Effect of the Operating Strategy of a Solid Oxide Fuel Cell on the

Ind. Eng. Chem. Res. 2011, 50, 1439–1452

1439

Effect of the Operating Strategy of a Solid Oxide Fuel Cell on the Effectiveness of Decentralized Linear Controllers P. Vijay,† M. O. Tade´,*,† and R. Datta‡ Centre for Process Systems Computations, Department of Chemical Engineering, Curtin UniVersity of Technology, Western Australia 6845, Australia, and Fuel Cell Centre, Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, Massachusetts 01609, United States

The effect of the operating strategy of the solid oxide fuel cell (SOFC) on the applicability of decentralized controllers is investigated in this work using a zero-dimensional model. Using frequency dependent interaction measures, three different operating modes resulting in 18 control structures are analyzed at different operating points. The results indicate that only 5 out of the 18 control structures are suitable for decentralized controller design. This is confirmed from the closed loop simulations using proportional, integral, and derivative (PID) controllers for the loops which are tuned sequentially. It is also shown through simulations that multivariable controllers such as decoupling controllers and model predictive controllers will be applicable for the remaining control structures with strong interactions. The results of this work will be useful in the design of plantwide controllers for the SOFC system. 1. Introduction The solid oxide fuel cell (SOFC) is expected to be an important power source in the future. The SOFC is expected to be employed in a variety of applications like stand-alone power systems, hybrid power systems, transportation appliances, and for supplying power to the power grid.1-7 An SOFC is a complex system with interactions and many operational requirements. A number of control strategies have been proposed for SOFC control2-5,8-14 which are based on multiloop control. In the literature, the usual method adopted for controlling the SOFC is multiloop control. The justification given for it is that the different processes in the SOFC (namely, electrochemical, thermal, electrical, and hydraulic) basically have different response times,6 and so, the interactions between the control loops will be minimum. While this is true for the particular operating modes and operating points considered in those works, the results from this work suggest it is not true in all cases. In the work of Stiller et al.,15 a multimode control strategy for the SOFC gas turbine hybrid system was presented whereby the SOFC operates in different modes including “normal operation”, “maximum Fuel Utilization (FU)”, “minimum FU”, and “minimum voltage”. However, rather than performing an interaction analysis for different operating modes, it was simply assumed that the interactions are minimum because of the different time constants of the processes. An SOFC must be combined with other balance of plant components such as reformers, heat exchangers, after burners, power conditioning systems, etc. in order to efficiently produce power for practical applications. Many of these balance of plant components also require automated control. In such situations, the coordination between the controllers must be considered while designing them. For example, in the work of Suh et al.,16 a control system is designed for the fuel cell/power converter combination, which took into consideration the coordination between the controllers. The inclusion of the power electronics into the SOFC system results in increase in the bandwidth of the system. It is necessary * To whom correspondence should be addressed. Phone: +61 8 9266 7581. Fax: +61 8 9266 2681. E-mail: [email protected]. † Curtin University of Technology. ‡ Worcester Polytechnic Institute.

to consider the RGA at all frequencies for such systems so that the inputs and outputs can be suitably paired. Steady state RGA analysis may result in wrong conclusions, which may lead to the pairing of wrong sets of variables for which, there may be considerable interactions at high frequencies. Therefore, it is essential to study the interaction over a range of frequencies. Also, the operating point of the system has an influence on the system interactions. For all these reasons, a comprehensive analysis of the system interactions must be performed before attempting to design a control system. A thorough interaction analysis of the SOFC considering the frequency, the operating points, and the operating modes was not performed in any of the previous works. In the literature, different operating strategies and decentralized control strategies have been recommended for the SOFC operation depending on specific applications. There is a need for identifying control structures suitable for decentralized control at the cell level considering the different operating modes of a cell encountered when it is put to different applications. The aim of this work is to investigate the applicability of decentralized linear controllers to the SOFC operated with different objectives. In the case of decentralized control, control structure refers to the input-output pairing that results in the best performance of the system. Different operating strategies necessitate different sets of manipulated and controlled variables and hence different control structures. Basically, three operation modes of the SOFC are considered in this work. The process interactions and the control loop interactions have been studied for the control structures corresponding to the three operation modes. For this purpose, the concepts of frequency dependent relative gain array (RGA) and relative gain array number have been utilized.17,18 Eighteen control structures corresponding to three different modes of operation of the SOFC are considered in this work. A nonlinear model of the SOFC is formulated which includes all the essential dynamics and at the same time not computationally taxing. The nonlinear SOFC is linearized at different operating points, and the process interactions corresponding to the 18 control structures are analyzed at these points. Control configurations of the SOFC that are suitable for decentralized control are identified from the RGA analysis. Feedback pro-

10.1021/ie100894m  2011 American Chemical Society Published on Web 12/01/2010

1440

Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011

The component balance for oxygen in the cathode channel control volume gives dnO K°canORT ) n˙Oi - n˙°O - n˙Or ) n˙icaxOca,in + dt Vca iνO K°canOpext nO + nN neF

(4)

The component balance of nitrogen in the cathode channel control volume gives dnN K°canNRT K°canNpext ) n˙Ni - n˙°N ) n˙icaxNca,in + dt Vca nO + nN

Figure 1. Schematic of the SOFC showing the basic operating principle and the manipulated and measured variables.

portional, integral, and derivative (PID) controllers are designed for the SOFC with the identified configurations using the sequential loop closing method. These controllers are then incorporated into the nonlinear model, and closed loop simulations are performed around the operating points in order to demonstrate the effectiveness of the decentralized controllers. It is also shown through simulations that multivariable control techniques such as decoupling control and model predictive control will be effective for the configurations which are not amenable to multiloop control. The results have significant implications for the control design. 2. SOFC Model for Control Purpose A simple model of the SOFC is adopted in this work for analyzing the applicability of decentralized control to the system. The aim is to represent all the essential dynamics and the interactions in the SOFC and at the same time avoid computational complexity. A zero-dimensional model of the SOFC has been considered for the analysis, which is based on the models in the work of Kandepu et al.8 and Murshed et al.10 Hydrogen is assumed to be the fuel, and the oxidant is air (79% nitrogen and 21% oxygen by volume). A schematic of the SOFC showing the basic working principle of the cell is shown in Figure 1. The variables which are shown with arrows in dashed lines are the input variables to the model. PCS in the figure refers to the power conditioning system the purpose of which is to condition the fuel cell power and make it suitable for the load. Note that the PCS is shown in the Figure 1 only for completeness and is not included in the model. The control volumes considered for writing the mass balances are the anode channel and the cathode channel control volumes of the cell. The whole cell (including the solid and the gaseous phases) is considered as the control volume for writing the energy balance. The dynamic model of the SOFC is formulated by writing the mole balances for the four gas species involved in the process (namely, hydrogen, oxygen, nitrogen, and water vapor) and the energy balance for the entire cell. The outlet flow of the components is assumed to be governed by a linear nozzle flow equation. Therefore, n˙°an ) K°an(pan - pext)

