Dynamic programming and an inverse problem in transport theory

Bellman, Richard; Kagiwada, H.; Kalaba, Robert E.

doi:10.1007/bf02235508

Cited by 84 publications

(112 citation statements)

References 3 publications

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“…The difficulty in solving the HJB equation in large-scale systems (like our microgrid) was known to Bellman itself, which coined the term 'curse-of-dimensionality' [19]: in order to overcome such difficulties, the PCAO (Parametrized Cognitive Adaptive Optimization) algorithm parametrizes the solution of the HJB equation (6) as V * (x) = z (x)Pz(x) and the optimal control strategy via u * = − 1 2 B ∂V * ∂ x , P is a positive definite matrix. More details for the function z(·) can be found in [20], [21]: in our specific microgrid case we found that a linear transformation z(x) = x is sufficient to achieve important improvements (as demonstrated in Section V).…”

Section: A Pcao Algorithmmentioning

confidence: 99%

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A supervisory approach to microgrid demand response and climate control

Korkas

Baldi

Michailidis

et al. 2016

2016 24th Mediterranean Conference on Control and Automation (MED)

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Abstract-Microgrids equipped with small-scale renewableenergy generation systems and energy storage units offer challenging opportunity from a control point of view. In fact, in order to improve resilience and enable islanded mode, microgrid energy management systems must dynamically manage controllable loads by considering not only matching energy generation and consumption, but also thermal comfort of the occupants. Thermal comfort, which is often neglected or oversimplified, plays a major role in dynamic demand response, especially in front of intermittent behavior of the renewable energy sources. This paper presents a novel control algorithm for joint demand response management and thermal comfort optimization in a microgrid composed of a block of buildings, a photovoltaic array, a wind turbine, and an energy storage unit. In order to address the large-scale nature of the problem, the proposed control strategy adopt a two-level supervisory strategy: at the lower level, each building employs a local controller that processes only local measurements; at the upper level, a centralized unit supervises and updates the three controllers with the aim of minimizing the aggregate energy cost and thermal discomfort of the microgrid. Comparisons with alternative strategies reveal that the proposed supervisory strategy efficiently manages the demand response so as to sensibly improve independence of the microgrid with respect to the main grid, and guarantees at the same time thermal comfort of the occupants.

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Section: A Pcao Algorithmmentioning

confidence: 99%

“…With such parametrization, the problem of solving the HJB equation is recast as the problem of finding the matrix P (and thus the strategy u) that better approaches the solution of the HJB equation. The PCAO algorithm defines the closeto-optimality index (mutated for the principle of optimality [19])…”

Section: A Pcao Algorithmmentioning

confidence: 99%

A supervisory approach to microgrid demand response and climate control

Korkas

Baldi

Michailidis

et al. 2016

2016 24th Mediterranean Conference on Control and Automation (MED)

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“…7 The formulation of Bellman functions in continuous time follows from taking limits of the associated discrete-period functional equations, presented e.g. in Bellman and Kalaba (1965). Derivations are straightforward; for a standard application reference which has had great influence on the labor market literature, and where also a simple derivation is offered, see Shapiro and Stiglitz (1984).…”

Section: The Basic Model Frameworkmentioning

confidence: 99%

“…The use of value functions of the Bellman type (see Bellman and Kalaba (1965)) in continuous time is by now widespread in economics, in particular within theoretical and empirical labor market analysis, see e.g. Mortensen and Pissardies (1999), Pissarides (2000) and van den Berg (2003).…”

Section: Introductionmentioning

confidence: 99%

Valuation of environmental improvements in continuous time with mortality and morbidity effects

Strand

2006

Resource and Energy Economics

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“…The coefficient sequence estimate h(p) corresponding to any given point ~1 on the residual efficiency frontier then solves a problem of the form 'minimize c,,,(i) with respect to 6 subject to ~~(6) = constant', and can be derived sequentially by means of recurrence relations which have the familiar Riccati equation form. This is hardly surprising; it has been known for decades [Bellman and Kalaba (1965)] that linear-quadratic minimization leads to recurrence relations of this type. Tucci (1990) introduces independence and normality restrictions for the model discrepancy terms under which these FLS recurrence relations for generating any particular estimate 6(p) along the frontier can be given a Kalman filter interpretation.…”

mentioning

confidence: 99%

A further note on flexible least squares and Kalman filtering

Kalaba

Tesfatsion

1990

Journal of Economic Dynamics and Control

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Tucci (1990) logically errs when he attempts to equate the flexible least squares (FLS) approach [Kalaba and Tesfatsion (KT) (1989)] with Kalman filtering. FLS addresses a multicriteria model specification problem which does not require probability assumptions either for its motivation or for its solution: the characterization of the set of all state sequence estimates which achieve vector-minimal incompatibility between imperfectly specified theoretical relations and process observations. Kalman filtering is a point estimation technique for determining the most probable state sequence estimate for a stochastic model assumed to be correctly and completely specified.To illustrate, consider the simple time-varying linear regression problem analyzed both in KT (1989) and in Tucci (1990). Scalar observations yi, . . . , yr have been obtained on an economic process which is not yet well understood. For simplicity, a linear relation is postulated between the observation y, and a vector X, of given regressor variables at each time t, of the form y, = x;bt, t=l 9 . . . , T. Although it is recognized that some systematic time variation in the coefficient vectors b, might have occurred over the observation period, it is anticipated that such evolution will have been gradual, so that the successive coefficient vectors satisfy b,+l = b,, t = 1,. . . , T -1. Each possible estimate 5 = (ii,. . . ,6,) for the sequence of coefficient vectors thus entails two conceptually distinct types of model specification errors: namely, measurement errors consisting of the discrepancies [v, -,x:6!] between the observation and the estimated linear regression at each time t, and dynamic errors consisting of the discrepancies [it+ 1 -h,] between the estimated coefficient vectors at successive times t and t + 1.The FLS approach acknowledges the multicriteria nature of this estimation problem, and the minimal nature of the availableAinformation concerning the true underlying process. A measurement cost cM( b) and a dynamic cost cD( b) are separately assessed for the two disparate types of model specification errors entailed by the choice of a coefficientnsequen:e estimate 6. Letting C denote the collection of cost vectors (c,(b), c,,,(b)) corresponding to all 0165-lSX9/90/$3.5Oi 1990, Elsevier Science Publishers B.V. (North-Holland)

show abstract

Dynamic programming and an inverse problem in transport theory

Cited by 84 publications

References 3 publications

A supervisory approach to microgrid demand response and climate control

A supervisory approach to microgrid demand response and climate control

Valuation of environmental improvements in continuous time with mortality and morbidity effects

A further note on flexible least squares and Kalman filtering

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