- Email: [email protected]
Sensitivity analysis of temperature uncertainty in an aircraft PEM fuel cell
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
Available online at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/he
Sensitivity analysis of temperature uncertainty in an aircraft PEM fuel cell G. Correa a,*, F. Borello b, M. Santarelli a a b
Politecnico di Torino, Dept. of Energetics, Corso Duca degli Abruzzi 24, 10129 Torino, Italy Politecnico di Torino, Dept. of Aerospace Engineering, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
article info
abstract
Article history:
Nowadays, growing interest in the application of renewable energy sources has led to
Received 12 June 2011
a rapid development of more environmentally sustainable and green thinking applications.
Received in revised form
The application of fuel cell technology to aircraft propulsion and/or auxiliary energy supply
1 August 2011
is becoming of great interest for undoubted advantages in terms of pollution emissions,
Accepted 10 August 2011
noise reduction and lower dependency on oil price and availability. In 2006 European
Available online 25 September 2011
Commission founded the ENFICA-FC project, coordinated by Politecnico di Torino, whose aim was to develop new all-electric and more-electric aircraft concepts and to directly
Keywords:
prove the feasibility of such designs by flying an all-electric general aviation aircraft
Aircraft Pem fuel cell
powered by hydrogen fuel cells. Reliability of fuel cell system is a very important aspect for
Sensitivity analysis
the safety of these aircraft, but experimental data concerning fuel cell systems in aero-
Temperature uncertainty
nautics are still unavailable to scientific community. This paper presents the research
Dynamic model
activity done by authors to support the design of “fuel cell aircrafts” from the failure analysis point of view; classic approaches to structural reliability are here applied to rank the importance of the failures of sensors used by fuel cell control logic in order to support the definition of accurate Failure Mode, Effects and Criticality Analysis (Risk matrix). Copyright ª 2011, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
1.
Introduction
Climate change and pollution are serious global concerns. Although it accounts for only 4.2% of the total global warming potential, the concern today is that aviation-generated C02 is projected to grow to approximately 5.7% by 2050 [12], which is accompanied by the current global travel demand increase.
Growing interest in the application of renewable energy sources has led to an increase of interest in the deployment of fuel cell (FC) technologies. The high efficiency and environmental advantages that fuel-cell technology could offer, have generated an increasing interest from the aviation community in fuel cell powered aircrafts, overall due to a successful experience along with a considerable
Abbreviations: Amb, Ambient; BoP, Balance of Plant; CFD, Cumulative Distribution Function; Conv, Convective; cv, Control volume; el, Electric; FC, Fuel cell; FHA, Functional Hazard Assessment; FMEA, Failure Mode and Effects Analysis; FMECA, Failure Mode, Effects and Criticality Analysis; FTA, Fault Tree Analysis; HEX, Heat Exchanger; MC, Monte Carlo; Mem, Membrane; NTU, number of transfer units; Nu, Nusselt number; PDF, Probability distribution function; RSM, Response surface method; st, Stack; VLA, Very Light Aircraft; wc, Coolant (deionizer water). * Corresponding author. Tel.: þ39 115644487. E-mail addresses: [email protected] (G. Correa), [email protected] (F. Borello), [email protected] (M. Santarelli). 0360-3199/$ e see front matter Copyright ª 2011, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijhydene.2011.08.036
14746
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
Symbols Ac C,Cp E0 E G g h hconv i I Kh le m nc OCV T U Vc Wel
2
Cell active surface, cm Specific heat, J/(kg K) Open Circuit Voltage, volts Internal energy of the stack, J Mass flow rate, kg/s Gibbs free energy, W Mass specific enthalpy of the mass flow, J/kg Convective heat transfer coefficient, W/(m2 K) Current density, A/cm2 Current, A Parameter of the heat transfer coefficient model Electric work produced by the stack, W Total mass, kg Number of cells in series in the PEMFC stack Open Circuit Voltage, V Temperature, K Overall heat transfer coefficient, W/(m2 K) Cell voltage, V Electric power produced by the stack, W
Greek letters F Heat transfer, W DH Enthalpy flow rate, W
development achieved in the ground transportation sector in the last ten years. Fuel cells system could become the main power source for small general aviation aircraft or could replace APU (Auxiliary Power Unit) on larger aircraft, to obtain all-electric or moreelectric machines. There are several potential advantages of using such a power source that range from environmental and economic issues to performance and operability aspects. The most stringent requirements derived from aeronautical applications are weight reduction, safety, and reliability. This determines tremendous challenges for the integration of fuelcell power plants in aircraft: maintaining the weight and balance of the aircraft, designing the thermal management system, managing the power flows of the hybrid power source when the fuel cell works in parallel with an additional power source, and ensuring a safe operation in a potentially flammable atmosphere. The main airframe manufacturers have started to investigate their applications because of their potential for improved performance and environmental compatibility. Regarding the use of fuel-cell systems for propulsion of manned aircraft, only three projects are currently underway. These are the Boeing fuel-cell demonstrator airplane [1,13], the Antares DLR-H2 fuel cell aircraft project [3] [4], and the Environmentally Friendly Inter-City Aircraft Powered by Fuel Cells (ENFICA-FC) project [23]. The introduction of this type of concept aircrafts with specific items and novel features installed on-board that increase the complexity of their system, became more important the analysis of the Safety (reliability) of these category airplanes. Conventionally, the system failure analysis in the energy system development is treated deterministically. The main
ε h
Efficiency, over-voltage, V
Abbreviations Amb Ambient BoP Balance of Plant CFD Cumulative Distribution Function Conv Convective cv Control volume el Electric FC Fuel cell FHA Functional Hazard Assessment FMEA Failure Mode and Effects Analysis FMECA Failure Mode, Effects and Criticality Analysis FTA Fault Tree Analysis HEX Heat Exchanger MC Monte Carlo Mem Membrane NTU number of transfer units Nu Nusselt number PDF Probability distribution function RSM Response surface method st Stack VLA Very Light Aircraft wc Coolant (deionizer water)
objective of this paper is mainly to show the possible application of an innovative methodology capable of quantifying the damage produced by the miss o malfunction of systems/ elements in the system response, through the application of the validated dynamic model of the FC-based plant, using analysis of uncertainties and sensitivity. In the first part of this paper is presented an analytical Proton Exchange Membrane (PEM) fuel cell stack model that describes its performance based on physical system inputs, taking into account the main physical concepts of the electrochemical, chemical and energy behavior. This model will include the fulfilled lumped capacitance model of the thermal management subsystem. This is developed to describe the temperature dynamics of the system based on the system inputs (power required and ambient temperature). The model is intended to be used for thermal control, performance simulations and future design of this kind of fuel cell aircrafts. The code is based on a dynamic model of the fuel cell coupled with a fluid dynamic simplified potential flow model taking into account cowl and propeller effects [22,24]. In the second part, the reliability/safety concepts for the fuel cell application in aviation will be defined. It is dedicated to all-electric general aviation airplanes in order to have some preliminary evaluation of the safety related issues in novel aircraft concepts. An uncertainty/sensitivity analysis will be carried out in order to know which of the components affects the reliability of the system most, and which component increases the system reliability more than others. Finally, the result of the applied sensitivity analysis in the failure analysis is shown. A summary of the conclusions that may be drawn from the results of this research work is given in the Conclusions
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
section, as well as outlining possible future developments and applications of the present work.
2.
Power system description
This section presents an overview of the main features and performances of each subsystem that contributed to generate flight power. The full description of the power system is reported in [22,24]. The motor is a brushless electric motor produced by Phase Motion Control. The brushless motor was chosen at the beginning of the project, to guarantee the necessary performance, it relies on air cooling and this has led to a saving in the weight as a water cooling system is no longer required. The motor-case was linked directly to the electronic boards (DC/AC inverter and DC/DC chopper). This can be considered an excellent solution, in terms of layout integration and cooling, because the air flow that runs along the motor wing tabs goes directly to the external surface of the converter case, where it continues to carry out its cooling action. The fuel cell system (developed by Intelligent Energy), which is able to provide 20 kW of net unregulated power, consists of: a Fuel Cell Stack & Electrochemical System, a Heat Exchanger System, an Air Delivery & Water recovery system, a Water Management Subsystem, an Electrical & Electronic Support System & Control and Internal Battery Subsystem. Two LiePo battery packs supply the additional energy that is necessary for take-off and climbing; packs are able to deliver 20 kW for about 18 min.
3.
Thermal model
3.1.
General
The model includes the air cathode supply system and accounts for the heat released by the electrochemical reaction (through the analysis of the physical and chemical enthalpy of every chemical species composing the anode and cathode streams), the heat generated by cell irreversibilities (mainly ohmic), the surface heat losses, the heat transferred to the deionized water of the coolant circuit and the heat losses from the shell. The transient phenomena captured in the model include the flow and inertia dynamics of the compressor. The fuel cell polarization model has been developed as a function of oxygen and hydrogen partial pressure, stack temperature and membrane water content, as shown in [2]. The heat associated with the electric power for this application (20 kW) cannot be passively dissipated by convection and radiation through the external surfaces of the FC. Therefore, it requires a cooling system similar to the one used in internal combustion engines [18]. As the PEMFC stack works at low temperature differences compared to the environment, the effectiveness of the heat transfer from the coolant to the ambient depends on the design of the heat exchanger.
3.2.
Stack thermal model
The schematic thermal management model of a fuel cell system is shown in Fig. 1. In order to do a first-law control volume analysis, all contributions to the energy balance have to be determined. The first law of thermodynamics is represented by the following equation: Fcv Wel ¼
The main goal of the model is to predict the system temperature as a function of the air cooling inlet conditions and the electrical power demand.
14747
X dE þ Gi $hi dt cv i
(1)
where ðdE=dtÞcv is the rate change of the internal energy of the control volume at time t, and Fcv is the heat transfer from the control volume (cv) of the stack and the environment.
Fig. 1 e Scheme of a control volume of a fuel cell.
