DOI: 10.1002/fuce.201400119
ORIGINAL RESEARCH PAPER
An Algorithm for On-line Measurement of the Internal Resistance of Proton Exchange Membrane Fuel Cell F. Chen1,2, Y. Gao1,2* 1 2
School of Automotive Studies, Tongji University, Shanghai 201804, China Clean Energy Automotive Engineering Center, Tongji University, Shanghai 201804, China
Received July 15, 2014; accepted January 28, 2015; published online April 02, 2015
Abstract The internal resistance of proton exchange membrane fuel cell (PEMFC) system is difficult to measure on-line due to its variation with time. The traditional electrochemical impedance spectroscopy (EIS) and its variants such as high frequency resistance (HFR) can be used to measure the resistance when the system is in steady state, but they fail in automotive applications where a change in speed or inclination modifications could lead to a sharp fluctuation in demand on power. In order to resolve this problem, a novel algorithm is proposed in this paper to estimate the resistance
1 Introduction Proton exchange membrane fuel cell (PEMFC) is a promising energy convert device and has been received increased attention over last two decades, especially in the field of automotive applications due to its high energy conversion efficiency (approximately 50%), low operating temperature (60 C~90 C), high power density, zero emissions and quick start-up in comparison with other traditional power sources [1–3]. The internal resistance of the PEMFC depends on current density, humidity and the temperature of the membrane. It is a key factor characterizing the ability of protons to transport across the membrane into the cathode catalyst layers. A good water management is important in the PEMFC to keep its good health and prolong its lifespan [4, 5]. However, it is difficult to actively control the water content in the membrane due to the difficulty associating with non-intrusively monitoring the humidity of the membrane [6]. Although some efforts had been made to estimate water content of the membrane [6, 7], because water content of membrane cannot be directly measured and it is highly related to internal resistance, the validation of the algorithm is tested based on internal resistance. In addition to this, the internal resistance can also be used to on-line diagnose a stack. For example, measuring the resistance of the cells is a viable method to assess elec-
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based on the alternating current (AC) impedance spectroscopy technique by adding an extra term to eliminate the errors caused by voltage variation or when the system is under unsteady state. Numerical simulations show that the proposed algorithm can not only accurately track the variation of the internal resistance, but is also robust against the noises caused by uncertainty and measurements. Keywords: Electrochemical Impedance Spectroscopy (EIS), Internal Resistance, On-line Measurement, Proton Exchange Membrane Fuel Cell (PEMFC)
trode hydration and flooding because a gradual increase in the hydration at the interface of the Pt/C electrical doublelayer leads to a drop in the resistance [8]. Currently, there are two basic methods to measure the internal resistance. One is based on the frequency domain such as the electrochemical impedance spectroscopy (EIS) and the high frequency resistance (HFR) methods, and the other one is the current interrupt (CI) method based on time domain. EIS has been widely used in fuel cell impedance analysis [9, 10] by imposing an AC sinusoidal perturbation featured with small amplitude (less than 5% of the original steady current) and a broad range of frequencies to the targeted fuel cell. The resulting variations in the cell voltage and current are then sampled synchronously, from which the impedance is obtained. The internal resistance of the fuel cell can also be derived based on a given equivalent circuit model, but the test is time-consuming and its errors increase if the voltage or current varies during the testing process. Therefore, the EIS is preferred in in-situ analysis in laboratory [11, 12]. In contrast, the HFR imposes a single high frequency, typically at the order of 1 kHz, on the fuel cell stack. The HFR is actually a special case
– [*] Corresponding author,
[email protected]
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ORIGINAL RESEARCH PAPER
Chen, Gao: An Algorithm for On-line Measurement of the Internal Resistance of PEM Fuel Cell of the EIS method, and is only to extract the internal resistance, i.e., the real component of the impedance. The CI method has been used in some areas including batteries, corrosions and fuel cells [13, 14]. Although this method is easy to be interpreted as it is based on the time domain, it has several drawbacks: It needs to impose a significant perturbation to the cell; it could fail when the system is in unsteady state; and it is sensitive to noises. Efforts have been made over the past few years in attempts to measure the internal resistance of PEM fuel cell [15, 16]. Based on the comparisons of different experiments, R. Cooper and M. Smith [15] concluded that the HFR method is more promising than the EIS and CI methods for on-line measurement. Min et al. [16] measured the resistance of a direct methanol fuel cell using the CI method and found that the errors are acceptable even when the cell worked in high current density region. However, in automotive applications, the current in the PEMFC varies and the output voltages also vary simultaneously due to change in speed and/or inclination modifications. Therefore, the above methods could fail in such cases. In order to overcome this problem, an algorithm is proposed in this paper to estimate the internal resistance when the system works under transient state. The paper is organized as follows. Section 2 presents the principal of the proposed method. In Section 3, the algorithm for calculating the resistance is given and some comments are provided to interpret the method. Numerical simulations are shown in Section 4 to demonstrate the effectiveness of the proposed algorithm. Some conclusion and remarks are given in Section 5.
