# Extension Sliding Mode Controller for Maximum Power Point Tracking of Hydrogen Fuel Cells

Authors: Meng-Hui Wang; Her-Terng Yau; Tai-Yuan Wang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Fuel cells (FCs) are characterized by low pollution, low noise, and high efficiency. However, the voltage-current response of an FC is nonlinear, with the result that there exists just one operating point which maximizes the output power given a particular set of operating conditions. Accordingly, the present study proposes a maximum power point tracking (MPPT) control scheme based on extension theory to stabilize the output of an FC at the point of maximum power. The simulation results confirm the ability of the controller to stabilize the output power at the maximum power point despite sudden changes in the temperature, hydrogen pressure, and membrane water content. Moreover, the transient response time of the proposed controller is shown to be faster than that of existing sliding mode (SM) and extremum seeking (ES) controllers.


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## Body

## 1. Introduction

With growing international concern regarding the depletion of the world’s natural resources and the effects of global warming, the need for renewable energy sources has emerged as a critical issue in recent years. Among the various clean energy technologies available, for example, wind power, solar energy, and fuel cells (FCs), FCs are particularly attractive due to their low pollution, virtually silent running, simple maintenance, long service lives, and so on. Moreover, FCs have a generating efficiency of approximately 40–60%; with an overall efficiency of more than 80% being possible through their integration with cogeneration technologies. Thus, as the cost of FC materials continues to decrease, the commercialization of FCs for stationary and standalone applications is becoming increasingly attractive [1, 2].The output voltage of an FC depends on the cell temperature, the air and hydrogen pressure, the oxygen partial pressure, and the membrane water content [3]. Notably, FCs have a nonlinear voltage-current characteristic, and thus there exists just one unique operating point which maximizes the output voltage and power under a particular set of operating conditions. Accordingly, a requirement exists for maximum power point tracking (MPPT) algorithms capable of controlling the stack current and fuel flow in such a way as to maximize the cell efficiency and output power under variable operating conditions [3].Many MPPT methods for photovoltaic systems have been proposed, including Hill-climbing/Perturb and Observe (P&O), incremental conductance, fractional open-circuit voltage, fractional short-circuit current, fuzzy logic control (FLC), neural networks (NNs), ripple correlation control, current sweep, DC-Link capacitor droop control, load current/load voltage maximization, and sliding mode control (SMC) [4]. Such methods provide an effective means of solving the MPPT problem, but vary widely in terms of their complexity, implementation hardware, convergence speed, and sensed parameters [4]. Various MPPT algorithms have been presented for maximizing the output power of FC modules, for example, P&O [5–9], adaptive MPPT control [10], motocompressor control [11], adaptive FLC control [12], resistance matching [13], voltage and current based MPPT [14], and adaptive extremum seeking (ES) control [15].As discussed previously, the FC efficiency and output power are highly sensitive to changes in the operating conditions. Previous studies have proposed reducing the effects of system disturbances on the FC performance by means of sliding mode (SM) [16] or fuzzy sliding mode (FSM) [17] controllers. In the present study, the output voltage of an FC is stabilized at the point of maximum power by means of an SM controller based on extension theory. The performance of the proposed extension sliding mode (ESM) controller is evaluated and compared with that of existing MPPT control schemes by means of Matlab/Simulink simulations.


