1. Introduction
An increase in worldwide energy demand and the quest for sustainability have led to major investments in renewable resources. The latter are eco-friendly, generally reliable, and have lower operating costs. While there are several disadvantages associated with renewable sources such as vulnerability or the inability to generate power in large quantities, the consensus among researchers is that energy in the future will primarily be generated by renewable sources. One of the latter is the fuel cell. This source has become very popular due to a higher efficiency compared to other renewable sources, low maintenance costs, and ubiquity of hydrogen. Fuel cells have been widely used in industrial settings (e.g., factories) [
1], residential units [
2], and the automotive industry (e.g., electric vehicles) [
3,
4], and are also commonplace in smart grid ecosystems where they are used as supplementary renewable sources in microgrids [
5,
6]. A major enabler that has aided in hydrogen mobility, and subsequently in the widespread use of fuel cells, has been the investment in infrastructure by several countries in North America, Europe, and Asia [
7,
8]. Government statistics in these countries [
8] show that demand for a hydrogen economy is increasing, hence possibly putting the fuel cell at the forefront of the clean energy revolution.
The device works by converting the chemical energy of hydrogen and oxygen into electricity through a pair of redox reactions [
9]. In addition to electricity, the chemical reaction produces water and heat [
9,
10]. While the method of operation is standard for all varieties of fuel cells, they have distinct features such as different material composition, efficiency, and operating temperatures. Some of the most common types that are currently in use and also heavily researched are the proton-exchange membrane fuel cell (PEMFC) [
9], the solid oxide fuel cell (SOFC) [
11], and the phosphoric acid fuel cell (PAFC) [
12].
It is worth noting that modeling and simulation play a critical role in fuel cell design. A high-fidelity model gives the designer the capability to modify parameters accordingly and use them in the final implementation, therefore leading to an optimal desired design. One such PEMFC model was originally developed in [
9]. This model has been investigated in a variety of domains over the years. For example, in [
13], the authors consider dynamic and controlled operation of a PEMFC integrated with a natural gas fuel processor system (FPS) and a catalytic burner (CB). Optimization is performed to generate the air and fuel flow intake setpoints to the fuel processor system for different load levels, and then linear quadratic regulator techniques are used to develop a controller to mitigate hydrogen starvation in the fuel cell. In [
9], analysis and design of air flow controllers that prevent oxygen deprivation of the fuel cell stack as the current changes was presented. Improvements in transient oxygen regulation when the fuel processing system voltage was included in the model were demonstrated. A combined PEMFC-FPS model was studied in [
14], where the authors used order-reduction techniques to show that similar controller performance can be obtained if the model size is reduced.
Unlike the work of [
9] and other aforementioned references, this research differs in that an augmented singularly perturbed model of the PEMFC-FPS is obtained. While the latter has been defined in [
14], it has not been studied in a singularly perturbed fashion. The technological context as well as the motivation behind this work lies in the fact that fuel cell systems can contain inherent parasitics such as small impedances or load perturbations. A standard dynamic model does not include such parasitics and that can lead to inaccuracies in simulations which, in turn, would affect the efficiency of the overall system design. Singular perturbation methods are typically used for such models and are very important from a practical aspect since they capture the model’s slow and fast dynamics, hence providing a more accurate model. Besides the singularly perturbed modeling of the fuel cell system, case studies of optimal controller design are presented. Namely, linear quadratic (LQ) controllers are designed for the individual time-scales to meet operational requirements. It is important to note that in this paper, the term “fuel cell system” strictly refers to a setup consisting of a PEMFC and an FPS and does not include additional components such as filters, converters, etc.
The contributions of this paper are threefold. First, the theory that enables one to convert an arbitrary linear system with two or more time-scales into a singularly perturbed model is developed. The algorithm leverages an ordered Schur decomposition that arranges the system’s eigenvalues along the main diagonal. Furthermore, the algorithm is computationally efficient, which is essential from a practical perspective (e.g., parametric studies). Next, the singularly perturbed model of the overall PEMFC-FPS is derived and formalized. Lastly, LQ controllers for each of the lower order models are designed so that the response meets operational requirements.
The paper is organized as follows. In
Section 2, the preliminaries of PEMFCs and singularly perturbed systems are introduced. In
Section 3, theoretical details on time-scale decoupling via the ordered Schur decomposition are presented.
Section 4 covers the singularly perturbed modeling of the PEMFC-FPS, followed by
Section 5 where LQ controller design is illustrated. The remaining sections, namely,
Section 6 and
Section 7, correspond to the Results and Discussion, and Conclusions, respectively.