(1)

n˙°ca ) K°ca(pca - pext)

(2)

The mole flow rates of the reactants and products are related to the current as i)

neFn˙rW νW

)-

neFn˙Hr νH

)-

neFn˙Or νO

(3)

(5) The component balance of hydrogen in the anode channel control volume gives dnH K°annHRT K°annHpext iνH ) n˙Hi - n˙°H - n˙Hr ) n˙Hi + dt Van nH + n W neF (6) The component balance of the water vapor in the anode channel control volume gives dnW K°annWRT K°annWpext iνW ) n˙iW - n˙°W + n˙rW ) + + dt Van nH + nW neF (7) The cell is assumed to be well-insulated so that the energy losses to the surroundings are neglected. Assuming constant specific heat capacities, the energy balance around the entire cell gives dT ) n˙icaxOca,incp,OTin + n˙icaxNca,incp,NTin + n˙Hicp,HTin dt K°capextnOTcp,O K°canNRT2cp,N K°canORT2cp,O + + Vca nO + nN Vca K°annHRT2cp,H K°anpextnHTcp,H K°capextnNTcp,N + nO + nN Van nH + nW K°anpextnWTcp,W (∆h)roνHi K°annWRT2cp,W - Vi (8) + + Van nH + nW neF

mscp,s

where (∆h)or is the enthalpy of the reaction. The cell voltage is given as V ) Vo - ηact - ηconc - ηohm

(9)

The open circuit voltage Vo is given by the Nernst equation as Vo ) E0 -

( )

0.5 RT pHpO neF pW

(10)

where the standard cell potential is given as10 E0 ) 1.2586 - 0.000252T

(11)

The Ohmic overpotential (ηohm) is approximated by Ohm’s law ηohm ) iRohm

(12)

The cell resistance as a function of cell temperature is given as10


((

Rohm ) r0 exp R

1 1 T T0

))

(13)

where r0 is the resistance at temperature T0, and R is a constant. The concentration overpotential is approximated by ηconc )

(

RT i 1neF iL

)

(14)

The activation overpotential is approximated by10 ηact ) a + b log i

(15)

where a and b are the Tafel constant and Tafel slope, respectively. The power produced by the fuel cell is given by P ) Vi

1441

8,10

Table 1. Parameters of the SOFC Model parameter

value

units

pext F ne K°an K°ca Van Vca R (∆h)°r r0 iL Tin ms Ac R

100000 96485 2 6.754 × 10-8 5 × 10-8 1.032 × 10-5 4.3 × 10-5 8.314 -0.2418 × 109 0.126 10 1100 50 0.01 -2870

Pa C mol-1 mol m2 s-1 N-1 mol m2 s-1 N-1 m3 m3 J mol-1 K-1 J kmol-1 Ω A cm-2 K kg m2

(16)

These equations (4-8) constitute the nonlinear five state model that is used to study the interactions between the control loops in this work. The fuel utilization (FU) is an important system variable which influences the life of the cell.9,15 The FU (ζf) is defined as the ratio of the mole flow rate of fuel utilized in the reaction to the inlet mole flow rate of the fuel. ζf )

nHr nHi

(17)

The parameters of the model are taken from the work of Kandepu et al.8 and Murshed et al.10 and are listed in Table 1. The operational requirements of the SOFC differ according to the application. In general, the cell current (i), the anode channel i ), and the cathode channel inlet mole inlet mole flow rate (n˙an flow rate (n˙ica) may be considered as the inputs to the system. i.e., these are the variables that are manipulated. The control objectives of the SOFC concern the variables such as the cell voltage (V), the cell temperature (T), the FU, and power (P). Therefore, these variables are considered as the output variables in this analysis. The cell voltage and the temperature can be directly measured, while the fuel utilization can be calculated using the hydrogen inlet flow rate and current measurements. The generalized formulation of the model for control purposes can be written as the following. The nonlinear state equations which constitute eqs 4-8 are represented as x˙ ) f(x, u)

(18)

The output y is given as y ) f(x, u)

(19)

i , where x ) [nH, nW, nO, nN, T]T, is a vector of states, u ) [n˙an i T T n˙ca, i] is the vector of inputs, and y ) [ζf, T, V, P] is the vector of outputs. Note that these are the generalized set of input and output variables, not all inputs are considered as the manipulated variables, and not all outputs are considered as the controlled variables for a particular operation mode of the SOFC. Model linearization is essential for applying the process interaction measures to the system. The linearized SOFC model, in terms of the deviation variables, is given by

∆x˙ ) A∆x + Bu∆u + Bw∆w

(20)

∆y ) C∆x + Du∆u + Dw∆w

(21)

where w denotes the disturbance inputs. The generalized block

Figure 2. General block diagram representation of the SOFC showing the sets of input, output, and disturbance variables considered in this work.