## 2. Mathematical Model of Fuel Cell

Figure1 presents a schematic illustration of the FC power generation process. As shown, the FC is fed with hydrogen and oxygen, and an electrical current is generated as a result of the subsequent oxidation/reduction reactions at the anode and cathode electrodes, respectively. The relevant chemical reactions are as follows [18, 19]:
(1)Anode:H2⟶2H++2e-,(2)Cathode:O2+4H++4e-⟶2H2O.Figure 1
FC power generation process.In developing the proposed MPPT control scheme, the present study considers the hydrogen FC developed by Ballard Power Systems (Canada). For a single FC, the output voltage is given as(3)VFC=En-Vact-Vohmic-Vcon,
where En is the thermodynamic reversible voltage exported from the FC and has the form
(4)En=1.229-8.5×10-4(T-298.15)+4.31×10-5Tln(PH2PO2),
in which PH2 and PO2 are the hydrogen and oxygen pressures, respectively, and T is the operating temperature.V
act in (3) is the activation overvoltage, that is, the combined cathode and anode activation overvoltage, and is given as
(5)Vact=-(δ1+δ2T+δ3TlnCO2+δ4TlnIFC),
where δn  (n= 1~4) are parametric coefficients, and CO2 is the concentration of dissolved oxygen at the gas/liquid interface and is computed as
(6)CO2=PO25.08×106×e(-408/T).Vohmic in (3) is the ohmic overvoltage, that is, the voltage loss within the FC resulting from the resistance of the various cell components and interconnects; that is,
(7)Vohmic=IFC(RM+RC),
where RC denotes that the resistance contacting and the electron stream is constant and RM is the impedance of the proton exchange membrane; that is,
(8)RM=(ρM×l)A,
in which A is the cell reaction area, l is the membrane thickness, and ρM is the membrane resistance coefficient given as
(9)ρM=181.6[1+0.03J+0.062(T/303)2(J)2.5][G-0.634-3J]exp[4.18(1-303/T)].Note that in (9), G is the membrane water content and J is the current density; that is,
(10)J=IFCA.
Finally, Vcon in (3) is the concentration overvoltage resulting from diffusion-limited mass transfer and is given as
(11)Vcon=-RTnFln(1-JJmax)=-B·ln(1-JJmax),
where R is the universal gas constant, F is Faraday’s constant, Jmax is the maximum current density, and B=RT/nF.For an FC cell stack comprisingN fuel cells, the output power is equal to
(12)Pstack=NVFC×IFC.As shown in (3)–(12), the output power of an FC varies in accordance with changes in the operating temperature, hydrogen pressure, reaction area, and membrane water content, respectively. Figure 2 presents the characteristic curve of the Ballard FC considered in the present study. As shown, in the ideal state, the open-circuit voltage of the FC is equal to approximately 1.244 V for any load current condition. In practice, however, the voltage drops with an increasing load due to a polarization effect. Thus, as described in the previous section, the present study proposes an SM control scheme based on extension theory to stabilize the output power of the FC at the point of maximum power.Figure 2
Characteristic curve of Ballard FC.


## 3. Overall Design of Maximum Power Point Tracking Controller

Figure3 illustrates the functional architecture of the proposed FC/control system. As shown, the main components include the FC, the extension sliding mode (ESM) controller, a DC-DC boost convertor, and a resistive load. In the proposed approach, the output voltage and current signals of the FC are taken as inputs to the ESM controller, which then regulates the DC-DC boost converter in such a way as to maintain the output power of the FC at the maximum power point by means of a pulse-width modulation (PWM) switching scheme [20].Figure 3
Schematic diagram of proposed FC system.The details of the PWM switching scheme are as follows.State 1.  When switch Q is on, the diode D is cut off, and the input voltage flows through the inductor L directly. Thus, the inductive current IL increases and capacitor C is required to supply energy to the load. As a result, current IL flows through the switch to form a loop. The corresponding voltage and current equations are as follows:
(13)I˙L(Q_on)=VFCL,(14)V˙o(Q_on)=-VoCR.State 2.  When switch Q is off, the diode D is switched on. Thus, the energy stored in the inductor L is discharged to the output end and IL reduces toward zero. The relevant current and voltage equations are as follows:
(15)I˙L(Q_off)=VFCL-VoL,(16)V˙o(Q_off)=ILL-VoCR.Combining (13)–(16), the current and voltage equations for the PWM controller are as follows:
(17)I˙L=(U-1)×VoL+VFCL,V˙o=(1-U)×ILC-VoCR,
where U is the duty cycle, Vo is the output voltage, and IL is the inductor current.


## 4. Proposed Extension Sliding Mode Controller

In the sliding mode control (SMC) method, the control rules are designed in such a way that the system slides continuously along a sliding surface comprising the boundaries of multiple control structures. In the MPPT control system developed in this study, the sliding surface is set asdp/dI=0, and the maximum power point is tracked via the appropriate manipulation of the DC-DC converter by the ESM controller [21, 22].