14748
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
WC WCst Tst T0WC jFWC Conv j ¼ Astack $hconv
Equation (1) can be developed as: mCst $
WC dTst X ¼ Gi $hph;i FAmb Conv FConv þ Fsource dt i
(2)
where mCst is the thermal capacity of the fuel cell body and hph [J/kg] is the physical contribution to enthalpy. This equation states that the dynamic of modification of the stack temperature depends on the reaction enthalpy rate, the rate of heat transfer to the coolant, the rate of heat transfer to the environment, and the heat generation. The energy exchange contributions at the boundaries are modeled by the following equations: a) mass flow into each volume X
Gi $hph;i ¼
i
X
Gi;IN $cp;i;IN $Ti;IN Gi;OUT $cp;i;OUT $Ti;OUT
(3)
i
b) heat convection to the environment: amb cellamb ðTst Tamb Þ jFAmb Conv j ¼ Astack $hconv
(4)
c) heat generation:
Fsource
2 3 X T $Ds st react þ ¼ nc $i$Ac $4 hj 5 2$F j
(6)
(5)
where, Dsreact [J/mol K] is the molar entropy of reaction P occurring in the stack, and the hj [V] is the sum of the overj voltages occurring in the cell. d) heat transfer from fuel cell body to the coolant (heat convection occurring between bipolar plates with the coolant, will be described in the Thermal Management Subsystem Model):
4.
Thermal management subsystem model
4.1.
General description
Products from the electrochemical reaction in a PEM fuel cell include not only electricity and water but also heat, which can raise the temperature of a PEM fuel cell causing damage to the stack. An appropriate temperature is critical in order to increase (and preserve) the performance of a PEM fuel cell, such as conductivity in the membrane which is directly related to the water content in the membrane. Therefore, an appropriate thermal management system plays an important role in the successful operation of fuel cell stacks. Fig. 2 shows the thermal management subsystem. It is composed by the coolant channels integrated into the fuel cell stack, a heat exchanger, a pump and a water tank reservoir. The fresh air flows through the engine cowl inlet, feeding heat exchanger, where it cools the fuel cell exhaust (watereair mixture) which is then partially recollected in water tank; air leaves the engine bay through an outlet at the bottom of engine cowl. The cooled waste water arrives at the water tank to be reused. The coolant tank consists of a water tank assembly, a water pump, the filters and a flow meter situated in the engine bay. The mass flow coolant will be computed in eq. (17) from the pump model. To solve this system it is necessary to know the balance of plant configuration (see Fig. 2), and from here to get every state in the model (inlet temperatures, outlet temperatures and mass flows of reactants, products, and coolant). In order to illustrate the main dynamics of the system, and the heat transfer processes occurring in the components of
Fig. 2 e Schematic diagram of Fuel cell stack.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
the thermal management subsystem, each component of the Balance of Plant models is described. In this context, the FC, the heat exchanger and the water reservoir are considered as lumped masses with associated dynamics, while the water pump is considered as a heat transfer element only. The coolant temperature is assumed to be constant within the connecting hoses. Then, it is possible to assemble each component of this system to compute the heat rates from the fuel cell body to the coolant, and consequently obtain the solution of the equation (2).
4.2.
Model of the coolant channels
Fig. 3 shows the control volume for the dynamic model of the coolant channels. The energy balance of the coolant inside the FC stack is governed by the following differential equation: jFWC Conv j ¼ mWC $Cwater $
dTWC X ðGi hi ÞWC dt
mWC $Cwater $
dT0WC WCst ¼AWC Tst T0WC þ GWC CpWCðlÞ stack $hconv dt outst Tinst WC TWC
ð10Þ
At the outlet, the temperature is considered as a function of the lumped temperature and is calculated as logarithmic mean temperature difference (LMTD): out 0 TWC T0WC Tin WC TWC ! DT ¼ Tout T0WC ln WC 0 Tin WC TWC
(11)
In this last equation it is possible to obtain the temperature at which it will be used to complete the terms of the eq.(2). The ambient temperature Tamb will be a system input. The other temperatures will be computed in the next section.
5.
(8)
where A is the surface area and h is the convective heat transfer coefficient. The heat transfer coefficient is a complex (usually empirical) parameter that incorporates into the heat transfer relationship the nature of the flow pattern near the surface, the fluid properties and the geometry: hconv ¼ f ðKh ; GWC Þ
(T0WC ) is a function of the enthalpy difference of the coolant flow, and of the convective heat transfer rate.
(7)
The rate of energy transfer from the surface to the coolant can be quantified by the Newton’s law: WC WCst Tst T0WC jFWC Conv j ¼ Astack $hconv
14749
(9)
The equivalent average convection coefficient hconv can be determined through experimental Nusselt number correlations [14]. The enthalpy difference of the coolant depends on the coolant mass flow rate and on the coolant temperatures at the inlet and outlet. The coolant mass flow rate is an input signal and it is taken from the pump model. The coolant inlet temperature is given from the outlet of the reservoir tank. As out the mass flow (GWC ¼ Gin WC ¼ GWC ), at the inlet is the same as in the outlet, the change in the lumped coolant temperature
Balance of plant model
All the auxiliary components constitute the balance of plant (BoP). These auxiliary components, such as the air supply system, and the thermal control system (including a coolant pump, a heat exchanger, and a water tank) are essential for the successful operation of the fuel cell system. A thermal system manages the heat produced from the cell stack and maintains the selected operating temperature which is essential for the performance and durability of a fuel cell [9]. The auxiliary systems are the link between the pilot and the operating conditions of the fuel cell stack. Understanding and modeling their dynamic behavior provides therefore a description of how the pilot actions affect the overall system performance.
5.1.
Heat exchanger model
The fresh air flows through the engine cowling inlet and the heat exchanger matrix, cools the waste watereair mixture from the fuel cell stacks and leaves the engine bay through the outlet opening at the front gear housing. For what concerns the heat exchanger, since the temperature of the exhaust air and of the exhaust water are both
Fig. 3 e Model of the coolant channels.
14750
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
unknown, the LMTD (logarithmic mean temperature difference) method cannot be applied. Instead, effectiveness-NTU method can be used following the method proposed by [14]. Finally the coolant exit temperature is: ToutHEX WC
¼
TinHEX WC
! εWC UAHEX $ TinHEX T0HEX WC CpWC GWC
(12)
And the fresh air exit is:
inHEX ToutHEX freshair ¼ Tfreshair þ
0 εfreshair UAHEX $ TinHEX freshair THEX
!
Cpair Gfreshair
(13)
where the heat exchanger lumped temperature (T0HEX ) is determined thus, mHEX CHEX $
dT0HEX WC ¼ FHEX dt
¼ εWC UAHEX $ TinHEX T0HEX WC 0 εfreshair UAHEX $ TinHEX freshair THEX
line, and a function of the convective heat transfer rate between the environment and the water tank, under the assumption that the presence of water tank walls can be ignored for what concerns heat transfer phenomena. Thus, the TWC is the same that the wall of the tank.
5.3.
The coolant pump plays a fundamental role in the thermal management subsystem. This role is to provide a cooling liquid flow at a desired flow rate, from the water tank to the stack. One of the operating requirements of a fuel cell stack is to limit the temperature rise across the fuel cell for practical reasons, considering that part of the heat produced during the fuel cell operation is removed by the coolant (as it was shown in the equation (2)). Therefore, the required mass flow rate is calculated using the calculated heat convection flow to the coolantjFWC Conv j, the type of coolant (heat capacity), and the temperature difference between inlet and outlet.
(14) GWC ¼
Where the AHEx is the heat transfer area, and it is proportional to the frontal surface of the heat exchanger (SHEx). The resolution of the heat exchanger model has been made with the discretization at n ¼ 20 elements within the HEX, where the output of one element is the input of the next one, as it is shown in Fig. 4 (where f-air: fresh air):
5.2.
Coolant tank model
This system consists of a water tank, a water pump, filters and a flow meter. In order to cope with the water tank temperature, the following differential equation must be solved: mWtank $Cwater $
mWtank $Cwater $
dT0WCTank ¼ DHWC þ FWtank Conv : dt
(15)
0 dT0WCtank tankamb ¼ AWC TWCtank Tamb Wtank $hconv dt out þ GWC CpWCðlÞ Tin WCtank TWCtank
Coolant pump model
5.4.
jFWC Conv j CpWCðlÞ DT
(17)
Overall coolant circuit model
To resolve the overall coolant circuit, the expressions (10), (11), (13)e(15), must be regrouped, and since the system is then complete (see Fig. 2), it is possible to know any temperature of the whole system. At the same time, as it is shown in Fig. 2, the temperature of the output of one element is the temperature of the input of the other, consequently: ¼ Tinst Touttank WC WC Toutst ¼ TinHEX WC WC ToutHEX ¼ Tintank WC WC
6. Uncertainty and sensitivity analysis methods (16)
6.1. Where the change in the lumped coolant temperature inside the coolant tank (T0WTank ) is a function of the enthalpy difference of the coolant flow at the water tank inlet and the outlet
General
Safety (reliability) problems related to GA (General Aviation) [6] category airplanes became more important with the
in HEX TWC
out TWC
HEX
WC HEX n
WC HEX i
T foutairHEX
T fin airHEX
Fig. 4 e Heat exchanger modular model.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
increase of complexity of airplane and their use. The introduction of new concept aircrafts with specific items and novel features installed on-board points out the need for general requirements dealing with reliability and safety. A not conventional power system based on PEMFC is very different from a conventional one; it requires many more components which usually have high volumes and weights due to the relatively recent technologies. RAPID200 has a standard configuration that was designed for a conventional power system and structural changes should be avoided as much as possible for safety/certification [23]. In a previous paper [8] an investigation has been performed to identify the main regulations and codes concerning certification and reliability/safety. Topics for special conditions or new regulations were proposed. The Rapid200 has ultra-light category certification, and Rapid-FC was applied for “permit to fly” according to the exemption in EC regulation (see Part A) within “National Authority procedure”. With the closest CS aircraft category being CS-VLA (Very Light Aircraft), there are no direct regulation requirements or recommendations for reliability/safety of systems. However, requirements and recommendations for the closest higher category may be used. In particular, a preliminary safety concept for GA aircraft powered by hydrogen fuel cells was defined. It consists of: A short description of novel items included in the GA aircraft, A preliminary functional analysis and FHA, A subsequent FMEA/FMECA based on data available from open literature or the know-how of the specific ENFICA-FC partners [23] The definition of a preliminary general risk matrix and a subsequent evaluation through a Fault Tree Analysis (FTA).
