Multilinear-Model Predictive Control of a Tubular ... - ACS Publications

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Multilinear-Model Predictive Control of a Tubular Solid Oxide Fuel Cell System Seyedahmad Hajimolana,† M. A. Hussain,†,* Masoud Soroush,‡ Wan A. Wan Daud,† and Mohammed H. Chakrabarti†,§ †

Department of Chemical Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia Department of Chemical and Biological Engineering, Drexel University, Philadelphia, Pennsylvania 19104, United States § Energy Futures Lab, Electrical Engineering Building, Imperial College London, South Kensington, London SW7 2AZ, U.K. ‡

ABSTRACT: As solid oxide fuel cells (SOFCs) are highly nonlinear systems, a single linear controller cannot perform satisfactorily over a wide range of operating conditions of the processes. This work studies multilinear-model predictive control of a tubular SOFC. The objective is to control the fuel cell outlet voltage over a wide range of operating conditions by manipulating inlet fuel pressure (flow rate). A first-principles model of an ammonia fed-tubular solid oxide fuel cell is used for the controller design. The model accounts for diffusion, inherent impedance, transport (momentum, heat and mass transfer), electrochemical reactions, activation and concentration polarizations, and the ammonia decomposition reaction. The servo and regulatory performances of the multimodel predictive controller (MMPC) are compared with those of a single-model predictive controller (SMPC) and a proportional-integral (PI) controller. For small load changes, the MMPC, SMPC, and PI controller all provide zero offset, and the MMPC yields the best closed-loop performance. However, for large load changes, the SMPC and PI controller fail to provide zero offset; under these two controllers the closed-loop system with the large load changes is unstable.

1. INTRODUCTION Solid oxide fuel cells (SOFCs) have several appealing features such as high energy-conversion efficiency, flexibility in fuel type, and ability to work with less expensive catalysts.1 A major application of SOFCs is in stationary power generation. SOFCs can also be integrated with a gas turbine, whereby the hot, high pressure exhaust gas stream of the fuel cell is used to spin the turbine thereby generating more electricity. An important advantage of SOFCs over other types of fuel cells, such as a proton exchange membrane fuel cell,2 is that they can generate electricity from a variety of fuels.3 Although hydrogen (H2) is usually the preferred fuel of choice as high power densities can be obtained, effective and economical production and storage of hydrogen as well as its refueling infrastructure are still facing major challenges.4 Therefore, there is interest to use alternative fuels such as bio fuels and ammonia. Ammonia is a good hydrogen carrier and may be an excellent substitute for hydrogen and hydrocarbons due to several reasons. First, the price of ammonia is as competitive as that of hydrocarbons. Second, ammonia can be easily liquefied under about 10 atm at ambient temperatures or at −33 °C under atmospheric pressure, and the volumetric energy density of liquefied ammonia is about 9 × 106 kJ/m3, which is higher than that of liquid hydrogen, making it useful for transportation and storage. Third, ammonia is less flammable compared with the other fuels and the leakage of ammonia can easily be detected by the human nose under 1 ppm. Fourthly and most importantly, there are no concerns about anodic coking, since all the byproducts of the electrode reaction are gaseous.4 All of the above imply that ammonia can be an ideal candidate as a liquid fuel for SOFCs, at least at the present stage when the coking problem of hydrocarbon fuels has not yet been resolved. © 2012 American Chemical Society

Although much experimental work has been done on ammoniafueled solid oxide fuel cells (NH3−SOFCs),4−8 only few research studies are available on mathematical modeling of NH3−SOFCs.9−13 Ni et al.11 developed an electrochemical model for studying NH3−SOFCs with proton-conducting and oxygen ion-conducting electrolytes. The performance of the H2−SOFC was also investigated for comparison. Ni (2011),12 based on previous models, presented a 2D thermo-electrochemical model to investigate the heat/mass transfer, chemical (ammonia thermal decomposition) and electrochemical reactions in a planar SOFC consuming ammonia as a fuel. Simulations were conducted to study physical-chemical processes occurring in NH3−SOFCs. However, the focus of these studies was mainly on mathematical models that mainly accounted for the electrochemistry of planar SOFCs. A detailed dynamic model can undergo model-reduction, leading to the development of a reduced-order model suitable for online control.18,51,52 SOFCs have challenging control problems due to their multitimescale dynamics, nonlinearity, and tight operational constraints.14 They are highly nonlinear because their controller design requires special attention.15−18 Localized or multiloop controllers have been used for different types of fuel cells, including polymer electrolyte membrane fuel cells (PEMFCs), molten carbon fuel cells (MCFCs), and SOFCs. Li et al.19 used an exact linearization state feedback approach to achieve nonlinear robust control of a PEMFC. Pukrushpan et al.20 reported a dynamic feed forward and state feedback controller along with a linear quadratic control method for controlling the SOFC power system. Received: Revised: Accepted: Published: 430

May 4, 2012 November 25, 2012 December 9, 2012 December 9, 2012 dx.doi.org/10.1021/ie301107r | Ind. Eng. Chem. Res. 2013, 52, 430−441

