Final Report: "Collaborative Proposal" on Optimal-Control Strategies Based on Comprehensive Modeling and System-Interaction Analyses for Energy-Efficient and Reduced-Emission Fuel-Cell-Energy-Systems: Power Electronics SubsystemsEPA Grant Number: R831581
Title: "Collaborative Proposal" on Optimal-Control Strategies Based on Comprehensive Modeling and System-Interaction Analyses for Energy-Efficient and Reduced-Emission Fuel-Cell-Energy-Systems: Power Electronics Subsystems
Investigators: Mazumder, Sudip K. , Nelson, Douglas , Leo, Donald J. , von Spakovsky, Michael R. , McIntyre, Chuck , Herbison, Dan
Institution: University of Illinois at Chicago , Synopsys Inc. , Virginia Polytechnic Institute and State University , University of California - Davis
Current Institution: University of Illinois at Chicago , Synopsys Inc. , University of California - Davis , Virginia Polytechnic Institute and State University
EPA Project Officer: Klieforth, Barbara I
Project Period: February 1, 2004 through January 31, 2007 (Extended to December 31, 2007)
Project Amount: $125,000
RFA: Technology for a Sustainable Environment (2003) RFA Text | Recipients Lists
Research Category: Nanotechnology , Pollution Prevention/Sustainable Development , Sustainability
Objective:In this report, we outline a mechanism for developing an optimal control for power electronic system to enhance the power-management efficiency (Chapter 3). However, it was realized that unlike conventional applications of power converters, which are typically fed by a stiff energy source, a fuel cell is a current-dependent voltage source and hence conventional control approaches may not work. As such, a systematic modeling (Chapter 1) and interaction analysis (Chapter 2) was carried out first to investigate this electrical coupling effect. What emerged is that transient change in load can affect the life of the fuel cell and so can potentially the current ripple. A buffer is required to mitigate the first problem and may require an additional power conditioner. The patent-filed optimal topological control alleviates the need for two separate power conditioner and hence saves footprint space, weight, and cost. It also increases the efficiency flatness via dynamic power management of the multi-modular architecture. The increase in efficiency has implications for the enhanced life of the fuel-cell stack as well as other emissions. Although not shown explicitly, the topological architecture can be modified for low-frequency-ripple mitigation that also directly addresses stack life, sizing, and efficiency. Overall, the theoretical work has been validated with detailed experimentations and two patents have been filed. Besides, 2 peer-reviewed international journal and 2 IEEE Conference papers have been published.
1. COMPREHENSIVE MODELING OF FUEL CELL POWER CONDITIONING SYSTEM
A comprehensive model of the planar PCS needs to meet the following objectives, i) resolve PSOFC power-system interactions and dynamics, and develop design insights towards the achievement of reliable system configurations, ii) enable study of interaction analysis, control, and optimization on a low cost modeling platform without the need of a supercomputer. Moreover, with faster simulation capability, this may lead to the real time simulation thereby enabling the study of long-term reliability.
Comprehensive mathematical models for each of the subsystems are developed, and integrated in a common platform to obtain the system model. The issues with the integration of the system model is identified and discussed. A reduced order system model is developed to address the issues of simulation.
Figure 2.1: Schematic shows the modeling framework for a comprehensive PSOFC based PCS modeling framework.
Figure 2.1, shows the proposed comprehensive modeling framework. The system models comprise PSOFC, BOPS, PES, and the battery bank. The flow of various parameters among the subsystems, as shown in the Figure 2.1, is ensured to ascertain proper system interaction. The comprehensive model requires two low-cost softwares for implementation (Simulink/Matlab including SimPowerSystem and gPROMS1 including gO:SIMULINK2).
B. PES Modeling
Residential PES Model
The PES topological model for a 5 kW residential application is shown in Figure 2.2. The model essentially consists of a dc-dc boost converter to step-up the PSOFC output voltage to an intermediate dc bus voltage (220 V). A dc-ac inverter is used to convert the dc bus voltage to feed power to the ac load.
The rated voltage of the PSOFC stack is between 70 – 90 V. As such, the front-end dc-dc converter for the 5 kW has to be unidirectional (because current should not flow into the fuel cell) and should step-up the voltage, which regulates the output voltage at 220 V. A single-switch unidirectional boost converter, as shown in Figure 2.2, would meet the requirements. The operational details of the boost converter are outlined in (Mohan et al., 1995). When the switch Sw is turned on, energy is built in the inductor L1, and when the switch is turned off the stored energy in the inductor is transferred to the output. The duty ratio of the switch is controlled to attain the desired output voltage.
For the dc-ac inverter a full-bridge voltage source inverter is chosen for its simpler design and operation. The operational details of the converter and the control are outlined in (Mohan et al., 1995). A sinusoidally-modulated switching sequence is generated for S1 (S3) and S2 (S4) to obtain an averaged sine wave at the output of the VSI. S1 (S2) and S3 (S4) operate in a complementary order.
PES Modeling for Vehicular Auxiliary Power Unit (APU)
The architecture of the PES topological model for the APU is shown in Figure 2.3. The model essentially consists of a dc-dc buck-derived converter to step-down the PSOFC output voltage to an intermediate dc bus voltage (42 V). An inverter is used to convert the dc bus voltage to feed the ac power bus. Further, the intermediate dc bus voltage is stepped down using a bidirectional converter to supply the 12 V dc loads and to charge the battery.
The rated voltage of the PSOFC stack is between 72–90 V. As such, the front-end 42V dc-dc converter for the 5 kW APU has to be unidirectional (because current should not flow into the fuel cell) and a step-down converter, which regulates the output voltage at 42 V. A full-bridge converter, as shown in Figure 2.4, is chosen to meet the requirements. The operational details and control design of the converter are outlined in (Redl et al., 1990). The switches S1 and S3 and S2 and S4 switch in complementary pairs, with a time delay between the switching of S1 (S3) and S2 (S4). This phase delay between the switching signals of S1 (S3) and S2 (S4) is controlled to obtain the desired output voltage.
Figure 2.5: Switching model of the dc-ac inverter which consists of two-module parallel Ćuk converters followed by a full-bridge VSI.
PSOFC Spatio-temporal Modeling
For accurate prediction of the effects of system interactions on the PSOFC, one needs to analyze the PSOFC internal parametric variations (Ferguson, 1991; Erdle et al., 1991; Hartvigsen et al., 1993; Yentekakis et al., 1993; Hendrikson et al., 1994; Ferguson et al., 1996). Because a transient model of the SOFC (e.g., (Mazumder et al., 2004)) cannot predict the spatial dynamics, a spatio-temporal electro-thermo-chemical model of the PSOFC (in Simulink) is developed, which provides spatial discretizations of the cell. This model is designed to accept required system inputs (reactant stream flow rates, compositions, and temperatures, cell geometric parameters, and cell current) and computes the corresponding spatial properties of a cell.
Figure 2.6: Spatial homogenous model for the PSOFC providing two-dimensional discretizations involving finite-difference method.
The cell temperature (T) in the two-dimensional (x and y) model, as shown in Figure 2.6, is computed from the time-dependent solution of the following equation:
where Vtn is the thermal neutral voltage, Vop is the operating voltage, ji is the current density, ρ is the mass density of the control volume, Cp is the combined specific heat at constant pressure, l is the cell thickness, ΔHshift is the enthalpy of the shift reaction in the control volume, ΔH is the enthalpy of the main reaction, n is the molar flow rate of the fuel and k is the thermal conductivity of the fuel cell. To approximate the second-order partial-differential equation, a finite-difference method using central differences (Constantinides, 1999) is used.
By approximating ∂T/∂t as a simple forward difference and substituting [(Tj+1-Tj)/Δt] for ∂T/∂t, substituting (1.2) into (1.1), and marching the solution through time yields the Euler’s explicitintegration scheme for the determination the spatial temperature variation. With the step
size Δx = Δy , the expression for temperature becomes
In the course of each of the iteration, the assumed operating voltage is used to determine the current density in each of the control volumes throughout the cell using
where ASRm,n is the local temperature-dependent area-specific resistance and Enm,n is the Nernst potential defined as
ΔG is the change in the Gibb’s free energy. The current density as in (1.6) is then summed to compute the total cell current:
These local current-flux values are used to determine the change in stream composition based on the reaction in each control volume:
The fuel exit-composition of each control volume is equilibrated with respect to the shift reaction before entering the downstream control volume:
where ΔGo is a temperature-dependent variable, pCO2, pH2, pCO , and H2O p are the partial pressures of CO2, H2, CO, and H2O, respectively, Kp is the constant of equilibrium, R is the universal gas constant, and ω is the adjust in the gas compositions due to the shift reaction. Equation (1.10) is used to adjust the compositions to enforce the shift equilibrium constraint at the entrance to each control volume
The configuration of the BOPS model for the PSOFC stack is shown in Figure 2.7. The analytical model is implemented in gPROMS. The comprehensive details of the initial configuration are provided in (Mazumder et al., 2004; Rancruel, 2005).
Figure 2.7: Schematic of the comprehensive BOPS model.
Briefly, the BOPS model comprises a set of first principle and semi-empirical equations for component and subsystem mass and energy conservation, system kinetics, and geometry for both design and off-design operation. A sampling of some of these equations for a number of principal BOPS components are outlined as follows.
Modeling of the Steam Methane Reformer
A set of simplifying assumptions are introduced to facilitate the modeling of the steam-methane reformer. A single reactor tube is analyzed as in Figure 2.8. Thus, all the tubes in the reactor behave independent of each other. Reforming and combustion gases are assumed to behave ideally in all sections of the reactor. The demethanation and water gas-shift reactions are kinetically controlled but are constrained by equilibrium considerations. It is assumed that no carbon deposition is allowed in the SMR reactor. The axial dispersion and radial gradients (plug flow conditions) are neglected.
Figure 2.8: Reactor tube model of the steam methane reformer.
The dynamic mass balance on the reformate gas side is given by
where C is the methane molar concentration (g-mole/m3), U the superficial velocity, Ra is the methane reaction rate per unit mass of catalyst, c ρ is the catalyst bed density and x is the axial direction.
The reformat gas side energy balance is as follows:
In this balance equation, Cp is the specific heat at constant pressure of the tube-side, Ac is the external surface area of particles per volume of catalyst bed, Tc is the catalyst temperature, hi is the inside heat transfer coefficient, hc is the catalyst-fluid heat transfer coefficient, ΔH1 the demethanation reaction enthalpy, and ΔH2 is the water-gas shift reaction enthalpy. In Equations 2.12 and 2.13, X1 and X2 represent the conversions of the demethanation and water-gas shift reactions, respectively. After spatial discretization, the reaction enthalpies are evaluated at each reactor segment’s average temperature.
Modeling of the Compact Heat Exchangers
The heat exchangers used in the BOPS configuration are all plate-fin type, compact heat exchangers with a single-pass, cross-flow arrangement. Their modeling details are presented in (Rancruel, 2005). The heat transfer and pressure drop models used are based on the work of Shah, (1981) and Kays and London, (1998). The effectiveness-NTU method is applied in order to relate the geometric models of the heat exchangers to the thermodynamic ones. The expression for the heat exchanger effectiveness is obtained as in (Mazumder et al., 2004) and is valid for single-pass, cross-flow arrangements with both fluids unmixed.
