Compact Multi-domain Models for Control of Fuel Cell ... - CiteSeerX

WORKSHOP ON MODELING AND CONTROL OF COMPLEX SYSTEMS, 2005, AYIA NAPA, CYPRUS

1

Compact Multi-domain Models for Control of Fuel Cell Systems Anna G. Stefanopoulou, Loucas S. Louca

Abstract— Power generation from Fuel Cells (FC) requires the integration of chemical, fluid, mechanical, thermal, electrical, and electronic subsystems. In this presentation we highlight the development of a physics based model that enables the accurate prediction of water generation and flow through the FC membrane. The missing model parameters are identified from measurements of controlled experiments. This model accurately predicts the behavior of the FC and can be used for control design, however, its complexity can be prohibitive. To this end, a review of model reduction procedures and how they can be implemented to this model in order to develop a simpler yet accurate model is included. Index Terms— Fuel Cell, Reaction, Diffusion, Partially Distributed System, Process Modeling, Physical Based Modeling, Model Reduction.

I. I NTRODUCTION

E

LECTROCHEMICAL power generation offers new opportunities for stationary power generation and mobile applications such back-up power units or vehicle propulsion. Proton Exchange Membrane Fuel Cells (also known as polymer electrolyte membrane fuel cells, PEMFCs) utilize the chemical energy from the reaction of hydrogen and oxygen to produce electricity, water and heat. The hydrogen is typically supplied through processing natural gas, or generated from electrolysis. PEMFC have high power density, a solid electrolyte, and long life, as well as low corrosion. PEM fuel cells operate in the temperature range of 50 to 100 C which allows fast start-up and shut-down. Their deployment in regions with abundant renewable energy sources such as solar, wind, and wave or specialized geo-thermal energy could be the first step towards a sustainable environment. All these opportunities come with a broad spectrum of new challenges. The first challenge arises from the increased dimensionality in the design space from the integration of chemical, fluid, mechanical, thermal, electrical, and electronic subsystems. Our ability to precisely control the reactant flow and pressure, stack temperature, and membrane humidity is critical for the viability, efficiency, and robustness of the fuel cell stack system in real world applications. These critical FC parameters should be controlled for a wide range of current,

This work was supported in part by the National Science Foundation under Grant CMS-021332 and in part by grants from the U.S. Army Automotive Research Center (ARC). Anna G. Stefanopoulou is with the Department of Mechanical Engineering at the University of Michigan, Ann Arbor, MI 48109, USA (e-mail [email protected]). Loucas S. Louca is with the Department of Mechanical and Manufacturing Engineering at the University of Cyprus, 1678 Nicosia, CYPRUS (e-mail [email protected]).

and thus power, by a series of actuators such as valves, pumps, compressor motors, expander vanes, fan motors, humidifiers and condensers. The coupling of multiple physical domains requires an enormous effort in order to develop models that can be seamlessly used for design, automation, and fault diagnosis. The second challenge arises from the high parametric uncertainty due to the nature of cutting edge technologies that are neither robust nor durable as mature technologies. Hence, systematic assessment of performance within certain confidence bounds and uncertainty propagation are two complex and crucial tasks that require flexible simulation-based tools and fault tolerant controllers. In this paper we first present the difficulties in parameterization of lumped-parameter models that can describe the reactant, vapor and liquid dynamics in a proton exchange membrane fuel cell (PEMFC) stack. Computational fluid dynamic models are typically used for the representation of such complex phenomena. While useful for understanding fuel cell design issues, computational fluid dynamic models are difficult to calibrate for large stacks and too complex for control design. Thus, any model-based control scheme used for water management must adequately trade-off simplicity while still capturing the dynamic behavior of electrode flooding and two phase flow. To this end we augment our previous fuel cell stack model [1] with capillary pressure to calculate the flow and accumulation of liquid water on the catalyst surface that block the reactants and cause hydrogen starvation. The overall low-order model has 29 physically-motivated state equations with two tunable parameters identified using the overall stack voltage characteristics. Preliminary results are shown from an experimental 1.5 kW stack of 24 fuel cells. Then, based on the challenges and questions raised during the development of this model, we proceed on the investigation of the modeling assumptions and model complexity. This step is necessary given that the model will be used for controller design and any possible reduction in the model size would be beneficial. Model-order reduction methodologies are reviewed with emphasis on the methods that retain the physical parameters of the original physical model. The advantage of the reduced order model with physical parameters is its applicability to control design and parameter identification. The parameters maintained in the reduced order model can be measured (or identified) with higher precision. This procedure will reduce the time and cost to tune the model, increase its range of validity, and improve the accuracy of the model predictions.

WORKSHOP ON MODELING AND CONTROL OF COMPLEX SYSTEMS, 2005, AYIA NAPA, CYPRUS _ e

II. N OMENCLATURE

III. F UEL C ELL O PERATION Fuel cell stack (FCS) systems require the integration of chemical, fluid, mechanical, thermal, electrical, and electronic subsystems. Understanding the important physical variables and their underlying interactions is indispensable for the system design and the overall performance. In this section we present the principles of fuel cell operation with the goal to highlight the mechatronics and cross-disciplinary aspects of FCSs.

Fig. 1.

Fuel Cell Types

There are different types of fuel cells [2] distinguished mainly by the type of electrolyte used in the cells, namely, polymer electrolyte fuel cell, (PEMFC, also known as proton exhange membrane fuel cells), alkaline fuel cell (AFC), phosphoric acid fuel cell (PAFC), molten carbonate fuel cell (MCFC), and solid oxide fuel cell (SOFC) as shown in Figure 1. The differences in cell characteristics, cell material, operating temperature, and fuel, make each type of fuel cell

_ e O2 Gas

+ H

+ H H2O

H2

_ e

O2

gr ap hit e)

CATHODE

Fig. 2.

