On the origins of approximations for stochastic chemical kinetics

This paper considers the derivation of approximations for stochastic chemical kinetics governed by the discrete master equation. Here, the concepts of (1) partitioning on the basis of fast and slow reactions as opposed to fast and slow species and (2) conditional probability densities are used to derive approximate, partitioned master equations, which are Markovian in nature, from the original master equation. Under different conditions dictated by relaxation time arguments, such approximations give rise to both the equilibrium and hybrid (deterministic or Langevin equations coupled with discrete stochastic simulation) approximations previously reported. In addition, the derivation points out several weaknesses in previous justifications of both the hybrid and equilibrium systems and demonstrates the connection between the original and approximate master equations. Two simple examples illustrate situations in which these two approximate methods are applicable and demonstrate the two methods' efficiencies

Stochastic simulation of catalytic surface reactions in the fast diffusion limit

The master equation of a lattice gas reaction tracks the probability of visiting all spatial configurations. The large number of unique spatial configurations on a lattice renders master equation simulations infeasible for even small lattices. In this work, a reduced master equation is derived for the probability distribution of the coverages in the infinite diffusion limit. This derivation justifies the widely used assumption that the adlayer is in equilibrium for the current coverages and temperature when all reactants are highly mobile. Given the reduced master equation, two novel and efficient simulation methods of lattice gas reactions in the infinite diffusion limit are derived. The first method involves solving the reduced master equation directly for small lattices, which is intractable in configuration space. The second method involves reducing the master equation further in the large lattice limit to a set of differential equations that tracks only the species coverages. Solution of the reduced master equation and differential equations requires information that can be obtained through short, diffusion-only kinetic Monte Carlo simulation runs at each coverage. These simulations need to be run only once because the data can be stored and used for simulations with any set of kinetic parameters, gas-phase concentrations, and initial conditions. An idealized CO oxidation reaction mechanism with strong lateral interactions is used as an example system for demonstrating the reduced master equation and deterministic simulation techniques

Two classes of quasi-steady-state model reductions for stochastic kinetics

The quasi-steady-state approximation (QSSA) is a model reduction technique used to remove highly reactive species from deterministic models of reaction mechanisms. In many reaction networks the highly reactive intermediates (QSSA species) have populations small enough to require a stochastic representation. In this work we apply singular perturbation analysis to remove the QSSA species from the chemical master equation for two classes of problems. The first class occurs in reaction networks where all the species have small populations and the QSSA species sample zero the majority of the time. The perturbation analysis provides a reduced master equation in which the highly reactive species can sample only zero, and are effectively removed from the model. The reduced master equation can be sampled with the Gillespie algorithm. This first stochastic QSSA reduction is applied to several example reaction mechanisms (including Michaelis-Menten kinetics) [Biochem. Z. 49, 333 (1913)]. A general framework for applying the first QSSA reduction technique to new reaction mechanisms is derived. The second class of QSSA model reductions is derived for reaction networks where non-QSSA species have large populations and QSSA species numbers are small and stochastic. We derive this second QSSA reduction from a combination of singular perturbation analysis and the Omega expansion. In some cases the reduced mechanisms and reaction rates from these two stochastic QSSA models and the classical deterministic QSSA reduction are equivalent; however, this is not usually the case

The stochastic quasi-steady-state assumption: Reducing the model but not the noise

Highly reactive species at small copy numbers play an important role in many biological reaction networks. We have described previously how these species can be removed from reaction networks using stochastic quasi-steady-state singular perturbation analysis (sQSPA). In this paper we apply sQSPA to three published biological models: the pap operon regulation, a biochemical oscillator, and an intracellular viral infection. These examples demonstrate three different potential benefits of sQSPA. First, rare state probabilities can be accurately estimated from simulation. Second, the method typically results in fewer and better scaled parameters that can be more readily estimated from experiments. Finally, the simulation time can be significantly reduced without sacrificing the accuracy of the solution

