Tracking the maximum efficiency point for the FC system ... - CiteSeerX

Tracking the maximum efficiency point for the FC system based on extremum seeking scheme to control the air flow

Nicu Bizon1)2) 1)

University of Pitesti, 1 Targudin Vale, Arges, 110040 Pitesti, Romania

2)

University Politehnica of Bucharest, 313 Splaiul Independentei, 060042 Bucharest, Romania

Tel +40 348 453 201, Fax +40 348 453 200; [email protected], [email protected]

Abstract. An advanced control of the air compressor for the Proton Exchange Membrane Fuel Cell (PEMFC) system is proposed in this paper based on Extremum Seeking (ES) control scheme. The FC net power is mainly depended on the air and hydrogen flow rate and pressure, and heat and water management. This paper proposes to compute the optimal value for the air flow rate based on the advanced ES control scheme in order to maximize the FC net power. In this way, the Maximum Efficiency Point (MEP) will be tracked in real time, with about 10 kW/s search speed and a stationary accuracy of 0.99. Thus, energy efficiency will be close to the maximum value that can be obtained for a given PEMFC stack and compressor group under dynamic load. It is shown that the MEP tracking allows an increasing of the FC net power with 3 - 12%, depending on the percentage of the FC power supplied to the compressor and the level of the load power. Simulations shows that the performances mentioned above are effective.

Keywords: Fuel cell system; Energy efficiency; Maximum Efficiency Point tracking; Extremum Seeking algorithm; Air compressor control

1. Introduction Polymer Electrolyte Membrane Fuel Cells (PEMFCs) are electrochemical devices that convert the chemical energy stored in hydrogen and oxygen (air), used as combustible and combustive reactants, directly into electricity [1, 2]. PEMFC is a promising alternative for transportation and distributed generation applications based-on stationary or portable FC Hybrid Power Sources (FCHPS). PEMFC has higher energy efficiency (which is in the range of 40 – 50% or close to 85% in the cogeneration mode) in comparison with the competing technologies (which is in the range of 30 – 35%) [2, 3]. Controlling the PEMFC system to further increase energy efficiency is a challenging action for the FCHPS designers. Three main control systems must to be designed for a PEMFC system [4, 5]: (1) the air/fuel supply, (2) the water supply, and (3) the heat management. The control problem presented in this paper is focused on the regulation of the air supply to the cathode. Besides the FC stack, which is the main component, the PEMFC system needs various auxiliary equipments (including air compressor, humidifier, pumps, cooling water circulation, and measurement and control equipment) to safe operate it. The power consumption of air compressor is the highest in comparison with the other auxiliary equipments (up to 80 % power of the overall auxiliary equipments) [6]. The compressor motor is powered by the PEMFC system itself and can consume up to 20% of the FC power [7]. So, maximizing the FC net power is one of the control goals related to energy efficiency. Thus, the PEMFC systems must operate safety close to the Maximum Efficiency Point (MEP) [8, 9]. In comparison with the Maximum Power Point (MPP), the MEP is difficult to be tracked because the FC operating point depends on a large numbers of related parameters, for instance, the feeding and the humidification

systems, cooling control circuit, and electrical interface, beside the dynamic load [10, 11]. Even if the energy efficiency of the PEMFC system operating at the MPP is a little bit lower than the maximum value obtained at the MEP, the MPP tracking control schemes are usually implemented because of their relative simplicity [12, 13]. In this paper we propose using a MEP tracking control based on advanced Extremum Seeking (aES) scheme [13]. This aES control scheme improves the basic performances of the ES control schemes [14, 15], assuring higher search speed and improved tracking accuracy of the MEP. Starting with the static feed-forward (sFF) and static feed-forward with PI control [7, 16] that are usually used as references, different MEP tracking control algorithms were proposed in the last decade based on the feedback linearization [17], dynamic feed-forward-feedback control techniques [18], sliding mode control [19], supper twisting algorithm [20], perturb and observe algorithm [9, 21], ES control schemes [22, 23, 24], model predictive control [25, 26], neural networks [27], fuzzy logic [28, 29], LQR/LRS strategies [30, 31], nonlinear differential flatnessbased control [32, 33], time delay control [34], and other adaptive control strategies [35, 36, 37]. Most of the previous studies cited are based on different types of feedback control loops and look-up tables that need different FC system state variables, which are acquired using many sensors or observers, and proper design of the controller to tolerate FC system uncertainties [38]. The designed controller could theoretically produce accurate results, but the complex computational algorithms are not suitable for implementation into the embedded controllers. If an air mass flow sensor will be used to evaluate or control the oxygen excess ratio (OER), then the following drawbacks appear: (1)slow response time (about 1 - 2 seconds), (2) less accuracy (with 1% - 10%), short life time (up to 3 years), and a high price. Thus, a sensor-less approach is of interest for controlling the Air Flow rate (AirFr) via a compressor. However, it is obvious that