Fie ld (

mb ran e Me

H2O

Fra me (p las tic )

Fra me (p las tic ) Flo w Fie ld (gr ap hit e)

ANODE H2 Gas

Pla te (gr ap hit e)

_ e

Pla te (gr ap hit e)

Time derivatives are denoted as d( )/dt. Spatial derivatives through the gas diffusion layer (GDL) thickness in the membrane direction (y) are denoted as ∂( )/∂y, and are approximated spatially with finite differences. The English letter a denotes water activity, c is molar concentration (mol/m 3), j is used as an index , i is current density (A/cm2 ), I is current (A), n is the mole number, N is molar flux (mol/s/m 2 ), p is pressure (Pa), R is the ideal gas constant, RH2 O is the evaporation rate (mol/s/m 3 ), s is the fraction of liquid water volume to the total volume, S is the reduced water saturation, T is temperature (K), u is voltage (V), V is volume (m 3 ), x is molar ratio, and y is mass ratio. The Greek letter  is used for porosity, λ for water content, µ for viscosity, ρ for density, φ for humidity. The subscript an denotes variables associated with the anode, c for capillary, ca for cathode, ch for channel, ct for catalyst, e for electrode (an or ca), l for liquid, mb for membrane, rc for reactions, H 2 for hydrogen, N 2 for nitrogen, O2 for oxygen, w for water, v for vapor, and st for stack variables.

Load

_ e

Flo w

2

Fuel cell component description

suitable for different applications. Operating below or near the boiling temperature of water PEMFCs and AFCs rely on protons or hydroxyl ions as the major charge carriers in the electrolyte, whereas in the high-temperature fuel cells (MCFC and SOFC) carbonate ions and oxygen ions are the charge carriers. The ability of MCFC and SOFC to operate on carbonate ions and oxygen ions makes them fuel flexible. On the contrary, the PEMFC dependency on high-purity hydrogen reactant requires novel hydrogen generation and storage technologies. PEMFC have high power density, a solid electrolyte, and long life, as well as low corrosion [3]. PEM fuel cells operate in the temperature range of 50 to 100 C which allows fast start-up and shut-down. Due to their benefits and advanced stage of development, we concentrate on PEMFC in this paper. PEMFCs utilize the chemical energy from the reaction of hydrogen and oxygen (called from now on as fuel) to produce electricity, water and heat. As shown in Figure 2, fuel travels through inlet manifolds to the flow fields. From the flow fields, gas diffuses through porous media to the membrane. The membrane, sandwiched in the middle of the cell, typically contains catalyst and microporous diffusion layers along with gaskets as a single integrated unit. One side of the membrane is referred to as the anode, the other the cathode. The anode and cathode are more generally referred to as electrodes. The catalyst layer at the anode separates hydrogen molecules into protons and electrons (2H 2 ⇒ 4H+ + 4e− ). The membrane permits ion transfer (hydrogen protons), requiring the electrons to flow through an external circuit before recombining with protons and oxygen at the cathode to form water (O 2 + 4H+ + 4e− ⇒ 2H2 O). This migration of electrons produces electricity, which is the useful work. The overall reaction of the fuel cell is therefore 2H 2 + O2 ⇒ 2H2 O + Heat. The electrical characteristics of fuel cells are normally given in the form of a polarization curve, shown in Figure 3, which is a plot of cell voltage versus cell current density (current per unit cell active area) at different reactant pressures and flows. Stack temperature and membrane water content also affect the fuel cell voltage. The difference between the actual voltage and the ideal voltage 1 represents the loss in the cell which turns into heat. As more current is drawn from the 1 The ideal standard voltage for a fuel cell in which H and O react is 2 2 1.18 V when the resulting water product is in gaseous form.

STEFANOPOULOU & LOUCA: FUEL CELL MODELS

3

Cell Voltage (V)

1.2 1 0.8 0.6 0.4

Flow

0.2 0

Fig. 3.

Pressure 3 bar 2 bar 1 bar

0.4

0.8 1.2 1.6 Current Density(A/cm2)

Polarization curves for different cathode pressures.

fuel cell, the voltage decreases, due to fuel cell electrical resistance, inefficient reactant gas transport, and low reaction rate. Lower voltage indicates lower efficiency of the fuel cell, hence low load (low current) operation is preferred. Operation at low load requires a large fuel cell stack and has detrimental consequences to the overall volume, weight, and cost. Instead of over-sizing the FC stack, a series of actuators such as valves, pumps, blowers, expander vanes, fan motors, humidifiers and condensers are used to control critical FC parameters for a wide range of current, and thus, power setpoints. The auxiliary actuators are needed to make fine and fast adjustments to satisfy performance, safety and reliability standards that are independent of age and operating conditions [4]. The resulting multivariate design and control synthesis task, also known as balance of plant (BOP), is complex because of subsystem interactions, conflicting objectives, and lack of sensors. The main control tasks are: (i) reactant supply system, (ii) heating and cooling system, (iii) water management, and (iv) power management system.

Cell Control Laboratory (FCCL) at the University of Michigan. A computer controlled system coordinates air, hydrogen, cooling, and electrical subsystems to operate the PEMFC stack. Dry pure hydrogen is pressure regulated to replenish the hydrogen consumed in the chemical reaction. The hydrogen stream is dead ended with no flow external to the anode. Using a purge solenoid valve, hydrogen is momentarily purged through the anode to remove condensed water accumulating in the gas diffusion layers and flow channels. Humidified air is flow controlled, in excess of the reaction rate, to provide a supply of water vapor and oxygen at the cathode. Deionized water is circulated through the system to remove heat produced due to the exothermic chemical reaction. Measurements of dry gas mass flow delivered to the electrodes are taken along with the electrode inlet and outlet temperature, pressure and relative humidity. The coolant temperature is measured leaving the cells. Figure 4, displays the major experimental components along with the measurement locations. The experimental set-up allows the design, testing, and integration of real-time time software for simulation, optimization, control, and diagnostics in transient load conditions. S

S V

Hydrogen Tank

*

*

Purge Valve

Reservoir HX from ambient

+

* Measure T, P, RH + Measure T v Measure Flow

*

*

Humidifier

V Compressor

MFC

to ambient

Fuel Cell Stack

Fig. 4.