The energy analysis of each discretized section of the heat exchanger is given by
Figure 2.9: Spatial discrete model of the heat exchanger.
where M is the mass in the control volume, subscripts C and H indicate the cold and hot sides, subscript W the wall, and x and y are the longitudinal and transverse directions, respectively. A refers to the heat transfer area while h refers to the heat transfer coefficients.
Modeling of the Methane and Air Compressors
In the compressor, heat flows from the fluid to the casing to the ambient as well as from the fluid to the impeller to the casing to the ambient through the bearings, seals, and shaft. The thermal capacitance of the casing, impeller, and inlet duct can be approximated by a single thermal mode at a temperature To such that
where m is the mass of the thermal mode, Cp is the thermal mode specific heat, ( )hA i is the inner conductance from the fluid to the thermal mode, ( )hA o is the outer conductance from the thermal mode to the ambient, and To is the thermal mode temperature.
The shaft component is used to compute the turbo machinery rotational speed (N) based on input values of turbine power output and compressor power input. A power balance yields
where I is the moment of inertia and ΔW is the power balance on the component. The transient heat transfer model for an expander is similar to that for compressors and scalable performance maps are used for both types of components to simulate off-design behavior.
C. System Integration
All the component models are built on their best possible modeling platform, and are optimized to yield best results during the individual subsystem simulation. However, several issues have been encountered during the integration of the individual subsystems. These can be classified into A) modeling issues and B) simulation issues.
1. Bulky BOPS model in gPROMS needs an gO:Simulink interface for data transfer between matlab/simulink and gPROMS. To ensure proper data exchange between the gPROMS and the matlab/simulink, input and output ports need to be created in the gPROMS based BOPS model and the input and output variables needed to be assigned to the respective ports.
2. To interface with the PES model, the PSOFC stack model needs to be a voltage source. However, conventional PSOFC stacks are modeled as voltage-controlled current source. Therefore, the PSOFC stack has to be remodeled to behave as a current-controlled voltage source. Now, the amount of current drawn should to be provided as information to the stack from the PES to calculate the stack voltage.
Issues with Simulation
1. The Bulky BOPS Model (very high order) significantly increases the computational overhead for the system simulation. Since the PES and the PSOFC stack models have significantly faster response and needed to be sampled at very high sampling rate (order of tens of microseconds). In the comprehensive model, the bulky BOPS model also gets sampled at this rate, which makes the system simulation extremely slow. To prevent this condition and hence to accelerate the simulation, each individual subsystem needs to sampled individually at its own rate depending on the response time.
2. With significant difference in the response time of the individual subsystem models, ensuring the integrity of data exchange among the subsystems running at their individual pace in their own platforms becomes an issue. Rate transition blocks are used to ensure date integrity in presence of multiple sampling rates.
3. The PES model consists of several switching converters. The switches with discontinuous states, lead to discontinuous-differential equations. Again the presence of the control feedbacks leads to algebraic loops in the model which need to be solved iteratively. A stiff solver and fast sampling is required for convergence and numerical stability. Numerical solver “ode23tb” is used to solve the crude error tolerances at the point of discontinuity and to solve the stiff differential equations with algebraic loops.
D. Reduced Order Modeling
The BOPS subsystem model consists of hundreds of components and subcomponents, and the order of the subsystem is very high. This order of the BOPS model (450 approx.) significantly increases the computational overhead for the system simulation. An added reason for higher computation time is the need for interfacing between the gPROMS and Simulink/Matlab models via the gO:Simulink interface. Again, in the integrated system, each of the subsystem needs to run at the maximum possible sampling time step, which is decided by the subsystem (the PES) that needs to run at the smallest time step. With such a small time step, the BOPS is found to be the bottleneck for the progress of the simulation.
As such, the prime need is to reduce the order of the BOPS model and implement the approximate model in the Simulink/Matlab platform to reduce computational time. Figure 2.10 shows the reduced-order modeling framework for the PSOFC PCS. The reduced order BOPS model is implemented in simulink using polynomial approximation making it all MATLAB/ Simulink system model. This enables the usage of Real Time Workshop (Mathworks) for realtime simulation.
PES Average Modeling
Because the PES switching model comprises discontinuous-differential equations, a stiff solver and fast sampling is required for convergence and numerical stability. For example, to solve the switching model of a PES converter operating at 20 kHz (i.e., a switching period of 50 μsec), sampling time as low as hundreds of nanoseconds may be required. Comparing that to the vastly different (typical) sampling times of PSOFC and BOPS models (which are around few milliseconds and hundreds of milliseconds, respectively), it is realized that the PES switching model is one of the key hindrances to the fast computation of PSOFC PCS model.
Figure 2.10: Reduced-order modeling framework for PSOFC based PCS, developed in Simulink /MATLAB platform, thereby enabling real-time simulation of the system.
Obviously, to increase the speed of the PCS simulation, the need is to avoid discontinuity, thereby reducing the sampling rate and bring it as close as possible to that required by the PSOFC and BOPS models without significantly compromising the accuracy. In one such approach (Khaleel et al., 2004), the PES is designed as a power converter with a constantefficiency module, where output of the converter is a fixed fraction of the input power. However, this model is inaccurate as it assumes that the efficiency of the PES is constant for all load conditions. Further, and more importantly, this model does not predict the transient response of the PES.
An alternate approach to avoid discontinuity of the PES model is to use a map (Mazumder et al., 2001). In a map, instead of solving the switching differential as a whole, the overall model is solved in finite time intervals by treating the PES as a piecewise-linear system (PLS). Subsequently, using the commutative property, the solution is pieced together in each of intervals to construct the final solution in terms of the initial conditions of the states of the PES model. Although a map is a powerful and accurate approach, it typically requires the knowledge of the analytical solution of the PLS and the complexity of the solution increases with the order of the PES model.
An averaged-modeling technique (Lee, 1990; Middlebrook and Ćuk, 1977) is adopted which enables us to analyze the behavior of the PES without significant computational overhead. Figures 2.11 and 2.12 shows the averaged models of the PES converters in equivalent-circuit form (Lee, 1990). One can notice that unlike the switching models, the averaged models do not have any discontinuous elements. Further, the averaged (circuit) models enable the usage of built-in circuit modules in Simulink.
Figure 2.11: Switching average model (AM) of the boost converter.
Figure 2.12: Average model of single-phase full-bridge VSI (L = La + Lb).
For a boost converter, the switching average model is obtained by averaging the state equations for the whole switching period.
where iL(t), vC(t) are the inductor current and capacitor voltage respectively. Vg(t) is the input
voltage of the converter and Io(t) is the output current of the converter.
Voltage Source Inverter
For a VSI, the switching average model is obtained by averaging the state equations for the complete switching period.
where ia(t) is the inductor current, vdc is the dc input voltage, Iin(t) is the input current, vac(t) is the output ac voltage of the VSI, and da(t) is the PWM duty-ratio of the inverter given as
From Figure 2.13, it can be noted that, an averaged model ideally provides the averaged dynamics of the PES and as such, it transforms the PES model from a discontinuous to a smooth form. The accuracy of the averaged model typically varies with the switching frequency, with higher accuracy for higher operating frequency.
Table I shows that the responses of the average models are significantly accurate, while the reduction in the simulation time is of the order of hundreds. The average models do not have any switching dynamics, and hence, it cannot predict the effect of high frequency switching ripple on the PSOFC stack. However, the average models, that accurately represent their switching models with such a level of increase in their simulation speed, enabled the long-term simulation of the complete PSOFC PCS.
TABLE I: Accuracy and Reduction in the Computational of the Average Models as Compared to the Switching Models
Figure 2.13: Accuracy of the PES average models, illustrated by comparing the averaged dynamics with their switching dynamics for (a) the dc-dc boost converter (when subjected to a load transient at t = 1 sec), and (b) voltage source inverter.
One-Dimensional PSOFC Model
The PSOFC subsystem, is usually designed as a voltage controlled current source, that is, it derives the current to be delivered to the load based on the output voltage of the cell. However, the PES, which consists of several power converters, are designed to be powered by voltage sources. Hence, the spatial domain cell subsystem is modified to affect the same. However, in doing so, the model has to solve for the voltage iteratively, which is again an overhead for the simulator. With spatial domain computation for each of the properties inside it for each finite time step (order of milliseconds), the simulation speed is reported to be significantly low.
Hence to hasten the simulation one-dimensional PSOFC model is derived from the two dimensional model. Figure 2.14 shows the homogenous discrete slab representation of the model by reduction in the (y) dimension. Using the homogeneous slab model for the PSOFC, the onedimensional spatio-temporal model of the PSOFC is derived from its two-dimensional form (i.e., equations (1.1)-(1.10)) by discretizing only in one dimension (x). The corresponding equations of the one-dimensional PSOFC model are tabulated in Table II. Figure 2.15 shows the comparison of the two dimensional (2D) model and the one dimensional (1D) model subjected to load transient. The accuracy of the one-dimensional model is quite appreciable (error of 5.2%) as compared to the two dimensional model. The significant reduction (approximately 57.15%) of the computational time for the 1D model speaks for its advantage for the interaction analysis study.
Figure 2.14: One-dimensional homogenous slab model for PSOFC providing discretizations involving finite-difference method.
TABLE II: Summary of the One-Dimensional Model (Derived from Spatial Model)
Figure 2.15: Comparison of the mean cell temperature of the PSOFC (when subjected to the load transient at t = 10 sec) obtained using its one- and two-dimensional models.
Polynomial-Approximation BOPS Model
To realize the polynomial-approximation model, first, the BOPS model is subjected to different sets of load transient and steady-state electrical feedbacks and stored the transient responses of all the output BOPS parameters (such as temperature, air and fuel flow rates, and air and fuel compositions). These parameters are used to interface the BOPS model to the PSOFC model. Subsequently, multi-order (starting from linear to seventh-order) polynomial approximations on each set of these data are applied to obtain the closest approximation that provides optimal compromise between speed of simulation and accuracy with regard to the data obtained using the comprehensive model. In (Khaleel et al., 2004), a similar approach is adopted to reduce the order of the BOPS model.
In Figure 2.16, a polynomial approximation of the BOPS response is illustrated. The order of polynomial is chosen, which maximizes the accuracy of the approximation model and minimizes time for simulation. The result of accuracy and the simulation with that of the time for different order of approximation as compared to the comprehensive model is given in Table III. A fifthorder polynomial approximation was chosen to achieve the optimal compromise between simulation speed and accuracy.
TABLE III: Computational Overhead And Accuracy Comparison Of The Polynomial Approximation
Figure 2.16: Polynomial approximation of the air- and fuel-flow rates of the BOPS model vs. net power.
E. Experimental Model Validation
The effect of electrical feedbacks on PSOFCs is studied on the stack prototype, as shown in Figure 2.17(a), consisting of 25 planar cells in series. The PSOFC stack is built at Ceramatec Inc. In the stack, all the planar cells are mounted in a single manifold. The cross-flow arrangement for the reactants in the stack is illustrated in Fig. 2.17(b). Each cell has an electro-active area of 64 cm2. The ASR of the individual PSOFC for the models at any core temperature are obtained using the empirical formula
where T is the stack core temperature, which depends on the flow rates and temperature of the inlet gases and the stack fuel utilization, the constants A and B are obtained experimentally from the stack characteristics.
Similarly, the 1-D and 2-D model calculates the specific heat capacity and cell density based on the core temperature and experimentally determined constants. The detail specification and the test condition of the stack are given in Table IV and V.