designing advanced AirFr controllers is an important issue for improving the energy efficiency of the PEMFC system. Hydrogen and oxygen must be supplied to the fuel cell in order to fulfill the stoichiometric relation required to produce the current demanded by a dynamic load. The recommended values for the OER are in range 1.5 – 3, but some of the cited approaches have considered as a reference OER=2 [7]. It can be noted that it is difficult and costly to estimate the OER during the stationary regime and some major errors could appear during the transitory regime [39].The cited works obtained good results in terms of AirFr, considering that the anode pressure follows the cathode pressure value [32]. Therefore, a fast electro-valve is required to control the anode pressure in order to maintain the pressure equilibrium between the cathode and anode. So, the air and fuel supply subsystems must be controlled simultaneously with the dynamic load to avoid oxygen starvation [7, 32]. A simple and cheap control technique is the sFFcontrol technique or one of the improved sFF control techniques that is based on FC current, but these control techniques are not robust and are showing slower responses to track the optimal value of OER [7, 21]. Because this study is partially focused on the stationary tracking accuracy of the MEP, the sFF control technique will be used as a reference in reporting the results and comparing these results with other studies that use the same control reference. All the cited studies, excepting the ES control techniques, need an accurate PEMFC model. The PEMFC system model is still under study in order to consider all the new phenomena observed under dynamic load [32]. In addition to the dynamic load, changes in operating conditions, such as temperature and humidity, or the nonlinearities of the compressor also influence the system parameters. Therefore it is necessary to consider parametric uncertainty for designing robust

controllers. Thus, if some parameters of the fuel cell air subsystem are considered constant, then the results are difficult to be implemented in practice. It is known that ES control scheme is an effective method for optimization problems when the system dynamics are not well known, offering a guaranteed convergence and a proved internal robustness. A novel constrained ES method for maximizing the FC net power is presented in [23]. The penalty functions effectively enforce the set constraints and enable the use of higher values for both closed loop gain and dither amplitude ES parameters to increase the search speed. The authors noted that this action leads to tracking accuracy depreciation: high overshoots and stationary ripple appear. The same conclusions are given in [24]. To overcome these drawbacks, the aES control scheme is proposed and improved here by adding a minimum dither signal. This assures the dither persistence during the stationary regime and improves the stationary tracking accuracy. The search speed during transitory regime could be set at safe limit for the PEMFC stack [39, 40]. The safe limit could be adapted to the FC current level based on constrained ES method [41]. To ensure the ES control scheme convergence to a neighborhood of the optimum, three assumptions are necessary [24, 42] : (1) the system is input-to-state stable, (2) a convex map exists between the control input and the performance output parameter, and (3) the dither persistence is assured. All these assumptions are satisfied for the PEMFC system controlled by the OER or AirFr parameter [24]. The OER is a lumped variable that cannot be measured directly, depending mainly on the AirFr, FC current, the relative humidity, and the cathode inlet pressure and temperature [25, 39].

Consequently, it is better to use the convex map that exists between FC net power and AirFr [21]. The feed forward control based on aES (FFaES) scheme proposed here, presents several advantages: (1) it is robust based on an integrator block that is included in the control loop(see Figure 1) [21, 43]; (2) It is not based on measurements of the PEMFC system’s states, being based only on measurement of the FC net power, which can be obtained with high accuracy, using inexpensive transducers for current and voltage; (3) it has high performances at a reduced computational power for the embedded controller [13].