Experimental hardware and measurement locations

V. WATER M ANAGEMENT IV. E XPERIMENTAL S ET UP One of the most challenging characteristic in fuel cells is its spatially varying behavior depending on the local temperature and the gas composition at the membrane surface. Due to the complexity inherent with distributed parameter analysis, the geometric complexity of the stack design, as well as the difficulty associated with taking measurements at the membrane surface or within the electrodes of large multi-cell stacks [5], we use lumped parameter models calibrated with experimental data. Unfortunately, experimental data necessary for understanding, predicting, and controlling the unique transient behavior of PEMFC stacks are not easy to obtain. It is not easy for example to obtain data from industry or laboratories due to the confidential and competitive nature of the information. Also commercial FC units are typically bundled with closed architecture controllers that obstruct system identification techniques. To address the need for transient data and experimental validation of models and controllers a laboratory was established with partial funding from the National Science Foundation (CMS-0219623). A 24-cell, 300 cm 2 , 1.4kW PEMFC stack was purchased from the Schatz Energy Research Center (SERC) at Humboldt State University and installed at the Fuel

The ability of the membrane to conduct protons is fundamental to the PEMFC operation. This ability is linearly dependent upon its water content [6]. On one hand, as membrane water content decreases, ionic conductivity decreases [7], resulting in a decreased cell electrical efficiency, observed by a decrease in the cell voltage. This decrease in efficiency causes increased heat production which evaporates more water, in turn lowering membrane water content even more. The interaction between high temperature and low humidity creates a positive feedback loop. On the other hand, excessive water stored in the electrodes obstructs fuel flow, resulting in cell flooding [6]. In both cases, managing the water concentration in the electrodes is very important for increasing optimal fuel cell efficiency and extending the FC life. A water injection or an evaporation mechanism is used to control the humidity of the reactants and eventually the membrane hydration. Although, passive (internal) humidification concepts are rigorously pursued [8], [9], external active control allows wider range of operation typically met in automotive applications [4]. As current is drawn from the FC, water is generated in the cathode and water molecules are dragged from the anode to the cathode. This transfer of vapor is known as electroosmotic drag. Additionally, the vapor concentration

4

WORKSHOP ON MODELING AND CONTROL OF COMPLEX SYSTEMS, 2005, AYIA NAPA, CYPRUS

gradient causes diffusion of water through the membrane and is referred to as back diffusion. The magnitude and direction of the net vapor flow through the membrane (anode to cathode or cathode to anode) is a function of the relative magnitudes of these two transport mechanisms. Hence, perturbation in the FC humidity can be caused by different mechanisms as characterized in [10]: (i) the water generated during the load increase(current drawn from the FC), (ii) changes in absolute and relative reactant pressure across the membrane, (iii) changes in the air flow out of the fuel cell that carries vapor, and dries the membrane, and (iv) changes of the FC temperature, and thus, the saturation pressure. These mechanisms indicate strong and nonlinear interactions among the humidity control task, the reactant flow management loop, the heat management loop and the power management loop. The interactions are so strong that part of the hydrogen flow subsystem is dedicated to the water management in the anode. The anode is particularly vulnerable to flooding since it is dead-ended so it is prone in accumulating vapor and inert gas. Various ingenious mechatronic solutions have been proposed to abate anode flooding [11]. These investigations aim to optimize the inefficient practice of purging or recirculating the anode contents utilizing a downstream anode valve and a pump. Pointing to the complexity of the humidification task the authors in [12] note that the humidification components account for 20% of stack volume and weight. The stack, on the other hand, under-performs with 20%-40% lower voltage if there is no proper humidication control. VI. F UEL C ELL M ODEL OVERVIEW Figure 5 shows a flow chart of the calculation algorithm used to implement a two-phase discretized 1-dimensional fuel cell model. In the anode channel, a mixture of hydrogen and water vapor flow through the gas diffusion layer (GDL). In the cathode channel a mixture of oxygen, nitrogen, and water vapor are flowing. The concentration of species in the channel are assumed to be homogeneous and spatially averaged. Lumped-parameter isothermal dynamics based on conservation of mass are used to calculate the concentration of species in the channels. These time varying channel concentrations provide one set of boundary conditions for the spatially varying diffusion process of the reactant gases through the GDL. The reactant gases must diffuse through the GDL to reach the catalytic layer. n p The time derivatives of gas concentrations c j = Vpj = RTj for two general gas species j=A, Bare a function of the local molar flux gradients (∇N A and ∇NB ), and the local reaction rates RA and RB of the particular gas species (as in the case of vapor condensation) forming two partial differential equations (PDEs): dcA ∂NA = ∇NA + RA = + RA , (1a) dt ∂y dcB ∂NB = ∇NB + RB = + RB . (1b) dt ∂y For two gases diffusing in a mixture with a bulk (convective) flow, we first define the molar ratio of gas species j being

Time-varying boundary channel conditions

Membrane water transport

Time-varying boundary membrane/catalyst reactions

Wl Cell Voltage Fig. 5.

Flow chart of model calculation algorithm

xj = cj /c and the average gas velocity V = (N A + NB )/c. Then the total molar flux is a function of the average gas velocity xj cV , as well as the diffusive flux, described by: ∂cA + xA (NA + NB ) , (2a) ∂y ∂cB + xB (NA + NB ) . (2b) NB = −DBA ∂y The effective diffusivity of the GDL is a function of the porosity of the diffusion layer, , as well as the volume of liquid water present, V l , [13] : NA = −DAB

Di  = Di f ()g(s),

s=

Vl , Vp

(3)

where s is the liquid water saturation ratio, and V p is the pore volume of the diffusion layer. The impact of liquid water saturation on the effective diffusion is modeled in [13] as g(s) = (1
− s)2 while the impact of porosity is modeled as 0.785


−0.11 . f () =  1−0.11 As the fuel cell is producing current, water is produced at the cathode catalytic layer based on the following reaction equation:  Ist ξ = 1 for H2 and H2 O with (4) N( ),rct = ξ = 2 for O2 2ξF

where Ist is the current drawn from the stack and F is the Faraday constant. The net flux of vapor through the membrane N v,mb depends on the relative magnitudes of electro-osmotic drag and back diffusion. i (cv,ca,mb − cv,an,mb ) Nv,mb = nd − αw Dw (5) F tmb where i is the fuel cell current density (I st /Af c ), nd (λmb ) is used to calculate the vapor molar flux from the anode to

STEFANOPOULOU & LOUCA: FUEL CELL MODELS

Fig. 6.

