Towards Reduced-Order Models of Solid Oxide Fuel Cell

The objective of this work is to find precise reduced-order discrete-time models of a solid oxide fuel cell, which is a multiple-input multiple-output dynamic process. At first, the full-order discrete-time model is found from the continuous-time first-principle description. Next, the discrete-time submodels of hydrogen, oxygen, and water pressures (intermediate variables) are reduced. Two model reduction methods based on observability and controllability Grammians are compared: the state truncation method and reduction by residualisation. In all comparisons, the secondmethod gives better results in terms of dynamic and steady-state errors as well as Nyquist plots. Next, the influence of the order of the pressure models on the errors of the process outputs (the voltage and the pressure difference) is studied. It is found that the number of pressure model parameters may be reduced from 25 to 19 without any deterioration of model accuracy. Two suboptimal reduced models are also discussed with only 14 and 11 pressure parameters, which give dynamic trajectories and steady-state characteristics that are very similar to those obtained from the full-order structure.


Introduction
There are three important reasons why renewable energy sources are becoming more and more popular.Firstly, burning of fossil fuels leads to air pollution and significant climate changes.Secondly, both mentioned phenomena badly affect public health, which has a very negative effect on the economy.Thirdly, fossil fuels are located in some countries only, whereas sources of renewable energy actually exist practically in all countries (of course not all of them are possible in all locations).Access to energy sources naturally increases national security.The countries that do not have fossil fuels may switch to renewable energy and become independent of other countries.As a result, the role of renewable energy is important and it is expected to grow fast in the future [1,2].Usually, renewable energy is obtained from wind turbines [3], geothermal systems [4], solar collectors [5], marine systems [6], and biofuels [7].Additionally, solid oxide fuel cells (SOFCs) are very promising sources of energy.SOFCs are electrochemical devices that are able to directly convert the chemical energy stored in hydrocarbon fuels into electrical energy [8,9].They have many advantages, namely, high electrical efficiency, fuel flexibility, low emission, quiet operation, and relatively low cost.That is why SOFCs are expected to become sound alternatives to conventional power generation schemes not only for domestic but also for commercial and industrial sectors.
Economically efficient and technologically safe operation of SOFCs requires well-designed control algorithms.Control of SOFCs is a challenging task, since they are nonlinear dynamic systems and it is essential to precisely satisfy some technological constraints that must be imposed on process variables [10].Hence, for controlling SOFCs, advanced Model Predictive Control (MPC) algorithms are preferred rather than classical Proportional-Integral-Derivative (PID) controllers.An important feature of MPC is the fact that a mathematical model of the controlled SOFC is used online to successively calculate the best possible sequence of manipulated variable(s).In the simplest case, for prediction in MPC, linear models of the process may be used [11,12] and the resulting control quality is better than that of the classical PID.However, in order to obtain very good control quality, a nonlinear dynamic model of the SOFC must be used in MPC.Different variants of nonlinear MPC algorithms for the SOFC are discussed in [13][14][15][16] (different model structures and optimisation algorithms are possible).Important applications 2 Complexity of the mathematical model of the SOFC also include process optimisation [17], fault tolerant control [15], and estimation [16,18].
In MPC, optimisation, fault tolerant control, and estimation, different model structures may be used.Firstly, the firstprinciple model based on technological laws may be used [13,16].Secondly, empirical (black-box) models are possible, for example, neural networks [14,19] or fuzzy systems [20,21].Although it may be easier to use empirical models than rigorous first-principle ones, it is necessary to point out three important disadvantages of black-box structures.Empirical structures make it possible to predict the sequence of the output variable(s) for a given sequence of the input and disturbance variable(s), but some intermediate process variables are usually not modelled.Moreover, frequently used blackbock models typically have numerous parameters, much more than the fundamental ones.Finally, accuracy of blackbox models may be good in typical operating conditions, but for other ones they are likely to generate output value(s) far from those calculated by the fundamental models (and the real process).
This work is concerned with the fundamental model of the SOFC, which is a multiple-input multiple-output nonlinear dynamic process.The objective is to find precise reduced-order discrete-time models.To achieve this goal, the discrete-time submodels of hydrogen, oxygen, and water pressures (intermediate variables) are reduced by means of two methods.In the first approach, the balanced Grammian of the state-space realisation is found and the state variables corresponding to small entries of the Grammian are removed.In the second reduction method, the model parameters are additionally adjusted in such a way that the steady-state gain of the reduced model is equal to that of the full-order one.The reduced models are compared in terms of dynamic and steady-state errors as well as Nyquist plots.Next, the influence of the order of the pressure models on the errors of the process outputs (the voltage and the pressure difference) is studied.The ideal reduced model and suboptimal ones, which give good compromise between accuracy and complexity, are discussed and compared with the full-order structure.
This work is structured as follows.The SOFC is shortly described in Section 2 and its full-order continuous-time model is detailed in Section 3. Section 4 derives the fullorder discrete-time model.Section 5 discusses two model reduction methods.The main part of the paper, presented in Section 6, at first details reduction of the hydrogen, oxygen, and water pressure models and next studies the influence of the reduced pressure models on modelling accuracy of two process outputs.Finally, Section 7 concludes the paper.

