The risk-return of the risk-free benchmark is used to evaluate the performance of the designed risk-free tranche. We find that our risk-free tranche for renewable wind and solar assets outperforms the risk-free benchmark in terms of their respective risk-return trade-offs. Hence renewable energy producers can leverage the designed risk-free tranche to place competitive bids and participate in the day-ahead market at par with the conventional generators. Adoption of such risk management strategies can help renewable producers to graduate from being mere price takers in the power markets.
Valuation of Flexible Energy Resources in a Nonbinding Commitment Transactive Energy Market
Abstract: Current distribution systems cannot support simultaneous and identical actions of a large number of distributed flexible energy resources reacting to an identical signal. This talk presents a transactive energy market framework when their access to transactions is restricted. A nonlinear pricing structure incentivizes small transactions spread out among arrivals of operation opportunities. A self-exciting point process expresses operation permissions. The problem of optimal operations in this market to maximize the cumulated revenue is modeled as a piecewise deterministic Markov decision process. Various properties of the optimal value and sensitivity to market parameters are studied. This is a joint work with Boris Defourny (Lehigh University).
Rare events in power grids: a Markov Chain Monte Carlo approach
Power systems are typically designed and controlled to ensure excellent standards of reliability. Probabilistic analysis of serious failures such as load shedding is therefore challenging. I will present the skipping sampler, a general-purpose Markov Chain Monte Carlo algorithm which has been developed for this purpose, and applications to rare event sampling in power grids. The talk is based on https://arxiv.org/abs/1905.09964 .
Tackling Hard Problems in Power Grid Optimization: Integer Programs and Human Choices
Abstract: In the power grid optimization literature, one often finds clean problem formulations with continuous decision variables and deterministic data. Reality is different. Specifically, this talk focuses on two tough problems my research lab has recently faced: (i) large-scale mixed integer programs, and (ii) power pricing and scheduling in the context of human choices. Specifically, large-scale mixed integer programs arise when managing large-populations of distributed energy resources with binary (on/off) control. We present a novel (yet historic) heuristic solution known as Hopfield methods. The problem of human choices in-the-loop is fundamental to our current Smart LeaRning Pilot for EV charging stations (SlrpEV). Specifically, we present a menu of differentiated charging service options to EV drivers, and optimize the pricing and charge scheduled based on their preferences, to maximize the operator's net profit. I close the talk with perspectives on tough problems that deserve increased attention for realizing sustainable and resilient power grids of the future.
Short Bio: Scott Moura is an Associate Professor in Civil & Environmental Engineering and Director of the Energy, Controls, & Applications Lab (eCAL) at the University of California, Berkeley. He is also a faculty member at the Tsinghua-Berkeley Shenzhen Institute. He received the B.S. degree from the University of California, Berkeley, CA, USA, and the M.S. and Ph.D. degrees from the University of Michigan, Ann Arbor, in 2006, 2008, and 2011, respectively, all in mechanical engineering.
Price formation and optimal trading in intraday electricity markets
We study price formation in intraday electricity markets in the presence of intermittent renewable generation. We use stochastic control theory to identify optimal strategies of agents with market impact and exhibit the Nash equilibrium in closed form for a finite number of agents as well as in the asymptotic framework of Mean field games, both in the setting of homogeneous agents and in the presence of a major producer acting strategically. We show that our model is able to reproduce the empirical facts observed in the market, such as price impact and price volatility. Joint work with Olivier Feron and Laura Tinsi.
Abstract:: The growth of renewables (wind and solar) have led to decreasing prices and increased volatility. We will discuss the needs of renewable asset owners for financial structures and some of the modeling challenges.