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Hajimolana and Soroush21 used two parallel (fully decentralized) PI controllers to maintain outlet voltage and fuel cell-tube temperature by adjusting the inlet air flow pressure and temperature, respectively. Kandepu et al.22 observed that the power and membrane temperature of a SOFC system can be controlled by a proportional-integral-derivative (PID) control system that manipulated the fuel and air feed flow rates. Aguiar et al.23 studied the temperature control of a SOFC stack using numerical simulations. They proposed a control structure with two feedback controllers. The first feedback controller was model-based controller that set the fuel and air flow rates proportional to the measured current while maintaining the fuel and air utilization ratio constant. The second feedback controller was a PID controller that controls the outlet fuel stream temperature by manipulating the inlet air flow rate around the value calculated by the model-based controller. Simulation results showed that for load changes of moderate magnitude, the PID controller can successfully control the outlet fuel stream temperature. However for larger load changes a different control method should be used to achieve good control performance. Kaneko et al.24 implemented a PID controller to regulate power of a SOFC by manipulating the inlet fuel flow rate. Chaisantikulwat et al.25 modeled a feedback PI controller to control the outlet voltage of a SOFC by manipulating the hydrogen concentration in the fuel stream. Sedghisigarchi and Feliachi26 proposed two completely decentralized PI controllers to control SOFC power and voltage by adjusting the firing angle and modulation index of a converter, respectively. The fuel cell inverter had the ability to adjust its firing angle quickly so that the inverter maintained a constant power output under fast transient disturbances. Strong interactions between the input and output variables of SOFCs and the physical and operating constraints of the cells limit the achievable performance of completely decentralized PID control systems; these control systems cannot handle the constraints optimally. Furthermore, despite the fact that PID control systems can often succeed in achieving their nominal control objective around nominal operating conditions and compensate for small disturbances, they are unable to provide optimal performance in the presence of constraints and are prone to failure under such conditions. Model predictive control (MPC) is a powerful control methodology. Several well established MPC methods are available in the literature.16,27−30,41,44 Wu et al.15 used a predictive controller based on a T−S fuzzy model to maintain the stack temperature. Vijay et al.31 proposed a predictive controller based on a bond graph SOFC model. Manipulated inputs were four air and fuel inlet and outlet flow rates. The control system achieved control objectives subject to constraints on the fuel utilization, air utilization, the cell operating temperature and the anode and cathode pressures. Li et al.32 proposed a nonlinear predictive controller that used genetic optimization to keep the fuel cell voltage and fuel utilization at desired values. Their model was a simple nonlinear model mainly representing the electrochemical process. They showed that in the presence of a 13% load change, the closed-loop performance was satisfactory. Zhang et al.27 developed a nonlinear predictive controller for a planar SOFC. The control objective was to keep the output power, fuel utilization, and operating temperature at constant values by manipulating the current density as well as inlet fuel and air flow rates. The states of the model were estimated using a moving horizon state estimator. A set of stochastic noise was artificially added to the output model of the SOFC to validate the robustness of the moving horizon estimator. The results showed satisfactory

performance of the moving horizon estimator. Wang et al.29 developed a subspace-based data-driven predictive controller for a SOFC stack. Their controlled variables were the stack voltage, fuel utilization, ratio of partial pressure of hydrogen to oxygen, and pressure difference between the anode and cathode, of which the fuel cell voltage was measured. The manipulated variables included the molar flow rates of hydrogen and oxygen, whereas the current demand was considered as a disturbance. Hua et al.30 developed a predictive controller based on a Hammerstein model of a SOFC to maintain the outlet voltage of a SOFC at a desired value by regulating the flow rate of an inlet natural gas stream. The model described the dependence of the cell outlet voltage on the inlet natural gas flow rate. Their simulation results showed that the Hammerstein model predicts the nonlinear dynamics of the SOFC satisfactorily. Sanandaji et al.33 developed a model predictive controller based on a reduced version of a complex physical model of a SOFC stack, with the low-complexity model being specifically tailored for real-time optimization. The model reduction was based on a method that continuously blended multiple linear models according to a scheduling parameter. Chen et al.53 presented a multilinear model predictive controller for a hybrid proton exchange membrane fuel cell/ultracapacitor system to ensure high cell outlet power and adequate supply of oxygen to the cell. The controller employs a set of fuzzy clustering-based models and an upper-layer adaptive switch to determine which of the models should be used at each time instant. Several other investigators29,34−36 used simplified models for model-based controller design. Obviously, a simplified model cannot perform better than a detailed model, as the former cannot predict the system behavior over a wide range of operating conditions accurately. As a SOFC is severely nonlinear and typically has several operation constraints, a single linear controller may not provide satisfactory performance over a wide range of operating conditions. In this work, the objective is to control the fuel cell outlet voltage using a multilinear-model predictive controller that manipulates the pressure of the inlet fuel stream. Such a control study for SOFCs has not yet been reported in the literature. A detailed model of an NH3−SOFC is used for the controller design herein, unlike in the previous studies where a simplified model was used. The performance of the controller is compared with those of a single-linear-model predictive controller and a PI controller. The organization of this paper is as follows. Section 2 describes the SOFC dynamic model. Section 3 briefly reviews model predictive control and describes the outlet voltage control using the multilinear-model predictive control approach. Section 4 presents and discusses the performances of the MMPC, SMPC, and PI controller. Finally, concluding remarks are presented in section 5.

2. SOFC DYNAMIC MODEL The mathematical model used in this study was adapted from a model reported earlier.21 The dynamic model describes an NH3−SOFC and is based on the first principles. The system under study here is a single tubular SOFC. The cell has two tubes, an outer and an inner tube, as shown in Figure 1. The outer one is a cell tube. The outer surface of the outer tube is the anode side of the cell and its inner surface is the cathode side. Between the anode and cathode sides (surfaces) lies a solid oxide electrolyte. The inner tube is an air injection and guidance alumina tube from which preheated air is injected into the bottom of the cell tube and flows over the cathode surface through the gap between the injection tube and the cell tube. 431

dx.doi.org/10.1021/ie301107r | Ind. Eng. Chem. Res. 2013, 52, 430−441


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Table 2. Operating Conditions of the Process parameter

21

Figure 1. Division of the single tubular SOFC into seven subsystems.