SOFC System Description
The literature concerned with first-principle modelling of SOFCs is rich, for example, [9,[22][23][24].The fundamental model of the SOFC introduced in [25] and next discussed in [12,26] is considered here.The following assumptions are made: (1) The gases are ideal.
(2) The stack is fed with hydrogen and air.
(3) The channels that transport gases along the electrodes have a fixed volume, but their lengths are small.Hence, it is only necessary to define one single pressure value in their interior.
(4) The exhaust of each channel is via a single orifice.The ratio of pressures between the interior and exterior of the channel is large enough to consider that the orifice is choked.
(5) The temperature is stable at all times.
(6) Because the working conditions are not close to the upper and lower extremes of current, the only source of losses is ohmic losses.(7) The Nernst equation can be applied.
The considered SOFC has two manipulated variables (the inputs of the process): the input gas flow rate  in f (mol s −1 ) and the input oxygen flow rate  in O 2 (mol s −1 ); one disturbance (the uncontrolled input)  in which is the external current load (A) and four controlled variables (the outputs of the process): the stack output voltage  r (V), fuel utilisation  f (−), the fuel cell pressure difference  dif (atm) between the hydrogen and oxygen passing through the anode and cathode gas compartments, and the ratio  HO (−) between hydrogen and oxygen flow rates.The partial pressures of hydrogen, oxygen, and water are denoted by  H 2 ,  O 2 , and  H 2 O , respectively (atm).The input hydrogen flow and the hydrogen reacted flow are denoted by  in H 2 and  r H 2 , respectively (mol s −1 ).

Continuous-Time Model
Figure 1 depicts the structure of the fundamental continuoustime model of the SOFC system.The pressure of hydrogen is where the input hydrogen flow is Hence, the pressure of hydrogen is where  H 2 ,  H 2 ,  f , and  e denote the valve molar constant for hydrogen, the response time of hydrogen flow, the fuel processor time constant, and the electrical time constant, respectively.The pressure of oxygen is where  O 2 and  O 2 denote the valve molar constant for oxygen and the response time of oxygen flow, respectively.The pressure of water is where the hydrogen flow that reacts is Hence, the pressure of water is where  H 2 O and  H 2 O denote the valve molar constant for water and the response time of water flow, respectively.Finally, outputs of the process are defined.Applying Nernst's equation and taking into account ohmic losses, the stack output voltage is where  0 ,  0 ,  0 ,  0 , and  0 denote the number of cells in series in the stack, the ideal standard potential, the universal gas constant, the absolute temperature, and Faraday's constant, respectively.Fuel utilisation is defined as the ratio between the hydrogen flow that reacts and the input hydrogen flow.
The values of the model parameters are given in Table 1.
Table 2 gives values of process variables for the initial operating point.The values of process inputs are constrained: 0 mol s −1 ≤  in f ≤ 1.7023 mol s −1 and 0 mol s −1 ≤  in O 2 ≤ 1.6134 mol s −1 ; the value of the disturbance is also limited: 200 A ≤  in ≤ 300 A.