FECM works well when the current density is the intermediate range, but not when the density is low or high. Reggiani et al. [17] proposed a nonlinear equivalent circuit (NEC) model as shown in Figure 2 represented by the part inside the dash rectangle. The capacitor C and resistance R are nonlinearly related to the current. In the figure, CS is a current-control voltage source to describe the active potential voltage, Rr is the internal resistance, and E is the open circuit voltage related to temperature and partial pressure of oxygen and hydrogen. Several experiments showed that the NEC model describes the system well under both transient and steady conditions [17]. For more complicated models, readers can refer to references [17–21]. In order to simplify the analysis of the relationship between the internal resistance and the AC impedance, we take the FECM as an example. The NEC model will be shown in numerical simulations in Section 4. As for FECM, if the current goes through the circuit, a voltage drop of uðtÞ will occur, and the transfer function defined by GðsÞ ¼ U ðsÞ=I ðsÞ can be written as follows: G ðsÞ ¼ R r þ
For completeness, we give a brief review on the equivalent circuit models for PEMFC before establishing their relationship and interpreting the principal of the proposed algorithm. A number of equivalent circuit models are available in the literature [17–21] to describe the dynamics of the PEMFC. The simplest one is the first-order equivalent circuit model (FECM) [17] as shown in Figure 1. In this model, the electrochemical double layer, which stores electrical energy and behaves like a super capacitor, is included. The resistance R parallel with the capacitor C is the equivalent resistance of the activation and concentration. The resistance Rr is the internal resistance, which represents the internal resistance of the cell stack, i.e., the resistance due to non-ideal electrodes and conductive plates to proton transport through the membrane.
Fig. 1 The first-order equivalent circuit model.
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(1)
where R, Rr, C are the circuit parameters as shown in Figure 1, s is the Laplace transform variable, I ðsÞ and U ðsÞ are the Laplace transform of the input iðtÞ and output voltage uðtÞ, respectively. Defining T ¼ RC t¼
2 Problem Formulation
R RCs þ 1
(2)
Rr RC ðRr þ RÞ
(3)
ts þ 1 Rr . Note that 0 < < 1, we Ts þ 1 Rr þ R thus have t < T. The amplitude frequency and phase frewe have GðsÞ ¼ ðRr þ RÞ
quency characteristic function are derived as follows:
jGðjwÞj ¼ jGðsÞjs¼jw
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðtwÞ2 þ 1 ¼ ðRr þ RÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðTwÞ2 þ 1
jðwÞ ¼ arctan tw arctan Tw
(4)
(5)
where w is the angular frequency. It is obvious that jðwÞ˛½p=2; 0, jGðjwÞjw¼0 ¼ R þ Rr , and jGðjwÞjwfi¥ ¼ Rr ; its Nyquist plot is shown in Figure 3. From above analysis, we know that if the frequency w is high enough, jGðjwÞj»Rr .