5

Capillary flow of liquid water through diffusion layer [13]

the cathode due to electro-osmosis, D w (λmb , Tst ) is used to calculate the vapor molar flux from the cathode to the anode due to diffusion, and t mb is the membrane thickness. The parameter αw is identified using experimental data. All remaining parameters and coefficients are defined in [7]. Although there are many efforts to quantify back diffusion, conflicting results suggest an empirically data-driven identification of diffusion might be a practical approach to this elusive subject. The membrane water transport algorithm, thus, depends on an unknown parameter αw that scales the diffusion model in [7]. The parts of the algorithm that depend on identifiable parameters are indicated by a dashed line in Figure 5. When the production or transport of vapor overcomes the ability of the vapor to diffuse through the GDL to the channel, the vapor supersaturates and condenses. The volume of liquid water Vl in the GDL is calculated through the capillary water flow Wl and the evaporation rate R H2 O : ρl

dVl = Wl,in − Wl,out − RH2 O Vp Mv /. dt

(6)

The condensed liquid accumulates in the diffusion layer until it has surpassed the immobile saturation limit s im at which point capillary flow will carry it to an area of lower capillary pressure (the GDL – channel surface). This process causes the liquid to flow across the diffusion media and be ejected into the channel (see Figure 6 taken from [13]). Liquid water flow through the gas diffusion layer is a function of the capillary pressure gradient [13], [14],    ∂S Af c nρl KKrl dpc Wl = − , (7) 10000µl dS ∂y  2 3 +1.263S ]/ (K/) is where pc = σ cos θc [1.417S −2.120S
 s−sim capillary pressure, S = max 0, 1−s is the reduced water im saturation and all other constants have been identified in [13]. Finally, the molar evaporation rate based on [13] is RH2 O = γ

pv,sat − pv , RT

pv = cv RT ,

(8)

where γ is the volumetric condensation coefficient. When the partial pressure of vapor is greater than the saturation pressure, RH2 O is negative, representing the condensation of water. A logical constraint must be included such that if no liquid water is present, RH2 O ≤ 0.

Liquid water in the GDL occupies some of the pore space, reducing the effective area through which reactant gas can diffuse and increasing the tortuosity of the diffusion path. This obstruction ultimately reduces the reactant partial pressures at the catalyst layers, in turn lowering the power output of the stack (voltage at a given current). For a given quantity of water, water condensing in the anode should have a less pronounced effect on the supply of hydrogen to the catalyst than water condensing in the cathode would have on the supply of oxygen because hydrogen has a greater diffusion coefficient than oxygen. If the liquid that reaches the surface of the GDL is not evaporated, it will cover the surface increasing the cell current density. This effect is not easily modeled because the surface roughness makes it difficult to predict how much surface area of the GDL is blocked by a given volume of water. For this reason we chose to identify through experimental data the thickness of the liquid film, and consequently determine the active catalyst area blocked by the liquid water flowing out of the GDL. The location of the second identifiable parameter within the overall model can be traced through the second dashed line in Figure 5. VII. D ISCRETIZATION AND PARAMETERIZATION The mass transport of gas and liquid water can be solved when the gas diffusion layer is split into discrete volumes. Refer to Figure 7 which shows an arbitrary 3rd order discretization. Each sub-volume in the diffusion layer is assumed to contain a homogenous solution of gases, and liquid water. The gradients used are then solved as difference equations, while the time derivativesmay still be solved with classical ODE solvers. For the purposes of model simplification, the concentration of nitrogen in the cathode diffusion layer is assumed to be identical to the concentration in the channel. The lumped-parameter two-phase flow model developed can be indirectly validated through model prediction of the effects of flooding on stack voltage. Specifically, we concentrate on model validation during anode flooding events. Specific operating conditions can be tested for conditions leading to cathode flooding. However, at moderate current densities (< 0.5 A/cm2 ) and cell operating temperatures (≈ 60 o C) along with the absence of humidification introduced in the hydrogen gas stream, back diffusion dominates drag in our experimental setup resulting in anode flooding. The accumulation of liquid water in the gas channel and diffusion layer on the anode is typically the dominant reason for voltage degradation. The occurrence of anode flooding is experimentally confirmed by a purging event. Following an anode purge, the voltage significantly recovers. Under the same testing conditions and voltage degradation, surging the cathode has little effect. Anode flooding is a result of the back diffusion dominating drag, resulting in a net vapor flux from the cathode to the anode through the membrane. However, implementing the mathematical model and the numerical values published by Springer et al [7], our model greatly under-estimates the diffusion effects and cannot be directly used to predict anode flooding. Instead, we introduce a parameter α w that multiplies

WORKSHOP ON MODELING AND CONTROL OF COMPLEX SYSTEMS, 2005, AYIA NAPA, CYPRUS

1

1 c O2,c(1) c v,c(1) c N2,c(1) V,L,c(1)

2

3

c O2,c(2) c O2,c(3) c O2,c(ch) c v,c(2) c v,c(3) c H2O,c(ch) c N2,c(2) c N2,c(3) c N2,c(ch) V,L,c(2) V,L,c(3)

N v,gen N v,mem N v,a(3)

N v,a(2) N v,a(1)

N v,c(1)

N v,c(2) N v,c(3)

N H2,a(3) N H2,a(2) N H2,a(1)

N O2,c(1) N O2,c(2) N O2,c(3)

W L,a(3) W L,a(2) W L,a(1)

W L,c(1) W L,c(2) W L,c(3)

N H2,rct

N O2,rct

W ca,out

Anode Channel

Fig. 7.