Model Discretisation
Discrete-time versions of the continuous-time transfer functions (3), (4), and (7) must be found.For this purpose, the continuous-time polynomials of the proper order of the variable  must be determined.From (3), one has From (4), From (7), Next, model equations are discretised using the first-order holder with the sampling period equal to 1 second.The discrete-time version of (12) is where  denotes the discrete sampling instant and  −1 is a unit delay operator.The discrete-time version of ( 13) is The discrete-time version of ( 14) is Additionally, the discrete-time version of (2) and the discrete-time version of ( 6) must be taken into account.The final form of the discretetime fundamental model is as follows.From the discrete transfer functions (15), ( 16), ( 17), (18), and (19), direct dependence of current values of process variables (i.e., for the current sampling instant ) as functions of corresponding variables in the previous instants is found.From (15), one has where Table 3: Parameters of the full-order discrete-time fundamental model.
To demonstrate effectiveness of the considered model discretisation method, two full-order dynamic models are simulated: the continuous-time one and the discrete-time one.The continuous-time model is simulated in Simulink; the differential equations are solved by the Runge-Kutta method of order 45 with a variable step size.The discrete-time model is implemented in MATLAB; the differential equations are solved with the constant step (equal to the sampling time).Both models are excited by the same series of step changes of the manipulated and disturbance variables depicted in Figure 2. Obtained trajectories of both dynamic systems (i.e., the sequences of three pressures and four process outputs) are compared in Figure 3 practically the same responses as the continuous-time one; no important differences are present.

Model Reduction Methods
The state-space representation of a discrete-time dynamic system is considered: It is assumed that matrix A is asymptotically stable.The controllability, W c , and observability, W o , Grammians are [27] To reduce the order of the dynamic system (29), a balanced state-space representation must be used.For such a system, both Grammians are equal and diagonal: The balanced representation of system ( 29) is found by setting matrix A to T −1 AT, setting matrix B to T −1 B, and setting matrix C to CT, where the transformation matrix T may be found by the procedure described in [28,29].
In this work, two model reduction algorithms [30] are considered: model reduction by state truncation (algorithm 1) and model reduction by residualisation (algorithm 2).In algorithm 1, the states for which the corresponding coefficients   are relatively "small" are simply removed.Model ( 29) is reformulated: where the states  1 and  2 are associated with "big" and "small" values of   , respectively.In the reduced model, the first state variables are only considered: When some of state variables are simply removed from the model, the steady-state properties of the resulting model may significantly differ from those of the original one.To solve the problem, in algorithm 2, the states of model (32) for which the corresponding coefficients   are "small" are also removed, but afterwards the model matrices are modified to guarantee that the steady-state gains of the full-order and reduced models are the same.Since state variables  2 are assumed to be significantly faster than  1 ones, it follows that  2 ( + 1) =  2 () and Hence, the reduced model becomes In the second algorithm in the output equation, there is a possible direct influence of the process input(s) on the Complexity 7 output(s), which is not necessary in the first algorithm.Therefore, the second algorithm leads to the difference equation for hydrogen pressure.
In contrast to the original full-order model and the reduced one obtained in the first algorithm (see (20)), the coefficients  1,1 0 and  1,2 0 may be necessary.Similarly, the second algorithm leads to the difference equation for oxygen pressure.
In contrast to the original full-order model and the reduced one obtained in the first algorithm (see ( 21)), the coefficients  2,1 0 and  2,2 0 may be necessary.Finally, the second algorithm leads to the difference equation for water pressure.
in which the coefficient  3 0 may be necessary in contrast to the original full-order model and the reduced one generated by the first algorithm (see (22)).

