Alexandre Street

Contingency-Constrained Unit Commitment With

This paper presents a new approach for the contingency-constrained single-bus unit commitment problem. The proposed model explicitly incorporates an n - K security criterion by which power balance is guaranteed under any contingency state comprising the simultaneous loss of up to K generation units. Instead of considering all possible contingency states, which would render the problem intractable, a novel method based on robust optimization is proposed. Using the notion of umbrella contingency, the robust counterpart of the original problem is formulated. The resulting model is a particular instance of bilevel programming which is solved by its transformation to an equivalent single-level mixed-integer programming problem. Unlike previously reported contingency-dependent approaches, the robust model does not depend on the size of the set of credible contingencies, thus providing a computationally efficient framework. Simulation results back up these conclusions.

n - K

This paper presents a new approach for energy and reserve scheduling in electricity markets subject to transmission flow limits. Security is imposed by guaranteeing power balance under each contingency state including both generation and transmission assets. The model is general enough to embody a joint generation and transmission n-K security criterion and its variants. An adjustable robust optimization approach is presented to circumvent the tractability issues associated with conventional contingency-constrained methods relying on explicitly modeling the whole contingency set. The adjustable robust model is formulated as a trilevel programming problem. The upper-level problem aims at minimizing total costs of energy and reserves while ensuring that the system is able to withstand each contingency. The middle-level problem identifies, for a given pre-contingency schedule, the contingency state leading to maximum power imbalance if any. Finally, the lower-level problem models the operators best reaction for a given contingency by minimizing the system power imbalance. The proposed trilevel problem is solved by a Benders decomposition approach. For computation purposes, a tighter formulation for the master problem is presented. Our approach is finitely convergent to the optimal solution and provides a measure of the distance to the optimum. Simulation results show the superiority of the proposed methodology over conventional contingency-constrained models.

Security Criterion: A Robust Optimization Approach

Renewable sources have recently emerged as a generation option for many countries in order to promote clean energy development. In the case of Brazil, small hydro plants and cogeneration from sugarcane waste (bagasse) have been attractive alternatives during the past years, with hundreds of MW installed since 2004. Despite their advantages, both alternatives are hindered by seasonal yet complementary availability. This forces producers to discount (or price) the risks faced when selling firm energy contracts and may ultimately lead to projects being commercially unattractive. We propose a stochastic optimization model that defines the optimal composition of a portfolio based on these two renewable sources in order to maximize the revenue of an energy trading company. At the same time, this model mitigates hydrological and fuel unavailability risks, thus allowing the participation of both sources in the forward market environment in a competitive manner. A case study is presented, based on data from the Brazilian system.

Energy and Reserve Scheduling Under a Joint Generation and Transmission Security Criterion: An Adjustable Robust Optimization Approach

This paper aims at resolving a major obstacle to practical usage of time-consistent risk-averse decision models. The recursive objective function, generally used to ensure time consistency, is complex and has no clear/direct interpretation. Practitioners rather choose a simpler and more intuitive formulation, even though it may lead to a time inconsistent policy. Based on rigorous mathematical foundations, we impel practical usage of time consistent models as we provide practitioners with an intuitive economic interpretation for the referred recursive objective function. We also discourage time-inconsistent models by arguing that the associated policies are sub-optimal. We developed a new methodology to compute the sub-optimality gap associated with a time-inconsistent policy, providing practitioners with an objective method to quantify practical consequences of time inconsistency. Our results hold for a quite general class of problems and we choose, without loss of generality, a CVaR-based portfolio selection application to illustrate the developed concepts.

Risk Constrained Portfolio Selection of Renewable Sources in Hydrothermal Electricity Markets

This paper presents a novel approach for the transmission network expansion planning under generalized joint generation and transmission n-K security criteria. The proposed methodology identifies the optimal expansion plan while modeling the power system operation under both normal and contingency states. An adjustable robust optimization approach is presented to circumvent the tractability issues associated with conventional contingency-constrained methods relying on explicitly modeling the whole contingency set. The adjustable robust model is formulated as a trilevel programming problem. The upper-level problem aims at minimizing the investment, operation, and system power imbalance costs. The middle-level problem identifies, for a given expansion plan, the contingency state leading to maximum power imbalance if any. Finally, the lower-level problem models the operators best reaction for a given contingency and investment plan by minimizing the system power imbalance. The resulting trilevel program is solved by a primal-dual algorithm based on Benders decomposition combined with a column-and-constraint generation procedure. The proposed approach is finitely convergent to the optimal solution and provides a measure of the distance to the optimum. Simulation results show the superiority of the proposed methodology over conventional contingency-constrained models.

