Complexity

Volume 2018, Article ID 6021249, 18 pages

https://doi.org/10.1155/2018/6021249

## Towards Reduced-Order Models of Solid Oxide Fuel Cell

Institute of Control and Computation Engineering, Faculty of Electronics and Information Technology, Warsaw University of Technology, ul. Nowowiejska 15/19, 00-665 Warsaw, Poland

Correspondence should be addressed to Maciej Ławryńczuk; lp.ude.wp.ai@kuzcnyrwal.m

Received 24 January 2018; Accepted 6 February 2018; Published 4 July 2018

Academic Editor: Sing Kiong Nguang

Copyright © 2018 Maciej Ławryńczuk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. 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 [13–16] (different model structures and optimisation algorithms are possible). Important applications 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 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 [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 () 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: .