diagram of the SOFC with the sets of input, output, and disturbance variables considered in this work is shown in Figure 2. 3. Operation Modes of the SOFC A generalized representation of the SOFC model for control purposes with the possible inputs and outputs was given in eqs 18-21. The SOFC has to be integrated with other components to achieve the objectives of the power system. Depending on the objectives of the power system (and hence the nature of the other system components including load, power conditioning, and the balance of plant components), the cell itself is required to be operated under different modes of operation. Some of the operating requirements of the SOFC stated in the literature are1-5,7,19-21 (1) Maintaining a constant cell temperature (2) Maintaining constant FU (3) Maintaining constant cell voltage (4) Tracking the power requirement Out of these, the requirement to maintain the cell temperature constant is an absolutely essential one regardless of other operational requirements. This is because constant temperature operation avoids thermal cracking of the cell.15,22-24 The requirements of maintaining constant FU and constant voltage cannot be realized simultaneously through manipulations at the cell level. For maintaining the FU constant despite changes in the load, either the anode inlet flow rate or the cell current have to be manipulated which results in a change in cell voltage. Similarly for maintaining a constant cell voltage, the inlet flow rate or cell current have to be manipulated, which in turn changes the FU. However, both requirements could be realized by including power electronics which will take care of the

1442


voltage requirement while the FU requirement can be achieved by manipulating anode inlet flow or current. Only the cell level process is considered in this work. There are primarily three input variables at the cell level which may be manipulated to control the cell performance. They are the anode and cathode inlet flow rates and the cell current. There are other variables such as the pressures and temperatures of the species at the inlet which have an influence on the operating conditions of the cell.11 However, they cannot be considered as manipulated variables because either their influence on the cell operating conditions is not very strong or they are not very conveniently manipulated. In the literature,6,9,15,19 maintaining constant FU and constant cell temperature is advocated as the desirable operating condition for an SOFC. Constant FU operation of the fuel cell is recommended because it minimizes the dynamics during load changes9 and also avoids uneven distribution of voltage and temperature within the cell.15 A suitable preset value of FU for a particular SOFC system can be chosen, provided that the range of output power is known. This preset FU value must be such that the reasonable variation in the fuel cell load must result in the system operation within a feasible operating range. In the case of variation in the load current, the condition of constant FU can be realized by manipulating the anode inlet flow rate.19 Similarly, the most convenient variable which can be manipulated so as to maintain a constant cell temperature is the cathode inlet flow rate.19 The fluctuation in the cell voltage as a result of manipulating the anode and the cathode inlet flow rates can be possibly handled by a power conditioning system so that the load is presented with a constant voltage. If the load is primarily an ac load, then it is feasible to operate the cell under a constant FU.2 In this case, the resulting voltage variation can be taken care of by the power conditioning system so that the load is supplied with a constant voltage. Also, the instantaneous load power demand can be taken care of by operating the cell in a hybrid system consisting of super capacitors or batteries. If the load is primarily dc type, it is necessary to maintain a constant voltage at the fuel cell terminals.2 The voltage in this case must be suitably chosen such that the FU lies in the desirable range. In Li et al.,2 the anode inlet flow rate was manipulated to maintain a constant cell voltage. For a cell that is being used to power dc loads or when the power conditioning unit inverter does not have voltage controllability, the ideal operating strategy would be to maintain the voltage and temperature of the cell constant. Sometimes the fuel cell is connected with a load whose power requirement is varying. This can happen for example to a fuel cell connected to a grid. As the fuel cell power cannot respond very fast to set point changes, some auxiliary storage devices are used along with the SOFC to satisfy the instantaneous power demand. Therefore, one of the operating modes of the SOFC is that of tracking the load power and maintaining a constant cell temperature. On the basis of the operational requirements and control objectives, the SOFC needs to be operated under three different operation modes. The operating modes of the SOFC considered in this work are (1) Constant FU and temperature (2) Constant voltage and temperature (3) Constant temperature and power tracking Maximizing operational efficiency of the system is one of the important objectives of the control system. Our previous studies have established that operating the SOFC in the constant

FU mode results in maximizing the system efficiency .25 The aim of the present work is to investigate which control strategies are suitable for realizing different control objectives of the SOFC. In the first mode of operation, the FU and the temperature are the output or measured variables. Out of the three inputs which can influence the operating point of the cell, two are considered as the manipulated variables, while the third input is considered as the disturbance to the system. The disturbances in the input variables are caused by the other components in the fuel cell system. The disturbance to the current may be due to the load, while the reformer is the origin of the disturbance in the anode inlet flow rate. Similarly, the disturbance to the cathode inlet flow rate may be caused by a blower upstream. The aim of the control system would then be to reject the disturbance. This leads to three control configurations (structures). The first configuration (denoted by OM11) corresponding to the first mode of operation is represented in terms of the transfer function matrix as follows. Note that the input vector is partitioned in order to segregate the disturbance which in this case is the anode inlet flow rate. Similarly, the transfer function matrix is also partitioned.

[] [

G11 G12 GD1 ζf ) G21 G22 GD2 T

[]

]

i n˙ica

(22)

n˙ian

The second control configuration corresponding to this operating mode (OM12) is represented in terms of the transfer function matrix as follows.

[] [

G11 G12 GD1 ζf ) G21 G22 GD2 T

[]

]

n˙ian n˙ica i

(23)

Similarly, the SOFC plant in the third control configuration, OM13, is represented in terms of the transfer function matrix as follows.

[] [

G11 G12 GD1 ζf ) G21 G22 GD2 T

[]

]

i n˙ian

(24)

n˙ica

In the second mode of operation considered for analysis, the voltage and the temperature are the output or measured variables. The first configuration corresponding to this operating mode (OM21) is represented in terms of the transfer function matrix as follows.

[] [

G11 G12 GD1 V ) G21 G22 GD2 T

[]

]

i n˙ica

(25)

n˙ian

The second control configuration corresponding to this operating mode (OM22) is represented in terms of the transfer function matrix as follows.

[] [

G11 G12 GD1 V ) G21 G22 GD2 T

]

[] n˙ian n˙ica i

(26)

Similarly, the SOFC plant in the configuration, OM23, is represented in terms of the transfer function matrix as follows.

[] [

G11 G12 GD1 V ) G21 G22 GD2 T

[] i n˙ian

]


(27)

n˙ica

The third mode of operation of the SOFC which has been investigated in this work is the power tracking mode. The objective of the controller is to track the changes in load power requirement. The requirement for maintaining a constant cell temperature is valid for this case also. This results in the following three control configurations. The SOFC plant in configuration OM31 is represented in terms of the transfer function matrix as follows.