Fuel gas flows over the anode surface through the gap among the cell outer tubes. To develop a first-principles model of the SOFC system, each single tubular fuel cell is divided into the seven subsystems (see Figure 1): Subsystem 1 (SS1) is the air inside the injection tube, subsystem 2 (SS2) is the solid injection tube, subsystem 3 (SS3) is air inside the space between the cell and injection tubes, subsystem 4 (SS4) is the diffusion layer inside the cathode side of the fuel cell tube, subsystem 5 (SS5) is the cell tube, subsystem 6 (SS6) is the diffusion layer inside the anode side of the fuel cell tube, and subsystem 7 (SS7) is the fuel flow channel. The fuel cell model is derived by writing mass, energy and/or momentum conservation equations for each of the five subsystems. Tables 1 and 2 present the parameters used in the simulation

value −4

ref

εcat τcat εano τano Δano Δcat Eactano

1.5 × 10 m 1.2 × 10−3 m 1.1 × 10−3 m 7.5 × 10−4 m 7 × 10−4 m 2.5 × 10−1 m 4.317 × 10−2 kg 8.735 × 10−3 kg 3 × 10−1 A·S/V 9 × 10−1 Ω 1 × 10−1 Ω 3.325 × 1012 kJ·kg−1·K−1 1.962 × 105 kJ/(kg·K) 7.4 × 10−1 kJ/(kg·K) 0.9768 + 0.000241 × Tit kJ/(kg·K) 4 × 10−1 5 4 × 10−1 5 5 × 10−5 m 1.3 × 10−4 m 1.1 × 105 kJ/kmol

Campanari Campanari Campanari Campanari Campanari Hall49 Hall49 Singhal47 Singhal47 Singhal47 Singhal47 Singhal47 Singhal47 Campanari

Eactcat

1.2 × 10 kJ/kmol

Campanari and Iora38

Aano rcto rcti rito riti L mct mit Cct Rtct Rto zr Er C̃ C̃ pit

5

2

Singhal47 Singhal47 Singhal47 Singhal47 Singhal47

L

L

and and and and and

reference

4Ω 1 atm

Hajimolana and Soroush21 Singhal47

Tano fuelin

1023 K

Singhal47

uano fuelin

6.42 m/s

ξ̅ano NH3in

0.939

Ni10

ξ̅ano H2in

0.03

Ni10

ξ̅ano N2in

0.001

Ni10

ξ̅ano H2Oin

0.03

Ni10

ξ̅ano O2in

0.2333

Ni10

Pinj airin

1 atm

Singhal47

Tinj airin

1173 K

Singhal47

uinj airin

450 m/s

Pressure drops caused by the injection pipe resistance over the distance L is negligible. For control applications, it is not necessary to have such great accuracy because the feedback will correct a considerable amount of error in the model. Therefore, a well-mixed subsystems is assumed in this model. 2.1. Model for Subsystem 1 (SS1). Subsystem 1 includes the air inside the injection tube. The mass/momentum and energy balances inside SS1 are given as follows:

Table 1. Model Parameter Values parameter

value

Rload Pano fuelin

dρairinj dt

inj inj inj inj = uair ρ − uair ρair in air

inj inj d(uair ρair )

dt

=

inj 2 inj (uair ) ρair in in

−

Iora38 Iora38 Iora38 Iora38 Iora38

L

(1)

in

inj inj d(H̃ air ρair )

∂t

−

inj 2 inj (uair ) ρair

+

inj ρairinj R *Tair in in

Mair

inj ρairinjR *Tair

Mair

(2)

inj inj inj inj inj ̃ inj = uair ρ Hairin − uair ρair Hair in air in

+

2Lh iti riti

inj (Tit − Tair )

(3)

This model assumes that the pressure drop caused by the injection pipe resistance over the distance L is negligible. Enthalpies of formation, heat capacities, viscosities, convection heat transfer coefficients, diffusion coefficients, and conductivities of the components of air and the fuel are the same as in ref 21. 2.2. Model for Subsystem 2 (SS2). SS2 is the solid injection tube. An energy balance for the injection tube takes the following form: σ 2πritoL dTit cat m it Cp̃ = − Tit) (Tct 4 − Tit 4) + 2πritoLh ito(Tair it dt R rad

and Iora38

and operating conditions, respectively. Several assumptions are made. The thicknesses of the gas boundary layers are very small relative to the radius of the outer tube, therefore, the equations governing the diffusion processes are written in the Cartesian coordinates. Fluid velocities, temperatures, and pressures are averaged along the radial direction. Specific properties such as conductivities, heat capacities, viscosities, and densities in each subsystem are also uniform. Furthermore, outlet partial pressures, temperatures and velocities of a subsystem are equal to the pressures, temperatures, and velocities inside the subsystem. The external load (load impedance) of the cell is a pure resistance.

inj + 2πrit iLh it i(Tair − Tit)

(4)

2.3. Model for Subsystem 3 (SS3). SS3 consists of the air (including oxygen and nitrogen) that flows inside the space between the injection tube outer surface and the cathode (inner) surface of the cell tube. A mass, momentum, and energy balance on the air inside SS3 takes the following form, respectively: 432



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Ni catalyst (typical Ni−YSZ cermet with a Ni loading of about 50 vol %) in SOFC:40

Mass balance on oxygen inside SS3: L

cat d(ρaircat ζÕ 2 )

dt

cat cat ̃cat cat cat ̃cat = uair ρ ζO2,in − uair ρair ζO2 in air

NH3 → 1.5H 2 + 0.5N2

in

⎛ 2r L ⎞ ct − NO2MO2⎜⎜ 2 i 2 ⎟⎟ ⎝ rct i − rito ⎠

⎛ E ⎞ rNH3 = zr exp⎜⎜ − r ⎟⎟PNH3 ⎝ R kjTct ⎠

(5)

(8)