TABLE IV: PES Specifications
TABLE V: PSOFC Stack Specifications and Test Conditions
Figure 2.17: a) Experimental setup of the 25 cell PSOFC stack with the PES and b) The PSOFC stack manifold and the arrangements for air and fuel flow.
Figure 2.18: Setup for the stack characterization and load transient test.
Figure 2.19: Steady-state I~V characteristics comparison of the planar SOFC stack.
Steady-state Characteristics of the PSOFC Stack
Figure 2.18 shows the setup for the characterization of the stack. The dc electronic load is varied in steps to draw different amount of current from the stack. The voltages of the stack at different current levels are noted using the oscilloscope. Comparison of I~V characteristics of the 25 cell stack model and the experimental unit, as shown in Fig. 2.19, shows the accuracy of the analytical model in the steady-state.
Model Validation during the Transients
To prove the accuracy of the model during the transient, the stack is subjected to electrical load transients and low frequency ripple. The connection arrangements and the response comparisons are discussed as follows. Response to the Load Transient
The effects of load transient on the output of the PSOFC stack are analyzed. The dc electronic load operating in constant current mode is connected across the stack output, as shown in Figure 2.18. The dc electronic load is programmed to apply a current transient from 2.2 A to 12 A and subsequently 12 A to 2.2 A, after 1400 seconds. Figs. 2.20(a) and 2.20(b) shows the drop in the output voltage of the 1-D stack model and the experimental stack prototype respectively. Hence, the 1-D stack model accurately emulates the electrical characteristics of the stack.
Due to the subjected load transient, the mean temperature in the stack increases as shown in Fig. 2.21. (The mean temperature profile near the no-load to full-load transient is skewed due to manual turning off of the electronic load at t = 2000 sec accidentally, and is emulated in the stack model during the simulation). From Figure 2.21, it can be noted that the 2D model more closely follows the experimental stack temperature as compared to the 1D Model.
Figure 2.20: Effect of load current transient (2.2 A to 12 A) on the voltage of a) the stack model and b) experimental planar stack, scope channel 1 (10 V /div) and channel 4 (2 A/div) measures the stack voltage and current respectively.
Response to the Low-Frequency Ripple
Figure 2.22 shows the setup to study the effect of the ripple on the stack. The PES consists of a bidirectional dc-dc boost converter. The switch of the converter is modulated using a biased sine wave reference at 60 Hz, generated using the signal generator. To observe various percentage of the ripple in the current, the bias and the amplitude of the sine wave is varied. The current drawn from the stack contains 120 Hz ripple. The load connected across the boost converter is adjusted so that the magnitude of the ripple is 40 percent of the mean stack current. Figures 2.23(a) and 2.23(b) shows the effect of a low frequency (120 Hz) ripple on the 1D stack model and the experimental prototype respectively, which validates the electrical response of the stack model.
Figure 2.21: Validation of the effect of load transient (no load (NL) – full load (FL) – no load (NL)) on the planar SOFC stack temperature. Actual no load to full load transient occurs at 2077 seconds.
Figure 2.22: Setup to study the effect of the ripple on the stack.
Fig. 2.23: a) Effect of 40 percent current ripple on the stack voltage of the model. b) Experimental validation of the ripple effect on planar SOFC stack, scope channels A and D show the stack voltage and current respectively.
2. SYSTEM INTERACTION ANALYSES
Planar solid-oxide fuel cells (PSOFCs) are promising candidates for future alternative-energy-based power systems because of their high energy-conversion efficiency, high fuel flexibility, and tolerance to the fuel impurities [1-3]. However, the high cost of production and issues with the reliability have been bottlenecks in the commercialization of the PSOFC and PSOFC based power-conditioning system (PCS) as well.
Several studies have addressed the issues related to the reliability of SOFC. Effects of higher operating temperature and high fuel utilization on the material properties of SOFC have been reported earlier [4, 5]. These studies on SOFC have primarily focused on studying the effects of material properties and electro-kinetics of chemical reactions on cell operating life and performance. However, there is a need to study the impact of the electrically-induced feedback effects (induced due to the PES and the ALs) on the PSOFC performance.
Achenbach  and Hartvigsen et al.  have demonstrated preliminary results on the impacts of linear electrical load impedance and their change on the dynamics of a SOFC. Acharya et al. [9, 10] and Mazumder et al.  have demonstrated effects on the performance and durability of a tubular SOFC (TSOFC) stack. Recently, Gemmen  attempted to estimate the effects of electrical loads and inverter current ripple on the performance of proton-exchange membrane (PEM) fuel cells using a simple first-order model of a PES. As investigated in [8, 9, 11], the dcac inverter ripple currents may have a degrading impact on the fuel-cell performance if not adequately controlled.
However, no study had been reported to estimate the effects of such electrical feedbacks on the planar SOFC. Again, the impact of several other electrical feedbacks such as the load power factor, harmonic distortion, and the nature of load transients on the performance and reliability of any fuel cell system are not investigated. Therefore, in this paper, using a transient system model comprising a two-dimensional and spatio-temporal PSOFC model and a nonlinear PES model, we investigate the impacts of PES and AL induced electrical effects on the performance of the PSOFC. The steady-state and transient predictions of the cell model are experimentally validated. Subsequently, using this validated model, parametric analyses on the impacts of transience, power factor, and distortion of the application load as well as low-frequency current ripple is conducted. Finally, we demonstrate (experimentally) the long-term impact of two most significant electrical feedbacks on the area-specific resistance and the corresponding loss of effective stack power. We consider two types of electrical feedbacks: those that are induced by the power electronics and those that are induced by the application load.
B. Power-electronics-induced Feedbacks
A single-phase fuel-cell dc-ac converter (also known as inverter as shown in Fig. 1) for stationary application feeds an ac load at line frequency, which is 60 Hz in USA and 50 Hz in Europe. The ac current drawn by the load from the inverter introduces ripple in the current drawn from the fuel cell. This low-frequency ripple current (with a frequency that is twice that of the line frequency) increases with an increase in the load current. Figure 2 illustrates the lowfrequency ripple in the PSOFC stack (PSOFCS) current. As shown later, the ripple current is also dependent on the power factor and load distortion. Because the electrochemical time constant of the PSOFCS is less than the time period of the ripple, the low-frequency ripple current may potentially affect the electrochemical properties of the stack, such as the fuel utilization in the stack and the current density.
Fig. 1: Schematic of a PSOFC based residential power conditioning system.
High-frequency ripple usually refers to the switching ripple of the PES and is typically over 10 kHz. It has an impact on the electrochemical impedance and efficiency of the PSOFC. However, because the time period of the high-frequency ripple is much smaller compared to the electrochemical time constant of the planar fuel cell, it may have negligible impact on the performance of the planar cell. Figure 2 illustrates high-frequency ripple in the current drawn from the PSOFC stack.
Fig. 2: Power-electronics induced high- and low-frequency ripples in the fuel cell current.
One of the most commonly observed load-induced feedbacks is the load-transient, which is attributed to sudden variation in the power demand of the load or its sudden isolation from the fuel cell stack. Other important feedback effects are the power factor of the load and its harmonic distortion, both of which are indicative of the quality of the load.
Figure 3 shows the load current transients at t = 2 and t = 5 seconds due to variations in the power demands of the load. Because PSOFC (like other fuel cells) is not a stiff voltage source, its voltage level decreases with an increase in the current drawn from it. This is due to an increase in the polarization loss at a higher current density. For the PSOFC, operation below a minimum voltage should be avoided because the mass-transport limitations of the electrochemical reaction can cause the anode of the cell to be re-oxidized. This degrades the cell performance and ultimately shortens the life of the stack. Hence, load transients, which may drop the voltage of the PSOFC below this minimum voltage, can degrade the performance and durability of the stack.
Load Power Factor
For a passive ac load, load power factor (which can vary between 0 and 1), as illustrated in Fig. 4a, represents the phase difference between the voltage across a load and the current drawn by it. When the load is resistive, it draws only active power from the source and hence, the power factor is unity. However, at non-unity power factors, an ac load draws both reactive and active powers from the source. The reactive power (Preactive(t)), as shown in Fig. 4b, circulates in the circuit. To support Preactive(t), additional reactive current is drawn from the source apart from the active current (which feeds active power to the load). Therefore, the current drawn by the reactive load increases
Fig. 3: Variation of load current due to the load transient.
Fig. 4: a) Voltage-current (V-I) characteristics of ac loads at different power factors, b) illustration of the circulating reactive power due to non-unity power factor of the load.
with an increase in the reactive power demand by the load. The higher the magnitude of this ac current the higher is the magnitude of the ripple in the current drawn from the dc source. As explained in the Appendix B, the lower the power factor the higher the magnitude of the fuel cell ripple current.
Total Harmonic Distortion (THD)
Total harmonic distortion signifies the harmonic content in an ac quantity, and is defined as the ratio of sum of powers of all harmonic frequency components in a signal above the fundamental frequency to the power of the fundamental frequency component. An ac-dc rectifier load is a typical example of the load which induces harmonic distortion to the ac current at the output of the inverter (refer to Fig. 1). As shown in Fig. 5a, current drawn from the inverter, i(t), is distorted. The harmonic analysis of the distorted current, as shown in Fig. 5b, reveals the presence of the significant odd-order harmonics that decreases with increase in the frequency. The analysis of the harmonics in the load in the Appendix D, concludes that the magnitude of the ripple in the current drawn from the bus, ibus(t), and therefore in the current drawn from the PSOFC stack, iFC(t), are dependent both on the power factor of the fundamental load current and the fractions of harmonic contents in the load.
Fig. 5: a) Current distortion due to a rectifier load. b) Fourier analysis of the distorted current.
For accurate estimation of the impact of the electrical feedbacks, a comprehensive modeling framework for a PSOFC PCS is built, which can potentially address the key issues including resolving the interactions among PSOFC, power electronics subsystem (PES), and application load (AL) and enable parametric system studies. The comprehensive numerical model requires low-cost Simulink/Matlab including SimPowerSystem for implementation, and comprises the spatio-temporal PSOFC stack subsystem implemented in Matlab/Simulink, and the PES and AL profile implemented in SimPowerSystem in Simulink.
Two-Dimensional PSOFC Model
For accurate prediction of the effects of system interactions on the PSOFC, one needs to analyze the PSOFC internal parametric variations [12-16]. A spatio-temporal electro-thermo-chemical model of the PSOFC is developed in Simulink, which provides spatial discretizations of the cell. This model is designed to accept required system inputs (reactant stream flow rates and compositions, temperatures, cell geometric parameters, and cell current) and computes spatial properties such as cell current density, across entire planar cell surface.
The cell temperature (T) in the two-dimensional (x and y) model, as shown in Fig. 6a, is computed from the time-dependent solution of the following equation:
where Vtn is the thermal neutral voltage, Vop is the operating voltage, ji is the current density, ρ is the mass density of the control volume, Cp is the combined specific heat at constant pressure, l is the cell thickness, ΔHshift is the enthalpy of the shift reaction in the control volume, ΔH is the enthalpy of the main reaction, n is the molar flow rate of the fuel and k is the thermal conductivity of the fuel cell. To approximate the second-order partial-differential equation, a finite-difference method using central differences  is used.