Design and Application of Distributed Economic Model Predictive Control for Large-Scale Building Temperature Regulation

Although recent research has suggested model predictive control as a promising solution for minimizing energy costs of commercial buildings, advanced control systems have not been widely deployed in practice. Large-scale implementations, including industrial complexes and university campuses, may contain thousands of air handler units each serving a multiplicity of zones. A single centralized control system for these applications is not desirable. In this paper, we propose a distributed control system to economically optimize temperature regulation for large-scale commercial building applications. The decomposition strategy considers the complexities of thermal energy storage, zone interactions, and chiller plant equipment while remaining computationally tractable. One of the primary benefits of the proposed formulation is that the low-level airside problem can be decoupled and solved in a distributed manner; hence, it can be easily extended to handle large applications. Peak demand charges, a major source of coupling, are included. The interactions of the airside system with the waterside system are also considered, including discrete decisions, such as turning chillers on and off. To deploy such a control scheme, a system model is required. Since using physical knowledge about building models can greatly reduce the number of parameters that must be identified, grey-box models are recommended to reduce the length of expensive identification testing. We demonstrate the effectiveness of this control system architecture and identification procedure via simulation studies

Closed-Loop Scheduling for Cost Minimization in HVAC Central Plants

In this paper, we examine closed-loop operation of an HVAC central plant to demonstrate that closed-loop receding-horizon scheduling provides robustness to inaccurate forecasts, and that economic performance is not seriously impaired by shortened prediction horizons or inaccurate forecasts when feedback is employed. Using a general mixed-integer linear programming formulation for the scheduling problem, we show that optimization can be performed in real time. Furthermore, we demonstrate that closed-loop operation with a moderate prediction horizon is not significantly worse than a long-horizon implementation in the nominal case, and that closed-loop operation can correct for inaccurate long-term forecasts without significant cost increase. In addition, we show that terminal constraints can be employed to ensure recursive feasibility. The end result is that forecasts of demand need not be extremely accurate over long times, indicating that closed-loop scheduling can be implemented in new or existing central plants

An Economic Model Predictive Control Framework for Distributed Embedded Battery Applications

Since building heating, ventilation, and air conditioning (HVAC) systems are significant consumers of primary energy, considerable efforts are being made to improve energy efficiency and decrease energy costs in these applications. Notably, substantial opportunities in the area of HVAC control exist for decreasing energy costs by shifting loads from peak periods to off-peak periods in the presence of time-varying utility prices. This load shifting is also beneficial for power companies since it results in a more constant total load allowing them to operate more efficiently. Economic model predictive control (MPC) has been shown to significantly decrease the energy costs of commercial HVAC systems via load shifting. Typically, thermal energy storage (TES) is used for this purpose by running HVAC equipment at higher rates during periods of low power prices to charge TES and at lower rates during periods of higher prices while discharging TES to meet building demand loads. However, with batteries becoming less expensive to manufacture, electrical energy storage in batteries is becoming a viable option for load shifting. Batteries can be used for both load shifting to decrease costs and revenue generation if the incentives on the electricity market are appropriate. In this work, embedded battery applications are considered. In embedded battery applications, the batteries are directly packaged with airside equipment such as air handler units (AHUs), roof-top units (RTUs), and variable refrigerant flow systems (VRFs). In this arrangement, the batteries are accessible only to the local unit and not to other units. In this paper, we propose a hierarchical control system framework for the economic optimization of distributed embedded battery units. The architecture considers both building mass storage as well as the electrical energy storage of the battery units. A high-level problem performs an economic optimization over the entire system using aggregate models. The low-level layer is broken into subsystems, each optimizing its local decisions with higher fidelity models. Advantages of this framework include no iterative communication required between subsystems, decreased computational complexity in the high-level problem allowing for real-time online implementation, and management of total demand across the entire system to reduce peak demand charges. We conclude with a simulation study demonstrating the benefits of the proposed control architecture

Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics

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