Fig. 2 The topology of the on-line measurement of the internal resistance.
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FUEL CELLS 15, 2015, No. 2, 337–343
Chen, Gao: An Algorithm for On-line Measurement of the Internal Resistance of PEM Fuel Cell
For simplification, we introduce and prove the following two lemmas: R nT R nT nT 2 2 Lemma 1: where 0 sin wtdt ¼ 0 cos wtdt ¼ 2 , 2p þ T ¼ , n ˛z . w Z T Z T Z T cos 2 wtdt sin 2 wtdt ¼ cos 2 2wtdt ¼ 0 Proof: 0
Z Z
To determine the criterion at which this approximation is 1 T valid, we consider (4). If ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi we take e ¼ wt > 0, k ¼ t > 1, qEq. ðtwÞ2 þ 1 and define y ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, then jGðjwÞj ¼ ðRr þ RÞy and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðTwÞ2 þ 1 1 þ e2 . If w is large enough such that k > 1 , e is small and y¼ k2 þ e2 can be neglected. We define the relative error ey as follows:
(6)
With some enlarge and narrow manipulations to the pffiffiffiffiffiffiffiffiffiffiffiffiffi e2 Eq. (6), we have 0 < ey £ 1 þ e2 1 then 0 < ey £ . The fol2 lowing inequality holds: DRr e2 R ¼ ðRr þ RÞ ey £ðRr þ RÞ 2 r
(7)
Generally, Rr and R are small, and we hence have DR 1 Rr þ R < 1. Taking w ¼ 10 , we have r < 0:5%. It is Rr t obvious that this is accurate for the problems we studied in this paper. Additionally, from Figure 3, one can see that Rr £ Re½GðjwÞ £ jGðjwÞj, meaning that we can estimate Rr from Re½GðjwÞ (i.e., the real part of GðjwÞ). In some cases, a preliminary estimation of C, R, Rr can be predetermined [17], and the value of the frequency w can then be estimated based on the relative error of Rr. According to previous discussions, we can calculate the internal resistance of the PEMFC from the Nyquist plot within the high frequency band. The purpose of this paper is to determine the internal resistance of the PEMFC from signals, such as stack voltage and disturbance current; its topology is shown in Figure 2. The whole system can be divided into 3 parts: PEMFC system, Resistance on-line estimation (ROE) and DC/DC, where the ROE is to generate the sinusoid disturbance current of id ðtÞ ¼ r sin wt with r being the amplitude, w the angular frequency; and the stack voltage is uðtÞ. In Section 3 we will show how to estimate the resistance using signal id ðtÞ and voltage uðtÞ.
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0
2
cos wtdt ¼
Fig. 3 Nyquist plot of the equivalent circuit model.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ e2 1 k 2 þ e2 k ey ¼ 1 k
T
0 T
Z
sin 2 wtdt þ
Z 0
T
0
Z
T
0
2
sin wtdt 0 T
cos 2 wtdt ¼
Z
Z
sin 2 wt þ cos 2 wt dt ¼ T
T
sin 2 wt þ cos 2 wt dt ¼ T
2
0 T
0
T sin wtdt ¼ 2 0 Z T Z T Z sin 2 wtdt þ cos 2 wtdt ¼ 0
T
0
