We present convergence rates for the error between the direct transcription solution and the true solution of an unconstrained optimal control problem. The problem is discretized using collocation at Radau points (aka Gauss-Radau or Legendre-Gauss-Radau quadrature). The precision of Radau quadrature is the highest after Gauss (aka Legendre-Gauss) quadrature, and it has the added advantage that the end point is one of the abscissas where the function, to be integrated, is evaluated. We analyze convergence from a Nonlinear Programming (NLP)/matrix algebra perspective. This enables us to predict the norms of various constituents of a matrix that is "close" to the KKT matrix of the discretized problem. We present the convergence rates for the various components, for a sufficiently small discretization size, as functions of the discretization size and the number of collocation points. We illustrate this using several test examples. This also leads to an adjoint estimation procedure, given the Lagrange multipliers for the large scale NLP.

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It is very common in chemical engineering applications to find optimal control problems whose optimality conditions do not provide information about the control over an interval. This type of problems is called partially singular, as the control switches between nonsingular and singular arcs. When direct transcription is applied, the resulting nonlinear programming problem is ill conditioned. Some mesh refinement and rigorous iterative methods have been developed to determine the control profile and switching points. This work presents a practical alternative that quickly produces accurate state and control profiles without adding nonconvex terms. The problem is first solved with a large number of equally spaced finite elements. Then, unnecessary elements are removed while keeping the solution structure. Finally, direct and indirect approaches are combined to apply a regularization scheme only to the singular part. Seven examples were solved to test our strategy. Results provide good approximations to the analytical switching points.

This study examines the performance of a solid oxide fuel cell- (SOFC-) based integrated gasification power plant concept at the utility scale (>100 MW). The primary system concept evaluated was a pressurized ∼150 MW SOFC hybrid power system integrated with an entrained-flow, dry-fed, oxygen-blown, slagging coal gasifier and a combined cycle in the form of a gas turbine and an organic Rankine cycle (ORC) power generator. The analyzed concepts include carbon capture via oxy-combustion followed by water knockout and gas compression to pipeline-ready CO2 sequestration conditions. The results of the study indicate that hybrid SOFC systems could achieve electric efficiencies approaching 66% [lower heating value (LHV)] when operating fueled by coal-derived clean syngas and without carbon dioxide capture. The system concept integrates SOFCs with the low-pressure turbine spool of a 50 MW Pratt & Whitney FT8-3 TwinPak gas turbine set and a scaled-up, water-cooled 20 MW version of the Pratt & Whitney (P&W) PureCycle ORC product line (approximately 260 kW). It was also found that a system efficiency performance of about 48% (LHV) is obtained when the system includes entrained-flow gasifier and carbon capture using oxygen combustion. In order to integrate the P&W FT8 into the SOFC system, the high-pressure turbine spool is removed which substantially lowers the FT8 capital cost and increases the expected life of the gas turbine engine. The impact of integrating an ORC bottoming cycle was found to be significant and can add as much as 8 percentage points of efficiency to the system. For sake of comparison, the performance of a higher temperature P&W ORC power system was also investigated. Use of a steam power cycle, in lieu of an ORC, could increase net plant efficiency by another 4%, however, operating costs are potentially much lower with ORCs than steam power cycles. Additionally, the use of cathode gas recycle is strongly relevant to efficiency performance when integrating with bottoming cycles. A parameter sensitivity analysis of the system revealed that SOFC power density is strongly influenced by design cell voltage, fuel utilization, and amount of anode recycle. To maximize the power output of the modified FT8, SOFC fuel utilization should be lower than 70%. Cathode side design parameters, such as pressure drop and temperature rise were observed to only mildly affect efficiency and power density.

in Wiley InterScience (www.interscience.wiley.com).A nonlinear programming (NLP) framework is developed to determine optimal operating policies for hybrid fuel cell/gas turbine power systems. The approach consists of a dynamic model of the power plant, reformulated as an index one differential algebraic equation (DAE) system. A dynamic optimization framework is developed where the constraints include the dynamic model of the plant. The system model is then discretized using Radau collocation on finite elements and formulated in the AMPL modeling environment. This allows for the straightforward solution of dynamic optimization problems using large-scale NLP solvers. IPOPT is the NLP solver used in this study. Program links were provided to Matlab/Simulink to visualize and interpret the results. The formulation of a dynamic optimization problem was focused on determination of optimal operating trajectories for tracking power plant load variations. Efficiency measures were also included as a part of the dynamic optimization problem to maximize efficiency while tracking the desired load profile. Results from 18 case studies show that the dynamic optimization can be performed quickly with excellent results. The applicability of the dynamic optimization framework for the estimation of feed fuel concentrations is also demonstrated.

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