Reduction of the Solid Oxide Fuel Cell Model
At first, separate reduction of hydrogen, oxygen, and water pressure models is considered.Next, the problem of finding the most appropriate orders of dynamics of these models is discussed taking into account two process outputs (the voltage and the pressure difference) whose modelling accuracy depends on the pressure models.Models of the remaining two process outputs, that is, fuel utilisation and the ratio between hydrogen and oxygen flow rates, are not reduced, since they are calculated from very simple equations, ( 26) and ( 27).
The last two diagonal entries of the Grammian are small in comparison with the first two ones, which suggests that the last two state variables may be reduced.However, for a thorough analysis, the reduced models of the first, the second, and the third orders are considered; in each case, two reduction algorithms are used.The parameters of the reduced-order hydrogen pressure model of order   H2 = 1, 2, 3 calculated by the first algorithm are given in Table 4, whereas the results of the second algorithm are given in Table 5.The general form of the reduced models calculated by the first algorithm is the same as that of the full-order model (see (20)), whereas the reduced model determined by the second method is characterised by (36).In the latter case, the coefficients  1,1 0 and  1,2 0 may be necessary, which manifest a direct influence of the model input  in f and the disturbance  in on the model output  H 2 .When the reduced model has the third order, the additional coefficients are not necessary to guarantee that the steady-state gain of the reduced model is the same as that of the full-order one.When the reduced model has the second order, only one of them is used, but in the case of the first-order model two of them are necessary.
To compare full-order and reduced models of the hydrogen pressure, it is convenient to consider their Nyquist plots.It is because Nyquist plots show both steady-state and dynamic behaviours of models.Figure 4 depicts the Nyquist plots of reduced-order and full-order models.As expected, the first algorithm does not guarantee that the stead-state gain of the reduced model is exactly the same as that of the original one (the gain may be read for  = 0).Furthermore, for other frequencies, the Nyquist plot of the reduced models is different from that of the full-order one.This observation is true for the simplest first-order model and, for the disturbance  in to output channel for the second-order structure, increasing the model order makes it possible to obtain precise steady-state gain.The second algorithm is much more efficient, since all reduced models, even the first-order one, have the same gain as the original one.Furthermore, the Nyquist plots of the reduced models are very similar to those of the full-order ones.
The models are also compared in a quantitative way.For this purpose, the dynamic model error is defined: The outputs of the full-order and reduced models for the sampling instant  are denoted by  full H 2 () and  reduced H 2 (), respectively.Both models are excited by the same series of Table 5: Parameters of the reduced-order hydrogen pressure model of order   H 2 calculated by the second algorithm.
step changes of the manipulated variables and steps of the disturbance depicted in Figure 2; that is,  dyn = 2000.Additionally, the steady-state model error is defined: In this case,  full H 2 () and  reduced H 2 () denote outputs of the full-order and reduced steady-state models for the data point , respectively.They are derived easily from the dynamic models.For the second algorithm, using (36), the dynamic model for hydrogen pressure is In the case of the reduced model obtained by the first algorithm (see (20)),  1,1 0 =  1,2 0 = 0.For the steady-state models,  ss = 4116.Dynamic and steady-state errors of the reduced models of hydrogen pressure are given in Table 6.As observed from the Nyquist plots, the second algorithm gives much better results (lower model errors).In particular, always  ss  H 2 = 0 for the second algorithm, since it always guarantees that the steady-state properties of the original and reduced models are the same.When the errors are lower than the machine accuracy (2.2204 × 10 −16 ), they are treated as 0. Both reduction algorithms give perfect models of order three.
Finally, Figure 5 compares the hydrogen pressure trajectories (model outputs) calculated by reduced-order (  H2 = 1, 2, 3) and full-order (  H2 = 4) models determined by both algorithms; all models are excited by the same series of step changes of the manipulated variables and steps of the disturbance depicted in Figure 2. As expected, taking into account the Nyquist plots and model errors, the second algorithm gives much better results for the same model order.For the sampling instants  = 230 and  = 231, the simplest first-order model obtained by the second algorithm gives negative pressure (of order 10 −2 ).Both reduction algorithms give perfect models of order three; that is, there is no difference between their outputs and the output of the full-order model.Reduced models of order one and two calculated by algorithm 2 give very precise trajectories, whereas the corresponding models found by algorithm 1 have bigger errors but they are still acceptable.T .(43)