T sin wtdt ¼ : 2 0 R nT R nT nT Therefore, 0 sin 2 wtdt ¼ 0 cos 2 wtdt ¼ . 2 Before introducing Lemma 2, we first present a basic principle in the linear time invariant (LTI) system. For a LTI system, if the sinusoid is the unique input signal, i.e., A sin ðwtÞ, the resulting output signal is also a sinusoidal that can be described by B sin ðwt þ jÞ in steady state. It differs from the input waveform only in its amplitude and phase angle [21]. The relationship between the amplitudes and phase angles in the input and the output is B ¼ AjGðjwÞj and j ¼ —GðjwÞ respectively, where GðjwÞ is the frequency characteristic function of the system. Lemma 2: For a LTI system, the input and its steady output signals are A sin ðwtÞ and B sin ðwt þ jÞ respectively, and its frequency characteristic function GðjwÞ can be expressed by 2Ry 2Rx GðjwÞ ¼ þj (8) n T A2 n T A2 R nT where Rx ¼ 0 A sin wt B sin ðwt þ jÞdt, R nT 2p Ry ¼ 0 A cos wt B sin ðwt þ jÞdt and T ¼ , n ˛zþ . w R nT Proof: Rx ¼ A B 0 sin 2 wt cos j þ sin wt cos wt sin j dt. R nT Since 0 sin wt cos wt sin jdt ¼ 0, R nT it gives Rx ¼ A B cos j 0 sin 2 wtdt. Using Lemma 1, we have nT A B cos j (9) Rx ¼ 2 Similarly, nT Ry ¼ A B sin j (10) 2 2Ry 2Rx þj . Since B ¼ AjGðjwÞj, it has GðjwÞ ¼ n T A2 n T A2 As shown in Figure 3, the inserted disturbing current is id ðtÞ ¼ r sin ðwtÞ, and the current extracted by DC/DC is i0 ðtÞ. Therefore, the current exerted on the stack isiðtÞ ¼ i0 ðtÞ þ r sin wt. Without loss of generality, the voltage of the stack can be assumed to be uðtÞ ¼ u0 ðtÞ þ rjGðjwÞj sin ðwt þ jÞ þ n0 (11) 2
where u0 ðtÞ is caused by i0 ðtÞ, rjGðjwÞj sin ðwt þ jÞ by the disturbance, GðjwÞ is the frequency characteristic function, and n0 is the random noise caused by measurement and uncertainty.
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ORIGINAL RESEARCH PAPER
3 Algorithm Development
ORIGINAL RESEARCH PAPER
Chen, Gao: An Algorithm for On-line Measurement of the Internal Resistance of PEM Fuel Cell We assumed that u0 ðtÞ ¼ a þ bt during a small time interval (i.e., 100 ms). Even u0 ðtÞ has a jump at time t, the errors can be viewed as a part of the noise n0 . Theorem 1: For every time interval of ½0; nT, if the input current disturbance is id ðtÞ ¼ r sin ðwtÞ, the voltage of the PEMFC system is described by Eq. (11), and w is high enough, the following relationships holds: 2R¢x Tb þ (12) r2 nT pr R nT 2p where R¢x ¼ 0 id u dt, T ¼ , n ˛zþ and is large w Tb is a motion compensation. enough, pr Proof: Z nT R¢x ¼ r sin wt ðu0 þ rjGðjwÞj sin ðwt þ jÞ þ n0 Þdt Rr ¼
¼ þ
0
Z Z
nT
u0 r sin wtdt þ
0
Z
nT
r2 jGðjwÞj sin wt sin ðwt þ jÞdt
(13)
0
nT
n0 r sin wtdt
0
Since n0 is a random noise, and r sin wt is a definite signal, their cross-correlation is zero and the third item on right-hand side of Eq. (13) is zero, i.e., Z nT n0 r sin wtdt ¼ 0 (14) 0
Considering
Z
nT
u0 r sin wtdt ¼ r
0
Z
¼ br
Z
ðkþ1ÞT
ða þ btÞ sin wtdt
kT
ðkþ1ÞT
t sin wtdt kT
with the theorem of integration by parts we have Z ðkþ1ÞT t sin wtdt br kT
Z
R
R udv ¼ uv vdu,
ðkþ1ÞT
2R¢x Tb þ . Notr2 nT pr ing that Rr £ RefGðjwÞg £ jGðjwÞj and w is large enough, we 2R¢ Tb have Rr ¼ 2 x þ . r nT pr Comment 1: Since u0 ðtÞ slowly changes with time, w is a high frequency signal, rjGj