[] [

G11 G12 GD1 P ) G21 G22 GD2 T

[] i n˙ica

]

(28)

n˙ian

The SOFC plant in configuration OM32 is represented in terms of the transfer function matrix as follows.

[] [

G11 G12 GD1 P ) G21 G22 GD2 T

[]

]

n˙ian n˙ica i

(29)

Similarly, the SOFC plant in the configuration OM33 is represented in terms of the transfer function matrix as follows.

[] [

G11 G12 GD1 P ) G21 G22 GD2 T

[]

]

i n˙ian

(30)

n˙ica

The control configurations considered in this work are schematically illustrated in Figure 3. As the SOFC is a nonlinear system, the linearized model is only valid at the point of its linearization. The SOFC is expected to operate within a particular operating range which is usually defined with respect to the range of the cell current. From the polarization curve of the cell, it is possible to define a range of current over which the power output is reasonable.19 For the cell considered, an operating range between 80A and 120A has been chosen. The controllability of the system has been studied at three different operating points within the operating range of the cell, which are named OP1, OP2, and OP3. The operating point OP1 corresponds to a current of 80 A, anode inlet flow rate of 4.6 × 10-4 mol s-1, and cathode inlet flow rate of 0.0025 mol s-1. The steady state values of the outputs corresponding to this operating point are a cell temperature of 1129.3 K, voltage of 0.8613 V, FU of 0.9, and power of 68.9 W. The operating point OP2 corresponds to a cell current of 100 A, anode inlet flow rate of 5.8 × 10-4 mol s-1, and cathode inlet mass flow rate of 0.0022 mol s-1. The steady state values of the outputs corresponding to this operating point are a cell temperature of 1203.8 K, voltage of 0.776 V, FU of 0.89, and power of 77.597 W. The operating point OP3 corresponds to a cell current of 120 A, anode inlet flow rate of 6.9 × 10-4 mol s-1, and cathode inlet mass flow rate of 0.002 mol s-1. The steady state values of the outputs corresponding to this operating point are a cell temperature of 1247 K, voltage of 0.6768 V, FU of 0.9, and power of 81.22 W. The nonlinear model was numerically linearized at the three different operating points (OP1, OP2, and OP3) with the specific set of inputs and outputs for each of the operation modes as discussed in this section. The model linearization of the non linear model implemented in Matlab

1443

Simulink has been performed using the control and estimation tools manager available in Simulink Control Design. 4. Process Interaction Analysis and Control Structure Selection for the SOFC In this section, the different control configurations of the SOFC presented in the section 3 are analyzed for process and control loop interactions using frequency dependent RGA and RGA number. This will help in identifying the control configurations suitable for decentralized control and also the favorable pairings of the manipulated and control variables. The relative gain array (RGA) is a measure of controllability and control loop interaction in multiloop control system design. The RGA is defined as the ratio of the open loop gain for a selected output when all the other loops of the process are open, to its open-loop gain when all the other loops are closed. The RGA indicates preferable variable pairing in decentralized control systems based on interaction considerations. Originally RGA was proposed in ref 26 as a steady state measure which ignores dynamic considerations. Later the RGA was interpreted in the frequency domain,26-29 which accounts for the process dynamics in deciding the optimal pairing. The frequency dependent RGA for a complex nonsingular square matrix (G) is given by Λ(jω) ) G(jω) X (G(jω)-1)T

(31)

where X denotes the Hadamard or Schur product (element-byelement multiplication). Note that the resulting RGA elements are complex quantities. The input-output pairings that have an RGA value close to unity are preferred, so that the interactions between the loops are minimum. Such pairing will prevent stability problems caused due to the interactions. Using only the magnitude of the RGA elements for the analysis might be misleading. Therefore, the phase part must also be considered. For this purpose, RGA number for diagonal pairing is defined as the following.18 RGA number } Λ(G) - I

|

|

sum

)

∑ |a | ij

(32)

i,j

where aij are the elements of the matrix. The RGA number for the off-diagonal pairing is obtained by subtracting 1 from it, i.e., Λ(G) -

[ ] 0 1 1 0

The pairing with a small RGA number is preferred. It is desirable to choose a pairing with an RGA number close to zero at crossover frequencies so as to avoid instabilities caused by interactions in the crossover region. The frequency range of interest is given by the bandwidth frequency. The bandwidth frequency is normally defined as the frequency up to which control is effective and it depends on the controller. Each loop of the multiloop controller must be stable over the range of operating points within the operating range of the cell. Therefore it is required to test for the controllability of the system at a number of operating points. The frequency dependent RGA for the diagonal and the offdiagonal pairings of the manipulated and the controlled variables for the nine control configurations corresponding to the three operating modes of the SOFC are shown in Figure 4. In these figures, “D” and “OD’ denote diagonal and off-diagonal pairings respectively of the manipulated and the controlled variables.

1444


Figure 3. Schematic representation of the control configurations corresponding to the three modes of operation of the SOFC: (a) OM11, (b) OM12, (c) OM13, (d) OM21, (e) OM22, (f) OM23, (g) OM31, (h) OM32, and (i) OM33.

Figure 4a shows the RGA for the diagonal and the off-diagonal pairing of variables corresponding to the control configuration OM11. Note that each control configuration shown in Figure 4 gives rise to two control structures: one corresponding to the diagonal pairing of the manipulated and the controlled variables and the other corresponding to the off-diagonal pairing of the variable. From Figure 4a, it is evident that the diagonal pairing of the manipulated and controlled variables has minimum interactions throughout the range of frequencies considered. Also, the interactions are not sensitive to the point of operation. Therefore, the diagonal pairing (n˙ica - T and i - ζf) is preferred for designing a decentralized controller. From Figure 4b, we can conclude that for the control i configuration OM12 the diagonal pairing (n˙ica - T and n˙an ζf) has minimum interactions through the frequency range considered and also the point of operation of the cell has no effect on the loop interactions. From Figure 4c, we can see that there are significant interactions between the control loops at lower frequencies if the cell is operated in the configuration OM13. However, at high frequencies, there are no significant interactions for the i - T and i - ζf). Operation of the cell at diagonal pairing (n˙an a high current density operating point (OP3) results in lesser loop interactions than the operation at a low current density