Momentum balance on the air inside SS3: L

cat cat d(uair ρair )

dt

=

cat 2 cat (uair ) ρair in in

−

−

cat 2 cat (uair ) ρair

+

cat ρaircat R *Tair cat Mair

where PNH3 is the partial pressure of NH3. It is considered that the reaction rate is mainly dependent on the partial pressure of NH3 and the operating temperature. Based on the reaction stiochiometry, the rate of hydrogen produced is 1.5 times of the rate of ammonia consumed, while the rate of nitrogen produced is half of the rate of ammonia consumed. It is assumed here that NH3, H2O and N2 diffuse into the anode at a negligible rate; only H2 gas diffuses into the anode (subsystem 4).12 An energy balance for this subsystem yields

cat ρaircat R *Tair in in

Mair (6)

Energy balance on SS3: L

cat cat d(H̃ air ρair )

∂t

cat cat ̃ cat cat cat ̃ cat = uair ρ Hairin − uair ρair Hair in air in

+ −

mct Cp

2ritoLh ito 2

rct i − rito

2

2rct iLhct i rct i 2 − rito 2

ct

cat (Tit − Tair )

cat (Tct − Tair )

⎛ 2r L ⎞ ct i cat ⎜ ⎟ − NO2H̃ air ⎜r 2 − r 2⎟ ⎝ ct i it o ⎠

dTct cat = H − 0.001VoutI + 2rct iLhct i(Tair − Tct) dt σ 2πrct iL ano + 2rctoLhcto(Tfuel − Tct) + R rad × (Tit 4 − Tct 4) + 2rctoLrNH3ΔHNH3

(9)

where H = 2πL(rct iNO2HOcat2 + rctoNH2HHano − rctoNH2OHHano ) 2 2O

(7)

(10)

2.4. Model Equations for Subsystem 4 (SS4). SS4 consists of the diffusion layer inside the cathode side of the fuel cell tube (Appendix A). 2.5. Model Equations for Subsystem 5 (SS5). This subsystem represents the fuel cell tube, which consists of the anode layer, the cathode layer, and the electrolyte layer between them. Electrochemical reactions occur in this subsystem (Appendix A). It is assumed that ammonia decomposition occurs on the anode surface layer inside the SS5. Therefore, this reaction absorbs the heat from subsystem 5. 2.5.1. Decomposition Reaction of Ammonia. The thermal decomposition of ammonia for producing hydrogen in the porous anode is solved using the chemical model described as follows. NH3 thermal decomposition takes place at the anodic surface inside the fuel channel as this process is favored at high temperatures. At a temperature of 1273 K, the equilibrium conversion of NH3 can be close to 100%. Even with effective catalysts (Ni−Pt or Ru), the conversion of NH3 into H2 and N2 is still considerably lower than the conversion predicted by means of thermodynamic equilibrium. Other studies also show that the rate of NH3 thermal decomposition (with no catalyst) is considerably lower than the thermal decomposition reaction with catalyst.50 Therefore, in the present study, the decomposition of NH3 without catalyst is assumed to be negligible and only catalytic thermal decomposition of NH3 in the porous anode layer is considered. However the model developed in this work does not account for the effect of the catalyst on the system.8 In the present study, it is considered that thermal decomposition takes place in the composite anode of the SOFC with typical catalyst (Ni) loading.40 On the basis of the experimental results,48 an analytical expression has been derived to describe the kinetics of thermal decomposition of NH3 on the surface of

which is the net rate of chemical energy entering and leaving SS4, Vout × I is the electric power supplied to the external load, the third and fourth terms on the right-hand side represent convective heat transfer from the anode and cathode side fluids to the fuel cell tube, and the fifth term denotes the heat transfer from the injection tube by radiation. In the temperature range of 600−1200 K, the thermal energy demand for the NH3 decomposition reaction can be approximately calculated as in ref . HrNH3 = 40265.095 + 24.23214Tct − 0.00946Tct 2

(11)

It can be assumed that the necessary heat flow (rNH3·ΔHNH3) is supplied by the SOFC and not from a decrease in the internal energy of the fuel gas. 2.6. Model Equations for Subsystem 6 (SS6). SS6 consists of the diffusion layer inside the anode side of the fuel cell tube (Appendix A). 2.7. Model Equations for Subsystem 7 (SS7). SS7 includes the fuel that flows in the space on the anode side of the cell tube. The fuel consists of H2, H2O, N2, and NH3 gases. The component mass balances on the H2, H2O, N2, and NH3 gases, as well as momentum and energy balances are given by eqs 12−17, respectively, L

ano ̃ano ζH2 ) d(ρfuel

dt

ano

ano

ano ano ̃ ano ano ̃ ρ ζH2in − u fuel ρfuel ζH2 = u fuel in fuel in

⎛ 2πrctoL ⎞ +⎜ ⎟M H [3/2R decomp − NH2] ⎝ A ano ⎠ 2 (12) 433


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ano ̃ano ζH2O) d(ρfuel

dt

Article

where k is the current sampling interval, k + i is a future sampling interval (within the prediction horizon), P is the prediction horizon, ny is the number of plant outputs and is equal to 1, wjy is the weight for output j, and [yi(k + i) − rj(k + i)] is the predicted deviation at future instant k + i. M is the control horizon, nmv is the number of manipulated variables (inputs) and is equal to 1, Δui(k + i − 1) is the predicted adjustment (i.e., move) in manipulated variable at future (or current) sampling interval (k + i − 1), and wjΔu is the weight for input j. The cost function can be subjected to inequality constraints on the manipulated and controlled variables as below:

ano ano ̃ano ano ano ̃ano ρ ζH2Oin − u fuel ρfuel ζH2O = u fuel in fuel in

⎛ 2πrctOL ⎞ +⎜ ⎟M H O[NH2O] ⎝ A ano ⎠ 2

(13)