By approximating ∂T /∂t as a simple forward difference and substituting [(Tj+1-Tj)/Δt] for ∂T/∂t, substituting Eq. 2 into Eq. 1, and marching the solution through time yields the Euler’s explicitintegration scheme for the determination the spatial temperature variation. With the step size Δx = Δy , the expression for temperature becomes
In the course of each of the iteration, the assumed operating voltage is used to determine the current density in each of the control volumes throughout the cell using
where ASRm,n is the local temperature-dependent ASR and Enm,n is the Nernst potential which is defined as
Fig. 6: a) Spatial homogenous model for the PSOFC providing two-dimensional discretizations. Electrolyte region signifies electro-active area of the cell. b) One-dimensional homogenous slab model for PSOFC providing discretizations involving finite-difference method.
ΔG is the change in the Gibb’s free energy. The current density as in Eq. 6 is then summed to compute the total cell current:
The local current-density values are used to determine the change in stream composition based on the electro-chemical reaction in each control volume. The fuel exit-composition of each control volume is equilibrated with respect to the shift reaction before entering the downstream control volume.
One-Dimensional PSOFC Model
To reduce the computational complexity of the spatial PSOFC model, a one-dimensional model based on homogeneous slab model for the PSOFC (as shown in Fig. 6b) is derived from its twodimensional form (i.e., equations (1)-(8)) by discretizations only in one dimension (x). The equation summary of the one-dimensional PSOFC model is tabulated in Table 1. The model based on co-flow of air and fuel computes various parameters at discrete segments along the length of the planar cell.
Table 1: Summary of the Derivation of the One Dimensional Model from the Two Dimensional Model.
The voltage of the planar SOFC stack varies with current drawn by the load, decreasing significantly at higher load currents. Therefore, a PES is needed to process the raw output power from the stack and provide power to the load at constant dc or ac voltage. The topological model of PES for a residential power system is shown in Fig. 7. The model consists of a dc-dc (boost) converter to step-up the PSOFC output voltage to a higher intermediate dc bus voltage. The operation and control of dc-dc boost converter is defined and explained in detail in . A dc-ac converter (inverter) is further used to convert the output of the dc-dc boost converter, to feed the ac load. For this purpose, a pulse-width-modulated voltage source inverter (VSI) is used, due to its simpler control scheme. The high frequency harmonic content of the output of the VSI is eliminated by the output filter. The operation and control of full-bridge VSI is defined and explained in detail in .
Fig. 7: Architecture of the PES for the residential PCS.
Typically, PES is a piecewise linear (PWL) systems, whose state-space equations can be expressed as
where x(t) ∈ ℜ n are the states of the system, u(t) ∈ ℜ n and v(t) ∈ ℜ m represent the inputs and the outputs of the system, respectively. The matrices Ai, Bi and Ci describe the dynamics of the statespace model for the ith switching state of the system. The sequence and the duration of the switching states are governed by the closed loop controller. As an illustration, the dc-dc boost converter can be modeled using Eq. 9 as
where, x1 represents the state vector, given as [iL vC]T and u1 is the input vector of the system, given as [Vstk 0]T and v1 is the output vector [0 ibus]T. When the switch S of the boost converter is turned ON, the energy content in the inductor L increases and the system of Eq. 10 becomes:
And when the switch S is turned OFF, the stored energy in the inductor is transferred to the output capacitor C of the converter through the diode D and Eq. 10 becomes:
Therefore, the dc-dc boost converter as shown in Fig. 7 can be completely described as
where the switching function s ( s = NOT(s)) represents the turn ON and turn OFF states of the switch S. The switching function is determined to obtain the required bus voltage. Similarly the VSI, as shown in Fig. 7, can be modeled as in Eq. 9
where, x2 represents the state vector, given as [ibus vout]T and u2 is the input vector of the system of equation and is given as [vc 0]T, and v2 is the output vector [0 iload]T. The switch pairs SW1-SW3 and SW2-SW4 switch in complement with a very small time delay between the switching of SW1 (SW3) and SW2 (SW4). When the switch pair SW1-SW3 turned ON, Eq. 13 becomes
When the switch pair SW2-SW4 turned ON, the system of equation Eq. 13 becomes
The overall VSI as shown in Fig. 7 can be described as
The switching function s of the VSI is given as s = [sa sb], where sa represent the ON state of the SW1-SW3 pair and sb represents the ON state of the SW2-SW4 pair. The ON time of sa and sb are generated by a sinusoidally modulated switching sequence to obtain an averaged sine wave ac at the output of the VSI. Combining Eq. 12 and Eq. 15, the complete PES system as in Fig. 7 can be expressed as:-
where X = [iL vC ibus vout]T is the state vector, U is the input vector of the system, given as [Vstk 0 0 0]T and V is the output vector, given as [0 0 0 iload]T.
D. Model Validation
The effect of electrical feedbacks on PSOFCs is studied on the stack prototype, as shown in Fig. 8a, consisting of 25 planar cells in series. In the stack, all the planar cells are mounted in a single manifold. The cross-flow arrangement for the reactants in the stack is illustrated in Fig. 8b. Each cell has an electro-active area of 64 cm2. Temperatures of the reactants (fuel and air) are maintained at 802 oC (1075 K). The detail specification and the test condition of the stack are given in the Appendix A.
Fig. 8: a) Experimental setup of the 25 cell PSOFC stack with the PES; the flow of air and fuel into the stack is kept constant b) A typical 7 cell PSOFC stack manifold and the arrangements for air and fuel flow.
Comparison of I-V characteristics of the 25 cell stack model and the 25 cell stack experimental unit, as shown in Fig. 9, shows the accuracy of the model in the steady-state. The response of the model is again verified during several electrical feedbacks to prove its accuracy even in the strictest of the transients.
Fig. 9: Steady-state I-V characteristics comparison of the planar SOFC stack.
We analyze the effects of load transient on the output of the PSOFC stack. A programmable electronic load operating in constant current mode is connected across the stack output, and a current transient of 2.2 A to 12 A and subsequently 12 A to 2.2 A is applied after 1400 seconds. Figures 10a and 10b shows the drop in the output voltage of the stack model and the experimental stack prototype respectively. The voltage drop is attributed to the enhanced polarization losses owing to a surge in the current density.
Due to the subjected load transient, the mean temperature in the stack increases as shown in Fig. 11a. (The mean temperature profile near the no-load to full-load transient is skewed due to manual turning off of the electronic load at t = 2000 sec accidentally). The increase in the stack temperature is attributed to the increased rate of the exothermic reaction. However, the thermal time constant of the stack being much larger than that of the SOFC electrochemistry/PES time constant, the cell temperature gradually reaches the steady state value after approximately 600 s. To investigate the effect of multiple load transient, we subject the system to multiple small duration loads. The duration of the load is kept fixed at 300 seconds followed by 300 seconds of no load condition. Due to multiple set of transient load, the temperature of the PSOFC does not have sufficient time to drop back to its initial value as shown in Fig. 11b. Hence the multiple load transients may lead to a higher temperature inside the PSOFC.
Fig. 10: a) Effect of load current transient (2.2 A to 12 A) on the voltage of the stack model. b) The experimental validation of its effect on the planar stack, scope channel 1 (10 V /div) and channel 4 (2 A/div) measures the stack voltage and current respectively.
Fig. 11: Validation of the effect of a) single load transient (no load (NL) – full load (FL) – no load (NL)) and b) multiple load transient on the stack temperature.
To study the effect of the ripple, the power electronics prototype, which consists of a bidirectional dc-dc boost converter, is connected across the stack. The duty ratio of switches of the converter is modulated using a sinusoidal signal of 60 Hz, producing 60 Hz ac voltage at the output. The current drawn from the stack contains 120 Hz ripple. The load connected across the boost converter is adjusted so that the magnitude of the ripple is 40 percent of the mean stack current. Figures 12a and 12b shows the effect of a low frequency (120 Hz) ripple on the stack model and the experimental prototype respectively.
Fig. 12: a) Effect of 40 percent current ripple on the stack voltage of the model. b) Experimental validation of the ripple effect on planar SOFC stack, scope channels A and D show the stack voltage and current respectively.
E. Parametric Analysis of the Effects of Electrical Feedbacks
To analyze the impacts of several electrical feedbacks, the validated model is subjected to the feedbacks of various amplitude and/ or frequency. And their effects on the performance and durability of the stack are evaluated based on their effect on the change in the mean temperature and the fuel utilization.
Effect of Load Transient
The level of increase in the mean temperature of the stack is found to be dependent only on the initial and the final load current drawn from the stack and is independent of the slew rate of the load current during the transient. Figure 13a shows the increase in the mean temperature, based on the percentage of the load current. However, as shown in Fig. 13b, the effect of the slew rate of the load transient on the temperature is negligible. As the frequency of the occurrence of the load transient increases, the stack gets less and less time to cool down, leading to a residual increase in the stack temperature. Figure 13c shows the effect of the multiple load transients on the stack temperature.
As has been investigated, an increase in the temperature in the stack leads to non-uniform temperature distribution in the stack. The degree of non-uniformity in the temperature increases with the severity of the load transient. A non-uniform increase in the cell temperature leads to a non-uniform expansion of the cell components due to the difference in their coefficients of thermal expansion. In a PSOFC, with strict binding among each cell components, a slight mismatch in the thermal expansion among the cell components can cause severe residual stress, which may degrade the performance and reliability of the cell . A preliminary thermal analysis of the rated load transient reveals that, significant residual tensile stress is developed at the interface of the electrolyte with the cathode, leading to a cell failure probability of approximately one percent. The reliability of a PSOFC stack depends on the reliability of the individual cells and the number of cells in the stack, and a slightest decrease in the reliability of the cells leads to a severe degradation in the stack reliability .
Fig. 13: Effect of a) magnitude of load transient, b) duration of load transient and c) the frequency of transients on the increase in the mean stack temperature.
Effect of Ripple
To draw maximum power from the stack and hence to achieve very high efficiency, the stack must be operated at a particular current level as shown in Fig. 14. However, due to the presence of low frequency ripple in the stack current, the operating mean stack current has to be decreased, to avoid zero reactant condition, as illustrated in Fig. 14. This decrease in the mean stack current decreases the stack fuel utilization.
Fig. 14: Effect of low frequency ripple on the performance and efficiency of the stack.
Figure 15a shows the decrease in the maximum operable fuel utilization with increase in the magnitude of the current ripple. Now, with decrease in the operating fuel utilization in the stack, the stack efficiency decreases as shown in Fig. 15b. Figure 15c shows the effect of the ripple on the stack temperature. Since the thermal time constant of the planar cell is much higher as compared to the time scale of the low frequency ripple, the stack temperature would not reflect any variation in a small time scale. The stack temperature depends on the power drawn from the stack. Since even with a 100 percent ripple the increase in the RMS current drawn from the stack is approximately 5.91 percent, the increase in the stack current in the steady state is negligible.
Fig. 15: Effect of low frequency ripple on a) operable fuel utilization b) achievable efficiency and c) mean temperature of the stack.
Effect of Load Power Factor
Due to non-unity power factor of the load, the ripple in the stack current increases, refer to the analysis in Appendix B. This ripple further depends on the output capacitance connected across the dc bus. Figure 16a shows the effect of load power factor variation on the stack current ripple at various capacitances and at a constant active power drawn by the load. As shown in Fig. 14, the available power from the stack decreases with an increase in the current ripple magnitude. Therefore, a decrease in the load power factor decreases the efficiency of the stack. However, as discussed under the effect of the ripple, and since the variation in the power drawn from the stack due to ripple is negligible at various power factor loads drawing the same active power, the increase in the mean temperature should be minimal. Figure 16b illustrates the effect of load power factor variation on the mean temperature of the stack.