point (OP1). In the SOFC, one control loop is a fast one having a high closed loop bandwidth frequency and the other loop is a slow one having a low closed loop bandwidth frequency. Therefore, the RGA across the frequency range is important while deciding the pairing with minimum interactions. This means that decentralized control is not a suitable option for the SOFC being operated in configuration OM13. For the cell operated in the control configuration OM21 (refer to Figure 4d), the variables in the diagonal pairing (n˙ica - T and i - V) result in minimum interactions between the control loops. Also, the operating point has minimum effect on the interactions. The operating point of the SOFC plays a part in deciding the interactions when the cell is operated in the control configuration OM22 as shown in Figure 4e. The interactions are minimal for the diagonal pairing (n˙ian - V and n˙ica - T) at lower frequencies. At higher frequencies, there are significant interactions except for the low current operating point OP1. This configuration is also not suitable for the use of decentralized control strategies. For the control configuration OM23 (refer to Figure 4f), the i - T and i - V) is favorable for designing diagonal pairing (n˙an a decentralized control system as it results in minimum interactions. The interactions between the loops are sensitive to the operating point of the cell. High current operating point (OP3) results in minimum interactions. In the control config-


1445

Figure 4. Frequency dependent RGA for the control configurations: (a) OM11, (b) OM12, (c) OM13, (d) OM21, (e) OM22, (f) OM23, (g) OM31, (h) OM32, and (i) OM33.

uration OM31, the diagonal pairing (n˙ica - T and i - P) leads to minimum interactions as shown in Figure 4g. The operating point of the cell influences the loop interactions only slightly. For the control configuration OM32, the operating point of the cell influences the loop interactions in a large way as shown in Figure 4h. While at lower frequencies, the diagonal pairing (n˙ica i - T and n˙an - P) results in minimum interactions, it is true only for the operating point OP1 at higher frequencies. In the i control configuration OM33, the diagonal pairing (n˙an - T and

i - P) gives minimum interactions only at higher frequencies as shown in Figure 4i. The RGA numbers for the diagonal and the off diagonal pairings of the nine control configurations pertaining to the three operating modes are shown in Figure 5. The RGA numbers for the various configurations substantiate the conclusions obtained from the RGA magnitudes. From the results of the analysis of the process interactions in the SOFC operated in nine different control configurations corresponding to the three operating

1446


Figure 5. Frequency dependent RGA numbers for the control configurations: (a) OM11, (b) OM12, (c) OM13, (d) OM21, (e) OM22, (f) OM23, (g) OM31, (h) OM32, and (i) OM33.

modes, it can be concluded that the interactions between the control loops are minimum for the diagonal pairing of the manipulated and controlled variables in five out of the nine control configurations, namely OM11, OM12, OM21, OM23 and OM31, and hence, they are suitable for the formulation of decentralized control strategies. For the other configurations, multivariable control strategies have to be considered. It can also be concluded that operating point has very little effect on the process interactions for these five configurations.

5. Open- and Closed-Loop Simulations The open-loop responses of the system to step changes in the disturbance variables are simulated in order to gain an understanding of the system dynamics. In this work, the SOFC is subjected to the disturbances in anode inlet flow rate, cathode inlet flow rate, and current when operating under various operation modes. The responses of the system operating around the operating point OP1 to step changes in the disturbances are


1447

Figure 6. System responses to step changes in (a and b) cell current, (c and d) anode inlet flow rate, and (e and f) cathode inlet flow rate.

shown in Figure 6, and the responses of the system to step increase in current from 80 to 85 A are shown in Figure 6a and b. The increase in the cell current results in the increase in the hydrogen consumption rate and hence increases the FU. The increased reaction rate results in the reduction in the partial pressures of the reactants and an increase in the partial pressures of the products of the reaction, which cause the open circuit voltage (Vo) and hence the cell voltage (V) to decrease. The net cell power also decreases. The increased reaction rate results in increase in reaction enthalpy and also increase in the over voltages. These lead to increase in the cell temperature. The responses are similar at other operating points. The responses of the system to step increase in the anode inlet flow rate from 4.6 × 10-4 to 4.8 × 10-4 mol s-1 are shown in Figure 6c and d. The increase in the anode inlet flow rate results in the decrease in FU (refer to eq 17). It also results in the increase in the partial pressure of hydrogen in the anode channel and hence an increase in the cell voltage. Increase in the flow of gases through the cell results in a net decrease in temperature. As a result of the increase in the cell voltage, the power output of the cell also increases. The responses of the system to step increase in the anode inlet flow rate from 0.0025 to 0.00252 mol s-1 are shown in Figure 6e and f. The increase in the cathode inlet flow rate results in the decrease in cell temperature because of the heat carried away by the air flow. As a result, there is a drop in the cell voltage and power. The FU remains same as the cathode inlet flow rate has no effect on it. The RGA analysis indicated that decentralized control is possible for operation of the SOFC in certain configurations.

On the basis of the results from section 4, the control structures corresponding to minimum loop interactions can be chosen for the control system design. In this section, the closed loop simulations of the nonlinear model operated under the operating mode OM11 is performed. For this purpose, PID controllers are implemented for each of the two controls loops of the cell. Although the results from the RGA analysis indicate that decentralized controllers are possible for these operation modes with certain pairings of the manipulated and the controlled variables, small amount of interactions will exist between the two control loops. Therefore, the controllers for the individual loops cannot be tuned independently. As the SOFC system considered is a MIMO system containing two feedback loops with controllers, the design of the controller for one loop will depend on the controller in the other loop and vice versa. This can be understood by obtaining the transfer function between Y1 and U1 (refer to Figure 2), assuming that the reference input for the second loop (R2(s)) and the disturbance input (D(s)) are zero. Y1 G12G21GC2 ) G11 U1 1 + G22GC2

(33)

Y1 G12G21GC2 ) G11 U1 1 + G22GC2

(34)

Therefore, the open loop transfer function that has to be considered for the design of the controller for this loop is

1448


Table 2. Tuning Scheme of the PID Controllers for Configuration OM11 at OP1 steps

loop

KP

KI

KD

S1 S2

i/ζf i n˙ca /T

25.177 -3.5029 × 10-6

5 -8.5569 × 10-8

5 -3.585 × 10-5

OLTF ) GC1G11 -

G12G21GC1GC2 1 + G22GC2

(35)