Anode GDL

Membrane

Cathode GDL

Cathode Channel

Mass Transport Diagram with discretization of diffusion layer

the diffusion of Springer et al. The parameter α w corresponds to the lumped “stack”-level membrane diffusion that needs to be identified using experimental data. Once anode flooding occurs, we postulate that the resulting voltage degradation arises from the accumulation of liquid mass in the GDL, mlan (3). The accumulated liquid mass is assumed to form a thin film, blocking part of the active fuel cell area Af c and consequently increasing the lumped current density, defined as apparent current density i app = Ist /Aapp in A/cm2 where the apparent fuel cell area A ap is approximated as the fuel cell area reduced by the liquid film generated at the interface of the anode channel and GDL Aapp = Af c (1 − mlan (3)/(nρl twl Af c /104 ))

Inputs for I90L300T60P3Dec18041714.xls 90 85

st

2

I (A)

3

c H2,a(ch) c H2,a(3) c H2,a(2) c H2,a(1) c v,a(1) c H2O,a(ch) c v,a(3) c v,a(2) V,L,a(3) V,L,a(2) V,L,a(1)

Ist . To fit anode and cathode variables in one plot, shifted inlet anode and cathode measured pressures (p an,ch (t)−pan,ch (t = 0) and pca,ch (t) − pca,ch (t = 0)) are shown in subplot 2. Similarly the shifted air and hydrogen flow are shown in subplot 3. The pressure and flow excursions observed in the anode occur after an anode purge is initiated by a solenoid valve downstream of the stack as shown in Figure 4. The purge is scheduled every 180 sec for 3 sec. The air mass flow in the cathode inlet Wca,in was controlled at 300% stoichiometry for this experiment. Finally, the coolant temperature out of the stack is shown in subplot 4. The coolant temperature is regulated thermostatically through a heat exchanger by an onoff fan around a desired set-point. At t=57 sec the desired set point was set from 50 C to 60 C and the stack heats up under its load. We name this experiment as I90L300T60P3 due to the fact that the current drawn I st =90 A (i = Ist /Af c =0.3 A/cm2 ) at 300% air stoichiometry, temperature T f c =60 C, and anode pressure pan,ch =3 psig for the majority of the time. Figure 9 shows the average current density i that is used to calculate the molar flux gradients in the GDL next to the catalyst. The solid line in the same subplot corresponds to the calculated apparent current density i app based on the effective area (9) that is not blocked by the liquid water film. The apparent current density is used to calculate the cell overpotential. Subplot 2 shows the voltage value for all 24 cells in the stack (thin lines) and the predicted model voltage (thick line). It is clear that when the apparent current density increases, the predicted voltage decreases matching the measured cell voltages.

∆ Pres (bar)

W c,in

80

with n=24 cells in the stack, ρ l =998 kg/m the water liquid density, and t wl the unknown liquid film thickness identified using experimental data. The liquid film thickness t wl and the diffusion multiplier α w are the tunable parameters which are identified by comparing the predicted voltage vˆf c (iapp , pO2 (1), pH2 (1), Tst , λmb ) from ( [1] eq.(3.21)) with the measured average cell voltage v¯ f c . We use one set of experiments to perform a nonlinear least t squares fitting technique that minimizes J = exp (¯ vf c (τ ) − T vˆf c (τ )) (¯ vf c (τ ) − vˆf c (τ ))dτ . A selected section of one experiment is used to identify the two parameters. A different experiment is shown in Figures 8-9 to demonstrate the model predictive capability. VIII. M ODEL P REDICTION The experimental data and model predictions are shown in Figures 8-9. The data collected are different from the data used for calibration. Figure 8 shows all the model inputs. In particular, subplot 1 shows the total current drawn from the stack

200

300

400

500

600

700

800

100

200

300

400

500

600

700

800

900

−0.02 −0.04

(9)

3

100 0

∆ Flow (g/s)

W a,in

900 CaIn AnIn

0.3 0.2 0.1 0

T (C)

6

100

200

300

400

500

600

700

800

900

100

200

300

400

500 time (s)

600

700

800

900

60 58 56 54 52

Fig. 8. Measurements used as model inputs for one experiment that exhibits anode flooding

Although the voltage prediction is an indirect means for evaluating the overall predictive ability of our model, voltage is a stack variable that combines the internal states of the stack provides an accessible, cheap, fast and accurate measurement. The model presented predicts the increase in liquid volume V l,an/ca that consequently decreases reactant diffusion, followed by an increase of the blocked active area, in turn increasing the apparent current density, finally reflected

STEFANOPOULOU & LOUCA: FUEL CELL MODELS

7

in a decrease in cell voltage. The model accurately predicts the voltage recovery after an anode purging event. Moreover the model predicts the increase in overpotential during a step change in current from 75 to 90 A in the beginning of the experiment. It is noteworthy that the predicted voltage shows the effects of (a) the instantaneous increase in current (static function) and (b) the excursion in partial pressure of oxygen due to the manifold filling dynamics as indicated by the voltage overshoot during the current step. Finally the model predicts the effects of temperature in the voltage as shown during the temperature transient from 50 to 60 C. Higher temperature improves the cell voltage through the static polarization function. At the same time, increase in temperature helps evaporate some of the formed liquid as indicated by the trend in the predicted iapp . Consequently, temperature affects the voltage through a dynamic path. I90L300T60P3Dec18041714.xls selected outputs i iapp

Current Dens (A/cm2)

0.36 0.34 0.32 0.3 0.28 0.26 0

100

200

300

400

500

600

700

800

900

100

200

300

400

500 time (s)

600

700

800

900

V Cell & Mdl

700 680 660 640 620

Fig. 9. Measurements used as model outputs for one experiment that exhibits anode flooding. The thin voltage lines correspond to the measured voltage of the 24 cells