Reduction of the
Although the last two diagonal entries of the Grammian are small in comparison with the first ones, which suggests that the last two state variables may be reduced, the reduced models of the first and the second orders are considered.The parameters of the reduced-order oxygen pressure model of order   H2 = 1, 2 calculated by the first algorithm are given in Table 7, whereas the results of the second algorithm are given in Table 8.The first algorithm finds the models of a general structure the same as the full-order model (see (21)), whereas the reduced model of the first order determined by the second method is characterised by (37).In this case, the coefficient  2,2 0 may be necessary, which manifests a direct influence of the disturbance  in on the model output  O 2 .For the second-order reduced model, the additional coefficient  10 Complexity Table 7: Parameters of the reduced-order oxygen pressure model of order   O2 calculated by the first algorithm.
Table 8: Parameters of the reduced-order oxygen pressure model of order   O2 calculated by the second algorithm.is not used, but it is necessary for the first-order model to modify its steady-state gain.Figure 6 depicts the Nyquist plots of reduced-order and full-order models of oxygen pressure.Both reduction methods give very precise models for the input  in O 2 to output channel; for all examined model orders, Nyquist plots are practically the same.Very important differences are present for the disturbance  in to output channel.For the firstorder model obtained by the first algorithm, the steady-state gain is different from that of the full-order model, but this discrepancy is not present for the second-order structure.For the second algorithm, modelling of the steady-state gain is excellent, but the first-order model gives wrong results as frequency increases.Similar to the first algorithm, this discrepancy is not present for the second-order structure.
For the oxygen pressure approximator, the dynamic model error,  , are defined similar to what is done in the case of hydrogen pressure (see ( 40) and (41), resp.).Errors of the reduced models of hydrogen pressure are given in Table 9.As suggested by the Nyquist plots, the second algorithm gives much better results (lower dynamic and steady-state errors).For the second algorithm, the steady-state error is 0. Both reduction algorithms give perfect models of order two.
Finally, Figure 7 compares the hydrogen pressure trajectories calculated by reduced-order (  O2 = 1, 2) and fullorder (  O2 = 3) models determined by both algorithms.As expected, taking into account the Nyquist plots and model errors, the second algorithm gives much better results for the first-order model; that is, the model obtained by the first algorithm does not have the correct steady-state properties, but this error is not significant.All models of the second order are perfect.T .

Reduction of the
( Although there is some difference in the order of the diagonal elements of the balanced Grammian, it is not so huge as in the case of the two previously considered pressure models.
For the water pressure model, the only option is to reduce the second state variable.The parameters of the reducedorder hydrogen pressure model of order   H 2 O = 1 calculated by the first algorithm are given in Table 10, whereas the Algorithm 2: disturbance I ＣＨ to output p ／ 2 Algorithm 2: input Figure 6: The Nyquist plots for the oxygen pressure models of reduced order (  O2 = 1, 2) and full order (  O2 = 3): reduction algorithm 1 (a) and reduction algorithm 2 (b).results of the second algorithm are given in Table 11.The general form of the reduced models calculated by the first algorithm is the same as that of the model of the full order (see (22)), whereas the reduced model determined by the second method is characterised by (38).In the latter case, the coefficient  3 0 is necessary, which manifests a direct influence of the disturbance  in on the model output  H 2 O .
Figure 8 depicts the Nyquist plots of reduced-order and full-order models.As it is observed in the case of hydrogen and oxygen models, the first algorithm gives slightly worse results for all frequencies.In particular, it does not guarantee that the steady-state gain of the reduced model is exactly the same as that of the original one.The second algorithm leads to a very good model.
For the water pressure approximator, the dynamic model error,  dyn  H 2 O , and the steady-state model error,  ss  H 2 O , are defined similar to what is done in the case of hydrogen pressure (see ( 40) and (41), resp.).Errors of the reduced model of water pressure are given in Table 12.As observed from the Nyquist plots, the model obtained by the second algorithm is characterised by much lower errors.For the second algorithm, the steady-state error is 0, but due to a relatively low dynamic order of the original model (only two), the dynamic error is greater than 0, although it is very small.Finally, Figure 9 compares the water pressure trajectories calculated by reduced-order (  H 2 O = 1) and full-order (  H 2 O = 2) models determined by both algorithms.As expected, taking into account the Nyquist plots and model errors, the second algorithm gives excellent results; the model generated by the first one does not have correct steady-state properties, but the error is relatively small.