14751
In the work carried out by [8], each one of the methods described in the regulation requirements are developed in order to do an analysis of failures. Nevertheless, in this work the analysis is limited to the methodology of the FMECA. The reason for this selection is that the main objective is mainly to show the possible application of an innovative methodology (as mentioned in the Introduction). In the work of [8] a first approach of calculation of FMECA was performed, using data from literature or supplied by the project partners of ENFICA-FC. It should be mentioned that only the results related to the FC system have been extracted. As shown in the [8], most of the elements that prove to be critical to the system are the sensors (i.e., the sensors on the anode, the cathode, the water and ambient temperature). Here, the goal is to carry out a methodology capable of quantifying the damage produced by the miss o malfunction of systems/elements in the system response, through the application of the validated dynamic model of the FC-based plant, using analysis of uncertainties and sensitivity. The idea essentially is to compare the propagation of uncertainties through the model, thus evaluating the effects of a sensor miss o malfunction. From the FC control point of view, the sensor signals could be treated as input parameters of the control system, as shown in Fig. 5. This means that if one or more sensors don’t work properly, the operation of the control system will be strongly affected so that the control process can cause malfunctioning or damage to the fuel cell. Probabilistic analysis can be applied to any model output quantity: in this paper, the analysis was limited to the stack temperature output (but the methodology is general enough to be extended to any other model output). In fact, while the power output of the system is one the most important parameter to control from an uncertainty analysis point of view (because it is the measure of the performance of the
Fig. 5 e General scheme of the sensor signals.
14752
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
whole system), another crucial aspect is the degradation of the FC system: the temperature control, as discussed in some papers [18,20] is essential for preserving the operative life of the fuel cell as it affects the membrane hydration. For this reason stack temperature was chosen as the system response to be studied by probabilistic investigation. Quantifying the measurement uncertainties then becomes an important task in the development of the failure analysis system. In order to quantify the influence of uncertainty in the model, a sensitivity analysis can be performed. The information given by the sensitivity analysis will be used as input to the damage data in the failure analysis. As mentioned, the focus is on the evaluation of the uncertainties in the system responses due to uncertainties in sensor signals.
6.2.
Methods
Conventionally, as shown in [8] the system failure analysis in the energy system development is treated deterministically, using a specific set of non-probabilistic input values (based on the experience) that produce a specific set of non-probabilistic output values. Even though these input values can have significant uncertainties that unavoidably propagate through the system to the outputs, such deterministic approaches are unable to directly quantify these uncertainties and their effect on the final system control. This deficiency can, of course, be overcome by treating the inputs and outputs probabilistically. Probabilistic analysis methodologies can rarely rely on analytical solutions and numerical approximations are usually adopted. In this paper, a typical reliability-oriented approach is used to carry out the probabilistic analysis. Reliability-oriented techniques can be divided in two wide categories: Numerical simulation (Monte Carlo method and related variance reduction techniques). Limit state approximation and variables transformation (First and Second Order Reliability Methods). Monte Carlo method is a numerical integration method based on the property of the expected value of a sample to converge in probability to the real expected value of the population the sample is extracted from; the method is basically an accumulation of deterministic evaluations of the model under analysis that is computed with a randomly generated set of inputs and a subsequent statistical analysis of the sample so generated [17]. The accuracy of Monte Carlo estimates, and hence the convergence of the method, depends on the square root of the number of samples, making the method very slow [16,26], above all when the computation of the analyzed system requires solution of complex models or when the number of variables is high. The advantage of Monte Carlo simulation is that no approximation of variables or function/model is needed. In order to optimize time required by a straight Monte Carlo simulation, variance reduction techniques may be applied, usually introducing a priori information about specific characteristics of the random variable space [5,10,21]; in this paper, since Monte Carlo is used as a reference solution for applications of other probabilistic
techniques, a crude Monte Carlo simulation is preferred because of the total lack of mathematical/numerical manipulation of the model. The most important disadvantage of numerical simulation, despite it’s great accuracy, is the computational effort it requires; for model of complex systems whose analysis is time consuming and accumulation of thousands of simulations for reliability analysis may become unacceptable (in this case it took around 420 min when only a single cycle of the 300s for only one current density was run). Rather than investigating the problem through virtual experiments, it’s possible to transform the analyzed generalized function (a numerical model in this case) and its basic random variables in order to exploit a special case (all the variables described as independent normal and standard with a linear function) for which peculiar properties hold; in this case probability of not crossing the function (i.e. cumulative distribution function) is directly related to the distance of the function from the origin of the random space so that the probabilistic analysis is transformed in a geometric minimum-finding problem. This kind of methods (First Order Reliability Method) [15] is usually very fast but introduces a large number of approximations so that accuracy strongly depends on similarity between the real case and the approximated one. A very important advantage of FORM is that, since the minimum-finding problem is solved by geometric gradient-based exploration of the random space, sensitivities are an immediate result obtained directly from probabilistic outcomes [15]. A good compromise between time saving computation and accuracy is represented by application of response surfaces: RSM is based on the approximation of the true function by an explicit mathematical expression (usually a second order polynomial) that can be obtained by best-fitting a proper number of function sample points [25]. The accuracy of the approximation is usually improved by different sample points relocation techniques or, as in this work, by high/adaptive order response surfaces. Once a response surface is computed, it can be used with any probabilistic analyses methodology to obtain the probabilistic information that are achievable with the method itself. In this paper, the high order response surface method presented in [7] is applied to the FC model; the response surface generation procedure can be summarized in three steps: single random variable analysis (the variables are considered independently), polynomials structure selection, final sampling and regression. The single random variable analysis is performed in order to evaluate the function degree of dependency with respect to each variable; after setting all variables to their mean values m, one variable at a time is perturbed to the values: me6s, m4s, m2s, m, mþ2s, mþ4s, mþ6s and for each variation an evaluation of the model is carried out. It’s then possible to perform a regression of the function with respect to one single variable, choose which terms of the single variable polynomials can be neglected by means of a variance reduction analysis and therefore evaluate the degree of dependency.
14753
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
The second step selects the regression function starting from a complete fourth order polynomial, and deleting the proper terms, according to the following rules: A term is deleted if it contains a variable with a power greater than the order evaluated at the first step A term is deleted if its own order is greater than the maximum order obtained by the single random variable analysis. Once the polynomials are selected, the procedure knows how many unknown coefficients must be evaluated so that it’s possible to perform a final sampling through Latin Hypercube Sampling (LHS) [11]. LHS is used to assure a full coverage of the random variable space. Finally the regression is performed via the Single Value Decomposition technique [19].
6.3.
Result and discussion
The aim of the study is to rank the different relative weight of sensor failure; so it’s important to understand how the uncertainty in the sensor measurement propagates to a fuel cell-related output (temperature in this case). Since sensitivities (defined by eq. (19) later on) measure the contribution of one random variable standard deviation to the overall output standard deviation, all the basic variables the problem depends on must have the same standard deviations, in order that sensitivities are unaffected by the choice
Table 1 e Spurious signal distribution. Variable Name Tan_in Pan_in Gan_in Pca_in Twc_in Gwc_in Tamb Gca_in Tca_in
Variable Name poly
Distribution
Mean Value
Standard Deviation
X1 X2 X3 X4 X5 X6 X7 X8 X9
Normal Normal Normal Normal Normal Normal Normal Normal Normal
1 1 1 1 1 1 1 1 1
0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015
of distribution parameters and can be ranked purely on their relative importance. For this reason, a Gaussian distribution with the same coefficient of variation has been chosen for all the basic random variables. Nevertheless, the methodology here applied (Fig. 6) is able to deal with any probability distribution. The testing of the method compared to a Monte Carlo process is made only in a fixed current density value (0.5 A/ cm2). After verifying that the methodology is reliable, the RSM analysis can be extended getting response surfaces for different current densities (0.3 A/cm2 and 0.7 A/cm2), as shown in Fig. 6. In a first step, the model has been running with Monte Carlo. Each input of the system has been multiplied by
Fig. 6 e General scheme of the sensitivity/uncertainly process.
14754
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
a source of uncertainty as shown in Table 1. The distribution of each input is shown in Fig. 7. The results of the model can be seen in Fig. 8, where it shows the CDF of the system output (stack temperature).
The second step of the procedure is to select the polynomials structure. In this case, it is only reported the polynomial corresponding to the power output of 0.5 A/cm2:
Fig. 7 e Signal distributions.
14755
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
Fig. 8 e CDF stack temperature comparison.
a random variable and the corresponding value of the cumulative distribution function, sensitivities can be seen as a function of CDF too. For this reason a set of sensitivities is obtained for every point the CDF is discretized in. For the sake of clarity the sensitivities are here presented in three
P ¼c0 þ c1 x1 þ c2 x2 þ c3 x3 þ c4 x4 þ c5 x5 þ c6 x6 þ c7 x7 þ c8 x8 þ c9 x9 þ c10 x1 x2 þ c11 x1 x3 þ c12 x1 x4 þ c13 x1 x5 þ c14 x1 x6 þ c15 x1 x7 þ c16 x1 x8 þ c17 x1 x9 þ c18 x2 x3 þ c19 x2 x4 þ c20 x2 x5 þ c21 x2 x6 þ c22 x2 x7 þ c23 x2 x8 þ c24 x2 x9 þ c25 x3 x4 þ c26 x3 x5 þ c27 x3 x6 þ c28 x3 x7 þ c29 x3 x8 þ c30 x3 x9 þ c31 x4 x5 þ c32 x4 x6 þ c33 x4 x7 þ c34 x4 x8 þ c35 x4 x9 þ c36 x5 x6 þ c37 x5 x7 þ c38 x5 x8 þ c39 x5 x9 þ c40 x6 x7 þ c41 x6 x8 þ c42 x6 x9 þ c43 x7 x8 þ c44 x7 x9 þ c45 x28 þ c46 x8 x9 ;
ð18Þ
In order to compute the 47 unknown coefficients, the code generates 86 sample points via Latin Hypercube Sampling and performs the regression through the Singular Value Decomposition technique. The results are reported in Table 2. In Fig. 8 the comparison between the CDF evaluated by Monte Carlo simulation (105 model evaluations) with the simulation with the response surface (model evaluations and 105 response surface evaluations) of the stack temperature output are reported; once the response surface is known, it’s possible to perform every probabilistic analysis. Finally, very important information that can be achieved by probabilistic analysis is the sensitivity of the random variable function to its input variables. The sensitivity of the stack temperature with respect to the main model parameters pi , is defined as: VTStack ðpi Þ S ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi$/Tstack ¼ f ðpi Þ Pn vTStack 2 i¼1 vpi
/
(19)
Sensitivities are not constant in the field of probability of a function of random variables. They indeed change as a function of the infinite realizations that a continuous random variable (as the ones it deals with) can assume. Since there is a univocal relationship between a possible realization of
Table 2 e Regression Coefficients. c0 47.46
c1 171.92
c2 102.89
c3 43.58
c4 98.53
c5 97.86
c6 0.01
c7 1.39
c8 144.35
c9 76.84
c10 266.72
c11 80.91
c12 194.84
c13 87.94
c14 0.01
c15 1.75
c16 450.09
c17 292.71
c18 61.92
c19 119.97
c20 151.64
c21 0.00
c22 0.05
c23 10.28
c24 117.58
c25 270.97
c26 261.03
c27 0.00
c28 1.57
c29 14.04
c30 59.46
c31 280.72
c32 0.01
c33 2.42
c34 260.58
c35 337.01
c36 0.01
c37 3.07
c38 337.86
c39 320.30
c40 0.00
c41 0.01
c42 0.00
c43 0.48
c44 1.05
c45 77.08
c46 122.51
14756
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
high prob
Tamb Gan_in
low prob
Tan_in
mean prob
Gan_in
low prob
Tan_in Pan_in
Pan_in
Pca_in
Pca_in
Tca_in
Tca_in
Gca_in
Gca_in
Gwc_in
Gwc_in
Twc_in
Twc_in 0
0,2
0,4
0,6
0,8
1
0
1,2
Fig. 9 e Sensitivity of the stack temperature @ 0.3 A/cm2.