Similarly, the transfer function between Y2 and U2 will involve the transfer function of the controller GC1 and hence the open loop transfer function to be considered for designing controller GC2 will involve GC1. As the control loops of the SOFC plant are not completely decoupled, and some interactions do exist between them, the tuning of the controllers is done by sequential loop closing method. The sequential loop closing method is one of the well-known methods to tune the multiloop control systems systematically.30,31 Multiloop controllers are designed sequentially using SISO design methods. The first loop is designed for the first pair of inputs and outputs considering all other inputs to be zero and it is closed. The second loop is designed with the first loop closed. The open-loop transfer function for the second loop will involve the controller of the first loop (refer to eq 35). Each controller is designed based on the transfer function between the paired input and output while former loops have been closed. The controllers for the two loops are tuned using the linearized models at the three operating points OP1, OP2, and OP3. The automated tuning capabilities (using the Ziegler-Nichols open-loop tuning algorithm and Singular frequency based tuning algorithm) of the Simulink control and estimation tools manager have been utilized for tuning the PID controllers sequentially for the two loops. In order to verify the performance of the controllers, they are then incorporated into the nonlinear model, which is initialized with the three operating points and the closed loop responses of the system are simulated around the corresponding operating points. Note that only PID feedback controllers are considered in the two control loops for obtaining the closed loop simulations. It is possible to improve the performance of the system further by introducing feed forward or cascade control methods. The tuning scheme for the two PID controllers for the configuration OM11 for the operating point OP1 is shown in Table 2. For this case, the optimal controller parameters are obtained in two steps. The responses of the controlled variables after each step of the tuning are shown in Figure 7. The closed-loop responses of the system in rejecting the disturbances while operated under control configuration OM11 at different operating points are shown in Figure 8. The responses correspond to a step change in the anode inlet flow rate from 4.6 × 10-4 to 4.8 × 10-4 mol s-1 applied at 2000 s. The objective of the control system in this case is to maintain

the FU and temperature of the cell at their set points despite the disturbance to the anode inlet flow rate. The controller tries to bring the FU back to the set point value by increasing the current, which results in increased reaction rate. The increased reaction rate results in a decrease in the partial pressures of the reactants in the cell and hence brings the voltage down. The increase in the cell current results in a net increase in the system power output. The closed-loop responses corresponding to the operating points OP2 and OP3 for step changes of the anode inlet flows from 5.8 × 10-4 to 6 × 10-4 mol s-1 and 6.9 × 10-4 to 7.1 × 10-4 mol s-1 applied at 2000 s are shown in Figure 8c and d and e and f, respectively. The same trend as explained for OP1 is seen in the closed loop responses of the cell at all these operating points also. From the figures, it may be seen that the controller results in satisfactory transient responses with little overshoot and quick enough settling times. It can be seen that the response time for the FU is much faster than that of the temperature response. From these closed loop simulations of the SOFC operated with diagonally paired control configurations OM11, it is clear that decentralized controllers designed using linearized model give satisfactory performance around the linearization points. It can also be concluded that the operating point has very little effect on the system interactions for these control configurations. Note that these controllers are valid when the cell is operated close to the operating points for which they are designed. The closed loop simulations for the other configurations (OM12, OM21, OM23, and OM31) have also been performed and the results indicate satisfactory performance of the decentralized PID controllers around the linearization points. However, for brevity, those curves have not been presented in this section. As the other configurations namely OM13, OM22, OM32, and OM33 are not amenable to decentralized control, one has to resort to multivariable control schemes. The configuration OM22, which has large interactions between the loops, especially at higher frequencies, as evident from Figure 4e, is chosen to demonstrate that multivariable control techniques will be effective for the configuration with large interactions. Note that it was not possible to design (tune) multiloop controllers for that configuration. A decoupling controller using dynamic feedforward and a model predictive controller (MPC) have been designed for the SOFC in configuration OM22 operating around OP1. Decoupling control is an early approach to multivariable control, where dynamic feedforward is provided so as to eliminate the coupling between the loops.17 Table 3 gives the tuning scheme for the loop PID controllers of the individual loops of the decoupled system. In this case also, the best attainable controller parameters were obtained in two steps. The responses of the controlled variables after each step of the tuning are shown in Figure 9. Note that the steady state values of the

Figure 7. Controlled variable responses after each tuning step for the control configuration OM11 operated at operating point OP1.


1449

Figure 8. Closed-loop system responses of the SOFC operating in configuration OM11 around: (a and b) OP1, (c and d) OP2, and (e and f) OP3. Table 3. Tuning Scheme for the PID Controllers of the Decoupled System steps

loop

KP

KI

KD

S1 S2

i n˙an /V i n˙ca /T

0.0825796 2.42588274e-6

0.0277547 -0.00431136

0.0036556198 -6.666505735

temperature and the voltage in Figure 9 are 1247 K and 0.6768 V, respectively, though they appear to be zero due to the large range of the ordinate. The closed-loop response of the controlled variables to a step change in cell current from 80 to 85 A, using the decoupling controller, is shown in Figure 10. Although the decoupling control method succeeds in eliminating the coupling between the loops, it has certain drawbacks. A perfect process model is required for developing effective decouplers, which is not always available. Also ideal decouplers may not be physically realizable. Moreover, by restricting the control loops to be noninteracting, overall multivariable control performance may suffer.17 A model predictive controller has also been designed for the SOFC operating in configuration OM22 around operating point OP1. The cell current is considered as a measured disturbance which can be fed forward to improve the controller response.

Suitable values of prediction ans control horizons are chosen (50 and 5, respectively). The simulations are performed using the model predictive control toolbox in Matlab. The responses of the controlled variables are shown in Figure 11. It can be concluded from the favorable characteristics of the responses (minimum overshoot and fast settling times) that model predictive control is well suited for SOFC applications. 6. Conclusions The applicability of decentralized control to the SOFC operating with different control objectives have been studied in this work using process interaction measures such as frequency dependent RGA and RGA number. Three operating modes of the SOFC are considered for the analysis: constant FU and temperature operation, constant voltage and temperature operation, and constant temperature and power tracking operation. Anode and cathode inlet flow rates and the cell current are the input variables that are manipulated to achieve the control objectives. The resulting 9 control configurations (18 control structures) have been analyzed for process and control loop interactions using frequency dependent RGA and RGA number.