Reduction of the Voltage and the Pressure Difference
Models.In the previous sections, hydrogen, oxygen, and water pressure models have been reduced separately.The pressures are intermediate model variables; the actual outputs variables are the voltage  r and the pressure difference  dif .Hence, it is necessary to select order of the pressure models in such a way that not only pressures but also outputs are approximated precisely.For this purpose, the dynamic and steady-state errors of both outputs are defined similar to what is done in the case of hydrogen pressure (see ( 40) and (41), resp.).For the output  r , the dynamic and steady-state errors are denoted by   output  dif , the errors are denoted by  dyn  dif and  ss  dif .Table 13 gives values of the dynamic and steady-state errors of the voltage for all possible combinations of orders of hydrogen, oxygen, and water pressure models (  H 2 ,   O 2 , and   H 2 ).As suggested by the partial results obtained for reduced models of the pressures, also when the errors of the output variable  r are taken into account, the second algorithm gives much precise models.In practice, all models obtained by the second algorithm have the steady-state error equal to 0. Considering the dynamic error, three models are selected for further analysis.10 compares dynamic trajectories of the voltage  r calculated by three reduced-order models and the full-order one in case of two compared algorithms.In general, all models work correctly.The perfect reduced model gives no errors, whereas the suboptimal structures lead to small errors.
From the comparison of the voltage trajectories, similar to the observations made so far for pressure models, it is clear that the second algorithm gives much more precise models.
Next, the second output variable is taken into account.Table 14 gives values of the dynamic and steady-state errors of the pressure difference for all possible combinations of orders of hydrogen and pressure models (  H 2 and   O 2 ); the order of the water pressure model does not influence the pressure difference.All the errors are very small, since modelling the pressure difference is much easier than modelling of the  25) is nonlinear, whereas the pressure difference depends linearly only on hydrogen and oxygen pressures (see (28)).The second observation is that the second algorithm in comparison with the first one gives smaller errors.In particular, all steady-state errors of the model generated by the second method are equal to 0. The same three reduced-order models are selected taking into account the errors of the pressure difference: the perfect reduced structure (with all errors equal to 0) and two suboptimal ones.Figure 11 compares dynamic trajectories of the pressure difference  dif calculated by three reduced-order models and the full-order one in case of two compared algorithms.As in the case of the voltage model, also all pressure difference models work correctly.The perfect reduced model gives no errors, whereas the suboptimal reduced structures lead to small errors.As always, in this study, the second algorithm gives more precise models.
To show good steady-state properties of the simplest reduced model ( , its steadystate characteristics obtained by two considered algorithms are compared with the ideal characteristics of the full-order model in Figure 12.When the voltage is considered, there are very small differences between the ideal characteristic and that of the reduced model generated by algorithm 1.The characteristic of the model produced by the second algorithm is very precise, as demonstrated numerically in Table 13.When the pressure difference is considered, the differences between the model obtained by the first algorithm and the full-order one are practically not visible; the second algorithm gives perfect characteristic, as demonstrated numerically in Table 14.
Finally, Table 15 gives a comparison of dynamic and steady-state model errors for the ideal full-order model and the chosen three reduced ones.The errors for two output variables are given and only the second algorithm is considered, since it always gives better results than the first one.The number of pressure model parameters is given.Although the perfect reduced-order model gives all errors equal to 0, it only reduces the overall number of parameters from 25 to 19.That is why in practical applications it may be more attractive to use suboptimal structures.The first suboptimal reduced model makes it possible to reduce the number of parameters to 14 but at the same time its dynamic and steady-state behaviours are practically the same as that of the full-order model.It is interesting to point out that even the simplest first-order reduced model, which makes it possible to reduce the number of parameters to 11, gives quite good results.Of course, selection of the models depends on specifications of the considered application.