representative points of the CDF: a low cumulative probability point (CDF ¼ 2.3262908e-04), the mean value (CDF ¼ 0.5) and a high cumulative probability point (CDF ¼ 9.9976737e-01). From Fig. 9e11, the sensitivities of stack temperature are shown at the density current of the 0.3, 0.5 and 0.7 [A/cm2]. In Table 3 a synthesis of the sensitivity at mean probability is reported. The sensor signal with the highest impact on the stack temperature is the cathode temperature inlet into the stack (Tca_in). This signal is visibly the most important sensor and its measurement has to be very accurate, since it shows an approximate 90% of the total sensitivity. The other significant signal is the coolant temperature inlet to stack (Twc_in) because it represents a 40% of impact on the stack temperature. This temperature has high impact on the energy balance of the fuel cell (eq.(2)) and hence a strong influence over the stack temperature control. Finally, the remaining signals show a not-negligible influence on the predicted value of stack temperature (Tst).
7.
high prob
Tamb
mean prob
Applied sensitivity result in FMECA
To translate the information provided by sensitivity analysis in quantitative measures that can be used in FMECA is a very complex matter; sensitivity analysis can provide a measure of the relative weight a certain parameter has on a system output level of knowledge, but only an expert of the system Tamb
mean prob
Gan_in
low prob
Tan_in Pan_in Pca_in Tca_in Gca_in Gwc_in Twc_in 0
0,2
0,4
0,6
0,8
1 2
Fig. 10 e Sensitivity of the stack temperature @ 0.5 A/cm .
0,4
0,6
0,8
1
Fig. 11 e Sensitivity of the stack temperature @ 0.7 A/cm2.
Table 3 e Sensitivity at expected value. Stack temperature sensitivity
Twc_in Gwc_in Gca_in Tca_in Pca_in Pan_in Tan_in Gan_in Tamb
0.3 A/cm2
0.5 A/cm2
0.7 A/cm2
0.362 0.033 0.017 0.924 0.061 0.000 0.011 0.002 0.088
0.386 0.028 0.007 0.916 0.061 0.006 0.006 0.004 0.079
0.381 0.019 0.004 0.918 0.062 0.0002 0.006 0.003 0.080
can evaluate the consequences of such a mismatching between real data and measured ones. As an example, consider the case of the sensor of the cathode inlet temperature. As shown in the sensitivity analysis (Table 3), this sensor is the most important, therefore a sensor error can cause a significant miscalculation of the temperature
Table 4 e Importance sensors ranking (Stack temperature). Component/System Tan
high prob
0,2
Inlet anode temperature controller Pan Inlet anode pressure controller Gan Inlet anode temperature controller Pca Inlet cathode pressure controller Twc-in Inlet coolant temperature controller Gwc Inlet coolant flow controller Tamb Ambient temperature sensor Gca Inlet cathode airflow controller Tca Inlet cathode temperature controller
Failure Mode
Importance
Spurious output
2
Spurious output
1
Spurious output
1
Spurious output
1
Spurious output
5
Spurious output
1
Spurious signal
3
Spurious output
1
Spurious output
9
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
at which the FC is working. This mismatching can lead to a scenario where the control system assumes that the fuel cell system is working at a temperature significantly higher or lower than the real temperature, with two completely different set of consequences; sensitivity analysis isn’t able to distinguish between the two cases, so it isn’t possible to use sensitivity as a direct measure of the “damage” reported in [8]. However, it is possible to create a ranking (see Table 4) of the importance of each sensor in order to allow the FMECA analyst to design the system with proper sensor quality and redundancies.
8.
Conclusion
The paper described how the development of a complete dynamic model of a Fuel Cell power aircraft could be used in order to precisely individuate the most significant signal failure in the control system of the aircraft itself. Moreover, the model has been validated through several real flight tests of an FC powered two-seat ultra-light aircraft, in the framework of the EU funded ENFICA-FC Project. The main outcomes of the paper are the following: 1. the description of a complete dynamic model of an FC powered two-seat ultra-light aircraft (with in insight on the thermal management model), validated through real flight tests; 2. the development of a innovative procedure of uncertainty/ sensitivity analysis of the input signals of the control system (state measurements through sensors) over the output variables representing the operation of FC powertrain of the aircraft; 3. the use of the proposed uncertainty/sensitivity analysis to improve the failure analysis: the sensitivity analysis provides useful information to engineers for developing a system as well as appropriate FMECA analysis and sensor control characteristics; 4. in particular, the proposed analysis can be used as a tool to rank the different sensors from a safety point of view: how uncertainty of a sensor will propagate to the performance of the FC system; 5. the results demonstrate that accurate temperature sensors and sensor calibration are of unavoidable importance for the control of the stack temperature, over systems such as PEM fuel cells; 6. as an example in terms of thermal control of the system: the sensor signal with the highest impact on the stack temperature is the cathode temperature inlet into the stack (Tca_in), while the coolant temperature inlet to stack (Twc_in) shows the highest impact on the energy balance of the fuel cell and hence a strong influence over the stack temperature control.
Acknowledgments The activity has been developed within the ENFICA-FC project. The authors acknowledge the important
14757
contribution of European Commission by the funding programmes ENFICA-FC, EC 6th FP e Contract No. AST5-CT2006-030779.
references
[1] Boeing, 2008, April, http://www.boeing.com/news/releases/ 2008/q2/080403a_nr.html. [2] Correa G. Dynamic modelling and comparison with experimental data of fuel cell systems for aeronautical applications. Ph.D. thesis. Politecnico di Torino;2011. [3] DLR. Fuel cells power first takeoff for dlr’s antares aircraft. Fuel Cells Bulletin 2009a;9. [4] DLR. 2009b, July, http://www.dlr.de/en/desktopdefault.aspx/ tabid-1/86_read-18278. [5] Deak I. Three Digits accurate Multiple normal Probabilities. Numerische Mathematik. 1980;38:369e80. [6] EASA. 2010, http://www.easa.europa.eu/flightstandards/gaba.html. [7] Frulla G, Borello F. In: Hayworth GI, editor. Structural reliability aspects for advanced composite material applications. Turin: Reliability Engineering Advances; 2009. [8] Frulla G, Correa G. Reliability/Safety concept definition allelectric general aviation (GA) airplane, propeller driven, powered by fuel cells. Turin: Proposal No. 30779 e ENFICAFC; 2008. [9] Gasser, F. An analytical, control-oriented state space model for a PEM fuel cell system. Ph.D. thesis. Lausanne: E´cole Polytechnique Fe´de´rale de Lausanne;2005. [10] Gray W.A. Generalization of the Directionl Simulation in the Load Space Approach to Structural Reliability Analysis. Ph.D. thesis;1990. [11] Iman RL, Davenport JM, Zeigler DK. Latin Hypercube sampling (program user’s guide). Technical Report SAND 79-1473. Albuquerque: Sandia National laboratories; 1980. [12] International Industry Working Group. Commercial aircraft design characteristics - trends and growth projections. 5th ed.; 2007. R1. [13] Lapen˜a-Rey N, Mosquera J, Bataller E, Ortı´ F. First fuel-cell manned aircraft. Journal of Aircraft 2010;47(6). [14] Lienhard J, Lienhard J. A heat transfer textbook. 3rd ed. Cambridge, Massachusetts: Phlogiston Press; 2000. [15] Madsen HO, Krenk S, Lind NC. Methods of structural rebiality. Prantice Hall; 1986. [16] Melchers RE. Structural reliability analysis and prediction. 2nd ed. John Wiley and Sons; 1999. [17] Metropolis N, Ulam S. The Monte Carlo method. Journal of the American Statistical Association 1949;44(247): 335e41. [18] Muller E, Stefanopoulou A. Analysis, modeling, and validation for the thermal dynamics of a polymer electrolyte membrane fuel cell system. Journal of Fuel Cell Science and Technology 2006;3(2):99e110. [19] Press W, Teukolsky S, Vetterling W, Flannery B. Numerical recipes in C: the art of scientific computing. 2nd ed. New York: Cambridge University Press; 1992. [20] Pukrushpan JT, Peng H, Stefanopoulou AG. Control-oriented modeling and analysis for automotive fuel cell systems. Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME 2004;126:14e25. [21] Ripley BD. Stochastic simulation. Wiley & Sons; 1987. [22] Romeo G, Borello F, Correa G. Set-up and test flights of an allelectric 2-seater aeroplane powered by fuel cells. Journal of Aircraft 2011a;49(8).
14758
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
[23] Romeo G, Borello F. Design and realization of a 2-seater aircraft powered by fuel cell electric propulsion. The Aeronautical Journal 2010;114(1155). [24] Romeo G, Cestino E, Borello F, Correa G. Engineering method for air-cooling design of 2-seat propeller driven aircraft powered by fuel cells. Journal of Aerospace Engineering 2011b;24:79e88.
[25] Santarelli M, Leone P, Cali M, Orsello G. Experimental analysis of the voltage and temperature behavior of a solid oxide fuel cell generator. Journal of Fuel Cell Science and Technology 2007;4(2):143e53. [26] Shooman M. In: Robert E, editor. Probabilistic reliability: an engineering approach. 2nd ed. Krieger Publishing Company; 1990.