1450


Figure 9. Controlled variable responses after each tuning step for the decoupling controller in control configuration OM22 operated at operating point OP1.

Figure 10. Closed-loop responses of the controlled variables for the configuration OM22 operated at OP1 using a decentralized controller.

Figure 11. Responses of the controlled variables for the configuration OM22 operated at OP1 using a model predictive control.

From the process interaction analysis, the diagonal pairing of the manipulated and the controlled variables corresponding to the configurations OM11, OM12, OM21, OM23, and OM31 are identified to have minimum process interactions. That means decentralized controllers can be effectively designed for the cell with constant FU and constant temperature strategy with the diagonal variable pairing corresponding to OM11 and OM12. For constant voltage and temperature operation, decentralized control is effective with the diagonal variable pairing corresponding to OM21 and OM23. Similarly, the diagonal variable pairing corresponding to OM31 is the best choice for designing decentralized controllers for constant temperature and power tracking operation of the SOFC. Also, the operating point has very little effect on the process interactions for these five configurations. PID controllers for the SOFC have been tuned for the configuration OM11 with diagonal pairing using sequential loop closing technique. The closed-loop simulations with the non-

linear model around the linearization points indicates that decentralized controllers designed using linearized model give satisfactory performance. Also, the results from the closed loop simulations of the configuration OM22 using decoupling and model predictive controllers demonstrate that multivariable control methods will be effective for the configurations identified to have strong interactions. In the literature, it is generally assumed that the SOFC control loops are decoupled. However, the results from this work indicate that this is not true for all configurations, at all frequencies and all operating points. For example, if the aim of the control system is to maintain a constant fuel utilization, by manipulating either the cell current or the anode inlet flow rate i ) and maintain constant temperature by manipulating the (n˙an cathode inlet flow rate, as is the case in configurations OM11 and OM12, it is obvious that there is not any interactions throughout the frequency range. If for some reason, the temperature is desired to be manipulated using the anode inlet


flow rate, it gives rise to some interactions though not much (configuration OM13). The scenario is not so simple in configuration OM22, where it is desired to control the cell voltage and the temperature by i , manipulating the anode and cathode inlet mass flow rates (n˙an i n˙ca). By looking at the governing equations, we can say that there may be interactions because the cell voltage could be influenced both by the anode and the cathode inlet flow rates (by means of modifying the species partial pressures, as is evident from the Nernst equation (eq 10)) and vice versa. The frequency dependent RGA results in Figures 4 and 5 show that the interactions are significant only at higher frequencies and also that the operating point of the cell has an effect on the interactions. Such a result has significance in control in the following scenarios. One scenario is when the control objective of the cell changes according to the requirement, i.e. a multimode controller is desired.15 Another scenario is that a combined controller needs to be designed for the cell and the power electronics. The inclusion of the power electronics into the SOFC system results in increase in the bandwidth of the system, which necessitates the controller design for high frequencies. Another scenario is when the cell is expected to operate over wide operating ranges and some adoptive controllers need to be considered. The study carried out in this work identifies the applicability of the decentralized controllers to the SOFC in terms of the cell operation modes, frequencies, and the cell operating points. The results reinforce the importance of performing a thorough interaction analysis before controllers could be designed for the system so as to obtain optimal controllers and provides a starting point for developing adaptive controllers or multimode controllers resulting in better system performance. A thorough understanding of the process interactions at the cell level is essential not only to design the controller for the stack but also to integrate the controllers of the power conditioning system, the balance of plant components, etc. (which will impose constraints on the cell control system) to obtain an effective control of the overall system. The results of this work will aid in making decisions pertaining to the selection of manipulated and control variables and selections of control method for plantwide control design of the SOFC system. The understanding gained as a result of this work will be used for designing appropriate gain scheduled controllers for the cell which are applicable for a wide operating range in the future. Also, plantwide controls will also be developed for the SOFC system in the future with the aim of maximizing system efficiency. Acknowledgment The authors acknowledge the Australian Research Council Discovery grant DP0880483 for supporting this research. Notation Ac ) area (m2) cp ) specific heat capacity at constant volume (J mol-1 K-1) F ) Faraday’s constant (C mol-1) G ) transfer function G ) transfer function matrix h ) specific enthalpy (J kg-1) i ) current (A) K ) valve coefficient m ) mass (kg) n˙ ) mole flow rate (mol s-1)

1451

n ) number of moles (mol) ne ) number of electrons participating in the reaction p ) pressure (Pa) P ) power (W) R ) universal gas constant (J mol-1 K-1) Rohm ) Ohmic resistance (Ω) T ) temperature (K) V ) voltage (V) Vo ) open circuit voltage (V) Van ) volume of anode channel (m3) Vca ) volume of cathode channel (m3) x ) mole fraction ν ) stoichiometric coefficient η ) overvoltage (V) Subscripts an ) anode side act ) anode activation c ) cell ca ) cathode side conc ) concentration ext ) external f ) fuel H ) hydrogen gas in ) inlet L ) limiting N ) nitrogen ohm ) Ohmic O ) oxygen s ) solid W ) water vapor 0 ) exchange Superscripts i ) inlet N ) nitrogen o ) outlet O ) oxygen r ) reaction 0 ) initial state

