Hamilton–Jacobi–Bellman framework for optimal control in multistage energy systems

Endoreversible Modeling and Optimization of a Multistage Heat Engine System with a Generalized Heat Transfer Law via Hamilton-Jacobi-Bellman Equations and Dynamic Programming

“…[5,7,11,, this paper has further investigated the multistage endoreversible Carnot heat engine system operating between a finite thermal capacity high-temperature fluid reservoir and an infinite thermal capacity low-temperature environment with a generalized heat transfer law [q ∝ (∆(T n )) m ], which includes the generalized convective heat transfer law [q ∝ (∆T ) m ] and the generalized radiative heat transfer law [q ∝ ∆(T n )]. For the fixed initial time and fixed initial temperature of the driving fluid, the continuous Hamilton-Jacobi-Bellman (HJB) equations related to the optimal fluid temperature configurations for maximum power output have been obtained by applying optimal control theory.…”

“…The control variable is T ≡ T 2 T 1 /T 2 , and the inequality T 1 > T 1 > T 2 > T 2 always holds for the heat engine, so one obtains T 2 ≤ T ≤ T 1 . This optimal control problem belongs to a variational problem whose control variable has the constraint of closed set, and the Pontryagin minimum value principle or Bellman's dynamic programming theory may be applied [5,7,11,67,68]. When the state vector dimension of the optimal control problem is small, the numerical optimization conducted by the dynamic programming theory is very efficient.…”

“…In the analyses of finite-time thermodynamics [1][2][3][4][5][6][7][8][9][10][11], the basic thermodynamic model is the so-called "endoreversible Carnot engine", in which only the irreversibility of the finite-rate heat transfer is considered. Curzon and Ahlborn [12] derived the efficiency η CA corresponding to the maximum power output of an endoreversible Carnot heat engine cycle with Newtonian heat transfer law [q ∝ ∆(T )].…”

“…Sieniutycz [5,7,11,[32][33][34][35][36][37][38], Sieniutycz and Spakovsky [39], and Szwast and Sieniutycz [40] investigated the maximum power output of Newtonian law multistage continuous endoreversible Carnot heat engine sys- (747) tem operating between a finite thermal capacity high--temperature fluid reservoir and an infinite thermal capacity low-temperature environment by applying HJB theory [5,7,11,[32][33][34][35][36]38], and the results were further extended to the multistage discrete endoreversible Carnot heat engine system [5,7,11,37,39,40].…”

“…[5,7,11,, this paper will further investigate the maximum power output of multistage endoreversible Carnot heat engine system with the finite thermal capacity high-temperature heat reservoir, in which the heat transfer between the reservoir and the working fluid obeys the generalized heat transfer law -61, 63-65]. Based on the universal optimization results, the analytical solution for the case with Newtonian heat transfer law will be further obtained, while for the non-Newtonian heat transfer laws, the continuous HJB equations will be discretized and the dynamic programming (DP) method will be adopted to obtain the complete numerical solutions of the optimization problem.…”