Figure 1. The FFaES scheme to control the AirFr

The FFaES control scheme is of adaptive closed-loop control type used for searching unknown MEP (see Figure 1). The stationary accuracy is defined as percentage based on relation 100⋅(Pnet/Pnet(max)), where Pnet(max) is the power of the FC system operating at MEP and Pnet is the FC net power extracted. The dynamic accuracy is dependent to search speed, which must be high in order to track the power profile of a dynamic load. The goal of this paper is to show that the proposed FFaES scheme can be used to accurately determine the unknown MEP of the FC system that supply a dynamic load. The experiments to analyze the both sFF and FFaES control schemes under dynamic load were performed by numerical simulation. Thus, this paper contributes to research on airflow control in PEMFC systems based on FFaES control scheme. The paper is organized as follows. Section 2 briefly presents the issues related to the system modeling and control, and gives details for relations and parameters used in simulation. Section 3 deals with the modeling and control of the compressor. The control of the air flow rate is shown in Section 4 based on aES control scheme proposed here. After a short review of the advantages of the aES control scheme in comparison with the classical ES control schemes, the aES control scheme is detailed. The performances on FC net power obtained using the FFaES and sFF control schemes are shown in Section 5. The comparative simulation results are mentioned for different load power sequences. First, it is shown the availabilities in maximizing the FC net power by regulating the air flow. Second, it is shown the dynamic performances of this selfoptimizing control scheme proposed here, which maximizes FC net power production and improves the system efficiency over all perturbations. The performances of the FFaES control

scheme under different load are clearly highlighted in Section 6. Last Section concludes the paper.

2. Modeling and control of the PEMFC system The PEMFC system is a complex system based on a PEMFC stack that usually is modeled using a 9th [7] or 6th [20] order system. In last decade a lot of improvements related to PEMFC stack modeling are proposed considering a model of one, two or three dimensions [4]. Because the purpose of this paper isn’t the PEMFC system modeling and the AirFr control proposed here (see Figure 1) is of ES control type (so it is not necessary to know or have an accurate model of the process controlled), a generic PEMFC model that is available in Matlab - Simulinkwill be used in all simulations. As it is mentioned above, three main control systems are involved in operating the PEMFC system at high energy efficiency. These systems must be designed to operate the PEMFC system in safe conditions considering all environmental perturbations and the load dynamics. This study is focused on the AirFr control, so only the AirFr control loop is presented in Figure 1. The compressor is supplied from the FC stack. Assuming that the power consumption of the other PEMFC auxiliaries is negligible versus the power consumption of the compressor, the FC net power, Pnet, can be estimated based on (1):

Pnet ≅ PFC − Pmc = VFC ⋅ I FC − Vmc ⋅ I mc

where the voltage and current of the FC stack and compressor (VFC and IFC, and Vmc and Imc, respectively) are measured.

(1)

In practice, the FC net power will be computed directly based on (2):

Pnet = VFC ⋅ I net

(2)

where Inet is the net FC current. The PEMFC system supplies the dynamic load via the controlled power interfaces, considering other renewable energy sources and energy storage devices to fulfill the power balance [33] and protect the FC stack against the sharp profiles of the load power [40].The load power profile is simulated by a current source controlled with different load sequences (Figure 2). To avoid oxygen starvation it is necessary to limit the FC current rate to 40 A/s for a limited step of about 50 A [39, 40]. Consequently, a rate limiter is included in both AirFr and FuelFr regulators used in the sFF control scheme of the PEMFC system 2, which is used as a reference.

Figure 2. The simulation diagram

The hydrogen and oxygen flow rates are regulated to stoichiometrically match the FC current required by the load. Thus, the FC current is a manageable variable to control both AirFr and FuelFr input variables in order to obtain the maximum FC net power [8]. The control of the FuelFr could be made for small PEMFC systems [44] or FC vehicle [24], where recycling the unreacted hydrogen may be unprofitable or impractical.

The sFF control of the FuelFr and AirFr is made based on (3) and (4):

FuelFr =

AirFr =

60000 ⋅ R ⋅ ( 273 + θ ) ⋅ N C ⋅ iFC 2F ⋅ (101325 ⋅ Pf ( H 2) ) ⋅ (U f ( H 2) / 100) ⋅ ( xH 2 / 100) 60000 ⋅ R ⋅ ( 273 + θ ) ⋅ N C ⋅ iFC 4 F ⋅ (101325 ⋅ Pf (O 2) ) ⋅ (U f (O 2) / 100) ⋅ ( yO 2 / 100)

(3)

(4)

where: R = 8.3145 J/(mol K); F = 96485 As/mol; NC represents the number of cells in series (65);

θ - operating temperature (65o Celsius) Uf(H2), Uf(O2)- nominal utilization of hydrogen (99.56%) and oxygen (59.3%); Pf(H2), Pf(O2)- pressure of the fuel (1.5 bar) and air (1 bar); xH2, yO2– composition of fuel (99.95%) and oxidant (21%).

The values mentioned in the brackets are for the preset model of the 6 kW – 45 V PEMFC stack (Figure 3). The maximum FC power, PMPP, delivered at nominal fueling conditions (AirFr = 300 lpm and FuelFr = 50 lpm) is obtained for the FC current (IMPP) of about 130 A (see Figure 3).