Available online at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/he
Sensitivity analysis of temperature uncertainty in an aircraft PEM fuel cell G. Correa a,*, F. Borello b, M. Santarelli a a b
Politecnico di Torino, Dept. of Energetics, Corso Duca degli Abruzzi 24, 10129 Torino, Italy Politecnico di Torino, Dept. of Aerospace Engineering, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
article info
abstract
Article history:
Nowadays, growing interest in the application of renewable energy sources has led to
Received 12 June 2011
a rapid development of more environmentally sustainable and green thinking applications.
Received in revised form
The application of fuel cell technology to aircraft propulsion and/or auxiliary energy supply
1 August 2011
is becoming of great interest for undoubted advantages in terms of pollution emissions,
Accepted 10 August 2011
noise reduction and lower dependency on oil price and availability. In 2006 European
Available online 25 September 2011
Commission founded the ENFICA-FC project, coordinated by Politecnico di Torino, whose aim was to develop new all-electric and more-electric aircraft concepts and to directly
Keywords:
prove the feasibility of such designs by flying an all-electric general aviation aircraft
Aircraft Pem fuel cell
powered by hydrogen fuel cells. Reliability of fuel cell system is a very important aspect for
Sensitivity analysis
the safety of these aircraft, but experimental data concerning fuel cell systems in aero-
Temperature uncertainty
nautics are still unavailable to scientific community. This paper presents the research
Dynamic model
activity done by authors to support the design of “fuel cell aircrafts” from the failure analysis point of view; classic approaches to structural reliability are here applied to rank the importance of the failures of sensors used by fuel cell control logic in order to support the definition of accurate Failure Mode, Effects and Criticality Analysis (Risk matrix). Copyright ª 2011, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
1.
Introduction
Climate change and pollution are serious global concerns. Although it accounts for only 4.2% of the total global warming potential, the concern today is that aviation-generated C02 is projected to grow to approximately 5.7% by 2050 [12], which is accompanied by the current global travel demand increase.
Growing interest in the application of renewable energy sources has led to an increase of interest in the deployment of fuel cell (FC) technologies. The high efficiency and environmental advantages that fuel-cell technology could offer, have generated an increasing interest from the aviation community in fuel cell powered aircrafts, overall due to a successful experience along with a considerable
Abbreviations: Amb, Ambient; BoP, Balance of Plant; CFD, Cumulative Distribution Function; Conv, Convective; cv, Control volume; el, Electric; FC, Fuel cell; FHA, Functional Hazard Assessment; FMEA, Failure Mode and Effects Analysis; FMECA, Failure Mode, Effects and Criticality Analysis; FTA, Fault Tree Analysis; HEX, Heat Exchanger; MC, Monte Carlo; Mem, Membrane; NTU, number of transfer units; Nu, Nusselt number; PDF, Probability distribution function; RSM, Response surface method; st, Stack; VLA, Very Light Aircraft; wc, Coolant (deionizer water). * Corresponding author. Tel.: þ39 115644487. E-mail addresses: [email protected] (G. Correa), [email protected] (F. Borello), [email protected] (M. Santarelli). 0360-3199/$ e see front matter Copyright ª 2011, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijhydene.2011.08.036
14746
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
Symbols Ac C,Cp E0 E G g h hconv i I Kh le m nc OCV T U Vc Wel
2
Cell active surface, cm Specific heat, J/(kg K) Open Circuit Voltage, volts Internal energy of the stack, J Mass flow rate, kg/s Gibbs free energy, W Mass specific enthalpy of the mass flow, J/kg Convective heat transfer coefficient, W/(m2 K) Current density, A/cm2 Current, A Parameter of the heat transfer coefficient model Electric work produced by the stack, W Total mass, kg Number of cells in series in the PEMFC stack Open Circuit Voltage, V Temperature, K Overall heat transfer coefficient, W/(m2 K) Cell voltage, V Electric power produced by the stack, W
Greek letters F Heat transfer, W DH Enthalpy flow rate, W
development achieved in the ground transportation sector in the last ten years. Fuel cells system could become the main power source for small general aviation aircraft or could replace APU (Auxiliary Power Unit) on larger aircraft, to obtain all-electric or moreelectric machines. There are several potential advantages of using such a power source that range from environmental and economic issues to performance and operability aspects. The most stringent requirements derived from aeronautical applications are weight reduction, safety, and reliability. This determines tremendous challenges for the integration of fuelcell power plants in aircraft: maintaining the weight and balance of the aircraft, designing the thermal management system, managing the power flows of the hybrid power source when the fuel cell works in parallel with an additional power source, and ensuring a safe operation in a potentially flammable atmosphere. The main airframe manufacturers have started to investigate their applications because of their potential for improved performance and environmental compatibility. Regarding the use of fuel-cell systems for propulsion of manned aircraft, only three projects are currently underway. These are the Boeing fuel-cell demonstrator airplane [1,13], the Antares DLR-H2 fuel cell aircraft project [3] [4], and the Environmentally Friendly Inter-City Aircraft Powered by Fuel Cells (ENFICA-FC) project [23]. The introduction of this type of concept aircrafts with specific items and novel features installed on-board that increase the complexity of their system, became more important the analysis of the Safety (reliability) of these category airplanes. Conventionally, the system failure analysis in the energy system development is treated deterministically. The main
ε h
Efficiency, over-voltage, V
Abbreviations Amb Ambient BoP Balance of Plant CFD Cumulative Distribution Function Conv Convective cv Control volume el Electric FC Fuel cell FHA Functional Hazard Assessment FMEA Failure Mode and Effects Analysis FMECA Failure Mode, Effects and Criticality Analysis FTA Fault Tree Analysis HEX Heat Exchanger MC Monte Carlo Mem Membrane NTU number of transfer units Nu Nusselt number PDF Probability distribution function RSM Response surface method st Stack VLA Very Light Aircraft wc Coolant (deionizer water)
objective of this paper is mainly to show the possible application of an innovative methodology capable of quantifying the damage produced by the miss o malfunction of systems/ elements in the system response, through the application of the validated dynamic model of the FC-based plant, using analysis of uncertainties and sensitivity. In the first part of this paper is presented an analytical Proton Exchange Membrane (PEM) fuel cell stack model that describes its performance based on physical system inputs, taking into account the main physical concepts of the electrochemical, chemical and energy behavior. This model will include the fulfilled lumped capacitance model of the thermal management subsystem. This is developed to describe the temperature dynamics of the system based on the system inputs (power required and ambient temperature). The model is intended to be used for thermal control, performance simulations and future design of this kind of fuel cell aircrafts. The code is based on a dynamic model of the fuel cell coupled with a fluid dynamic simplified potential flow model taking into account cowl and propeller effects [22,24]. In the second part, the reliability/safety concepts for the fuel cell application in aviation will be defined. It is dedicated to all-electric general aviation airplanes in order to have some preliminary evaluation of the safety related issues in novel aircraft concepts. An uncertainty/sensitivity analysis will be carried out in order to know which of the components affects the reliability of the system most, and which component increases the system reliability more than others. Finally, the result of the applied sensitivity analysis in the failure analysis is shown. A summary of the conclusions that may be drawn from the results of this research work is given in the Conclusions
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
section, as well as outlining possible future developments and applications of the present work.
2.
Power system description
This section presents an overview of the main features and performances of each subsystem that contributed to generate flight power. The full description of the power system is reported in [22,24]. The motor is a brushless electric motor produced by Phase Motion Control. The brushless motor was chosen at the beginning of the project, to guarantee the necessary performance, it relies on air cooling and this has led to a saving in the weight as a water cooling system is no longer required. The motor-case was linked directly to the electronic boards (DC/AC inverter and DC/DC chopper). This can be considered an excellent solution, in terms of layout integration and cooling, because the air flow that runs along the motor wing tabs goes directly to the external surface of the converter case, where it continues to carry out its cooling action. The fuel cell system (developed by Intelligent Energy), which is able to provide 20 kW of net unregulated power, consists of: a Fuel Cell Stack & Electrochemical System, a Heat Exchanger System, an Air Delivery & Water recovery system, a Water Management Subsystem, an Electrical & Electronic Support System & Control and Internal Battery Subsystem. Two LiePo battery packs supply the additional energy that is necessary for take-off and climbing; packs are able to deliver 20 kW for about 18 min.
3.
Thermal model
3.1.
General
The model includes the air cathode supply system and accounts for the heat released by the electrochemical reaction (through the analysis of the physical and chemical enthalpy of every chemical species composing the anode and cathode streams), the heat generated by cell irreversibilities (mainly ohmic), the surface heat losses, the heat transferred to the deionized water of the coolant circuit and the heat losses from the shell. The transient phenomena captured in the model include the flow and inertia dynamics of the compressor. The fuel cell polarization model has been developed as a function of oxygen and hydrogen partial pressure, stack temperature and membrane water content, as shown in [2]. The heat associated with the electric power for this application (20 kW) cannot be passively dissipated by convection and radiation through the external surfaces of the FC. Therefore, it requires a cooling system similar to the one used in internal combustion engines [18]. As the PEMFC stack works at low temperature differences compared to the environment, the effectiveness of the heat transfer from the coolant to the ambient depends on the design of the heat exchanger.
3.2.
Stack thermal model
The schematic thermal management model of a fuel cell system is shown in Fig. 1. In order to do a first-law control volume analysis, all contributions to the energy balance have to be determined. The first law of thermodynamics is represented by the following equation: Fcv Wel ¼
The main goal of the model is to predict the system temperature as a function of the air cooling inlet conditions and the electrical power demand.
14747
X dE þ Gi $hi dt cv i
(1)
where ðdE=dtÞcv is the rate change of the internal energy of the control volume at time t, and Fcv is the heat transfer from the control volume (cv) of the stack and the environment.
Fig. 1 e Scheme of a control volume of a fuel cell.