Fig. 16: Effect of power factor of the load on a) the magnitude of stack current ripple and b) stack temperature.
Effect of THD
An increase in the THD of the ac load increases the distortion in the output ac current due to increase in the magnitude of the harmonic components as well their phase. This distortion in the ac current also introduces distortion in the current drawn from the planar stack. Figure 17a shows the effect of the variation of the THD at different power factors on the ripple magnitude in the stack current. At a fixed power factor of the fundamental current, the percentage ripple in the stack current decreases with an increase in the THD of the load (refer to the analysis and calculation in the Appendix C). However, due to minimal variation in the power drawn from the stack, the effect of the THD on the increase in the mean temperature is minimal as shown in Fig. 17b.
Fig. 17: Effect of THD on a) the stack current ripple and b) stack temperature.
F. Long-Term Degradation Experiment
To study the effect of two of the important electrical feedbacks in the long term, two set of experimental test beds are built. The degradation study of ripple is conducted on a 5 cell stack connected to a boost converter, and a 5 cell stack connected to the constant load. The duty ratio of the switch of the boost converter is modulated sinusoidally at 60 Hz. The average current drawn from both the stacks are kept at approximately 13 A. The open circuit voltage of the 5 cell stack is 5.087 V. To study the long term degradation effect due to the load transient, a 25 cell planar stack is connected to a programmable dc-dc converter followed by a load resistance. The dc-dc converter is programmed to draw 13 A current for first 20 minutes and 2.2 A for the next 10 minutes in every half an hour. Therefore, the average current drawn from the stack is 9.4 A. The open circuit voltage (Voc) of the stack is 24.75 V.
Figure 17 shows the percentage degradation of the ASR (area specific resistance) of the stacks after approximately 900 hrs. It shows that, the degradation of the ASR due to the load transient is very high as compared to the stack carrying constant current. Similarly, increase in the ASR of the stack with low-frequency ripple current is higher as compared to the stack feeding constant current. The degradation in the ASR due to the ripple is found to be 0.06 Ωcm2 higher as compared to the constant current case after 880 hrs of operation. The increase in the ASR degradation leads to a drop in the power output of the stack. The percentage drop in the output power is given as
where Ifc is the average current drawn from the fuel cell, Acell is the electro-active area of the planar cell, ASRb is the base ASR of the stack, and ASR/ is the ASR of the stack after degradation. Comparing the efficiencies of the stacks, the stack carrying hundred percent ripple in the current is found to have an additional drop of 1.72% to that of the stack carrying constant current. And the efficiency of the stack with the load transient drops by 10.32% as compared to the constant current stack.
The analysis of the long-term degradation experiment above indicates that the ASR of the cell degrades with time, when a current is drawn from the stack. However, this degradation is enhanced due to the presence of the ripple in the current and due to the load transients. The degradation of the ASR of the cell deteriorates its efficiency. The load transient, even though at a smaller rate as compared to the frequency of the ripple poses higher threat to the performance and the efficiency of the planar stack. Figures 18a and 18b show the drop in the available power from a planar cell due to the load transients and the low frequency ripple current.
Fig. 17: (a) Comparison of long-term ASR degradation due to low-frequency ripple, constant current and load transient. (b) and (c) The drop in the output power of the PSOFC due to ASR degradation caused by load transient and low-frequency ripple, respectively. The solid lines and the dotted lines refer to the profiles before the start of the degradation study, and after 912 hrs of the study respectively.
3. OPTIMAL CONTROL FORMULATION AND IMPLEMENTATION
Fuel-cell based power system comprises a fuel-cell-stack subsystem (FCSS), a balance-of-plant subsystem (BOPS) that controls the flow rate of fuel and air to the FCSS and maintains the temperature of the stack, and a power-electronics subsystem (PES) that provides the power interface between the FCSS and application load(s). The slow response time of the BOPS mechanical system as compared to the electrochemical and electrical time constants of the fuelcell and the PES has been a major concern for fuel-cell power system designers (Achenbach, 1995; Gemmen, 2001; Mazumder et al., 2007; Pradhan et al., 2007). During a sudden increase in the load demand, the fuel utilization increases rapidly (Mazumder et al., 2007) and several works are in progress to enhance the response time of the BOPS to mitigate this problem that can have degrading effect on the performance of the cell (Hsiao and Selma, 1997). However, currently, the most widely used approach is the use of an energy-buffering device (e.g. a battery), which provides the additional energy to the load during the load transient, thereby preventing a sudden change in stack power flow. However, for energy buffering, an energy-management system is necessary to control the energy flow between the energy generator (fuel-cell stack) and the storage device (battery) and the application load during the transient and steady states. Several researchers are working on this issue related to the control of the fuel-cell/battery based hybrid energy-management system. A control system is proposed in (Hochgraf and Singh, 2001), which controls the state-of-the charge (SOC) of the battery by manipulating the voltage of the stack via dynamic system modeling of predetermined parameters for the stack and the battery. This control strategy tries to eliminate the need for any input power converter, resulting in the cost reduction of the power system. However, this strategy neither provides any option to alleviate the degrading effects of the load transient on the stack nor it considers the efficiency of the power system.
A fuel-cell/ battery hybrid energy-management system with microprocessor-based control is proposed in (Early and Werth, 1990), which attempts to alleviate the degrading effects of the load transients by enabling the fuel-cell stack to be taken out of the system when the load requirement exceeds a fixed maximum energy output of the stack. Because the efficiency and the energy density of the battery are small as compared to the fuel- cell stack, at higher loads (when the battery needs to supply increasingly larger current) the efficiency of the system goes down. As such, the required energy storage capacity of the battery increases, leading to an increase in the space and cost of the battery and the power system.
In some of the prior works for the design of fuel-cell hybrid energy-management system only one power converter was used either at the output of the battery or the stack. For instance in (Jossen et al., 2005; Gao, Jiang, and Dougal, 2005), a DC-DC converter is connected at the stack output to deliver a more stable output voltage to the load while the battery is connected at the converter output. However, during a load transient, battery handles the full load current until the battery voltage goes below the bus voltage, which leads to an oscillation. Moreover, uncontrolled charging may damage the battery. Finally, as the required bus voltage increase, the number of batteries required to support the higher bus will also increase, leading to higher cost of the system.
In another approach (Droppo et al., 2003), the power system uses a power converter to control the power flow from the battery, which provides additional power to the load when the stack voltage goes below a certain minimum. However, as experimentally investigated in this work, the battery current does not respond to an abrupt load increase immediately and hence cannot prevent the zero-reactant condition in the stack unless the operating fuel utilization is inefficiently low.
The dual-power-converter approach used by Jang et al. (2004), avoids the limitations of the aforementioned approaches. However, with either the battery or the stack able to provide the full-load power independently, the converter redundancy of the system is two that leads to higher cost, footprint space, and weight of the power system.
Finally, in an effort to reduce the PES redundancy, Kambouris and Bates (2006) used an IGBT six-pack to implement three bidirectional dc-dc converters to control the power from the battery and the fuel cell. The bidirectional converters are connected selectively to the battery and /or the fuel-cell stack based on the load demand and the fuel-cell-stack capability. However, this approach is not suitable for mitigation of the load-transient since the stack and the battery are in series. Further, the architecture is not modular and hence, not suitable for higher power.
To address the pending challenges outlined above, a power-management control system for a distributed PES (comprising multiple modules of DC-DC boost converters) for a fuel-cell power system is outlined in this paper. The distributed PES serves as the power-electronic interface for both the fuel-cell stack as well as the battery, thereby reducing the cost of the PES as well as its weight and footprint space. This, in turn, alleviates the high cost of the fuel-cell power system, which is currently one of the bottlenecks with regard to the commercialization of such systems. Moreover, due to the dynamic power management of the modules of the distributed PES, shortcoming of conventional PESs (with regard to typical characteristics of drooping efficiency with reducing output power) is alleviated. This, in turn, optimizes the performance and efficiency of the overall power system while nullifying the effects of load transients on the fuel-cell stack. Further, using a composite-Lyapunov-function-based methodology, the stability of the PES undergoing dynamic change in the number of active power converter modules with varying load conditions is formulated. Finally, validation of the PES concept is outlined by interfacing a multi-module bidirectional DC-DC boost converter with fuel-cell stacks.
B. Description of the PES and Control System
Fig. 1 shows the topology of the distributed PES, which interfaces to the fuel-cell stack and the battery. It consists of multiple bidirectional DC-DC (boost) converter modules. The number of these modules (N) depends on the maximum load demand and the rated capacity of the individual converter modules. Assuming that the rated power of the individual converter is Prated and the maximum overall power demand of the load (including battery charging) is Pmax, the total number of the modules (N) is given by N = ceil(Pmax/Prated) + 1, where the ceil function returns the smallest integer, greater than the fraction Pmax/Prated. Further, the redundancy of the distributed PES architecture is given by N/(Pmax/Prated) ≈ N/N-1. The outputs of all the modules are connected in parallel to a bus capacitor while the inputs of the N-1 modules are connected (using transfer switches TS1 through TSN-1) to either the fuel-cell stack or the battery depending on the load demand and the operating efficiency. A dedicated bidirectional (Nth) converter is connected directly to the battery to facilitate the charging of the battery even when the power system is delivering full power to the load. It is noted that, for part-load operation, other modules of the PES can be used for charging the battery as well.
Fig. 2 shows the control-system architecture for the distributed PES. The BOPS controller outlined in (Pradhan, 2007) generates the stack reference current I* FC. The linear bus-voltageerror compensator (Gbus), outlined in (Pradhan, 2007), generates the total current reference Itotal by comparing the bus-voltage reference with feedback bus-voltage. The difference between Itotal and I* FC is the required battery current reference I* bat. Subsequently, using I* FC and I* bat, the dynamic power management unit (DPMU) generates the transfer switch signals and the current references I* 1 through I* N that is fed to the inner current loops for generating switching signals for the converters.
The dynamic power management strategy is as follows. In the steady state, when the load demand is met by the fuel-cell stack, the DPMU determines the number of DC-DC converter modules (m ≤ N-1) to be connected to the fuel-cell stack and the module current reference signals I* 1 through I* m to ensure optimal power sharing that will maximize the overall efficiency of the distributed PES. The optimal condition for this efficient power strategy is described in Section II.A. Further, the DPMU uses the battery voltage (vbat) and its set point (V*bat) as inputs and controls the charging current to the battery (provided by the fuel-cell stack) using the Nth dedicated DC-DC converter with a current reference of I* N (= I* bat). To enhance the response of the bidirectional DC-DC boost converters, current-mode control is used for the PES which is based on the control of the input current for the desired output bus voltage. The PES control, using the outputs of the linear compensators Gi1 through Gim and GiN, generates the switching signals S1 through Sm and SN using the modulators M1 through Mm and MN for the m active converter modules and the dedicated Nth converter. The structure of the jth current-loop compensator is represented by the transfer function Gij(s) = Kij (1+s/ωzij)/s(1+s/ωpij). The compensator structure for the voltage-loop compensator is the same as well; that is, Gbus(s) = Kv (1+s/ωzv)/s(1+s/ωpv). The choice of the gains and placements of the poles and zeros are described in (Pradhan, 2007). The DPMU also generates the on/off signals for the transfer switches TS1…TSN-1 based on the inputs from the controller such that the transfer switches (TS1…TSm) for the m active modules connect the modules to the fuel-cell bus while the remaining N-1-m transfer switches (TSm+1…TSN-1) are turned off along with switches Sm+1 through SN-1 for additional protection.