Figure 3. The parameters and power characteristics for the 6 kW (bottom) and 1.2 kW (top) FC stacks

To compare the results relative to FC stack power, a preset model of the 1.26 kW – 24 V PEMFC stack is also used in the simulations. The 1.26 kW PEMFC stack has42 cells in series, resulting PMPP at about IMPP = 52 A in nominal fueling conditions (AirFr = 2400 lpm and FuelFr = 12.2lpm). The other parameters are mentioned in Figure 3. A proper design of the control loops needs a FC model that sufficiently represents its dynamics [37]. So, the dynamics of the FC stack and compressor must be considered. The FC time constant (tFC) was set to 2 seconds for both preset models used in simulation. The partial pressures of hydrogen and oxygen follow the FC current changes with delays in range from few seconds to several of hundreds of seconds. The dynamic of the partial pressures 1

pH 2 =

Nc Vanode kH 2 (QHin 2 − 2 I FC ), tH 2 = 1 + tH 2 ⋅ s 4 FU H 2 R ⋅ (273 + θ ) ⋅ k H 2

(5)

1

pO 2 =

kO 2 Nc Vcatode (QOin2 − I FC ), tO 2 = 1 + tO 2 ⋅ s 4 FU O 2 R ⋅ (273 + θ ) ⋅ kO 2

(6)

where: tH2 and tO2 are the hydrogen and oxygen time constants (s); Vanode , Vcathode - volume of the anode and cathode (m3); -1 QHin2 , QOin2 - hydrogen and oxygen input flow (kmol s or l/min);

kH2 , kO2 –hydrogen and oxygen valve molar constants [kmol (atm s)-1]; Also, the thermodynamic time constant is short (the order of minutes), and this will be neglected in this study. The OER value is estimated based on (9) [39]: OER ≡ λO 2 =

3 2 a3 ⋅ I FC + a2 ⋅ I FC + a1 ⋅ I FC + a0 b1 ⋅ I FC + b0

wherea0 = 402.4, a1 = 402.4, a2 = -0.8387, a3 = 0.027, b0 = 1, and b1 = 61.4.

(6)

3. Modeling and controlling the compressor

The dynamic of the compressor is less than that of the FC stack and this is usually neglected in studies of the PEMFC system behavior[21] or is considered as a 2nd order system [39]. Gd ( cm ) =

2 ωcm 2 s + 2ξ cmωcm ⋅ s + ωcm

(7)

2

where tcm = 2π / ωcm and ξ cm will set the time constant and the surge level of the compressor. The static model of the compressor is nonlinear [45] and is usually given through a look-up table. Because the ES control will search the MEP, to prove its functionality it is not necessary to have an accurate model of the compressor. Furthermore, the same model is used for both compressors that feed with air the reference PEMFC system 2 and the PEMFC system 1 under FFaES control. The performance in increasing the FC net power 1, Pnet1, will be given using the FC net power2, Pnet2, as reference and the same FuelFr (see Figure 3).Thus, the errors in modeling the compressor will be canceled in the evaluation of ∆Pnet = Pnet1 − Pnet 2 . So, the static gain will be given by (8) [21]: Gs ( cm ) =

AirFr = K cm Vcm

(8)

The power of the air compressor, Pcm, is computed based on (9) [39]. Pcm = I cm ⋅ Vcm = ( c2 ⋅ AirFr 2 + c1 ⋅ AirFr + c0 ) ⋅

Vcm (max) 100

(d ⋅ I 1

FC

+ d0 )

wherec0 = 0.6, c1 = 0.04, c2 = -0.00003231, d0 = 0.9987, and d1 = 46.02.

4. The aES control of the air flow rate

(9)

The AirFr of the reference PEMFC system 2 is set by the sFF regulator (4) based on the FC current. The AirFr of the PEMFC system 1 is set by the sFF regulator based on the reference current 1, Iref1. This is estimated in the aES control loop considering the pnet = p net ( AirFr ) as a convex map between the control input (AirFr) and the performance output parameter (pnet).

The diagram to draw the net power characteristics is shown in Figure 4 and the results are given in Figure 5 and 6 for the 6 kW and 1.26 kW PEMFC systems. In practice, the optimum AirFr value will be set via the Iref1 current, which is the command variable for the PWM controller of the compressor DC-DC power converter (see Figure 1). In the simulation diagram (Figure 2), the optimum AirFr value will be set directly by the Iref1 current that is generated by the aES control scheme.