14748
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
WC WCst Tst T0WC jFWC Conv j ¼ Astack $hconv
Equation (1) can be developed as: mCst $
WC dTst X ¼ Gi $hph;i FAmb Conv FConv þ Fsource dt i
(2)
where mCst is the thermal capacity of the fuel cell body and hph [J/kg] is the physical contribution to enthalpy. This equation states that the dynamic of modification of the stack temperature depends on the reaction enthalpy rate, the rate of heat transfer to the coolant, the rate of heat transfer to the environment, and the heat generation. The energy exchange contributions at the boundaries are modeled by the following equations: a) mass flow into each volume X
Gi $hph;i ¼
i
X
Gi;IN $cp;i;IN $Ti;IN Gi;OUT $cp;i;OUT $Ti;OUT
(3)
i
b) heat convection to the environment: amb cellamb ðTst Tamb Þ jFAmb Conv j ¼ Astack $hconv
(4)
c) heat generation:
Fsource
2 3 X T $Ds st react þ ¼ nc $i$Ac $4 hj 5 2$F j
(6)
(5)
where, Dsreact [J/mol K] is the molar entropy of reaction P occurring in the stack, and the hj [V] is the sum of the overj voltages occurring in the cell. d) heat transfer from fuel cell body to the coolant (heat convection occurring between bipolar plates with the coolant, will be described in the Thermal Management Subsystem Model):
4.
Thermal management subsystem model
4.1.
General description
Products from the electrochemical reaction in a PEM fuel cell include not only electricity and water but also heat, which can raise the temperature of a PEM fuel cell causing damage to the stack. An appropriate temperature is critical in order to increase (and preserve) the performance of a PEM fuel cell, such as conductivity in the membrane which is directly related to the water content in the membrane. Therefore, an appropriate thermal management system plays an important role in the successful operation of fuel cell stacks. Fig. 2 shows the thermal management subsystem. It is composed by the coolant channels integrated into the fuel cell stack, a heat exchanger, a pump and a water tank reservoir. The fresh air flows through the engine cowl inlet, feeding heat exchanger, where it cools the fuel cell exhaust (watereair mixture) which is then partially recollected in water tank; air leaves the engine bay through an outlet at the bottom of engine cowl. The cooled waste water arrives at the water tank to be reused. The coolant tank consists of a water tank assembly, a water pump, the filters and a flow meter situated in the engine bay. The mass flow coolant will be computed in eq. (17) from the pump model. To solve this system it is necessary to know the balance of plant configuration (see Fig. 2), and from here to get every state in the model (inlet temperatures, outlet temperatures and mass flows of reactants, products, and coolant). In order to illustrate the main dynamics of the system, and the heat transfer processes occurring in the components of
Fig. 2 e Schematic diagram of Fuel cell stack.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
the thermal management subsystem, each component of the Balance of Plant models is described. In this context, the FC, the heat exchanger and the water reservoir are considered as lumped masses with associated dynamics, while the water pump is considered as a heat transfer element only. The coolant temperature is assumed to be constant within the connecting hoses. Then, it is possible to assemble each component of this system to compute the heat rates from the fuel cell body to the coolant, and consequently obtain the solution of the equation (2).
4.2.
Model of the coolant channels
Fig. 3 shows the control volume for the dynamic model of the coolant channels. The energy balance of the coolant inside the FC stack is governed by the following differential equation: jFWC Conv j ¼ mWC $Cwater $
dTWC X ðGi hi ÞWC dt
mWC $Cwater $
dT0WC WCst ¼AWC Tst T0WC þ GWC CpWCðlÞ stack $hconv dt outst Tinst WC TWC
ð10Þ
At the outlet, the temperature is considered as a function of the lumped temperature and is calculated as logarithmic mean temperature difference (LMTD): out 0 TWC T0WC Tin WC TWC ! DT ¼ Tout T0WC ln WC 0 Tin WC TWC
(11)
In this last equation it is possible to obtain the temperature at which it will be used to complete the terms of the eq.(2). The ambient temperature Tamb will be a system input. The other temperatures will be computed in the next section.
5.
(8)
where A is the surface area and h is the convective heat transfer coefficient. The heat transfer coefficient is a complex (usually empirical) parameter that incorporates into the heat transfer relationship the nature of the flow pattern near the surface, the fluid properties and the geometry: hconv ¼ f ðKh ; GWC Þ
(T0WC ) is a function of the enthalpy difference of the coolant flow, and of the convective heat transfer rate.
(7)
The rate of energy transfer from the surface to the coolant can be quantified by the Newton’s law: WC WCst Tst T0WC jFWC Conv j ¼ Astack $hconv
14749
(9)
The equivalent average convection coefficient hconv can be determined through experimental Nusselt number correlations [14]. The enthalpy difference of the coolant depends on the coolant mass flow rate and on the coolant temperatures at the inlet and outlet. The coolant mass flow rate is an input signal and it is taken from the pump model. The coolant inlet temperature is given from the outlet of the reservoir tank. As out the mass flow (GWC ¼ Gin WC ¼ GWC ), at the inlet is the same as in the outlet, the change in the lumped coolant temperature
Balance of plant model
All the auxiliary components constitute the balance of plant (BoP). These auxiliary components, such as the air supply system, and the thermal control system (including a coolant pump, a heat exchanger, and a water tank) are essential for the successful operation of the fuel cell system. A thermal system manages the heat produced from the cell stack and maintains the selected operating temperature which is essential for the performance and durability of a fuel cell [9]. The auxiliary systems are the link between the pilot and the operating conditions of the fuel cell stack. Understanding and modeling their dynamic behavior provides therefore a description of how the pilot actions affect the overall system performance.
5.1.
Heat exchanger model
The fresh air flows through the engine cowling inlet and the heat exchanger matrix, cools the waste watereair mixture from the fuel cell stacks and leaves the engine bay through the outlet opening at the front gear housing. For what concerns the heat exchanger, since the temperature of the exhaust air and of the exhaust water are both
Fig. 3 e Model of the coolant channels.
14750
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
unknown, the LMTD (logarithmic mean temperature difference) method cannot be applied. Instead, effectiveness-NTU method can be used following the method proposed by [14]. Finally the coolant exit temperature is: ToutHEX WC
¼
TinHEX WC
! εWC UAHEX $ TinHEX T0HEX WC CpWC GWC
(12)
And the fresh air exit is:
inHEX ToutHEX freshair ¼ Tfreshair þ
0 εfreshair UAHEX $ TinHEX freshair THEX
!
Cpair Gfreshair
(13)
where the heat exchanger lumped temperature (T0HEX ) is determined thus, mHEX CHEX $
dT0HEX WC ¼ FHEX dt
¼ εWC UAHEX $ TinHEX T0HEX WC 0 εfreshair UAHEX $ TinHEX freshair THEX
line, and a function of the convective heat transfer rate between the environment and the water tank, under the assumption that the presence of water tank walls can be ignored for what concerns heat transfer phenomena. Thus, the TWC is the same that the wall of the tank.
5.3.
The coolant pump plays a fundamental role in the thermal management subsystem. This role is to provide a cooling liquid flow at a desired flow rate, from the water tank to the stack. One of the operating requirements of a fuel cell stack is to limit the temperature rise across the fuel cell for practical reasons, considering that part of the heat produced during the fuel cell operation is removed by the coolant (as it was shown in the equation (2)). Therefore, the required mass flow rate is calculated using the calculated heat convection flow to the coolantjFWC Conv j, the type of coolant (heat capacity), and the temperature difference between inlet and outlet.
(14) GWC ¼
Where the AHEx is the heat transfer area, and it is proportional to the frontal surface of the heat exchanger (SHEx). The resolution of the heat exchanger model has been made with the discretization at n ¼ 20 elements within the HEX, where the output of one element is the input of the next one, as it is shown in Fig. 4 (where f-air: fresh air):
5.2.
Coolant tank model
This system consists of a water tank, a water pump, filters and a flow meter. In order to cope with the water tank temperature, the following differential equation must be solved: mWtank $Cwater $
mWtank $Cwater $
dT0WCTank ¼ DHWC þ FWtank Conv : dt
(15)
0 dT0WCtank tankamb ¼ AWC TWCtank Tamb Wtank $hconv dt out þ GWC CpWCðlÞ Tin WCtank TWCtank
Coolant pump model
5.4.
jFWC Conv j CpWCðlÞ DT
(17)
Overall coolant circuit model
To resolve the overall coolant circuit, the expressions (10), (11), (13)e(15), must be regrouped, and since the system is then complete (see Fig. 2), it is possible to know any temperature of the whole system. At the same time, as it is shown in Fig. 2, the temperature of the output of one element is the temperature of the input of the other, consequently: ¼ Tinst Touttank WC WC Toutst ¼ TinHEX WC WC ToutHEX ¼ Tintank WC WC
6. Uncertainty and sensitivity analysis methods (16)
6.1. Where the change in the lumped coolant temperature inside the coolant tank (T0WTank ) is a function of the enthalpy difference of the coolant flow at the water tank inlet and the outlet
General
Safety (reliability) problems related to GA (General Aviation) [6] category airplanes became more important with the
in HEX TWC
out TWC
HEX
WC HEX n
WC HEX i
T foutairHEX
T fin airHEX
Fig. 4 e Heat exchanger modular model.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
increase of complexity of airplane and their use. The introduction of new concept aircrafts with specific items and novel features installed on-board points out the need for general requirements dealing with reliability and safety. A not conventional power system based on PEMFC is very different from a conventional one; it requires many more components which usually have high volumes and weights due to the relatively recent technologies. RAPID200 has a standard configuration that was designed for a conventional power system and structural changes should be avoided as much as possible for safety/certification [23]. In a previous paper [8] an investigation has been performed to identify the main regulations and codes concerning certification and reliability/safety. Topics for special conditions or new regulations were proposed. The Rapid200 has ultra-light category certification, and Rapid-FC was applied for “permit to fly” according to the exemption in EC regulation (see Part A) within “National Authority procedure”. With the closest CS aircraft category being CS-VLA (Very Light Aircraft), there are no direct regulation requirements or recommendations for reliability/safety of systems. However, requirements and recommendations for the closest higher category may be used. In particular, a preliminary safety concept for GA aircraft powered by hydrogen fuel cells was defined. It consists of: A short description of novel items included in the GA aircraft, A preliminary functional analysis and FHA, A subsequent FMEA/FMECA based on data available from open literature or the know-how of the specific ENFICA-FC partners [23] The definition of a preliminary general risk matrix and a subsequent evaluation through a Fault Tree Analysis (FTA).