ano

L

ano ̃ ζN2 ) d(ρfuel

dt

ano

ano

ano ano ̃ ano ano ̃ ρ ζN2in − u fuel ρfuel ζN2 = u fuel in fuel in

⎛ 2πrctoL ⎞ +⎜ ⎟M N [R decomp] ⎝ A ano ⎠ 2

(14)

umin ≤ u(k + i − 1) ≤ umax

ano

L

ano ̃ d(ρfuel ζNH3)

dt

ano ano ano u fuel ρ ζ̃ in fuel in NH3in

=

−

ano ano ano ̃ u fuel ρfuel ζNH 3

⎛ 2πrctOL ⎞ +⎜ ⎟MNH3[−2R decomp] ⎝ A ano ⎠

Δumin ≤ u(k + i − 1) ≤ Δumax

L

dt

(15)

ymin ≤ y(k + i − 1) ≤ ymax

in

+

−

ano * ano ρfuel R Tfuel ano M fuel

(16)

assuming that the wall resistance of the channel is negligible. An energy balance for the fuel gas inside SS5 results in L

ano ̃ ano d(ρfuel Hfuel )

ano ano ̃ ano ano ano ̃ ano = u fuel ρ Hfuel in − u fuel ρfuel Hfuel in fuel

∂t

in

+

2rctoLhcto A ano

ano (Tct − Tfuel )

⎛ 2πrctoL ⎞ +⎜ − NH2HHano ) ⎟(NH2OHHano 2O 2 ⎝ A ano ⎠ (17)

3. MODEL PREDICTIVE CONTROL Model predictive control (MPC) can ensure optimal operation in the presence of constraints. MPC uses a dynamic model of the process under consideration to predict and optimize the future behavior of the process over a moving horizon. The open-loop optimization calculates a sequence of control actions at each time instant, but only the first control move is implemented at each time instant. As new process measurements arrive at each sampling instant, state estimates are calculated, model parameters are updated, and then the open-loop optimization is conducted. The ability of MPC to handle constraints and plants with multiple inputs and multiple outputs systematically, as well as its flexibility in using almost any form of models has made MPC an attractive tool for the process industries. The MPC usually minimizes the cost function V(k) in the form:42 P

V (k ) =

ei(k) = Vout(k) − Vî (k)

i = 1, 2, ..., M

∑ γ= k − q + 1

(22)

α3k − iei2(γ ) (23)

α1 ≥ 0 and α2 > 0 represent the weights on the present and past errors, respectively; q is an integer, and α3 (0 < α3 < 1) is a specific constant to weight down older data. On the basis of the performance index Ji(k), the supervisor selects the candidate SMPC that is most suitable for the current process conditions. At each time k, the performance index of the currently chosen model, Jc(k), is compared with those of the other models. If at the present time instant

i=1 j=1 nmv

∑ ∑ {wjΔuΔuj(k + i − 1)}2 i=1 j=1

(21)

k−1

Ji (k) = α1ei 2(k) + α2

ny

M

i = 1, ..., P

where V̂ i(k) is the value of the cell outlet voltage predicted by model i at time k, Vout is the measure value of the cell outlet voltage at time k, and M is the number of linear models. The following objective function is then used for selecting the appropriate model for each operating condition:

∑ ∑ {wjy[yj (k + i) − rj(k + i)]}2 +

(20)

The controller tunable parameters include the control horizon, prediction horizon, and weighting matrices. If a single linear model is used to describe a process around a nominal operating point and the model is used in model-based control, for robust model-based control of the process the model has to be updated every time the process moves away from the nominal operating point. Such adaptation/updating can take a long time and result in a large transient error. One way to handle this problem is to use a set of linear models together with a switching approach for model-based control.43 Each model represents the plant dynamics in a specific range of operating conditions. The switching criterion is crucial in designing the multiple linear model system. Indeed, a MMPC consists of several SMPC and a switching mechanism. At any instant, one of the SMPCs controls the process under consideration (“inner loop”), and in an “outer loop”, a “supervisor’ selects the right SMPC among the set of SMPCs and decides when to switch to a different SMPC, on the basis of a performance criterion and input-output data. The best model for a given operation range is obtained by minimization of an objective function. Let the error in the predicted value of cell outlet voltage by the ith model be defined as

ano 2 ano ano 2 ano = (u fuel ) ρfuel − (u fuel ) ρfuel in ano * ano ρfuel R Tfuel in in ano M fuel in

i = 1, ..., M

(19)

Output variable constraint is given as

An axial momentum balance for the fuel yields ano ano ρfuel ) d(u fuel

i = 1, ..., M

(1 + h) min Ji (k) Jc (k)

(18)

i

434

(24)



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then the controller whose model has the lowest Ji(k) value is selected and implemented. In eq 24, h > 0 is a tunable hysteresis parameter that prevents excessive switching. Both in theory and in practice, it is important that excessive switching is avoided. The model switching algorithm is summarized in the flowchart shown in Figure 2.

of the model can be based on the prediction error described by eq 22. To implement the MMPC, three linear models that represent linear approximations of the nonlinear dynamic fuel cell system are employed; each model represents the cell dynamics in a specific range of operating conditions. Table 3 gives the nominal operating point and range of operating conditions for each model. In each specific range, one Table 3. Operating Points for Linearization of the Models and the Range of the Validity of Each Model operating range of dynamic system: 0.84 V ≤ Vout ≤ 0.89 V model

operating point

range of operating conditions

model 1 model 2 model 3

0.89 0.873 0.856

0.873 V ≤ Vout 0.856 V ≤ Vout < 0.873 V Vout < 0.856 V

Table 4. Tuning Parameters of the MMPC and SMPC MMPC tuning parameters

MPC1

MPC2

MPC3

SMPC

control interval (time units) prediction horizon (intervals) control horizon (intervals)

1 10 2

1 10 2

1 10 2

2 5 1

Figure 2. Flowchart of the switching algorithm.