### 4.1. Mathematical Derivation

For the proposed ESM control scheme, the maximum power point occurs whendp/dI=0. In other words, the sliding surface to be controlled is defined as follows:
(18)S=dPdIFC.SubstitutingP=IV into (18) and simplifying, the sliding surface is obtained as
(19)S=dIFCVFCdIFC=VFC+IFC·∂VFC∂IFC,
where
(20)∂VFC∂IFC=∂∂IFCEn-∂∂IFCVact-∂∂IFCVohmic-∂∂IFCVcon,(21)∂∂IFCEn=0,(22)∂∂IFCVact=-δ4TIFC,(23)∂∂IFCVohmic=(RM+RC)+IFC×∂RM∂IFC,(24)∂RM∂IFC=lAC×∂ρM∂IFC,(25)∂∂IFCρM=M×(0.03A+N×2.5×J1.5×1A-O),(26)M=181.6[G-0.634-3(IFC/A)]exp[4.18(1-303/T)],(27)N=0.062×(T303)2(28)O=-3(1+0.03J+N×J2.5)(G-0.634-3J),(29)∂Vcon∂IFC=B×1AJmax-IFC.As described previously, the aim of the ESM controller is to track the maximum power point, that is, to drive the system toward the following sliding surface:(30)S=VFC+IFC·∂VFC∂IFC=0.From Figure3, it is seen that IL=IFC. Thus, the PWM control rule required to satisfy (30) can be expressed as the following (31) for the system dynamics to remain on the switching surface
(31)S˙=∂S∂IFC∂IFC∂t=∂S∂IFC×[(U-1)×VoL+VFCL]=0.In the ideal case,I˙FC=0. Thus, the duty cycle in (31) has the form
(32)Ueq=1-VFCVo.
The duty cycle U must have a value in the range of 0 to 1. Thus, in implementing the proposed ESM control scheme, the duty cycle is derived as follows:
(33)U={1Ueq+ηS≥0Ueq+ηS0<Ueq+ηS<10,Ueq+ηS≤0,
where η>0 is the control gain which is to be determined. In previous studies, the control gain required to minimize steady-state chattering was determined using a fuzzy sliding mode scheme. By contrast, in the present study, the optimal value of η is determined using extension theory.


### 4.2. Proposed Gain Tuning Method

Extension theory was originally proposed by Cai in 1983 and contains two main disciplines, namely, matter-element theory and extension set theory [23, 24].(1)Definition of Matter-Element.  Defining the name of the matter as N, the characteristics of the matter as C, and the value of C as U, the matter-element can be expressed as follows:
(34)R={NC1U1⋮⋮CnUn},
where C=[c1,c2,…,cn] is a characteristic vector and U=[u1,u2,…,un] is the value vector of C.(2)Definition of Extension Set.  Let Ω be the universe of discourse. The extension set A~ on Ω  is then defined as the following set of ordered pairs:
(35)A~={(x,y)∣x∈Ω,y=K(x)∈(-∞,∞)},
where y=K(x) is referred to as the correlation function for extension set A~ and has the form shown in Figure 4.Figure 4
Extended correlation function.As shown in Figure4, K(x) maps each element of Ω  to a membership grade between -∞ and ∞. The higher the degree of the membership grade, the more the number of elements which belong to the set. It is noted that the particular case of 0≤K(x)≤1 corresponds to a normal fuzzy set. Moreover, the condition K(x)≤1 implies that element x has no chance of belonging to the set. By contrast, the condition -1<K(x)<0 represents the extension domain; that is, element x has a chance of belonging to the set.The gain tuning method proposed in this study can be summarized as follows [10].Step 1.
Let the duty cycle of the PWM switch be redefined as(36)U={ηS,S∈EUeq+ηS,S∉E,
where E is the extension field (see Figure 4). Appropriate control laws are designed in the rational region and extension region, respectively, in order to ensure system stability. The extension correlation function and corresponding weights are then calculated in accordance with the defined characteristic values of S and S˙, respectively.Step 2.
Partition the unknown matter element into various grade set characteristics, where the number of grade sets is equal to the number of sets in the classical domain and joint domain; that is,(37)R=(N,Ci,Vi)=[NC1〈a1,b1〉C2〈a2,b2〉⋮⋮Cn〈an,bn〉],
where N is the matter element which is to be divided, Ci are the characteristics of the matter element, and Vi is the characteristic value range, that is, the distribution range of the ith characteristic. Finally, ai indicates the maximum value of the ith characteristic among the matter element grade sets, while bi denotes the minimum value of the ith characteristic.Step 3.
Assign weights to the various characteristics such that(38)∑i=1nλi=1.In the ESM control scheme proposed in the current study, the weightings are determined via the following entropy-based approach; that is,(39)λi=[∑i=1n(1-ς×Vi)]-1,0<ς<1.Step 4.
Calculate the degree of association between the data to be measured in accordance with(40)K(xi)={-ρ(xi,Vi)|Vi|,-1≤ρ(xi,Vi)≤0orρ(xi,Vi)<-1ρ(xi,Vi)ρ(xi,Vt)-ρ(xi,Vi),otherwise,
where
(41)ρ(xi,Vi)=|xi-ai+bi2|-bi-ai2,ρ(xi,Vt)=|xi-ci+di2|-di-ci2.Step 5.
Calculate the relative values of the degree of association of the various grade sets using the normalization equation shown in (42) such that the degree of association of each grade set falls within 〈1,-1〉.
(42)K(q)*=2K(q)-K(q)min-K(q)maxK(q)max-K(q)min.Step 6.
Determine which grade set the matter element belongs to. IfK(q)*=1, the degree of correlation is of the evaluation result of type i. The probability of other set types is determined according to the degree of association. The outcome of this step is used to identify the gain value η which ensures that the output power of the FC tracks the maximum power point.