Conclusions
This work discusses reduction of the dynamic model of the solid oxide fuel cell.Firstly, the discrete-time representation of the continuous-time model is found.It is shown that the discrete-time model gives process trajectories practically the same as the original continuous-time one.Next, the discretetime submodels of hydrogen, oxygen, and water pressures (intermediate variables) are reduced by means of state truncation and residualisation methods.The reduced pressure models are assessed taking into account both dynamic and steady-state errors as well as Nyquist plots.Next, the influence  of the order of the pressure models on the dynamic and steady-state errors of the process outputs (the voltage and the pressure difference) is studied.Three reduced models are chosen: with 19, 14, and 11 pressure parameters (the original one has 25 of them).In general, in all cases, the second reduction method is more precise.
It is important to summarise the advantages of the described reduced models: (1) The reduced models calculate not only the output variables (as the black-box models) but also the intermediate variables (the pressures).
(2) The reduced models are selected taking into account dynamic and static errors as well as Nyquist plots.(3) The first reduced model (with 19 pressure parameters) does not have any steady-state and dynamic errors; it is able to calculate pressures and process outputs exactly in the same way it is done by the full-order model.(4) Although the second and the third reduced models (with 14 and 11 pressure parameters, resp.) have some errors, they are quite small and the reduced models give dynamic trajectories and steady-state characteristics that are very similar to those obtained by the full-order structure.
The reduced-order model of the solid oxide fuel cell may be further used in all model-based methods.Firstly, it may be used for design and implementation of MPC algorithms [31], in particular computationally efficient MPC schemes [32].In addition to classical MPC algorithms, robust versions with guaranteed stability may be considered [33].Secondly, it may be also used in online process optimisation [31] cooperating with MPC.Thirdly, the model may be used in fault diagnosis of the process and fault-tolerant control [34].The presented model reduction approach may be also used for different variants of dynamic models of fuel cells [9, 22-24, 35, 36].

Figure 1 :
Figure 1: The structure of the fundamental continuous-time model of the SOFC system.

2 Figure 2 :
Figure 2: The sequences of the manipulated variables and the disturbance used for dynamic simulations.

Figure 3 :
Figure 3: The comparison of the pressures and the outputs calculated from continuous-time and discrete-time models of full order.

dyn 𝑝 O 2 ,
and steady-state model error,  ss  O 2 Water Pressure Model.The diagonal of the balanced Grammian of the state-space realisation corresponding to the discrete-time difference equation (22) describing water pressure is  = [3.6014× 10 −3 5.4358 × 10 −5 ]

Figure 9 :
Figure 9: The water pressure trajectories calculated by reduced order (  H 2 O = 1) and full order (  H 2 O = 2) models determined by algorithm 1 (a) and algorithm 2 (b).

Table 1 :
Parameters of the fundamental continuous-time model of the SOFC system.

Table 2 :
The values of process variables for the initial operating point.

Table 4 :
Parameters of the reduced-order hydrogen pressure model of order   H 2 calculated by the first algorithm.

Table 6 :
The dynamic error  dyn  H 2 and the steady-state error  ss  H 2 of the reduced hydrogen pressure models (  H 2 = 1, 2, 3).

Table 9 :
The dynamic error  dyn  O 2 and the steady-state error  ss  O 2 of the reduced oxygen pressure models (  O 2 = 1, 2).

Table 10 :
Parameters of the reduced-order water pressure model of order   H 2 O = 1 calculated by the first algorithm.

Table 11 :
Parameters of the reduced-order water pressure model of order   H 2 O calculated by the second algorithm.

Table 12 :
The dynamic error  dyn  H 2 O and the steady-state error  ss  H 2 O of the reduced water pressure model (  H 2 O = 1).
The perfect model (all errors are 0) that has the lowest number of parameters is characterised by the orders   H 2 = 3,   O 2 = 2, and   H 2 O = 2. Furthermore, two suboptimal models (their errors are greater than 0) are chosen: in the first case,   H 2 = 2,   O 2 = 1, and   H 2 O = 1; in the second case,   H 2 =   O 2 =   H 2 O = 1.The first suboptimal model offers a reasonable compromise between accuracy complexities; the second one has the minimal order.Figure

Table 13 :
The influence of the order of hydrogen, oxygen, and water pressure models (  H 2 ,   O 2 , and   H 2 O ) on the dynamic error  dyn  r and the steady-state error  ss  r of the output variable  r .
voltage.It is because the Nernst equation (

Table 14 :
The influence of the order of hydrogen and oxygen pressure models (  H 2 and   O 2) on the dynamic error  dyn  dif and the steady-state error  ss  dif of the output variable  dif .

Table 15 :
Comparison of the ideal full-order model (  H 2 = 4,   O 2 = 3, and   H 2 O = 2) and the chosen reduced ones calculated by the second algorithms in terms of the number of pressure parameters as well as the dynamic and steady-state errors for two process output variables (algorithm 2).  H 2   O2   H 2 O Parameters