Fig. 2: Dynamic power management unit (DPMU) for the PES to realize optimal energyconversion efficiency and transient ride-through. Symbols Gbus and Gi1 through GiN represent the bus-voltage compensator and the current-loop compensators (for the N DC-DC converter modules). Only m (≤ N-1) modules are active at any time while the rest of the (N-1-m) modules are turned off. The Nth module is always connected to the battery. Each of the N-1 transfer switches (TS1…TSN-1) also receives input from the DPMU.
During the transient state the dynamic power management strategy uses any converter module that was inactive before the transient to transfer the power difference between the new load demand and the existing fuel-cell-stack input power. Thus, for an increase in the load demand, assuming m active converters were connected to the fuel-cell stack before the load transient, the power-management control system activates the remaining N-m-1 converters (and transfer switches TSm+1…TSN-1) such that additional power is transferred from the battery to the load by equally distributing the I* bat among the N-m-1 converters. The reason all the N-m-1 converters are activated together is to have a fast dynamic response and mitigate any effect of load variation on the stack by fast battery buffering. The transient state continues until the BOPS adjusts the air- and fuel-flow rates for the fuel-cell stack for the new load demand. At that time, the net load power is again provided by the stack and the energy-efficient scheme outlined in Section II.A is applied to determine the optimal number of converter modules that need to be activated to support the new load demand. It is important to ensure the stability of the distributed PES during this structural change of the system. Therefore, in Section II.B, using a recently-developed novel multiple Lyapunov approach (Mazumder and Acharya, 2006), a reachability condition is outlined, which provides the condition for convergence of the PES dynamics from one steadystate condition to another.
C. Efficient Power Sharing Strategy with Fuel-cell Stack Meeting the Power Demand
The efficiency of the jth DC-DC boost converter module is described by
PSWj is the total losses in the switches of the converter, Pinductor j accounts for the loss in the core and the copper loss in the inductor, Pcap j is the parasitic loss in the output capacitor, PTSj is the conduction loss in the transfer switches, and Plumpj corresponds to the additional losses in the converter due to the various unaccounted parasitics in the converter. The expressions for the individual losses for the DC-DC bidirectional boost converter is given by
where the key parameters are defined as follows:
Thus, it can be concluded that, for a converter operating at a particular switching frequency, the total loss is a function of I j for a given Vin and Vbus. It is noted that, under steady-state conditions, the input voltage is the same as the voltage of the fuel-cell stack since the battery buffers the stack only under transient condition; that is, Vin = Vstack . Therefore, the steady-state efficiency of the individual stack connected converter using (1a) can be modified as
The overall efficiency of the distributed PES (including N DC-DC modules) is given by
To maximize the efficiency of the multi-converter system with m (≤ N-1) number of active modules (it is noted that the Nth converter is always connected to the battery) the objective function (Jm) for a given Vstack is defined as
It is noted that I* bat is negative when the battery is being charged. In (3.5b) and (3.5c), Irated is the current capacity of the individual converter modules, Itotal (= I I* ) FC * bat + is the current demand for a given load condition (and is less than or equal to Imax, which is the maximum current corresponding to Pmax), Vstack, OC is the stack voltage when no current is drawn from the stack, and RASR is the equivalent resistance of the stack. Equations (3.5a) - (3.5c) represent a constrained optimization problem with both equality and inequality constraints. It is noted that, the inequality in (3.5b) represent a box constraint that is represented as a 2m one-sided constraint. The Lagrangian ( L : ℜm × ℜ2m × ℜ → ℜ ) for the primal problem defined by (3.5a) –(3.5c) is defined by
where λ ∈ℜ2m is the Lagrange multiplier (or dual variable) for the jth inequality constraint (3.5b) and v ∈ ℜ1 is the Lagrange multiplier (or dual variable) associated with the equality constraint (3.5c). The Lagrange dual function w : ℜ2m × ℜ → ℜ corresponding to (3.6) is defined by (Boyd and Vandenberghe, 2004)
and yields the lower bound on the optimal value p*of (3.5) with corresponding optimal I*. Subsequently, the optimization of the Lagrange dual function using
yields optimal value d* (with corresponding dual optimal Lagrange multipliers λ*, v*) such that d* ≤ p*, which represents the weak duality condition that always hold.
On the other hand, the strong duality condition, which is met under certain specific conditions, ensures an optimal duality gap of zero yields d* = p* thereby yielding optimal primal-dual operating points of (I*, λ*, v*). It turns out that, if the primal problem (e.g. as described by (3.5)) is convex with gj and h, respectively, convex and affine and with differentiable objective and constraint functions, the Karush-Kuhn-Tucker (KKT) condition that is outlined below yields primal and dual optimal points with zero duality gap; that is
An additional supplementary test for the optimal operating (I*, λ*, v*) is the second-order sufficient condition, which requires that the Hessian of the Lagrangian in (3.6) be a positive definite matrix; that is
The solution to the optimization problem in (3.5) determines the optimal distribution of the input currents among the m DC–DC converters. The overall system objective is to find the number of modules to be connected (i.e. m*) such that the cost function Jm as defined in (3.5a) is
Case Illustration With m = 3
Following (2b) – (2f), we express (5a) as
with the constraints described by (3.5b) and (3.5c) for m = 3. In (3.10a), coefficients αj, βj, and γj can be determined based on power-stage parameters or can be determined by experimentally mapping the loss function of the jth module fj(Ij) as a function of the input current. Using the Lagrangian in (3.6) to the primal problem defined by (3.10a), (3.5b), and (3.5c), we obtain following (3.9) the following KKT optimality conditions described by:
Equations (3.11a) – (3.11e) are satisfied only when
Using (3.13) and (3.11b), we obtain the optimal solutions for I* to be
For the special case, when all the converter modules have identical parameters (i.e. α1 = α2 = α3 = α and β1 = β2 = β3 = β), (3.14) and (3.15) reduce to the following:
Further, the Hessian of the Lagrangian in (3.11e) yields
which is a positive definite matrix given that α1> 0, α2 > 0, and α3 > 0. Therefore, the optimality condition (for N > 1) is expected to be achieved when the connected m DC-DC converters share the current equally among them.
D. Reaching Criterion for Post-load-transient Stability: Analysis with Fuel-Cell Stack and Battery Meeting the Power Demand
While Section II.A outlines the issue of optimal power management under steady-state conditions for different load demands, an issue of equal importance is ensuring the transient stability of the distributed PES when the number of converter modules may change following a variation in the power demand of the application load. Conventional stability analyses of PES using average models or nonlinear maps assume orbital existence (Mazumder, Nayfeh, and Boroyevich, 2001). However for global stability, convergence of the reaching dynamics of the PES to its orbit for a given initial condition is essential. Therefore, reaching criterion based analysis (Mazumder and Acharya, 2006) of the distributed PES is needed to ensure its posttransient stability, as outlined below.
To obtain the reaching condition, first, the distributed PES is described by the following piecewise linear (PWL) state-space equation:
where i is an integer that represents the switching states of the PES, x (t )∈ ℜ n represent the states of the PES, n n Ai ∈ℜ × are the matrices and n Bi ∈ ℜ are the column vectors for each of the switching states of the PES. Appendix A provides two case illustrations on the derivation of the matrices Ai and Bi for a closed-loop PES operating with m = 2 and m = 3 active modules after the transient. Now, dropping the notation of time (here on) and translating (3.19) into error coordinates using e = x − x* , where e represents the error vector and x* represents the vector of steady-state values of the states of the PES, (3.19) can be rewritten as
where * Bi = Bi + Aix . Subsequently, to determine the reaching criterion of the PES, a convex combination of multiple positive-definite and quadratic Lyapunov function, Vk (e) > 0 (for the kth switching sequence) is defined as follows:
Using (3.20) and (3.21), (3.22) can be transformed to the following:
the matrix inequality in (3.24) can be represented as a conventional convex optimization problem with linear-matrix-inequality constraints. This convex optimization problem is of the class of feasibility problems, which involves obtaining a matrix Pki such that the linear-matrix-inequality in (3.24) is satisfied. These problems can be solved by using computationally efficient interior-point algorithms (Nesterov and Nemirovskii, 1995) and are available in common mathematical tools like Matlab. However, if there are no solutions of Pki for (3.24) (which is automatically indicated in MATLAB when the total number of iterations exceed a default threshold), the dual of Vk (e) is investigated to confirm that the state error trajectories of the PES states do not converge to the orbit. In that case, one needs to find the dual of Vk (e) which is defined as
is a positive-definite matrix. To confirm that the stateerror trajectories of the SPC do not converge to the orbit for the kth switching sequence, VDk (e) has to satisfy the following criteria:
Following (3.21) and (3.22), (3.27) can be reduced similar to (3.24) to the following form for the
If there are no solutions of Pki for (3.24) but there exist solutions of Qki for (3.28), the stateerror trajectories of the PES do not converge to an orbit, which implies that after a load transient, the PES dynamics will not converge to the new steady-state.
E. PES Prototype Design and Results
The PES power stage for the distributed hybrid power system prototype is designed with 4 bidirectional DC-DC boost converters (i.e. N = 4), for a 2 kW application, with each module rated for 700 W with a rated current of 14 A. The input voltage range is chosen as 55 V to 70 V. The regulated output DC bus voltage of the PES is 120 V. The switching frequency of the boost converters is chosen to be 20 kHz. Fig. 3a shows the experimental distributed PES with parameters provided in Table 1. The design is implemented using two boards: one for the power stage and the other for the controller. The controller interface receives the current- and voltagesense feedback signals from the power board using a multi-strand cable and provides the switching signals for the converter modules and the transfer switches using the same interface. A spectrum digital DSK TMS3206713 along with a high-speed Altera FPGA (EPF10K50VRC240-2) is used for implementing the compensators and generating the digital control signals for the converter switches and the transfer switches. To deactivate a particular bidirectional converter, both the switches need to be turned off simultaneously and hence, control signal of each of the switches are generated using the high-speed IR2103 that provides very efficient cross-conduction prevention logic. Regarding the transfer switches, since they undergo turn-on and turn-off much less frequently compared to the converter switches, their switching losses are negligible. However, they need to have very small on resistance (rds_ON) to reduce their conduction losses.
Fig. 3: Experimental prototype of the (a) distributed PES and (b) a setup for the overall (twostack) fuel-cell-stack based power system. (c) The polarization curve for each stack.
As such, a three-phase MOSFET bridge MSK 4401 (http://www.mskennedy.com/client_images/catalog19680/pages/files/4401re.pdf) with integrated gate drive is used to activate/deactivate the converter modules. The on resistance of the each of the transfer switches in the bridge is only 13 mΩ, which yields a maximum loss of 2.5 W even at full load. Using the experimental PES, we now evaluate the transient stability and optimal steady-state performance of the PES, following the methodology described earlier.
