(314f) Optimal off-Line Trajectory Planning for Load Ramping of Hybrid Fuel Cell/Gas Turbine Power Generating Plants

The dynamic operation and control of Fuel Cell/Gas Turbine (FC/GT) Hybrid power plants requires a synergy of operation among subsystems, increased reliability of operation, and reduction in maintenance and downtime. The control strategy plays a significant role in system stability and performance as well as ensuring the protection of equipment for maximum plant life [1]. The main goal of this study is to develop and implement a dynamic optimization framework for FC/GT power generating plants. Optimal control of load changes requires dynamic scheduling of setpoints and feedforward control moves for the system's controllers. To build feedforward action into the conventional proportional plus integral based feedback control, a technique is used where, given a desired load profile Pd, the setpoints and feedforward control signals are found via dynamic optimization [2].

The attached figure shows the internally reforming solid oxide fuel cell/gas turbine system selected for this study. In this system, the feed water humidifies natural gas and is pre-heated to anode inlet conditions. Methane is reformed in the fuel cell and its chemical potential is converted to electrical energy. The anode exhaust, which contains some unreacted fuel, is mixed with the cathode exhaust and sent to a catalytic oxidizer. Before being sent to the humidifier, the hot oxidizer exhaust passes through a heat recovery unit in which it preheats the compressed air before it enters the turbine. The hot compressed air is expanded through the turbine section, driving a high-speed generator.

The base system has been modeled in MATLAB and Simulink via dynamic lumped capacitance models. As this is a truly large-scale optimization problem, this model is reformulated in AMPL [3], a mathematical programming modeling language well suited for such problems. AMPL also provides the necessary first and second derivatives which aid in convergence of nonlinear programming (NLP) solvers. The specific NLP solver that we will use for dynamic optimization studies is the interior point solver IPOPT [4]. The main reason for choosing IPOPT over an active set sequential quadratic programming (SQP) solver is that this power generation system has numerous inequality and bound constraints, and the number of such constraints increases with finer discretization. This necessitates the use of an interior point solver which overcomes the combinatorial bottleneck of choosing the correct active set. IPOPT is a state-of-the-art interior point NLP solver that has been tested extensively. It can also be tailored to handle complementarity constraints which arise in the actuator and the heat exchanger units.

We will also present ways of handling numerical difficulties that are frequently encountered in such systems. Some of these include consistent reformulation of index-2 differential algebraic equations that arise through chemical equilibrium constraints in the stack model, complementarity constraints that arise in modeling saturation blocks (e.g., actuator output is constrained to [0,1]) and heat exchangers (via number of transfer units formulations), and numerical schemes suited for stiff differential equations. The problem formulation is discretized via orthogonal collocation at Radau points. This is known to handle stiff differential equations well. Recently we have proved that approximations of optimal control problems using Radau collocation converge rapidly to the true solution as the discretization size is made finer [5]. As the models deal with quantities of different magnitudes (e.g. compositions, temperature, enthalpies), scaling is necessary for efficient optimization.

We set up generalized optimization models in AMPL for specified dynamic load changes at a given rate. The optimization is constrained by plant dynamics, as well as input and output restrictions. For the optimization purposes, the process's actual controllers are open loop. The control moves and outputs are determined by the optimizer, and then used as feedforward control moves and setpoints for the individual control loops. The objective of the optimization is to minimize the deviations of the control move and output from the desired values. The desired setpoint trajectory for the net plant power is a function of time.

Using this framework, optimization is performed and feedforward control moves and setpoints are scheduled based on the off-line optimization results. The final aim is to create a database of load change trajectories. We will present results for a few cases and demonstrate that this work can facilitate optimal trajectories for closed loop control, and aid in achieving higher control performance than conventional control.

References:

[1] H. Ghezel-Ayagh and J. M. Daly. Progress in Development of Direct Fuel Cell/Turbine Systems. The Proceedings of 27th International Technical Conference on Coal Utilization & Fuel Systems. Clearwater, Florida, March 4-7, 2002.

[2] C. K. Weng. Robust Wide-Range Control of Electric Power Plants. Ph.D. Thesis, The Pennsylvania State University, December, 1994.

[3] R. Fourer, D. Gay, and B. Kernighan, AMPL, The Scientific Press, South San Francisco,1993.

[4] A. Wächter and L.T. Biegler, On the Implementation of a Primal-Dual Interior Point Filter Line Search Algorithm for Large-Scale Nonlinear Programming, Math. Program., to appear, 2004.

[5] S. Kameswaran and L.T. Biegler, Convergence Rates for Direct Transcription of Optimal Control Problems using Collocation at Radau Points, submitted for publication, 2005.