3. Time-Scale Decoupling via the Ordered Schur Decomposition
This section serves as the basis for the singularly perturbed modeling of the PEMFC-FPS augmented system. Namely, the algorithm that achieves complete decoupling of an arbitrary singularly perturbed model is developed. This method is then applied in the following section. Theoretical techniques presented here were briefly introduced in [
26]. This section contains a thorough explanation of the method.
3.1. Ordered Schur Transformation
As noted earlier, many real physical systems contain small parasitics when they are modeled. This forces parts of the system to operate in different time-scales. Quite often, it may be difficult to distinguish between the time-scales. Methods such as permutation matrices or other similarity transformations [
19] have been proven successful to obtain a standard singularly perturbed form for two time-scale systems but it is challenging when additional time-scale dynamics are present.
In this paper, the aforementioned issue is addressed by developing a method that brings an implicit singularly perturbed system into its explicit form, where the perturbation parameters are either known or can be easily determined. is associated with the slowest state variable and is associated with the fastest, namely, .
Consider a general implicit multiple time-scale system without inputs, as shown in (
10).
where
is the state vector. To simplify the problem, an
ordered Schur decomposition is employed to transform the model into a well-conditioned form. This is followed by the extraction of the perturbation parameter and sequential decoupling to obtain the individual time scales. The Schur decomposition is an efficient method used to find the system’s eigenvalues by utilizing the
QR algorithm [
27].
For a matrix
, there exits a unitary matrix
such that
is upper quasi-triangular.
Matrix blocks can be or . blocks correspond to real eigenvalues while blocks correspond to complex eigenvalues.
The eigenvalues appearing along the diagonal of
can be arbitrarily ordered. An additional transformation has to be employed to achieve desired reordering (descending) of the system matrix [
27,
28,
29]. A transformation
[
30] can be found such that the unitary matrix
T decomposes the system into the Schur form. Upon applying the ordered Schur algorithm, the dynamic Equation (
10) takes the following form
where
Diagonal block matrices , , contain the system’s eigenvalues in descending order. are vectors each representing each time scale and are matrix blocks of appropriate dimensions. Note that blocks , , represent individual time-scales rather than individual eigenvalues.
3.2. Parameter Extraction and Time-Scale Decoupling
Prior to decoupling the transformed system, it is essential to convert it to an explicit singularly perturbed form by extracting the perturbation parameters from the system matrix. This is achieved by defining the perturbation parameters. For two time-scale stable systems with clearly separated eigenvalues (real or complex with small imaginary parts),
is commonly evaluated as
[
31]. However, referring to examples such as the PEMFC-FPS under consideration, it can be inferred that the imaginary parts of the eigenvalues are indispensable for calculating the perturbation parameter. Hence, the latter is evaluated as the ratio of the magnitudes of the largest eigenvalue of the slowest time-scale with the smallest eigenvalue of the next fastest time-scale
Since the system is in ordered Schur form, the perturbation parameters can be easily evaluated using (
14) and extracted to put the system into the explicit singularly perturbed form. The explicit multi-time-scale system now looks as follows:
where the elements of the system matrix have been scaled in accordance with parameter extraction.
The explicit system can now be decoupled into multiple distinct time-scale systems by successively applying the Chang transformation in (
6).
To initiate the decoupling, the perturbation parameter is extracted from the fastest time-scale and (
12) is rewritten as a standard two time-scale singularly perturbed form.
In (
16), matrices
and
represent the last row of the system’s matrix in (
13) with
extracted.
represents the fastest time-scale, while
contains the rest of the matrix blocks which happen to be all zero in this case.
and
are matrices of appropriate dimensions containing the rest of the system matrix in (
12).
Utilizing (
6), the system in (
16) is initially decoupled into two subsystems, where the fast subsystem represents the fastest time-scale available and the slow subsystem contains the rest of the time-scales.
As a reminder to the reader, matrices
L and
H satisfy (
9). A modified Newton’s method developed in [
25] follows.
The new slow subsystem (
17a) is partitioned again, as in (
19), where
is now extracted from the second fastest time-scale.
The algorithm is applied sequentially till all the perturbation parameters have been extracted and the system is in explicit singularly perturbed form. The relation between the system in original coordinated
and the new one is given as
where
T is defined as
and
is
Matrix
represents the linear transformation for each time-scale
i. On the other hand, matrix
is an augmentation of
with an identity matrix
I of appropriate dimensions so that its size is the same as
T. Unlike in [
31], the system matrix in ordered Schur form simplifies the computations. For a quasi-triangular system such as (
12),
in (
16) is
. Then, equations for the solution of matrices
L and
H in (
9) simplify to the following.
An additional simplification comes due to the new system matrix structure.