Literature Cited (1) Jurado, F.; Valverde, M.; Cano, A. Effect of a SOFC plant on distribution system stability. J. Power Sources 2004, 129, 170–179. (2) Li, Y. H.; Choi, S. S.; Rajakaruna, S. An Analysis of the Control and Operation of a Solid Oxide Fuel-Cell Power Plant in an Isolated System. IEEE Trans. Energy ConVers. 2005, 20 (2), 381–387. (3) Li, Y. H.; Rajakaruna, S.; Choi, S. S. Control of a Solid Oxide Fuel Cell Power Plant in a Grid-Connected System. IEEE Trans. Energy ConVers. 2007, 22 (2), 405–413. (4) Mueller, F.; Jabbari, F.; Brouwer, J.; Roberts, R.; Junker, T.; GhezelAyagh, H. Control Design for a Bottoming Solid Oxide Fuel Cell Gas Turbine Hybrid System. ASME J. Fuel Cell Sci. Technol. 2007, 4, 221– 230. (5) Mueller, F.; Jabbari, F.; Gaynor, R.; Brouwer, J. Novel solid oxide fuel cell system controller for rapid load following. J. Power Sources 2007, 172, 308–323. (6) Singhal, S. C.; Kendall, K. High Temperature Solid Oxide Fuel Cells: Fundamentals, Design and Applications; Elsevier Ltd.: New York, 2003. (7) Thounthong, P.; Rae¨l, S.; Davat, B. Energy management of fuel cell/ battery/supercapacitor hybrid power source for vehicle applications. J. Power Sources 2009, 193, 376–385. (8) Kandepu, R.; Imsland, L.; Foss, B. A.; Stiller, C.; Thorud, B.; Bolland, O. Modeling and control of a SOFC-GT-based autonomous power system. Energy 2007, 32, 406–417. (9) Mueller, F.; Brouwer, J.; Jabbari, F.; Samuelsen, S. Dynamic Simulation of an integrated Solid Oxide Fuel Cell System Including CurrentBased Fuel Flow Control. ASME J. Fuel Cell Sci. Technol. 2006, 3, 144– 154.

1452


(10) Murshed, AKM. M.; Huang, B.; Nandakumar, K. Control relevant modeling of planer solid oxide fuel cell system. J. Power Sources 2007, 163, 830–845. (11) Qi, Y.; Huang, B.; Luo, J. Dynamic modeling of a finite volume of solid oxide fuel cell: The effect of transport dynamics. Chem. Eng. Sci. 2006, 6, 6057–6076. (12) Vijay, P.; Samantaray, A. K.; Mukherjee, A. Bond graph model of a solid oxide fuel cell with a C-field for mixture of two gas species. Proc. Inst. Mech. Eng., Part I: J. Syst. Control Eng. 2008, 222 (4), 247–259. (13) Vijay, P.; Samantaray, A. K.; Mukherjee, A. On the rationale behind constant fuel utilization control of solid oxide fuel cells. Proc. Inst. Mech. Eng., Part I: J. Syst. Control Eng. 2009, 223 (2), 229–252. (14) Vijay, P.; Samantaray, A. K.; Mukherjee, A. Bond Graph Modelbased Evaluation of a Control Scheme to Improve the Dynamic Performance of a Solid Oxide Fuel Cell. Mechatronics 2009, 19 (4), 489–502. (15) Stiller, C.; Thorud, B.; Bolland, O.; Kandepu, R.; Imsland, L. Control strategy for a solid oxide fuel cell and gas turbine hybrid system. J. Power Sources 2006, 158, 303–315. (16) Suh, K. W.; Stefanopoulou, A. G. Coordination of converter and fuel cell controllers. Int. J. Energy Res. 2005, 29, 1167–1189. (17) Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process Dynamics and Control, second ed.; John Wiley and Sons, Ltd: MA, 2004. (18) Skogestad, S.; Postlethwaite, I. MultiVariable Feedback Control: Analysis and Design, second ed.; John Wiley and Sons, Ltd: England, 2005. (19) Aguiar, P.; Adjiman, C. S.; Brandon, N. P. Anode-supported intermediate-temperature direct internal reforming solid oxide fuel cell II. Model-based dynamic performance and control. J. Power Sources 2005, 147, 136–147. (20) Sedghisigarchi, K.; Feliachi, A. Control of Grid-Connected Fuel Cell Power Plant for Transient Stability Enhancement. IEEE Power Engineering Society Winter Meeting, New York Hilton & Towers, New York, January 27-31, 2002; pp 383-388. (21) Zhu, Y.; Tomsovic, K. Development of models for analyzing the load-following performance of microturbines and fuel cells. Elect. Power Syst. Res. 2002, 62, 1–11.

(22) Inui, Y.; Ito, N.; Nakajima, T.; Urata, A. Analytical investigation on cell temperature control method of planar solid oxide fuel cell. Energy ConVers. Manage. 2006, 47, 2319–2328. (23) Nakajo, A.; Stiller, C.; Ha¨rkegård, G.; Bolland, O. Modeling of thermal stresses and probability of survival of tubular SOFC. J. Power Sources 2006, 158, 287–294. (24) Selimovic, A.; Kemm, M.; Torisson, T.; Assadi, M. Steady state and transient thermal stress analysis in planar solid oxide fuel cells. J. Power Sources 2005, 145, 463–469. (25) Vijay, P.; Samantaray, A. K.; Mukherjee, A. Constant Fuel Utilization Operation of a SOFC System: An Efficiency Viewpoint. ASME J. Fuel Cell Sci. Technol. 2010, 7 (4), 041011-1–041011-7. (26) Bristol, E. H. On a New Measure of Interaction for Multivariable Process Control. IEEE Trans. Autom. Control 1966, AC-11, 133–134. (27) Choi, J. Y.; Lee, J.; Jung, J. H.; Lee, M.; Han, C. Sequential loop closing identification of multivariable process Models. Comput. Chem. Eng. 2000, 24, 809–814. (28) Hiiggblom, K. E. Analytical approach to evaluation of distillation control structures by frequency-dependent relative gains. Comput. Chem. Eng. 1997, 21 (12), 1441–1449. (29) Hovd, M.; Skogestad, S. Simple Frequency-dependent Tools for Control System Analysis, Structure Selection and Design. Automatica 1992, 28 (5), 989–996. (30) Chiu, M. S.; Arkun, Y. A methodology for sequential design of robust decentralized control systems. Automatica 1992, 28, 997–1001. (31) Hovd, M.; Skogestad, S. Sequential Design of Decentralized Controllers. Automatica 1994, 30 (10), 1601–1607.

ReceiVed for reView April 16, 2010 ReVised manuscript receiVed November 2, 2010 Accepted November 15, 2010 IE100894M

Effect of the Operating Strategy of a Solid Oxide Fuel Cell on the

Recommend Documents