14751
In the work carried out by [8], each one of the methods described in the regulation requirements are developed in order to do an analysis of failures. Nevertheless, in this work the analysis is limited to the methodology of the FMECA. The reason for this selection is that the main objective is mainly to show the possible application of an innovative methodology (as mentioned in the Introduction). In the work of [8] a first approach of calculation of FMECA was performed, using data from literature or supplied by the project partners of ENFICA-FC. It should be mentioned that only the results related to the FC system have been extracted. As shown in the [8], most of the elements that prove to be critical to the system are the sensors (i.e., the sensors on the anode, the cathode, the water and ambient temperature). Here, the goal is to carry out a methodology capable of quantifying the damage produced by the miss o malfunction of systems/elements in the system response, through the application of the validated dynamic model of the FC-based plant, using analysis of uncertainties and sensitivity. The idea essentially is to compare the propagation of uncertainties through the model, thus evaluating the effects of a sensor miss o malfunction. From the FC control point of view, the sensor signals could be treated as input parameters of the control system, as shown in Fig. 5. This means that if one or more sensors don’t work properly, the operation of the control system will be strongly affected so that the control process can cause malfunctioning or damage to the fuel cell. Probabilistic analysis can be applied to any model output quantity: in this paper, the analysis was limited to the stack temperature output (but the methodology is general enough to be extended to any other model output). In fact, while the power output of the system is one the most important parameter to control from an uncertainty analysis point of view (because it is the measure of the performance of the
Fig. 5 e General scheme of the sensor signals.
14752
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
whole system), another crucial aspect is the degradation of the FC system: the temperature control, as discussed in some papers [18,20] is essential for preserving the operative life of the fuel cell as it affects the membrane hydration. For this reason stack temperature was chosen as the system response to be studied by probabilistic investigation. Quantifying the measurement uncertainties then becomes an important task in the development of the failure analysis system. In order to quantify the influence of uncertainty in the model, a sensitivity analysis can be performed. The information given by the sensitivity analysis will be used as input to the damage data in the failure analysis. As mentioned, the focus is on the evaluation of the uncertainties in the system responses due to uncertainties in sensor signals.
6.2.
Methods
Conventionally, as shown in [8] the system failure analysis in the energy system development is treated deterministically, using a specific set of non-probabilistic input values (based on the experience) that produce a specific set of non-probabilistic output values. Even though these input values can have significant uncertainties that unavoidably propagate through the system to the outputs, such deterministic approaches are unable to directly quantify these uncertainties and their effect on the final system control. This deficiency can, of course, be overcome by treating the inputs and outputs probabilistically. Probabilistic analysis methodologies can rarely rely on analytical solutions and numerical approximations are usually adopted. In this paper, a typical reliability-oriented approach is used to carry out the probabilistic analysis. Reliability-oriented techniques can be divided in two wide categories: Numerical simulation (Monte Carlo method and related variance reduction techniques). Limit state approximation and variables transformation (First and Second Order Reliability Methods). Monte Carlo method is a numerical integration method based on the property of the expected value of a sample to converge in probability to the real expected value of the population the sample is extracted from; the method is basically an accumulation of deterministic evaluations of the model under analysis that is computed with a randomly generated set of inputs and a subsequent statistical analysis of the sample so generated [17]. The accuracy of Monte Carlo estimates, and hence the convergence of the method, depends on the square root of the number of samples, making the method very slow [16,26], above all when the computation of the analyzed system requires solution of complex models or when the number of variables is high. The advantage of Monte Carlo simulation is that no approximation of variables or function/model is needed. In order to optimize time required by a straight Monte Carlo simulation, variance reduction techniques may be applied, usually introducing a priori information about specific characteristics of the random variable space [5,10,21]; in this paper, since Monte Carlo is used as a reference solution for applications of other probabilistic
techniques, a crude Monte Carlo simulation is preferred because of the total lack of mathematical/numerical manipulation of the model. The most important disadvantage of numerical simulation, despite it’s great accuracy, is the computational effort it requires; for model of complex systems whose analysis is time consuming and accumulation of thousands of simulations for reliability analysis may become unacceptable (in this case it took around 420 min when only a single cycle of the 300s for only one current density was run). Rather than investigating the problem through virtual experiments, it’s possible to transform the analyzed generalized function (a numerical model in this case) and its basic random variables in order to exploit a special case (all the variables described as independent normal and standard with a linear function) for which peculiar properties hold; in this case probability of not crossing the function (i.e. cumulative distribution function) is directly related to the distance of the function from the origin of the random space so that the probabilistic analysis is transformed in a geometric minimum-finding problem. This kind of methods (First Order Reliability Method) [15] is usually very fast but introduces a large number of approximations so that accuracy strongly depends on similarity between the real case and the approximated one. A very important advantage of FORM is that, since the minimum-finding problem is solved by geometric gradient-based exploration of the random space, sensitivities are an immediate result obtained directly from probabilistic outcomes [15]. A good compromise between time saving computation and accuracy is represented by application of response surfaces: RSM is based on the approximation of the true function by an explicit mathematical expression (usually a second order polynomial) that can be obtained by best-fitting a proper number of function sample points [25]. The accuracy of the approximation is usually improved by different sample points relocation techniques or, as in this work, by high/adaptive order response surfaces. Once a response surface is computed, it can be used with any probabilistic analyses methodology to obtain the probabilistic information that are achievable with the method itself. In this paper, the high order response surface method presented in [7] is applied to the FC model; the response surface generation procedure can be summarized in three steps: single random variable analysis (the variables are considered independently), polynomials structure selection, final sampling and regression. The single random variable analysis is performed in order to evaluate the function degree of dependency with respect to each variable; after setting all variables to their mean values m, one variable at a time is perturbed to the values: me6s, m4s, m2s, m, mþ2s, mþ4s, mþ6s and for each variation an evaluation of the model is carried out. It’s then possible to perform a regression of the function with respect to one single variable, choose which terms of the single variable polynomials can be neglected by means of a variance reduction analysis and therefore evaluate the degree of dependency.
14753
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
The second step selects the regression function starting from a complete fourth order polynomial, and deleting the proper terms, according to the following rules: A term is deleted if it contains a variable with a power greater than the order evaluated at the first step A term is deleted if its own order is greater than the maximum order obtained by the single random variable analysis. Once the polynomials are selected, the procedure knows how many unknown coefficients must be evaluated so that it’s possible to perform a final sampling through Latin Hypercube Sampling (LHS) [11]. LHS is used to assure a full coverage of the random variable space. Finally the regression is performed via the Single Value Decomposition technique [19].
6.3.
Result and discussion
The aim of the study is to rank the different relative weight of sensor failure; so it’s important to understand how the uncertainty in the sensor measurement propagates to a fuel cell-related output (temperature in this case). Since sensitivities (defined by eq. (19) later on) measure the contribution of one random variable standard deviation to the overall output standard deviation, all the basic variables the problem depends on must have the same standard deviations, in order that sensitivities are unaffected by the choice
Table 1 e Spurious signal distribution. Variable Name Tan_in Pan_in Gan_in Pca_in Twc_in Gwc_in Tamb Gca_in Tca_in
Variable Name poly
Distribution
Mean Value
Standard Deviation
X1 X2 X3 X4 X5 X6 X7 X8 X9
Normal Normal Normal Normal Normal Normal Normal Normal Normal
1 1 1 1 1 1 1 1 1
0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015
of distribution parameters and can be ranked purely on their relative importance. For this reason, a Gaussian distribution with the same coefficient of variation has been chosen for all the basic random variables. Nevertheless, the methodology here applied (Fig. 6) is able to deal with any probability distribution. The testing of the method compared to a Monte Carlo process is made only in a fixed current density value (0.5 A/ cm2). After verifying that the methodology is reliable, the RSM analysis can be extended getting response surfaces for different current densities (0.3 A/cm2 and 0.7 A/cm2), as shown in Fig. 6. In a first step, the model has been running with Monte Carlo. Each input of the system has been multiplied by
Fig. 6 e General scheme of the sensitivity/uncertainly process.
14754
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
a source of uncertainty as shown in Table 1. The distribution of each input is shown in Fig. 7. The results of the model can be seen in Fig. 8, where it shows the CDF of the system output (stack temperature).
The second step of the procedure is to select the polynomials structure. In this case, it is only reported the polynomial corresponding to the power output of 0.5 A/cm2:
Fig. 7 e Signal distributions.
14755
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
Fig. 8 e CDF stack temperature comparison.
a random variable and the corresponding value of the cumulative distribution function, sensitivities can be seen as a function of CDF too. For this reason a set of sensitivities is obtained for every point the CDF is discretized in. For the sake of clarity the sensitivities are here presented in three
P ¼c0 þ c1 x1 þ c2 x2 þ c3 x3 þ c4 x4 þ c5 x5 þ c6 x6 þ c7 x7 þ c8 x8 þ c9 x9 þ c10 x1 x2 þ c11 x1 x3 þ c12 x1 x4 þ c13 x1 x5 þ c14 x1 x6 þ c15 x1 x7 þ c16 x1 x8 þ c17 x1 x9 þ c18 x2 x3 þ c19 x2 x4 þ c20 x2 x5 þ c21 x2 x6 þ c22 x2 x7 þ c23 x2 x8 þ c24 x2 x9 þ c25 x3 x4 þ c26 x3 x5 þ c27 x3 x6 þ c28 x3 x7 þ c29 x3 x8 þ c30 x3 x9 þ c31 x4 x5 þ c32 x4 x6 þ c33 x4 x7 þ c34 x4 x8 þ c35 x4 x9 þ c36 x5 x6 þ c37 x5 x7 þ c38 x5 x8 þ c39 x5 x9 þ c40 x6 x7 þ c41 x6 x8 þ c42 x6 x9 þ c43 x7 x8 þ c44 x7 x9 þ c45 x28 þ c46 x8 x9 ;
ð18Þ
In order to compute the 47 unknown coefficients, the code generates 86 sample points via Latin Hypercube Sampling and performs the regression through the Singular Value Decomposition technique. The results are reported in Table 2. In Fig. 8 the comparison between the CDF evaluated by Monte Carlo simulation (105 model evaluations) with the simulation with the response surface (model evaluations and 105 response surface evaluations) of the stack temperature output are reported; once the response surface is known, it’s possible to perform every probabilistic analysis. Finally, very important information that can be achieved by probabilistic analysis is the sensitivity of the random variable function to its input variables. The sensitivity of the stack temperature with respect to the main model parameters pi , is defined as: VTStack ðpi Þ S ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi$/Tstack ¼ f ðpi Þ Pn vTStack 2 i¼1 vpi
/
(19)
Sensitivities are not constant in the field of probability of a function of random variables. They indeed change as a function of the infinite realizations that a continuous random variable (as the ones it deals with) can assume. Since there is a univocal relationship between a possible realization of
Table 2 e Regression Coefficients. c0 47.46
c1 171.92
c2 102.89
c3 43.58
c4 98.53
c5 97.86
c6 0.01
c7 1.39
c8 144.35
c9 76.84
c10 266.72
c11 80.91
c12 194.84
c13 87.94
c14 0.01
c15 1.75
c16 450.09
c17 292.71
c18 61.92
c19 119.97
c20 151.64
c21 0.00
c22 0.05
c23 10.28
c24 117.58
c25 270.97
c26 261.03
c27 0.00
c28 1.57
c29 14.04
c30 59.46
c31 280.72
c32 0.01
c33 2.42
c34 260.58
c35 337.01
c36 0.01
c37 3.07
c38 337.86
c39 320.30
c40 0.00
c41 0.01
c42 0.00
c43 0.48
c44 1.05
c45 77.08
c46 122.51
14756
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
high prob
Tamb Gan_in
low prob
Tan_in
mean prob
Gan_in
low prob
Tan_in Pan_in
Pan_in
Pca_in
Pca_in
Tca_in
Tca_in
Gca_in
Gca_in
Gwc_in
Gwc_in
Twc_in
Twc_in 0
0,2
0,4
0,6
0,8
1
0
1,2
Fig. 9 e Sensitivity of the stack temperature @ 0.3 A/cm2.