3.1. Outlet Voltage Control. An effective control strategy should not steer a SOFC to failure conditions. Common requirements for operating SOFCs involve the control of outlet voltage. Load changes cause changes in the outlet voltage. To circumvent an unwanted loss in voltage that could possibly damage electrical equipment, an advanced control system is required for the system. One approach involves the development of multiple linear models that account for the anticipated operating range, design a controller based on each model and then develop a criterion which the control system switches from one controller to another. This approach is used here; a multilinear-model predictive controller is implemented to control the outlet voltage of a SOFC by manipulating the fuel flow rate. For comparison purposes, a single-model predictive controller and a conventional PI controller are also implemented. The servo and regulatory performances of the MMPC, SMPC, and PI controller are compared. In these simulation studies the output cell voltage is limited to 0.865 V ≤ Vout ≤ 0.875 V for small set-point tracking changes and to 0.84 V ≤ Vout ≤ 0.89 V for large set-point tracking changes. The number of linear models depends on the size of the operating region of the cell and the degree of the nonlinearity of the cell within the operating region; the larger is the region or the more nonlinear is the cell in the region, the higher should be the number of the linear models. Quantification of the size of the operating region and the degree of nonlinearity

Figure 3. Responses under the MMPC, SMPC, and PI controller to a series of small set-point changes in Vout: (a) controlled variable; (b) manipulated variable profiles corresponding to panel a.

of the MPCs (MPC1, MPC2, or MPC3) is implemented. There are six disturbances: air pressure, temperature, and velocity, fuel 435



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Figure 4. Models selected and used by MMPC during the simulation.

Figure 6. Manipulated variable profile corresponding to Figure 5.

of the outlet voltage: 0.84 and 0.89 V, respectively. An output variable weight of 1 and a manipulated variable weight of 0.1 are used. The conventional PI controller is first tuned using the standard Ziegler−Nichols (Z−N)46 guidelines. The parameters are fine-tuned by trial and error, leading to the controller gain K = 2 and the integral time τ = 5. This approach was also used for the internal model control guidelines,56 which led to comparable controller parameter values. The model predictive controllers used in the studies presented herein are mixed state- and output feedback controllers; in addition to requiring a measurement of the controlled variable/ outputwhich is easily measurable using a digital voltmeter they require online measurements of all state variables of the fuel cell. In practice, the state variables that cannot be measured online or are not measured online, should be estimated using a state estimator/observer.

Figure 5. The performance of PI controller and SMPC in tracking a series of large set point changes.

4. RESULTS AND DISCUSSION 4.1. Small Set-Point Changes. Figure 3a shows a comparison of the outlet voltage responses of the SOFC to a series of small set-point changes under the PI controller, SMPC, and MMPC. To perform a sensible comparison, all three controllers are exposed to the same model. Upon comparing the responses of the three controllers one can conclude that all three controllers are capable of providing satisfactory control. However, the response under the PI controller has larger response time. The SMPC and MMPC have very close performances. To compare

temperature and velocity, and external load resistance. The control system varies the fuel flow pressure to regulate the SOFC outlet voltage. To ensure good performance of the MMPC, the tuning parameters must be set to appropriate values. The parameters were initially set according to the recommendation of Shridhar and Cooper45 and they were then fine-tuned based on actual control performance. The controller tunable parameter values are listed in Table 4. The upper and lower limits of the manipulated input are 0.2 and 4 atm, respectively, and those 436



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input profile; as expected, the fuel flow pressure is always less than 4 atm. Figure 8 reveals how two switch coordinates between three controllers (MPC1, MPC2, and MPC3) in series based on the

the performances of the controllers quantitatively, we use the integral of time absolute error (ITAE). The ITAE is 1.03 × 105 under the MMPC, 1.4015 × 105 under the SMPC, and 1.5762 × 105 under the PI controller. As expected, these indicate that the MMPCs provide the best performance. The corresponding manipulated variable profiles are given in Figure 3b. Figure 4 shows the sequence of switching of MMPC between three models. When the outlet voltage changes from 0.870 to 0.875 V, the MMPC calculates the prediction error and then Ji(k) for each model. Since J1(k) is the minimum value among the others, model 2 switches to model 1 for implementing the process. For the next changes of outlet voltage J2(k) is the minimum value among the other models. Therefore model 1 switches to model 2 as can be seen in Figure 4. 4.2. Large Set-Point Changes. Figure 5 shows the outlet voltage responses under the three controllers when there are large set-point changes. As can be seen in Figure 5, the SMPC and PI controller performances are unsatisfactory. Also, the corresponding manipulated variable profiles in Figure 6 shows vigorous fluctuations, which indicates that these two controllers are not able to provide satisfactory control when there are large load changes. The results indicate the inadequacy of PI control and single-linear-model predictive control for the process under consideration. On the other hand, as shown in Figure 7a, the

Figure 8. Strategy of switching for synchronizing three controllers (MPC1, MPC2, and MPC3 in that series).

lowest Ji(k). The results explain why the MMPC has the best performance among SMPC and PI controller. 4.3. Simulation Results of MMPC for ±30% Load Changes. Figure 9a depicts the closed-loop outlet voltage

Figure 7. (a) Performance of MMPC in tracking a series of large set point changes. (b) Manipulated variable profiles corresponding to panel a. Figure 9. (a) Closed-loop response of the SOFC under MMPC to step changes of ±30% in the load resistance. (b) Manipulated variable profile corresponding to panel a.