Figure 4. The diagram to draw the Pnet = Pnet ( AirFr )

Figure 5. The characteristics Pnet = Pnet ( AirFr ) for different FC currents set to 6 kW PEMFC system

Figure 6. The characteristics Pnet = Pnet ( AirFr ) for different FC currents set to 1.26 kW PEMFC

system

Two important classes of ES control approaches are defined [46]: perturbation-based and modelbased methods. The aES control scheme is from the first class. Despite many favorable properties related to the robustness and convergence, the basic ES control schemes also comes with several drawbacks: (1) the periodic perturbation may produce dangerous peaks during the searching phase; (2) the ripple is high during the tracking phase; (3) if a small dither amplitude is used to reduce that ripple, then the convergence time will increase too much; (4) the tracking accuracy depends by the dither frequency with a factor of O(ω2) [46]; (5) if a small dither frequency is used to improve that accuracy, then the convergence time will increase too much. Thus, the ES designing is system dependent. Consequently, improved ES control schemes were proposed to solve these drawbacks: (1) the peaks are limited by saturation blocks; (2) the dither amplitude decays to zero using an exponential technique [47] or amplitude modulation with the first harmonic magnitude (H1) [13]; (3) H1 is normally big during the searching phase (furthermore, it is not necessary to define a starting criteria for the exponential technique, which is usually too complex or system sensitive); (4) an additional low frequency component of the dither (LF dither, d LF = Am ⋅ sin(ωdl t ) ) could be included with a minimum amplitude (see Figure 1); (5) H1 decays normally to zero during the stationary phase, so the high frequency component of the dither (HF dither, d HF = k2 ⋅ H1 ⋅ sin(ωdht ) ) is negligible and does not affect the stationary accuracy. This accuracy can be improved based on LF dither. It is known that dither shape is not important [23, 48]. So, if the HF dither is a rectangular signal, then the LF dither can be obtained easily by means of a frequency divider.

In conclusion, the system equilibrium, which in this case is a point close to the MEP, is dependent on the amplitude and the frequency of the excitation signal (the dither). Thus, it is important to set these parameters in relation to the dynamics of the system in order to extract the gradient estimation [43]. The diagram of the aES control is shown in Figure 1. The ES control problem can be summarized mathematically as:

Maximize:

(10) Pnet = J ( x, AirFr , I FC )

Subject to: x& = f ( x, AirFr , I FC ), x ∈ X

(11)

where the objective J(x,AirFr, IFC) is the system output function and f is a smooth function describing the dynamics of the FC stack, relating the model states x, control input AirFr, and disturbance input IFC to the objective function value. Note that the objective function can be directly measured in practice based on (2). A high pass filter (HPF) with cutoff frequency ωh isolates the variations of inet variable from its average value. The HPF state variable is denoted by ihpf. This i2 signal is then low pass filtered (LPF) with the cutoff frequency ωl, resulting ibpf signal. During the searching phase, this signal is modulated by the HF dither (which has the same frequency as the first harmonic H1), resulting i3 that contains the required gradient, dJ/d(AifFr). During the stationary phase, the LF harmonics of the ibpf signal are filtered and then are modulated by the LF dither. Finally an integral controller with gain k1 drives this estimated gradient to zero during the searching phase. The equations of the aES control scheme are the following:

pnet = J ( x, AirFr , I FC ) , i1 = inet = pnet / VFC •

•

(12)

iHPF = −ω h iHPF + ωh inet , i2 = i1 − iHPF , iBPF = −ωl iBPF + ωl i2 ,

(13)

i3 = iBPF ⋅ sin(ω dh t ) + (iBPF ∗ L−1{G BPF }) ⋅ sin(ω dl t )

(14)

•

i4 = i3

(15)

i5 = k1i4

(16)

d hf = k2 H1 sin(ωdht ), dlf = Am sin(ωdl t )

(17)

iref = i5 + d hf + dlf

(18)

where equations (12), (13), (14), (15), (16), (17) and (18) represent the FC net power characteristics, the band pass filter, the modulator, the integrator, the gained gradient, the HF and LF dithers, and the current reference. The following notations have been used (see also the Figure 1): k1 is the ES loop gain; k2 - the dither gain; H1 - the magnitude of first harmonic of the FC power; ∗- the convolution operator.

If the band pass filter (BPF) has an enough large frequency band, then the dither persistence is assured. Thus, the cutoff frequencies must to be set at ωl=αlωdh, 3