IX. I S THIS A G OOD M ODEL ? The validation described above indicates that the model captures the system dynamical behavior between all controllable inputs and important measured outputs. Model-based control techniques for air and hydrogen flow can now be designed and tuned. Optimal control of the reactant flow can manage the channel vapor concentration, thus affect the evaporation rate, and consequently mitigate flooding or dry conditions. Although such a model-based control design is indispensable, there are many questions that arise regarding the model complexity and sensitivity. In particular, the developed model consists of 29 continuous ordinary differential equation after the discretization of the GDL shown in detail in [15]. Although the computational burden is not excessive, it would be better to have a lower order model. Moreover, it is not clear at this point if the model predictive ability improves if more GDL volumes are

included in the discretization. Finally, the parameterization results indicate a small redundancy in the two identified parameters αw and twl . Simpler model could point to better parameterization or experiments that ensure identifiability. X. M ODEL O RDER R EDUCTION Physics-based models provide accurate representation and are important for the understating of the fuel cell behavior. Such approach usually results in complicated models that may include physical phenomena and details that are unnecessary for specific tasks. More specifically, the complexity of models becomes an important consideration when the model is to be used in everyday engineering tasks like control design or design optimization. Models with large number of states or high computational cost become prohibitive to use in such cases. Model complexity is a major consideration in the area of automatic control. Controller design methods favor low order plant models (number of states, measurements, etc.) in order to design robust controllers, and therefore, simple and low order models are desired. When a low order mathematical model (typically a transfer function) that accurately predicts the system behavior is found, the model complexity problem is considered solved. A successful example can be found in [16] where the power output of a fuel cell stack was approximated by a static nonlinearity and an impedance characteristic. This simple model was then used for coordination the power from the fuel cell and a battery in a military truck auxiliary power unit application. A reduced model with physical description (states and/or parameters) provides useful information during the controller tuning since we have a better understanding of the plant behavior. Such models would also be desirable for design studies where the model parameters need to be related to design parameters and state variables related to the performance specifications. Various approaches are used in the controls community to generate low order models from larger order models. The premise is that simple but accurate models can be determined from a fully augmented baseline model [17]–[19]. These techniques use linear coordinate transformations of the state vector in order to obtain a smaller size state vector (reduced) that results the same input-output characteristics. Very often the physical meaning of the model could be lost, i.e., the states and parameters of reduced models do not retain their physical meaning. Examples were a balanced realization pointed to the important states and lead to an efficient and robust controller calibration for hydrogen generation in fuel cell systems can be found in [1], [20] following the methodology in [21]. Linear matrix inequalities (LMIs) have also been successfully used to approximate multidimensional and uncertain systems [22]. The truncation of discretized partially distributed systems (PDS) is studied in [23]. Proper orthogonal decomposition (POD) in [24] and Galerkin projections in [25] with different time scales as in [26] can produce significant reduction in the size (number of states) of nonlinear partially distributed models. A variety of algorithms have been developed to help automate the production of dynamic system models that are accu-

8

WORKSHOP ON MODELING AND CONTROL OF COMPLEX SYSTEMS, 2005, AYIA NAPA, CYPRUS

rate yet simple but maintain their physical meaning (Proper Models). The most recent algorithm is a model reduction approach that uses an energy-based metric to produce proper models [27]–[29]. This approach can be applied to nonlinear as well as linear dynamic systems and considers the importance of energetic elements (inertial, compliant and resistive). The concept of a proper model was introduced in [30] who defined it as the model with the minimal set of physical parameters required to predict dominant system dynamics - along with an algorithm that deduces the appropriate system model complexity. Additional work on generating proper models has produced a series of algorithms for addressing different aspects of the model complexity problem [31]–[33]. The approach is a specialization of the more general reduction approach of inertial manifolds [34] and balancing for nonlinear systems [35]–[37]. Proper models are generally computationally more efficient, which is beneficial for design optimization where models are evaluated numerous times. In addition, they provide insight into the system for identifying the critical parameters and assisting the design process. A. Activity-Based Model Reduction The original work on energy-based metrics for model reduction [27], [29] is briefly presented in order to provide motivation for the implementation of this methodology to the fuel cell model. The reduction procedure starts with a high-fidelity (full) model, and then based on an energy-based metric removes physical phenomena that are identified to be insignificant. To maintain the proper model properties the reduction proceeds by removing physical phenomena or/and quantities from the model (e.g., damper, mass, etc.). By operating on the physics of the model, the process yields a physically meaningful system model even after reduction. This approach is in contrast with previously mentioned model reduction techniques that are primarily used in the automatic control area, i.e., states and parameters of the reduced models retain their physical significance. This enhances the usefulness of the reduced model for design analyses since the design parameters are explicit in the reduced model. The elimination of physical phenomena requires the identification of those elements that are not important, and therefore do not have a significant contribution to the overall system behavior. The power associated with each element in the system provides an indication of an elements contribution to the total behavior of the system (at least as far as energy is concerned). An element with high power associated with it, stores (energy storage element) or absorbs (dissipative element) a sizeable portion of the power that is supplied into the system, and therefore contributes significantly to the system behavior. However, because power is time dependent, its use as a modeling metric would lead to varying instantaneous estimates of elements importance. In order to eliminate time dependency, the absolute value of power is integrated over time to a scalar metric:  T |P(t)|dt (10) A= 0

where P(t) is the element power and T is the time window over which the activity is calculated. This scalar metric has units of energy and represents the amount of energy that flows in and out of the element over the time window T . The energy that flows in and out of an element is a measure of how active this element is (how much energy passes through it), and consequently the quantity in Equation (10) is termed activity. It is assumed that the higher the activity, the higher the contribution of the element to the overall system behavior. The activity is calculated for each energy element based on the system response. The activity, as defined above, provides a measure of absolute importance of the elements. To provide a relative measure, the total activity (A total ) of the model, which represents the total amount of energy that flows in and out of the system over the period T , is defined. The total activity is then used to calculate a normalized metric called element activity index or just activity index. The activity index AI i , as shown in Equation (11), is calculated for each element in the model and represents the portion of the total system energy that is flowing through that element. T |Pi (t)|dt Ai AIi = = k 0  T Atotal |Pj (t)|dt

i = 1, . . . , k

(11)