representative points of the CDF: a low cumulative probability point (CDF ¼ 2.3262908e-04), the mean value (CDF ¼ 0.5) and a high cumulative probability point (CDF ¼ 9.9976737e-01). From Fig. 9e11, the sensitivities of stack temperature are shown at the density current of the 0.3, 0.5 and 0.7 [A/cm2]. In Table 3 a synthesis of the sensitivity at mean probability is reported. The sensor signal with the highest impact on the stack temperature is the cathode temperature inlet into the stack (Tca_in). This signal is visibly the most important sensor and its measurement has to be very accurate, since it shows an approximate 90% of the total sensitivity. The other significant signal is the coolant temperature inlet to stack (Twc_in) because it represents a 40% of impact on the stack temperature. This temperature has high impact on the energy balance of the fuel cell (eq.(2)) and hence a strong influence over the stack temperature control. Finally, the remaining signals show a not-negligible influence on the predicted value of stack temperature (Tst).
7.
high prob
Tamb
mean prob
Applied sensitivity result in FMECA
To translate the information provided by sensitivity analysis in quantitative measures that can be used in FMECA is a very complex matter; sensitivity analysis can provide a measure of the relative weight a certain parameter has on a system output level of knowledge, but only an expert of the system Tamb
mean prob
Gan_in
low prob
Tan_in Pan_in Pca_in Tca_in Gca_in Gwc_in Twc_in 0
0,2
0,4
0,6
0,8
1 2
Fig. 10 e Sensitivity of the stack temperature @ 0.5 A/cm .
0,4
0,6
0,8
1
Fig. 11 e Sensitivity of the stack temperature @ 0.7 A/cm2.
Table 3 e Sensitivity at expected value. Stack temperature sensitivity
Twc_in Gwc_in Gca_in Tca_in Pca_in Pan_in Tan_in Gan_in Tamb
0.3 A/cm2
0.5 A/cm2
0.7 A/cm2
0.362 0.033 0.017 0.924 0.061 0.000 0.011 0.002 0.088
0.386 0.028 0.007 0.916 0.061 0.006 0.006 0.004 0.079
0.381 0.019 0.004 0.918 0.062 0.0002 0.006 0.003 0.080
can evaluate the consequences of such a mismatching between real data and measured ones. As an example, consider the case of the sensor of the cathode inlet temperature. As shown in the sensitivity analysis (Table 3), this sensor is the most important, therefore a sensor error can cause a significant miscalculation of the temperature
Table 4 e Importance sensors ranking (Stack temperature). Component/System Tan
high prob
0,2
Inlet anode temperature controller Pan Inlet anode pressure controller Gan Inlet anode temperature controller Pca Inlet cathode pressure controller Twc-in Inlet coolant temperature controller Gwc Inlet coolant flow controller Tamb Ambient temperature sensor Gca Inlet cathode airflow controller Tca Inlet cathode temperature controller
Failure Mode
Importance
Spurious output
2
Spurious output
1
Spurious output
1
Spurious output
1
Spurious output
5
Spurious output
1
Spurious signal
3
Spurious output
1
Spurious output
9
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
at which the FC is working. This mismatching can lead to a scenario where the control system assumes that the fuel cell system is working at a temperature significantly higher or lower than the real temperature, with two completely different set of consequences; sensitivity analysis isn’t able to distinguish between the two cases, so it isn’t possible to use sensitivity as a direct measure of the “damage” reported in [8]. However, it is possible to create a ranking (see Table 4) of the importance of each sensor in order to allow the FMECA analyst to design the system with proper sensor quality and redundancies.
8.
Conclusion
The paper described how the development of a complete dynamic model of a Fuel Cell power aircraft could be used in order to precisely individuate the most significant signal failure in the control system of the aircraft itself. Moreover, the model has been validated through several real flight tests of an FC powered two-seat ultra-light aircraft, in the framework of the EU funded ENFICA-FC Project. The main outcomes of the paper are the following: 1. the description of a complete dynamic model of an FC powered two-seat ultra-light aircraft (with in insight on the thermal management model), validated through real flight tests; 2. the development of a innovative procedure of uncertainty/ sensitivity analysis of the input signals of the control system (state measurements through sensors) over the output variables representing the operation of FC powertrain of the aircraft; 3. the use of the proposed uncertainty/sensitivity analysis to improve the failure analysis: the sensitivity analysis provides useful information to engineers for developing a system as well as appropriate FMECA analysis and sensor control characteristics; 4. in particular, the proposed analysis can be used as a tool to rank the different sensors from a safety point of view: how uncertainty of a sensor will propagate to the performance of the FC system; 5. the results demonstrate that accurate temperature sensors and sensor calibration are of unavoidable importance for the control of the stack temperature, over systems such as PEM fuel cells; 6. as an example in terms of thermal control of the system: the sensor signal with the highest impact on the stack temperature is the cathode temperature inlet into the stack (Tca_in), while the coolant temperature inlet to stack (Twc_in) shows the highest impact on the energy balance of the fuel cell and hence a strong influence over the stack temperature control.
Acknowledgments The activity has been developed within the ENFICA-FC project. The authors acknowledge the important
14757
contribution of European Commission by the funding programmes ENFICA-FC, EC 6th FP e Contract No. AST5-CT2006-030779.
references
[1] Boeing, 2008, April, http://www.boeing.com/news/releases/ 2008/q2/080403a_nr.html. [2] Correa G. Dynamic modelling and comparison with experimental data of fuel cell systems for aeronautical applications. Ph.D. thesis. Politecnico di Torino;2011. [3] DLR. Fuel cells power first takeoff for dlr’s antares aircraft. Fuel Cells Bulletin 2009a;9. [4] DLR. 2009b, July, http://www.dlr.de/en/desktopdefault.aspx/ tabid-1/86_read-18278. [5] Deak I. Three Digits accurate Multiple normal Probabilities. Numerische Mathematik. 1980;38:369e80. [6] EASA. 2010, http://www.easa.europa.eu/flightstandards/gaba.html. [7] Frulla G, Borello F. In: Hayworth GI, editor. Structural reliability aspects for advanced composite material applications. Turin: Reliability Engineering Advances; 2009. [8] Frulla G, Correa G. Reliability/Safety concept definition allelectric general aviation (GA) airplane, propeller driven, powered by fuel cells. Turin: Proposal No. 30779 e ENFICAFC; 2008. [9] Gasser, F. An analytical, control-oriented state space model for a PEM fuel cell system. Ph.D. thesis. Lausanne: E´cole Polytechnique Fe´de´rale de Lausanne;2005. [10] Gray W.A. Generalization of the Directionl Simulation in the Load Space Approach to Structural Reliability Analysis. Ph.D. thesis;1990. [11] Iman RL, Davenport JM, Zeigler DK. Latin Hypercube sampling (program user’s guide). Technical Report SAND 79-1473. Albuquerque: Sandia National laboratories; 1980. [12] International Industry Working Group. Commercial aircraft design characteristics - trends and growth projections. 5th ed.; 2007. R1. [13] Lapen˜a-Rey N, Mosquera J, Bataller E, Ortı´ F. First fuel-cell manned aircraft. Journal of Aircraft 2010;47(6). [14] Lienhard J, Lienhard J. A heat transfer textbook. 3rd ed. Cambridge, Massachusetts: Phlogiston Press; 2000. [15] Madsen HO, Krenk S, Lind NC. Methods of structural rebiality. Prantice Hall; 1986. [16] Melchers RE. Structural reliability analysis and prediction. 2nd ed. John Wiley and Sons; 1999. [17] Metropolis N, Ulam S. The Monte Carlo method. Journal of the American Statistical Association 1949;44(247): 335e41. [18] Muller E, Stefanopoulou A. Analysis, modeling, and validation for the thermal dynamics of a polymer electrolyte membrane fuel cell system. Journal of Fuel Cell Science and Technology 2006;3(2):99e110. [19] Press W, Teukolsky S, Vetterling W, Flannery B. Numerical recipes in C: the art of scientific computing. 2nd ed. New York: Cambridge University Press; 1992. [20] Pukrushpan JT, Peng H, Stefanopoulou AG. Control-oriented modeling and analysis for automotive fuel cell systems. Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME 2004;126:14e25. [21] Ripley BD. Stochastic simulation. Wiley & Sons; 1987. [22] Romeo G, Borello F, Correa G. Set-up and test flights of an allelectric 2-seater aeroplane powered by fuel cells. Journal of Aircraft 2011a;49(8).
14758
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 1 4 7 4 5 e1 4 7 5 8
[23] Romeo G, Borello F. Design and realization of a 2-seater aircraft powered by fuel cell electric propulsion. The Aeronautical Journal 2010;114(1155). [24] Romeo G, Cestino E, Borello F, Correa G. Engineering method for air-cooling design of 2-seat propeller driven aircraft powered by fuel cells. Journal of Aerospace Engineering 2011b;24:79e88.
[25] Santarelli M, Leone P, Cali M, Orsello G. Experimental analysis of the voltage and temperature behavior of a solid oxide fuel cell generator. Journal of Fuel Cell Science and Technology 2007;4(2):143e53. [26] Shooman M. In: Robert E, editor. Probabilistic reliability: an engineering approach. 2nd ed. Krieger Publishing Company; 1990.