MMPC track the set-point changes without any fluctuations or overshoots. Figure 7b shows the corresponding manipulated 437



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response of the fuel cell to step changes of ±30% in the external load resistance at t = 200 s (two separate simulation runs). It can be seen that the MMPC is able to reject the disturbances asymptotically. The positive step change initially increases the voltage. After the disturbance affected the controlled output, the control system decreases the inlet fuel flow pressure and pushes the controlled output back to the set-point value (see Figure 9b). Figure 10 panels a and b show the controlled output and the manipulated input respectively when two separate (not

Figure 11. Closed-loop response of the SOFC under the MMPC to step changes of ±30% in the input air pressure, temperature, and velocity.

disturbance (see Figure 11b). As seen in Figure 11b the needed adjustments are small. The closed-loop results indicate that in all cases, MMPC outperforms SMPC and PI controller in terms of set-point tracking and load disturbance rejection.

5. CONCLUSIONS A multiple-linear-model predictive controller, a single linear model predictive controller, and a proportional-integral controller were implemented on a tubular SOFC, and their performances were compared. The simulation results showed that the SMPC provides better control than the classical PI controller. However, for higher load changes (±30%), both the PI controller and SMPC failed to control the voltage, while the MMPC performed satisfactorily. Also, the MMPC can regulate the process when subjected to greater load disturbances that does not surpass the design capacity of the fuel cell. To prevent unnecessary frequent switching in the MMPC, a hysteresis switching algorithm was employed. A future study of ours is to implement the MMPC in real-time on a lab-scale tubular solid oxide fuel cell system and to investigate the issues involved in the real-time implementation of the MMPC.

Figure 10. (a) Closed-loop response of the SOFC under MMPC to step changes of ±30% in the inlet fuel temperature and velocity. (b) Manipulated variable profile corresponding to panel a.

simultaneous) ±30% step changes are made at t = 200 s. When the inlet fuel temperature changes stepwise by ±30% the outlet voltage changes. The controller then adjusts the inlet fuel flow pressure to reject this disturbance asymptotically. These results indicate that the controller can reject the two disturbances effectively. Figure 11 panels a and b depict the controlled output and the manipulated input respectively for six separate (not simultaneous) step changes of ±30% in the inlet air pressure, temperature and velocity at t = 100 s. When the air inlet pressure is increased by 30% the outlet voltage increases. The controller then adjusts the manipulated input and bring the controlled output back to its setpoint value. When the inlet air temperature changes stepwise by +30%, the outlet voltage increases. The controller then decreases the inlet fuel flow pressure to reject this disturbance asymptotically. The positive step change in inlet air flow velocity leads to increases in the outlet voltage and the cell-tube temperature. The controller then adjusts the inlet fuel pressure to reject the

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APPENDIX A

Model Equation for Subsystem 4 (SS4)

A mole balance on oxygen inside the cathode-side diffusion layer yields 438


Industrial & Engineering Chemistry Research TPB ⎛ 1 ⎞ Δcat d ⎡⎢ PO2 ⎤⎥ ⎟⎟R O = NO2 − ⎜⎜ 2 R dt ⎢⎣ Tct ⎥⎦ ⎝ 2πrct iL ⎠

Article

must diffuse. Pore volume percentage, as well as diffusion length, can be varied to optimize these properties. The ohmic loss, ηohm, is another loss that should be accounted for in a SOFC. As the electrolyte resistivity at high temperatures vary slightly within a small range of electrolyte temperature, the electrolyte resistance has been assumed to be constant in several studies.39,54,55,57 Wang et al.10 ignored ohmic polarization entirely due to a very thin electrolyte layer that was considered. In this study, we assume that the cell ohmic resistance is constant, and the ohmic loss is accounted for through the equivalent circuit considered in the next paragraph. An approximate equivalent circuit of an SOFC that consists of two internal resistances and one internal capacitance can be found in ref 39. According to the equivalent circuit approximation, the cell outlet voltage is governed by:

(A-1)

Electrochemical Model Inside Subsystem 5 (SS5)

Electrochemical reaction occurs inside the fuel cell at the triple phase boundaries (TPBs). The SOFC electrochemical reactions that occur inside the TPBs are Anodic side reaction: H 2 + O2 − → H 2O + 2e

Cathodic side reaction: O2 (g) + 4e → 2O2 −

The Nernst potential for these reactions is given by E H2 = E H02

⎡ PTPB(PTPB)1/2 ⎤ H O ⎥ ln⎢ 2 TPB2 + ⎥⎦ ⎢⎣ 2F PH2O

⎛ 1 ⎞ ⎞ dVtl 1 ⎛ 1 1 =⎜ + ⎟E − ⎜ ⎟Vtl dt Cct ⎝ R tct R to + R load ⎠ ⎝ R tctCct ⎠

RJTct

(A-2)

⎛ R load ⎞ Vout = ⎜ ⎟Vtl ⎝ R to + R load ⎠

However, the actual cell voltage (E) is less than its theoretical open circuit voltage, because it is strongly affected by several losses including activation losses, concentration losses due to mass transport resistance in the electrodes, and ohmic losses due to ionic and electronic charge transfer resistances. Actual voltage is thus given by E = E H2 − ηact − ηact − ηohm H2

O2

⎛ ⎞ 1 I=⎜ ⎟Vtl ⎝ R to + R load ⎠ where Rto is the total ohmic resistance in the inherent impedance of the cell, Rtct is the total charge transfer resistance of the cell, Cct is the charge transfer capacitance of the cell, I is the current through the external resistive load, Vout is the fuel cell outlet voltage (voltage across the external load) and Vtl is the voltage across the total ohmic resistance and the load resistance in series.