j=1 0

Ai is the activity of the i th element as given by Equation (10) and k is the total number of energy elements in the model. Note that for highly nonlinear systems the time response and quantities needed for calculating the activity index must be obtained numerically. First, numerical integration of the model produces the time response of the system states along with the necessary outputs (generalized effort and flow of all energy elements) for calculating the power. Then, the activities can be calculated using the expressions in Equation (11). With the activity index defined as a relative metric for addressing element importance, the Model Order Reduction Algorithm (MORA) is constructed. The first step of MORA is to calculate the activity index for each element in the model for a given system excitation and initial conditions. Next, the activity indices are sorted to identify the elements with high activity (most important) and low activity (least important). With the activity indices sorted, the model reduction proceeds given some engineering specifications, which are defined by the modeler who then converts them into a threshold of the total activity (e.g., 99%) that he/she wants to include in the reduced model. This threshold defines the borderline between the retained and eliminated model elements. The elimination process is shown in Figure 10 where the sorted activity indices are summed starting from the most important element until the specified threshold is reached. The element which, when included, increments the cumulative activity above the threshold, is the last element included in the reduced model. The elements that are above this threshold are eliminated from the model. The elimination of the low activity elements is done by removing these low activity elements from the model description.

STEFANOPOULOU & LOUCA: FUEL CELL MODELS

9

100

Threshold

we can start with a higher number of segments and retain only the important segments based on the activity metric.

Act ivit y Index [ %]

80

Removed Element s 60

ACKNOWLEDGEMENTS We thank the graduate research assistants D. McKay and B. McCain of the Fuel Cell Control Laboratory at the University of Michigan.

Cumulat ive Index 40

Sort ed Index 20

R EFERENCES 1

2

3

k-2

k-1

k

Element Ranking

Fig. 10.

Activity Index Sorting and Element Elimination

B. Discussion While some outcomes of the model reduction procedure may seem intuitive to an expert in a specific area, they provide critical information to a modeler with less domain expertise. In addition, identifying the elements that contribute the most for a highly transient maneuver is in general not a trivial task. Generating the proper model with MORA provides critical information to the engineer. Perhaps the most readily assessable benefit is that the activity identifies the important parameters (elements) relative to a particular scenario. Thus, even if the model is not reformulated into reduced form, having a rank ordered list of the parameter importance directs the designer towards the design features that can produce the greatest effect on the system. Knowledge of parameter importance eliminates the need to perturb otherwise unimportant model parameters. Specifically to the fuel cell design where we have a complicated systems with many parameters, the activity metric can identify only a smaller number of parameters (i.e., have high activity) that have the highest contribution to the fuel cell behavior, e.g., efficiency, transients, etc. Another benefit of ranking parameter importance is that the time and effort (cost) of obtaining model parameters can be reduced. A full model can be created with low precision parameter values. Then, upon learning which parameters are important, resources can be directed into obtaining higher precision values of the important parameters. This is again of particular importance to the fuel cell modeling, where parameters are very difficult to measure or even estimate. This reduces the time, effort and cost in order to develop an accurate model that at the same time has the minimum needed complexity. Another modeling issue that needs to be addressed more systematically is the discretization of the GDL. For the solution and analysis of the gas and liquid water transport in the diffusion layer a 3rd order discretization is selected. This was an ad hoc decision that was based on experience and not any modeling methodology. The appropriate number of segments that the GDL has to be divided into, can be addressed with a model reduction or deduction approach. With the deduction we can identify the appropriate number of segments by starting simple and then increase the number of segments until we reach the desired accuracy. Also, with the reduction approach

[1] J. T. Pukrushpan, A. G. Stefanopoulou, and H. Peng, Control of Fuel Cell Power Systems: Principles, Modeling, Analysis and Feedback Design. New York: Springer, 2000. [2] U.S. Department of Energy, Office of Fossil Energy, and National Energy Technology Laboratory, Fuel Cell Handbook. EG&G Technical Services, Inc, Science Application International Corporation, 2002. [3] J. Larminie and A. Dicks, Fuel Cell Systems Explained. West Sussex, England: John Wiley & Sons Inc, 2000. [4] W.-C. Yang, B. Bates, N. Fletcher, and R. Pow, “Control challenges and methodologies in fuel cell vehicle development,” SAE Paper 98C054, 1998. [5] M. M. Mench, Q. L. Dong, and C. Y. Wang, “In situ water distribution measurements in a polymer electrolyte fuel cell,” Journal of Power Sources, 2003. [6] T. Zawodzinski, C. Derouin, S. Radzinski, R. Sherman, V. Smith, T. Springer, and S. Gottesfeld, “Water uptake by and transport through nafion 117 membranes,” Journal of the Electrochemical Society, 1993. [7] T. E. Springer, T. A. Zawodzinski, and S. Gottesfeld, “Polymer electrolyte fuel cell model,” Journal of the Electrochemical Society, vol. 138, no. 8, pp. 2334–2342, 1991. [8] D. Bernardi, “Water-balance calculations for solid-polymer-electrolyte fuel cells,” Journal of Electrochemical Society, vol. 137, no. 11, pp. 3344–3350, 1990. [9] M. Watanabe, H. Uchida, Y. Seki, M. Emon, and P. Stonehart, “Self hymidifying polymer electrolyte membranes for fuel cells,” Journal of the Electrochemical Society, vol. 143, no. 12, pp. 3847–3852, 1996. [10] D. A. McKay and A. G. Stefanopoulou, “Parameterization and validation of a lumped parameter diffusion model for fuel cell stack membrane humidity estimation,” in IEEE Proceedings of 2004 American Control Conference, 2004. [11] P. Rodatz, A. Tsukada, M. Mladek, and L. Guzzella, “Efficiency improvements by pulsed hydrogen supply in PEM fuel cell systems,” in Proceedings of the 15th IFAC Triennial World Congress, 2002. [12] F. B¨uchi and S. Srinivasan, “Operating proton exchange membrane fuel cells without external humidification of the reactant gases,” Journal of Electrochemical Society, vol. 144, no. 8, pp. 2767–2772, 1997. [13] J. H. Nam and M. Kaviany, “Effective diffusivity and water-saturation distribution in single- and two-layer pemfc diffusion medium,” Int. J Heat Mass Transfer, vol. 46, pp. 4595–4611, 2003. [14] M. Kaviany, Principles of Heat Transfer in Porous Media, second ed. New York: Springer, 1999. [15] W. T. Ott, D. A. McKay, and A. G. Stefanopoulou, “Modeling, parameter identification, and validation of water dynamics for a fuel cell stack,” to appear 2005 Proc of ASME International Mechanical Engineering Congress & Exposition IMECE2005-81484. [16] A. S. J. P. B. K. Z. Filipi, L. Louca and H. Peng, “Fuel cell apu for silent watch and mild electrification of a medium tactical truck,” SAE Transactions - Journal of Engines, 2004. [17] B. C. Moore, “Principle component analysis in linear systems: Controllability, observability, and model reduction,” IEEE Transactions on Automatic Control, vol. AC-26/1, pp. 17–32, 1981. [18] R. E. Skelton and A. Yousuff, “Component cost analysis of large scale systems,” International Journal of Control, vol. 37, no. 2, pp. 285–304, 1983. [19] D. G. Meyer and S. Srinivasan, “Balancing and model reduction for second-order form linear systems,” IEEE Transactions on Automatic Control, vol. 41, no. 11, pp. 1632–1644, 1996. [20] V. Tsourapas, A. G. Stefanopoulou, and J. Sun, “Dynamics, optimization and control of a fuel cell based combined heat power (chp) system for shipboard applications,” Proc.of American Control Conference, 2005. [21] A. H. Chaudhry, B. A. Francis, and G. J. Rogers, “Retaining states of the original system in reduced order model,” IEEE Engineering Society, Winter Meeting, vol. 1, pp. 532–536, 1999.