Theorem 1. Due to the structure of (15), matrix of the Chang transformation evaluates to a zero matrix. Proof. Upon applying the recursive algorithm (
18), it is easy to show that matrix
in (
22a) evaluates to a zero matrix by solving the Sylvester equation for the first iteration
Matrix M is defined as and is full rank. Therefore, , which implies . Since and , then by induction, for all other iterations . This applies to matrix , for the remaining time-scales. □
In the absence of matrix
L, (
22b) then becomes just a Sylvester equation.
Likewise, transformation (
6a) simplifies to (
25). Note that the state variables in (
25) are arbitrary.
After the process is repeated for all the
N time-scales available in the system, the individual subsystems are then given as follows.
System (
26) is now completely decoupled into a standard explicit singularly perturbed form. Note that if control is considered, the input matrix for each time-scale would be obtained sequentially in a similar fashion. The slow and fast input matrices for a two time-scale system are evaluated as
and
, respectively.
The overall process discussed in this section can be summarized as follows.
Algorithm 1 Time-Scale Decoupling of Implicit Singularly Perturbed Systems |
- 1:
Input: Implicit singularly perturbed system - 2:
Apply Schur decomposition - 3:
If Eigenvalues are not ordered then - 3:
apply swapping algorithm [ 27] or [ 29] - 4:
Evaluate for each time-scale and form explicit system - 5:
Use Chang transformation [ 22] to get individual time-scales - 6:
Output: Completely decoupled system
|
4. Singularly Perturbed Modeling of the PEMFC-FPS
The general nonlinear models of the PEMFC and FPS systems studied in this paper were originally derived and used for controller design in [
9]. The fuel cell considered in [
9] has been used for research in the automotive industry and has a stack size of approximately 40 kW. Its operation temperature varies from 50
C to 100
C. The associated FPS is typically used for PEMFCs with stack size of 100 kW and is made up of a hydrodesulfurizer, a catalytic partial oxidation reactor, a water gas shift reactor, and a preferential oxidation reactor [
9]. Original PEMFC and FPS models have been linearized around a nominal operation point, where the system net power is
kW and oxygen excess ratio is
[
9]. Both linear and nonlinear model responses are compared in [
9] and it is validated that the errors are insignificant.
This paper does not elaborate any further on the nonlinear systems but instead uses the linearized system for modeling and controller design. The reader is referred to [
9,
10] for additional details.
4.1. PEMFC and FPS Linear Model
Variables that are used in determining the model of the PEMFC fall into two main categories: masses of the gases that are used in the chemical reaction and the pressure of these gases. The state vector for the linear PEMFC model is defined as
. Corresponding descriptions of state variables is provided in
Table 1.
Likewise, for the FPS, masses and pressures of gases are essential for modeling. The state vector in this case is
.
Table 2 describes the state variables of the tenth order FPS model.
Using information from the PEMFC and FPS models, an augmented linear state-space model is created as in (
27) [
14].
State-space system in (
27) can be written as follows:
where
,
,
, and
represent matrices in (
27). Numerical values of
,
,
,
,
, and
are available in [
9] and also included in
Appendix A for reference. Note that the feedthrough matrix
D is zero in both the PEMFC and FPS, therefore
.
In the following subsection, a singularly perturbed linear model of (
27) is derived using methods from
Section 3.
4.2. Singularly Perturbed Model of the PEMFC-FPS
The eigenvalues of the augmented system are key in determining the time-scale composition of the model.
Figure 2 depicts graphically the sorted eigenvalues of the PEMFC-FPS. Recall from
Section 3.2 that the magnitude of the eigenvalues is used to evaluate the singular perturbation parameter
. Based on the distribution shown in
Figure 2, a three time-scale model is chosen. The first nine modes represent the slow subsystem, the following six represent the second time-scale, and the last three constitute the fastest time-scale. Note that this decision is not necessarily strict.
At this point, the singular perturbation parameters of the three time-scale model can be calculated using (
14). The magnitude of the largest eigenvalue of the slow subsystem is
. The magnitude of the smallest eigenvalues of the second and third time-scales are
and
. Hence,
,
, and
are as follows.
The singularly perturbed model then becomes
State variables belonging to each time-scale are presented in
Table 3.
After applying the algorithms developed in
Section 3 to the original augmented PEMFC-FPS model in (
27), the matrices that help form the singularly perturbed model (
30) are obtained. The state matrix
, control matrix
, and output matrix
are available in
Appendix B. Linear mapping
T represents the transformation from the original augmented model (
27) to the ordered Schur model. It is also important to note that matrices
and
, specified in
Appendix B, do not represent the state and control matrices in (
30) because the singular perturbation parameters have not been extracted yet. To obtain model (
27),
and
are simply multiplied by the
of the corresponding time-scale.
Lastly, the Chang transformation [
22] is applied sequentially to obtain a completely decoupled system (refer to Algorithm 1).