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Research Article | Open Access
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 second method 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.
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 , geothermal systems , solar collectors , marine systems , and biofuels . 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 . 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 [13–16] (different model structures and optimisation algorithms are possible). Important applications of the mathematical model of the SOFC also include process optimisation , fault tolerant control , and estimation [16, 18].
In MPC, optimisation, fault tolerant control, and estimation, different model structures may be used. Firstly, the first-principle 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 black-bock models typically have numerous parameters, much more than the fundamental ones. Finally, accuracy of black-box 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 full-order 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.
2. SOFC System Description
The literature concerned with first-principle modelling of SOFCs is rich, for example, [9, 22–24]. The fundamental model of the SOFC introduced in  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 () and the input oxygen flow rate (); one disturbance (the uncontrolled input) which is the external current load (A) and four controlled variables (the outputs of the process): the stack output voltage (V), fuel utilisation (−), the fuel cell pressure difference (atm) between the hydrogen and oxygen passing through the anode and cathode gas compartments, and the ratio (−) between hydrogen and oxygen flow rates. The partial pressures of hydrogen, oxygen, and water are denoted by , , and , respectively (atm). The input hydrogen flow and the hydrogen reacted flow are denoted by and , respectively ().
3. Continuous-Time Model
Figure 1 depicts the structure of the fundamental continuous-time model of the SOFC system. The pressure of hydrogen iswhere the input hydrogen flow isHence, the pressure of hydrogen iswhere , , , and 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 iswhere and 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 isHence, the pressure of water iswhere and 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 iswhere , , , , and 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 ratio between hydrogen and oxygen flow rates isThe pressure difference between the hydrogen and oxygen passing through the anode and cathode gas compartments isAll things considered, the continuous-time fundamental model consists of (2), (3), (4), (6), (7), (8), (9), (10), and (11). 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: and ; the value of the disturbance is also limited: .
4. 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 hasFrom (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) iswhere denotes the discrete sampling instant and is a unit delay operator. The discrete-time version of (13) isThe discrete-time version of (14) isAdditionally, the discrete-time version of (2)and the discrete-time version of (6)must be taken into account. The final form of the discrete-time 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 haswhere , , , , , , , , , , , and . From (16), one haswhere , , , , , , , , and . From (17), one haswhere , , , and . From (18), one haswhere and . From (19), one haswhere and . The discrete-time version of the Nernst equation (8) isIn the discrete-time domain fuel utilisation (9) isand the ratio between hydrogen and oxygen flow rates (10) isand the pressure difference (11) isAll things considered, the full-order discrete-time model is defined by (20), (21), (22), (23), (24), (25), (26), (27), and (28). The order of model dynamics is defined by the integer numbers: , , and . Parameters of the full-order fundamental discrete-time model are given in Table 3.
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. The discrete-time model gives practically the same responses as the continuous-time one; no important differences are present.
5. Model Reduction Methods
The state-space representation of a discrete-time dynamic system is considered:It is assumed that matrix is asymptotically stable. The controllability, , and observability, , Grammians are  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 to , setting matrix to , and setting matrix to , where the transformation matrix may be found by the procedure described in [28, 29].
In this work, two model reduction algorithms  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 and 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 are assumed to be significantly faster than ones, it follows that and Hence, the reduced model becomesIn the second algorithm in the output equation, there is a possible direct influence of the process input(s) on the 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 and 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 and may be necessary. Finally, the second algorithm leads to the difference equation for water pressure.in which the coefficient may be necessary in contrast to the original full-order model and the reduced one generated by the first algorithm (see (22)).
6. 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).
6.1. Reduction of the Hydrogen Pressure Model
The diagonal of the balanced Grammian of the state-space realisation corresponding to the discrete-time difference equation (20) describing hydrogen pressure is 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 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 and may be necessary, which manifest a direct influence of the model input and the disturbance on the model output . 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 ). 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 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 and , respectively. Both models are excited by the same series of step changes of the manipulated variables and steps of the disturbance depicted in Figure 2; that is, . Additionally, the steady-state model error is defined:In this case, and 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)), . For the steady-state models, . 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 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 (), they are treated as 0. Both reduction algorithms give perfect models of order three.