10

WORKSHOP ON MODELING AND CONTROL OF COMPLEX SYSTEMS, 2005, AYIA NAPA, CYPRUS

[22] C. L. Beck, J. Doyle, and K. Glover, “Model reduction of multidimensional and uncertain systems,” IEEE Transactions on Automatic Control, vol. 41, no. 10, pp. 1466–1477, October 1996. [23] M. J. Balas, R. Quan, R. Davidson, and B. Das, “Low-order control of large aerospace structures using residual mode filters,” ASME, Dynamic Systems and Control Division (Publication) DSC, vol. 10, pp. 157–165, 1988. [24] S. Y. Shvartsman, C. Theodoropoulos, R. Rico-Martinez, I. G. Kevrekidis, E. S. Titi, and T. J. Mountziaris, “Order reduction for nonlinear dynamic models of distributed reacting systems,” Journal of Process Control, vol. 10, no. 2, pp. 177–184, April 2000. [25] A. Armaou and P. D. Christofides, “Computation of empirical eigenfunctions and order reduction for nonlinear parabolic PDE systems with time-dependent spatial domains,” Nonlinear Analysis, Theory, Methods, and Applications, vol. 47, no. 4, pp. 2869–2874, August 2001. [26] V. R. Saksena, J. O’Reilly, and P. V. Kokotovic, “Singular perturbations and time-scale methods in control theory: Survey 1976-1983,” Automatica, vol. 20, no. 3, pp. 273–293, May 1984. [27] L. S. Louca, J. L. Stein, G. M. Hulbert, and J. K. Sprague, “Proper model generation: An energy-based methodology,” in Proceedings of the 1997 International Conference on Bond Graph Modeling, Phoenix, AZ, January 1997, pp. 44–49, ISBN 1-56555-103-6, SCS, San Diego, CA. [28] L. S. Louca, “An energy-based model reduction methodology for automated modeling,” Ph.D. dissertation, The University of Michigan, Ann Arbor, MI, 1998. [29] L. S. Louca, J. L. Stein, and G. M. Hulbert, “A physical-based model reduction metric with an application to vehicle dynamics.” Enschede, The Netherlands: The 4th IFAC Nonlinear Control Systems Design Symposium (NOLCOS 98), 1998. [30] B. H. Wilson and J. L. Stein, “An algorithm for obtaining proper models of distributed and discrete systems,” in Transactions of the ASME, Journal of Dynamic Systems, Measurement, and Control, vol. 117, no. 4, New York, NY, 1995, pp. 534–540. [31] J. B. Ferris, J. L. Stein, and M. M. Bernitsas, “Development of proper models for hybrid systems,” in Proceedings of the 1994 ASME International Mechanical Engineering Congress and Exposition - Dynamic Systems and Control Division, Symposium on Automated Modeling: Model Synthesis Algorithms, Chicago, IL, November 1994, pp. 629– 636, book No. G0909B. [32] J. B. Ferris and J. L. Stein, “Development of proper models for hybrid systems: A bond graph formulation,” in Proceedings of the 1995 International Conference on Bond Graph Modeling, Las Vegas, NV, January 1995, pp. 629–636, ISBN 1-56555-037-4, SCS, San Diego, CA. [33] D. G. Walker, J. L. Stein, and A. G. Ulsoy, “An input-output criterion for linear model deduction,” in Transactions of the ASME, Journal of Dynamic Systems, Measurement, and Control, vol. 122, no. 3, 2000, pp. 507–513, ISSN 0022-0434, New York, NY. [34] P. D. Christofides and P. Daoutidis, “Finite-dimensional control of parabolic PDE systems using approximate inertial manifolds,” Journal of Mathematical Analysis and Applications, vol. 216, no. 2, pp. 398–420, December 15 1997. [35] J. M. A. Scherpen, “Balancing for nonlinear systems,” Systems & Control Letters, vol. 21, no. 2, pp. 143–153, August 1993. [36] J. Hahn and T. F. Edgar, “Balancing approach to minimal realization and model reduction of stable nonlinear systems,” Industrial and Engineering Chemistry Research, vol. 41, no. 9, pp. 2204–2212, May 1 2002. [37] S. Lall, J. E. Marsden, and S. Glavaski, “A subspace approach to balanced truncation for model reduction of nonlinear control systems,” International Journal of Robust and Nonlinear Control, vol. 12, no. 6, pp. 519–535, May 2002.