Stability Analysis of Wholesale Electricity Markets under Dynamic Consumption Models and Real-Time Pricing

This paper analyzes stability conditions for wholesale electricity markets under real-time retail pricing and realistic consumption models with memory, which explicitly take into account previous electricity prices and consumption levels. By passing on the current retail price of electricity from supplier to consumer and feeding the observed consumption back to the supplier, a closed-loop dynamical system for electricity prices and consumption arises whose stability is to be investigated. Under mild assumptions on the generation cost of electricity and consumers' backlog disutility functions, we show that, for consumer models with price memory only, market stability is achieved if the ratio between the consumers' marginal backlog disutility and the suppliers' marginal cost of supply remains below a fixed threshold. Further, consumer models with price and consumption memory can result in greater stability regions and faster convergence to the equilibrium compared to models with price memory alone, if consumption deviations from nominal demand are adequately penalized.Comment: 8 pages, 7 Figures, accepted to the 2017 American Control Conferenc

Constraint algebra in LQG reloaded : Toy model of a U(1)^{3} Gauge Theory I

We analyze the issue of anomaly-free representations of the constraint algebra in Loop Quantum Gravity (LQG) in the context of a diffeomorphism-invariant gauge theory in three spacetime dimensions. We construct a Hamiltonian constraint operator whose commutator matches with a quantization of the classical Poisson bracket involving structure functions. Our quantization scheme is based on a geometric interpretation of the Hamiltonian constraint as a generator of phase space-dependent diffeomorphisms. The resulting Hamiltonian constraint at finite triangulation has a conceptual similarity with the "mu-bar"-scheme in loop quantum cosmology and highly intricate action on the spin-network states of the theory. We construct a subspace of non-normalizable states (distributions) on which the continuum Hamiltonian constraint is defined which leads to an anomaly-free representation of the Poisson bracket of two Hamiltonian constraints in loop quantized framework.Comment: 60 pages, 6 figure

Systematic reduction of complex tropospheric chemical mechanisms, Part I: sensitivity and time-scale analyses

International audienceExplicit mechanisms describing the complex degradation pathways of atmospheric volatile organic compounds (VOCs) are important, since they allow the study of the contribution of individual VOCS to secondary pollutant formation. They are computationally expensive to solve however, since they contain large numbers of species and a wide range of time-scales causing stiffness in the resulting equation systems. This paper and the following companion paper describe the application of systematic and automated methods for reducing such complex mechanisms, whilst maintaining the accuracy of the model with respect to important species and features. The methods are demonstrated via application to version 2 of the Leeds Master Chemical Mechanism. The methods of Jacobian analysis and overall rate sensitivity analysis proved to be efficient and capable of removing the majority of redundant reactions and species in the scheme across a wide range of conditions relevant to the polluted troposphere. The application of principal component analysis of the rate sensitivity matrix was computationally expensive due to its use of the decomposition of very large matrices, and did not produce significant reduction over and above the other sensitivity methods. The use of the quasi-steady state approximation (QSSA) proved to be an extremely successful method of removing the fast time-scales within the system, as demonstrated by a local perturbation analysis at each stage of reduction. QSSA species were automatically selected via the calculation of instantaneous QSSA errors based on user-selected tolerances. The application of the QSSA led to the removal of a large number of alkoxy radicals and excited Criegee bi-radicals via reaction lumping. The resulting reduced mechanism was shown to reproduce the concentration profiles of the important species selected from the full mechanism over a wide range of conditions, including those outside of which the reduced mechanism was generated. As a result of a reduction in the number of species in the scheme of a factor of 2, and a reduction in stiffness, the computational time required for simulations was reduced by a factor of 4 when compared to the full scheme

Systematic reduction of complex tropospheric chemical mechanisms, Part II: Lumping using a time-scale based approach

This paper presents a formal method of species lumping that can be applied automatically to intermediate compounds within detailed and complex tropospheric chemical reaction schemes. The method is based on grouping species with reference to their chemical lifetimes and reactivity structures. A method for determining the forward and reverse transformations between individual and lumped compounds is developed. Preliminary application to the Leeds Master Chemical Mechanism (MCMv2.0) has led to the removal of 734 species and 1777 reactions from the scheme, with minimal degradation of accuracy across a wide range of test trajectories relevant to polluted tropospheric conditions. The lumped groups are seen to relate to groups of peroxy acyl nitrates, nitrates, carbonates, oxepins, substituted phenols, oxeacids and peracids with similar lifetimes and reaction rates with OH. In combination with other reduction techniques, such as sensitivity analysis and the application of the quasi-steady state approximation (QSSA), a reduced mechanism has been developed that contains 35% of the number of species and 40% of the number of reactions compared to the full mechanism. This has led to a speed up of a factor of 8 in terms of computer calculation time within box model simulations

Systematic reduction of complex tropospheric chemical mechanisms using sensitivity and time-scale analyses

International audienceExplicit mechanisms describing the complex degradation pathways of atmospheric volatile organic compounds (VOCs) are important, since they allow the study of the contribution of individual VOCS to secondary pollutant formation. They are computationally expensive to solve however, since they contain large numbers of species and a wide range of time-scales causing stiffness in the resulting equation systems. This paper and the following companion paper describe the application of systematic and automated methods for reducing such complex mechanisms, whilst maintaining the accuracy of the model with respect to important species and features. The methods are demonstrated via application to version 2 of the Leeds Master Chemical Mechanism. The methods of local concentration sensitivity analysis and overall rate sensitivity analysis proved to be efficient and capable of removing the majority of redundant reactions and species in the scheme across a wide range of conditions relevant to the polluted troposphere. The application of principal component analysis of the rate sensitivity matrix was computationally expensive due to its use of the decomposition of very large matrices, and did not produce significant reduction over and above the other sensitivity methods. The use of the quasi-steady state approximation (QSSA) proved to be an extremely successful method of removing the fast time-scales within the system, as demonstrated by a local perturbation analysis at each stage of reduction. QSSA species were automatically selected via the calculation of instantaneous QSSA errors based on user-selected tolerances. The application of the QSSA led to the removal of a large number of alkoxy radicals and excited Criegee bi-radicals via reaction lumping. The resulting reduced mechanism was shown to reproduce the concentration profiles of the important species selected from the full mechanism over a wide range of conditions, including those outside of which the reduced mechanism was generated. As a result of a reduction in the number of species in the scheme of a factor of 2, and a reduction in stiffness, the computational time required for simulations was reduced by a factor of 4 when compared to the full scheme