To test whether the dynamics of the distributed PES converges in the presence of a load transient, we demonstrate a two-fold validation. First, using the reaching condition outlined in (3.24), we evaluate the post-transient stability of the PES for a given m. This is ascertained by plotting the minimum eigenvalue of Pki, which if positive proves that the post-transient PES dynamics will converge to the equilibrium for any arbitrary initial condition. Subsequently, we conduct experimental validation of convergence of PES dynamics in parametric and time domains after the transient. Fig. 4c illustrates the transient stability of the PES (for N = 3 and m≤ N – 1) when the load demand varies from 0.6 kW to 1 kW. Before the transient, only one converter feeds the load from the fuel-cell stack. However, immediately after the transient, the other available converter for transient condition is activated, which ensures that while the stack current remains at the pre-transient level, the battery picks up the additional current. Fig. 4b illustrates the convergence of the post-load-transient dynamics in parametric domain. The initial oscillations in the battery and the stack currents are due to the interaction between the two converters. Fig. 4a demonstrates the generalized results based on the reaching condition (3.24) for a resistive load with modules sharing current equally and having the same nominal parameters. It shows that the minimum eigenvalue of Pki in (3.24) is greater than zero, thereby establishing that Pki is positive definite and implying that for m = 2 the dynamics of the distributed PES will converge to the equilibrium for all initial power demand. In other words, following the load transient when the two modules are activated, the system dynamics will stabilize (for arbitrary initial condition) since the reaching condition is satisfied. Along the same lines as above, Figs. 5a – 5c provide stability results for the load transient (0.6 kW → 1.4 kW). However, in this scenario, following the load transient, two additional modules are activated instead of one for the previous case. Thus, after the transient, of the three modules (m = 3), one module continues to provide the (pre-transient-level) power from the stack while the two converters activated after the transient provide the additional power from the battery. We note that, the second battery converter is activated 20 ms after the first and this is attributed to the dynamics of the reference battery current and slight discrepancy of the two current references due to slight mismatch among the power-stage parameters of the two converter modules. However, after this initial time lag, the converters share the current equally among them. Overall, post-transient stability is achieved and experimental results and reaching-condition prediction are in harmony.
With regard to the optimal performance in the equilibrium condition, Fig. 6 shows the improvement in the PES efficiency (demonstrated by the top trace) when the number of active modules is varied (from one through three) as a function of the stack current compared to when three equal-current-sharing modules are always activated. Clearly, using optimal number of PES modules leads to flatter efficiencies for larger power range. This is achieved because as the load demand drops and so does the stack current requirement, without power management, the efficiency of the PES when it is always operated with three (equal-current-sharing) active modules drops; however, when power management is implemented, the control system determines the optimal number of modules needed for a given load demand and yet maximize the efficiency. This is because reducing the number of active modules pushes the power requirement from the remaining modules, thereby yielding higher efficiency since typically converters yield highest efficiency near their rated output power.
1. gPROMS is a software package used for nonlinear process modeling and optimization
2. gO: SIMULINK is a software used for interfacing the gPROMS model with Simulink.
3. ASR of each of the cells modified to take care of the stack equivalent ASR.
In this chapter, a comprehensive spatio-temporal model of the PSOFC PCS is developed which enables the study of interactions among various subsystems. However, the bulky BOPS model with hundreds of components and subcomponents turn out to be a major bottleneck for the system simulation. The PES model, being a high frequency switching discontinuous system, slows down the speed of simulation of the integrated system model. Hence to increase the speed of simulation reduction of the subsystem models to obtain simpler models with lower order and devoid of switching discontinuity are essential. The reduced PES model obtained by averaging the switching states leading to a continuous subsystem is found to follow the response of the switching model with appreciable accuracy, while increasing the speed of the simulation significantly. The stack model is reduced by reducing the spatial model in one of the dimension to obtain 1D discrete model. A multi-order polynomial approximation of the BOPS model closely follows the steady state response of the analytical model. The resulting reduced-order model has improved the simulation speed of the system with very high accuracy.
The model validation experiments show that the response of the analytical models closely follow the experimental result. While the faster 1D model electrical characteristics emulate the response of the experimental prototype in the steady state and in the transient, the 2D model is highly accurate in predicting the thermal response of the PSOFC stack. Therefore, these models when used selectively, can accurately predict the dynamics of an actual PSOFC stack, enabling the interaction analysis and thereby the control design and optimization an achievable goal in a finite time.
We delineate several different electrical feedbacks induced due to the power electronics subsystem (PES) and the application load (AL), which may potentially affect the performance and durability of planar solid-oxide fuel cell stack (PSOFCS). To analyze the impact of such feedbacks, a comprehensive, spatio-temporal, and numerical system model is developed. The accuracy of the model and their ability in determining the effects of several electrical feedbacks on PSOFC during the transient and in the steady state are experimentally validated. Using the validated model accurate estimations of the impacts of several electrical feedback effects on the performance and durability of the PSOFC PCS is conducted using parametric study. An experimental degradation study is done to estimate the long-term effects of the load transient and the ripple on the performance of the stack. Specifically, we conclude the following:
· The no-load to full-load transient increases the current density in the planar fuel cell abruptly and immediately. The higher level of current density increases the fuel utilization and the polarization voltage leading to a drop in the cell voltage. This change in the fuel utilization is detrimental to the cell performance and efficiency;
· The load transient not only increases the mean temperature but also changes the spatial distribution of stack temperature. This variation depends on the magnitude of the current transient and is independent of the slew rate of the transient;
· The load transients accelerate the degradation of the area specific resistance (ASR) of the planar cell. Therefore, they deteriorate the efficiency of the stack. To prevent this, suitable energy buffering techniques should be available to eliminate the effect of the load transient from the stack;
· The higher ripple current magnitude in the stack current forces to decrease the operating fuel utilization of the stack, and hence, lowers the stack efficiency. However, this has negligible impact on the stack temperature;
· In the long term, the ripple current accelerates the degradation of the ASR, deteriorating the efficiency of the stack;
· Lower power factor of the load increases the magnitude of current ripple drawn from the stack. And this reduces the efficiency of the stack. The effect of the load power factor on the stack temperature is minimal;
· Higher THD of the ac load decreases the magnitude of current ripple drawn from the stack. However, it has negligible impact on the stack temperature.
The parametric study in the paper provides a detailed insight into the effects of several electrical-feedback effects on the planar solid-oxide fuel cell (PSOFC) stack and the PSOFC power conditioning system (PCS) as a whole. This will facilitate the design of control and optimization of PSOFC PCS parameters towards the achievement of a highly-efficient and reliable power system.
A power-management control system based on distributed power-electronics subsystem (PES) for a fuel-cell based energy system is outlined. Unlike the conventional approach, which typically uses a lumped PES unit for interfacing the stack to the application load, the distributed PES has multiple modules, which can be controlled and selectively activated depending on the load demand. Of the N distributed modules, up to N-1 can be connected to the fuel-cell-stack or the battery under steady-state or transient condition, while the Nth converter is connected to the battery for charging even under full-load condition. For part-load operation, additional modules can be used for battery charging as well. Thus, the need for a dedicated full-power-rating battery converter (as in conventional approach) is significantly minimized. As the load demand increases, an optimal criterion determines the number of such modules that need to be activated for maximum PES energy-conversion efficiency that in return leads to enhanced stack utilization. This optimal criterion has been developed using a convex optimization framework. Although this paper outlines an analytical formulation of the optimization function (representing the overall loss of the PES), the coefficients for the optimization function can be determined experimentally as well by simply mapping the loss (with varying power demand) as a function of the input current. Further, since the power-management scheme relies on selection of optimal number of active modules following change in load demand, a reaching criterion has been developed to ensure the post-load-transient stability of the PES. The reaching criterion uses a multiple-Lyapunov-function based methodology and determination of the convergence of PES dynamics simply requires solving a matrix inequality. Predictions of both the optimal criterion and the reaching criterion have been validated. The results show that the optimal power
Fig. 4: Post-transient stability of the distributed PES after a load transient from 0.6 kW to 1 kW. Initially one module feeds the load from the stack and subsequently after the transient, a second module is activated that feeds the additional power from the battery. (a) Minimum eigenvalue of Pki > 0 implies that positive Pki is positive definite and that reaching condition (24) is satisfied for all initial power demand for m = 2, thereby ensuring convergence of PES dynamics after the second module is activated following the load transient. (b) and (c) Experimental validations of reachability and stabilization of the currents.
Fig. 5: Post-transient stability of the distributed PES after a load transient from 0.6 kW to 1.4 kW. Initially one module feeds the load from the stack and subsequently after the transient, two modules are activated that feeds the additional power from the battery. (a) Minimum eigenvalue of Pki > 0 implies that positive Pki is positive definite and that reaching condition (24) is satisfied for all initial power demand for m = 3, thereby ensuring convergence of PES dynamics after the second module is activated following the load transient. (b) and (c) Experimental validations of reachability and stabilization of the currents.
Fig. 6: Experimental comparison of the PES efficiency with varying stack current level. (top trace) Efficiency when m is varied between 1 through 3 following the optimal criterion outlined in Section II.A. When m = 2 or m = 3, the current is shared equally among the modules. (bottom trace) Efficiency when m is always 3 (i.e. no optimal power management implemented) and all the modules share current equally. Clearly, the former demonstrates flatter efficiency profile of the PES leading to better fuel-cell-stack utilization in steady-state.
management strategy leads to flatter and higher efficiency of the PES for most part along with convergence of the post-load-transient PES dynamics for varying load conditions. Overall, the distributed-PES-based power-management-control scheme leads to enhanced source utilization due to better PES efficiency profile and reduces (as compared to typical conventional approach) the requirements of footprint space, weight, and cost for battery-buffering by significantly reducing the requirement for a dedicated full-power-rating converter for the battery.
A. Specifications and Test Conditions Of The Experiment
REFERENCES FOR CHAPTER 1
 Erdle, E., Gross, J., Muller, H.G., Müller, W.J.C., Reusch, H.J., and Sonnenschein, R., 1991, Proceedings of Second International Symposium on SOFCs, Edited by Singhal et al., pp. 265.
 Ferguson, J.R., 1991, “Analysis of temperature and current distributions in planar SOFC designs,” Proceedings of Second International Symposium on SOFCs, Edited by Singhal et al., pp. 305.
 Ferguson, J.R., Fiard, J.M., and Herbin, R., 1996, “A mathematical model of solid oxide fuel cells”, Journal of Power Sources, vol. 58, pp. 109-122.
 Hartvigsen, J., Elangovan, S., and Khandkar, A., 1993, Science and Technology of Zirconia V, Edited by Badwal, S., Bannister, M., and Hannink, R., Technomic Publishing Company, Inc., Lancaster, Pennsylvania.
 Kays, W.M., and London, A.L., 1998, Compact Heat Exchangers, Krieger Publishing Company, Malabar, Florida.
 Khaleel, M., Koeppel, B., and Moorehead, S., 2004, “Modeling and control of a solid oxide fuel cell auxiliary power unit”, 2004 DOE Hydrogen, Fuel Cells & Infrastructure Technologies. http://www.eere.energy.gov/hydrogenandfuelcells/pdfs/review04/fc_48_khaleel_04.pdf.
 Lee, F.C., 1990, “Modeling, analysis, and design of PWM converter”, Virginia Power Electronic Center Publications Series, Blacksburg, Virginia.
 Mazumder, S.K., Nayfeh, A.H., and Boroyevich, D., 2001, “Theoretical and experimental investigation of the fast- and slow-scale instabilities of a dc-dc converter”, IEEE Transactions on Power Electronics, vol. 16, no. 2, pp. 201–216.