Time consistency and risk averse dynamic decision models: Definition, interpretation and practical consequences

The worldwide development of the natural gas industry resulted in an integration process between electrical and gas sectors in several countries. In Brazil, this process has been taking place in a consistent manner, especially on account of the increase in gas consumption for industrial use and of the installation of thermoelectric plants. Due to the predominance of hydro plants in electric power generation, thermoelectric energy production is basically dependent on hydrology and, as a result, presents a wide annual variability. Consequently, the investment applied to gas production and transportation infrastructure may become under-utilized during a large part of the time; thus, it is important to find mechanisms apt to improve its utilization. In this respect, the present work investigates the creation of a flexible market for gas, where contracts for flexible gas supply would be offered to industrial users, who would receive the gas assigned to thermal power plants when the latter are not dispatched, and would resort to an alternate fuel when these plants are dispatched. The attractiveness of such a contract would depend, of course, on its price. The purpose of this work is to develop a stochastic model for pricing flexible gas supply contracts, taking into account the uncertainty associated to the supply - dependent on the dispatch of the thermal power plants, which have the priority of use of the gas - and the risk profile of potential consumers.

An Adjustable Robust Optimization Approach for Contingency-Constrained Transmission Expansion Planning

We present a new methodology to support an energy trading company (ETC) to devise contracting strategies under an optimal risk-averse renewable portfolio. The uncertainty in the generation of renewable energy sources is accounted for by exogenously simulated scenarios, as is customary in stochastic programming. However, we recognize that spot prices largely depend on unpredictable market conditions, making it difficult to capture its underlying stochastic process, which challenges the use of fundamental approaches for forecasting. Under such framework, industry practices make use of stress tests to validate portfolios. We then adapt the robust optimization approach to perform an endogenous stress test for the spot prices as a function of the buy-and-sell portfolio of contracts and renewable energy generation scenarios. The optimal contracting strategy is built through a bilevel optimization model that uses a hybrid approach, mixing stochastic and robust optimization. The proposed model is flexible to represent the traditional stochastic programming approach and to express the ETCs uncertainty aversion in the case where the price distribution cannot be precisely estimated. The effectiveness of the model is illustrated with examples from the Brazilian market, where the proposed approach is contrasted to its stochastic counterpart and both are benchmarked against observed market variables.

Pricing Flexible Natural Gas Supply Contracts Under Uncertainty in Hydrothermal Markets

Strategies for investing in renewable energy projects present high risks associated with generation and price volatility and dynamics. Existing approaches for determining optimal strategies are based on real options theory, that often simplify the uncertainty process, or on stochastic programming approaches, that simplify the dynamic aspects. In this paper, we bridge the gap between these approaches by developing a multistage stochastic programming approach that includes real options such as postponing, hedging with fixed (forward) contracts and combination with other sources. The proposed model is solved by a procedure based on the Stochastic Dual Dynamic Programming (SDDP) method. The framework is extended to the risk averse setting. A specific case study in investment in hydro and wind projects in the Brazilian market is used to illustrate that the investment strategies generated by the proposed approach are efficient.

Contracting Strategies for Renewable Generators: A Hybrid Stochastic and Robust Optimization Approach

Renewable sources play an important role in the current climate world policy, emerging as an efficient way to reduce greenhouse gas emissions that cause global warming. Despite their appeal, renewable sources bring to the fore important challenges on the economic side. In Brazil, the three main renewable sources are wind power, small run-of-river hydro and cogeneration from sugarcane waste. Their highly seasonal yet complementary availability makes individual energy selling through contracts a dangerous option. By taking advantage of the resource mix, the optimal joint risk-adjusted trading strategy creates financial surplus value that can be studied using cooperative game theory. Therefore, the objective of this work is twofold: first, to propose a risk-averse renewable energy hedge pool to jointly sell a single complementary renewable generation portfolio and, second, to analyze different schemes of sharing the financial gains, namely quotas, between the members of such a pool from a cooperative game theory point of view. Results using realistic data from the Brazilian system are discussed and four different quota allocation strategies are analyzed: Energy Proportional, Shapley value, Nucleolus and Proportional Nucleolus.

Risk neutral and risk averse approaches to multistage renewable investment planning under uncertainty

The objective of this work is to discuss the modeling of auctions of long-term electricity supply contracts for new capacity in Brazil. The modeling of risks such as price-quantity (hydrological) risk, project completion risk, environmental constraints, climate change and regulatory risks are discussed. An analytical model will be developed to price these risks and case studies will be presented for real projects that have participated in the Brazilian contract auctions for new capacity. We also discuss selection of projects with different risks.