(A-3)

The activation polarizations are the result of the kinetics involved with the electrochemical reactions. It becomes an important loss when the current is low, because at low currents the reactants must overcome an energy barrier named activation energy (Eact) to drive the electrochemical reactions at the electrode−electrolyte interface and this barrier leads to the polarization. The activation barrier is the result of many complex electrochemical reaction steps where, typically, the rate-limiting step is responsible for the polarization. One can account for the anode and cathode activation polarizations using the Butler−Volmer correlation:37 ηact

ano

=

ηact = cat

RJTct F

RJTct F

⎛ I ⎞ ⎟⎟ sin h ⎜ ⎝ 2I0ano ⎠

Model Equation for Subsystem 6 (SS6)

Mole balances on hydrogen and water vapour inside the anodeside diffusion layer are given as follows: TPB Δano d ⎡⎢ PH2O ⎤⎥ = −NH2O + R dt ⎢⎣ Tct ⎥⎦

−1⎜

−1⎜

I0ano = 14 × 10

) πrctoL(PHTPB 2

⎛ Eact ⎞ ano ⎟⎟ exp⎜⎜ ⎝ R kJTct ⎠

)1/4 I0cat = 14 × 109πrct iL(POTPB 2

⎛ Eact ⎞ cat ⎟⎟ exp⎜⎜ R T ⎝ kJ ct ⎠

(A-9)

(A-10)

The rates of consumption of hydrogen and oxygen by the electrochemical reactions to generate an electric current of I are given by

(A-5)

where I0ano and I0cat are the anode and cathode exchange currents, respectively, which are given by38 9

⎛ 1 ⎞ ⎟⎟R H O ⎜⎜ 2 ⎝ 2πrctoL ⎠

TPB ⎛ 1 ⎞ Δano d ⎡⎢ PH2 ⎤⎥ ⎟⎟R H = NH2 − ⎜⎜ 2 R dt ⎢⎣ Tct ⎥⎦ ⎝ 2πrctoL ⎠

(A-4)

⎛ I ⎞ ⎟⎟ sin h ⎜ ⎝ 2I0cat ⎠

(A-8)

(A-6)

R H2 =

⎛ 1 ⎞ ⎜ ⎟I ⎝ 2F ⎠

(A-11)

R O2 =

⎛ 1 ⎞ ⎜ ⎟I ⎝ 4F ⎠

(A-12)

The consumption of the reactants is accompanied by the production of water at the following rate:

(A-7)

Concentration losses are those associated with concentration variation of the critical species due to mass transport processes. There are usually two sources of losses that are due to mass transport: (i) diffusion between the bulk phase and cell surfaces, and (ii) transport of reactants and products through electrodes. Therefore, the concentration polarization is highly dependent on the gases used, as well as the distance through which the gases

■

⎛ 1 ⎞ R H 2 O = − ⎜ ⎟I ⎝ 2F ⎠

(A-13)

AUTHOR INFORMATION

Corresponding Author

*Tel: +603-79675206. Fax: +603-79675319. E-mail: mohd_azlan@ um.edu.my. 439



Article

τ = tortuosity ζ̃ = mass fraction

Notes

The authors declare no competing financial interest.

■

Subscripts and Superscripts

ACKNOWLEDGMENTS The authors are grateful for the UM/MOHE High Impact Research Grant (UM.C/HIR/MOHE/ENG/18). The authors acknowledge the UM Bright Sparks Unit {Bsp_App462/ 11(K)} for providing financial assistance during this work.

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NOMENCLATURE Aano = anode-side flow-channel cross-sectional area (m2) Cct = charge transfer capacity (F = A s/V) Cp = specific heat capacity at constant pressure (kJ/ (kmol K)) C̅ p = specific heat capacity at constant pressure (kJ/ (kg K)) Cv = specific heat capacity at constant volume (kJ/ (kmol K)) C̅ v = specific heat capacity at constant volume (kJ/ (kg K)) D = total diffusion coefficient (m2/s) E = irreversible cell voltage (V) Erev = reversible cell voltage (V) Eact = activation energy (kJ/kmol) Er = activation energy of ammonia decomposition (kJ/kmol) e = model prediction error h = convective film heat-transfer coefficient (kJ/ (s m2 K)) H = enthalpy of formation (kJ/kmol) H̅ = enthalpy of formation (kJ/kg) I = cell current (A) Jj = Error in outlet voltage prediction by the jth model k = time L = length of the tubular fuel cell (length of the cell and injection tubes) (m) m = mass (kg) M = molecular weight (kg/kmol) N = rate of mass transfer (kmol/(s m2)) p = partial pressure (atm) P = total pressure (atm) r = radius (m) R = universal gas constant; R = 8.20575 × 10−2 m3 atm/ (K kmol) R* = universal gas constant; R*= 8.31447 × 103 m2 kg/ (K kmol s2) RJ = universal gas constant; RJ =8.31447 × 103 J/ (K kmol) RkJ = universal gas constant; RkJ =8.31447 kJ/ (K kmol) Rtct = total charge transfer resistance (Ω) Rload = external load resistance (Ω) Rto = total ohmic resistance (Ω) Rrad = radiation heat transfer resistance Rr = rate of reforming reaction (kmol/s m2) RH2 = rate of consumption of H2 (kmol/s) RH2O = rate of consumption of H2O (kmol/s) RO2 = rate of consumption of O2 (kmol/s) T = temperature (K) u = fluid velocity Vout = fuel-cell outlet voltage (V) zr = frequency factor of the ammonia decomposition (kmol/ atm0.5 m2 s)

■

ano = anode cat = cathode inj = injection tube TPB = triple phase boundary air = air ct = cell tube j = jth model o = outer fuel = fuel it = injection tube i = inner in = inlet load = load to = total ohmic tct = total charge transfer out = output

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Greek Letters

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