 Mazumder, S.K., Acharya, K., Haynes, C.L., Williams, R., von Spakovsky, M.R., Nelson, D.J., Rancruel, D.F., Hartvigsen, J., and Gemmen, R.S., 2004, "Solid-oxide-fuel-cell performance and durability: resolution of the effects of power-conditioning systems and application loads", Special Issue on Distributed Generation, IEEE Transactions of Power Electronics, vol. 19, pp. 1263 - 1278.
 Middlebrook, R.D., and Cuk, S., 1977, “A general unified approach of modeling switching-converter power stages”, IEEE Power Electronics Specialists Conference, pp. 521-550.
 Mohan, N., Undeland, T.M., and Robbins, W.P., 1995, Power electronics: converters, applications & design, John Wiley & Sons, Second Edition, ISBN: 0471584088.
 Rancruel, D., 2005, Dynamic Synthesis/Design and Operation/Control Optimization Approach Applied to a Solid Oxide Fuel Cell based Auxiliary Power Unit under Transient Conditions, Ph.D. Dissertation, Virginia Polytechnic Institute and State University, Blacksburg, VA.
 Redl, R., Sokal, N.O., and Balogh, L., 1990, “A novel soft-switching full-bridge dc-dc converter; analysis, design considerations, and experimental results at 1.5kW, 100kHz”, IEEE Applied Power Electronics Conference, pp. 162-172.
 Shah, R.S., 1981, “Compact heat exchanger design procedure”, in Kakaç, S., Bergles, A. E., and Mayinger, F. (Editors), Heat Exchangers, Thermal-Hydraulic Fundamentals and Design, Hemisphere Publishing Corporation, Washington, pp. 495-536.
 Yentekakis, I.V., Neophytides, S., Seimanides, S., and Vayenas, C.G., 1993, “Mathematical modeling of cross-flow, counter-flow and cocurrent-flow solid oxide fuel cells: theory and some preliminary experiments”, Proceedings of 2nd International Symposium on Solid Oxide Fuel Cells, Athens, Greece, Official. Publication of the EEC, Luxembourg, pp. 281-288.
REFERENCES FOR CHAPTER 2
 “Assessment of fuel cells as auxiliary power systems for transportation vehicles”, Hydrogen and Fuel cells Merit Review Meeting, Berkeley, CA, 2003, http://www.tiax.biz/industries/pdfs/fuelcells/fc-apus_0114-03.pdf.
 Kim, J. W., Virkar, A. V., Mehta, K., Fung, K. Z., and Singhal, S.C., 1999, “Low Temperature, High Performance Anode Supported Solid Oxide Fuel Cells”, Journal of Electrochemical Society, 146(1), pp. 69-78.
 Khaleel, M.A., Recknagle, K.P., Lin, Z., Deibler, J.E., Chick, L.A., and Stevenson, J.W., 2001,“Thermo-Mechanical and Electrochemistry Modeling of Planar SOFC Stacks”, Electrochemical Society Proceedings, 2001-16, pp. 1032-1041.
 Hsiao, Y. C. and Selman, J. R., 1997, “The Degradation of SOFC Electrodes,” Proceedings of Solid State Ionics, 98, pp. 33-38.
 Huang, X. and Reifsnider, K., 2001, “Modeling Long-Term Performance of Solid Oxide Fuel Cells: A Phenomenological Approach,” 15th Annual Conference of Fossil Energy Materials, Knoxville, TN.
 Achenbach, E., “Response of a Solid Oxide Fuel Cell to Load Change,” Journal of Power Sources, 57, pp. 105 –109, 1995.
 Hartvigsen, J., Elangovan, S., and Khandkar, A., 1993, Science and Technology of Zirconia V, Edited by Badwal, S., Bannister, M., and Hannink, R., Technomic Publishing Company, Inc., Lancaster, Pennsylvania.
 Gemmen, R. S., 2003, “Analysis for the Effect of Inverter Ripple Current on Fuel Cell Operating Condition,” Journal of Fluids Engineering, 125(3), pp. 576 - 585.
 Acharya, K., Mazumder, S. K., Burra, R. K., Williams, R., and Haynes, C. L., 2003, “System-Interaction Analyses of Solid Oxide Fuel Cell (SOFC) Power-Conditioning System,” Proceedings of the IEEE IAS Conference, pp. 2026 – 2032.
 Acharya, K., Mazumder, S. K., Burra, R. K., 2004, “Impact of Power-Electronics Systems on the Performance and Durability of Tubular Solid-Oxide Fuel Cell”, Proceedings of the IEEE Applied Power Electronics Conference, 3, pp. 1515 - 1520.
 Mazumder, S.K., Acharya, K., Haynes, C.L., Williams, R., von Spakovsky, M. R., Nelson, D. J., Rancruel, D. F., Hartvigsen, J., and Gemmen, R. S., 2004, “Solid-Oxide-Fuel-Cell Performance and Durability: Resolution of the Effects of Power-Conditioning Systems and Application Load,” Special Issue on Distributed Generation, IEEE Transactions of Power Electronics, 19, pp. 1263 - 1278.
 Ferguson, J.R., Fiard, J.M., and Herbin, R., 1996, “A Mathematical Model of Solid Oxide Fuel Cells,” Journal of Power Sources, 58, pp. 109 – 122.
 Yentekakis, I. V., Neophytides, S., Seimanides, S., and Vayenas, C. G., 1993, “Mathematical Modeling of Cross-Flow, Counter-Flow and Cocurrent-Flow Solid Oxide Fuel Cells: Theory and Some Preliminary Experiments”, Proceedings of 2nd International Symposium on Solid Oxide Fuel Cells, Athens, Greece, Official. Publication of the EEC, Luxembourg, pp. 281- 288.
 Erdle, E., Gross, J., Muller, H.G., Müller, W.J.C., Reusch, H.J., and Sonnenschein, R., 1991, Proceedings of Second International Symposium on SOFCs, Edited by Singhal et. al., pp. 265-267.
 Ferguson, J. R., 1991, Proceedings of Second International Symposium on SOFCs, Edited by Singhal et. al., pp. 305-309.
 Hendrikson, P. V., 1994, Second Nordic Symposium on High Temperature Fuel Cells, Edited by T. Nordby and F. W. Poulsen, The Nordic Energy Research Programme and University of Oslo, Norway.
 Constantinides, A., and Mostoufi, N., 1999, Numerical Methods for Chemical Engineers with Matlab Applications, Prentice Hall PTR, Upper Saddle River, NJ.
 Erickson, R. W. and Maksimovic, D., 2004, Fundamentals of Power Electronics, Second Edition, Kluwer Academic Publishers, Massachusetts.
 Lee, F. C., 1990, Modeling, Analysis, and Design of PWM Converter, Virginia Power Electronic Center Publications Series, Blacksburg, Virginia.
 Yakabe, H., Ogiwara, T., Hishinuma, M., and Yasuda, I., 2001, “3D Model Calculation for Planar SOFC,” Journal of Power Sources, 102, pp. 144 -154.
 Stolten, D., 2004, “Comparison of SOFC and High Temperature Electrolysis,” High Temperature Electrolysis IEA Meeting, San Antonio
REFERENCES FOR CHAPTER 3
 Achenbach, E.A., 1995, “Response of a solid oxide fuel cell to load change” Journal of Power Sources, vol. 57, pp. 105-109, 1995.
 Boyd, S. and Vandenberghe, L., 2004, Convex Optimization, Cambridge University Press, New York.
 Brogan, W.L., 1990, Modern Control Theory, Prentice Hall, Upper Saddle River, New Jersey.
 Droppo, G.W., Schienbein, L.A., Harris, B.E., and Hammerstorm, D.J., 2003, “DC to DC converter and power management system”, U.S. Patent 6 628 011.
 Early, J. and Werth, J., 1990, “Fuel cell/battery control system”, U.S. Patent 4 961 151.
 Gemmen, R., 2003, “Analysis for the effect of inverter ripple current on fuel cell operating condition,” Journal Fluids Engineering, vol. 125, no. 3, pp. 576–585.Gao, L., Jiang, Z., and Dougal, R.A., 2005, “Evaluation of active hybrid fuel cell/battery power sources”, IEEE Transactions on Aerospace and Electronic Systems, vol. 41, issue 1, pp. 346-355.
 Hochgraph, C. and Singh, P., 2004, “Method and system for fuel cell control”, U.S. Patent 6 794 844.
 Hsiao Y.C. and Selma, J. R., 1997, “The degradation of SOFC electrodes”, Proceedings of Solid State Ionics, vol. 98, pp. 33-38.
 International Rectifier, IR2103 datasheet, www.irf.com/productinfo/datasheets/data/ir2103.pdf.
 Jang, S.J., Lee, T.W, Lee, W.C, and Won, C.Y., 2004, “Bi-directional dc-dc converter for fuel cell generation system”, IEEE Power Electronics Specialists Conference, vol. 6, pp. 4722-4728.
 Jossen, A., Garche, J., Doering, H., Goetz, M., Knaupp, W., and Joerissen, L., 2005, “Hybrid systems with lead–acid battery and proton-exchange membrane fuel cell”, Journal of Power Sources, vol.144, pp. 395-401.
 Kambouris, C.A. and Bates, J.T., 2006, “Dc-dc converter for a fuel cell system”, U.S. Patent 7014 928 B2.
 M.S. Kennedy Corporation, MSK 4401 datasheet, http://www.mskennedy.com/client_images/ catalog19680/pages/files/4401re.pdf.
 Mazumder, S.K., Nayfeh, A.H., and Boroyevich, D., 2001, “Theoretical and experimental investigation of the fast- and slow-scale instabilities of a dc/dc converter”, IEEE Transactions on Power Electronics, vol. 16, no. 2, pp. 201-216.
 Mazumder, S.K. and Acharya, K., 2006, “Multiple Lyapunov function based reaching condition analyses of switching power converters,” IEEE Power Electronics Specialists Conference, pp. 2232-2239.
 Mazumder, S.K., Pradhan, S., Hartvigsen, J., Rancruel, D., and von Spakovsky, M.R., 2007, “Investigation of load-transient mitigation techniques for planar solid-oxide fuel cell (PSOFC) power-conditioning systems”, IEEE Transaction on Energy Conversion, vol. 22, no. 2, pp. 457-466.
 Nesterov, Y. and Nemirovskii, A., 1995, “An interior-point method for generalized linear-fractional problems,” Springer Journal on Mathematical Programming: Series B, vol. 69, no. 1, pp. 177-204.
 Pradhan, S., Mazumder, S.K., Hartvigsen, J., and Hollist, M., 2007, “Effects of electrical feedbacks on planar solid-oxide fuel cell”, ASME Journal of Fuel Cell Science and Technology, vol. 4, no. 2, pp. 154-166.
 Pradhan, S., 2007, Modeling, analysis and control of effects of the electrical feedbacks on PSOFC power conditioning system, Doctoral Dissertation, Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, Illinois.
Journal Articles on this Report : 2 Displayed | Download in RIS Format
|Other project views:||All 3 publications||2 publications in selected types||All 2 journal articles|
||Balci K, Yapar G, Akkaya Y, Akyuz S, Koch A, Kleinpeter E. A conformational analysis and vibrational spectroscopic investigation on 1,2-bis(o-carboxyphenoxy) ethane molecule. Vibrational Spectroscopy 2012;58:27-43.||
||Mazumder SK, Pradhan S. Efficient and robust power management of reduced cost distributed power electronics for fuel-cell power system. Journal of Fuel Cell Science and Technology 2009;7(1):011018. (11 pp.).||