## 4.1. Mathematical Derivation

For the proposed ESM control scheme, the maximum power point occurs whendp/dI=0. In other words, the sliding surface to be controlled is defined as follows:
(18)S=dPdIFC.SubstitutingP=IV into (18) and simplifying, the sliding surface is obtained as
(19)S=dIFCVFCdIFC=VFC+IFC·∂VFC∂IFC,
where
(20)∂VFC∂IFC=∂∂IFCEn-∂∂IFCVact-∂∂IFCVohmic-∂∂IFCVcon,(21)∂∂IFCEn=0,(22)∂∂IFCVact=-δ4TIFC,(23)∂∂IFCVohmic=(RM+RC)+IFC×∂RM∂IFC,(24)∂RM∂IFC=lAC×∂ρM∂IFC,(25)∂∂IFCρM=M×(0.03A+N×2.5×J1.5×1A-O),(26)M=181.6[G-0.634-3(IFC/A)]exp[4.18(1-303/T)],(27)N=0.062×(T303)2(28)O=-3(1+0.03J+N×J2.5)(G-0.634-3J),(29)∂Vcon∂IFC=B×1AJmax-IFC.As described previously, the aim of the ESM controller is to track the maximum power point, that is, to drive the system toward the following sliding surface:(30)S=VFC+IFC·∂VFC∂IFC=0.From Figure3, it is seen that IL=IFC. Thus, the PWM control rule required to satisfy (30) can be expressed as the following (31) for the system dynamics to remain on the switching surface
(31)S˙=∂S∂IFC∂IFC∂t=∂S∂IFC×[(U-1)×VoL+VFCL]=0.In the ideal case,I˙FC=0. Thus, the duty cycle in (31) has the form
(32)Ueq=1-VFCVo.
The duty cycle U must have a value in the range of 0 to 1. Thus, in implementing the proposed ESM control scheme, the duty cycle is derived as follows:
(33)U={1Ueq+ηS≥0Ueq+ηS0<Ueq+ηS<10,Ueq+ηS≤0,
where η>0 is the control gain which is to be determined. In previous studies, the control gain required to minimize steady-state chattering was determined using a fuzzy sliding mode scheme. By contrast, in the present study, the optimal value of η is determined using extension theory.


## 4.2. Proposed Gain Tuning Method

Extension theory was originally proposed by Cai in 1983 and contains two main disciplines, namely, matter-element theory and extension set theory [23, 24].(1)Definition of Matter-Element.  Defining the name of the matter as N, the characteristics of the matter as C, and the value of C as U, the matter-element can be expressed as follows:
(34)R={NC1U1⋮⋮CnUn},
where C=[c1,c2,…,cn] is a characteristic vector and U=[u1,u2,…,un] is the value vector of C.(2)Definition of Extension Set.  Let Ω be the universe of discourse. The extension set A~ on Ω  is then defined as the following set of ordered pairs:
(35)A~={(x,y)∣x∈Ω,y=K(x)∈(-∞,∞)},
where y=K(x) is referred to as the correlation function for extension set A~ and has the form shown in Figure 4.Figure 4
Extended correlation function.As shown in Figure4, K(x) maps each element of Ω  to a membership grade between -∞ and ∞. The higher the degree of the membership grade, the more the number of elements which belong to the set. It is noted that the particular case of 0≤K(x)≤1 corresponds to a normal fuzzy set. Moreover, the condition K(x)≤1 implies that element x has no chance of belonging to the set. By contrast, the condition -1<K(x)<0 represents the extension domain; that is, element x has a chance of belonging to the set.The gain tuning method proposed in this study can be summarized as follows [10].Step 1.
Let the duty cycle of the PWM switch be redefined as(36)U={ηS,S∈EUeq+ηS,S∉E,
where E is the extension field (see Figure 4). Appropriate control laws are designed in the rational region and extension region, respectively, in order to ensure system stability. The extension correlation function and corresponding weights are then calculated in accordance with the defined characteristic values of S and S˙, respectively.Step 2.
Partition the unknown matter element into various grade set characteristics, where the number of grade sets is equal to the number of sets in the classical domain and joint domain; that is,(37)R=(N,Ci,Vi)=[NC1〈a1,b1〉C2〈a2,b2〉⋮⋮Cn〈an,bn〉],
where N is the matter element which is to be divided, Ci are the characteristics of the matter element, and Vi is the characteristic value range, that is, the distribution range of the ith characteristic. Finally, ai indicates the maximum value of the ith characteristic among the matter element grade sets, while bi denotes the minimum value of the ith characteristic.Step 3.
Assign weights to the various characteristics such that(38)∑i=1nλi=1.In the ESM control scheme proposed in the current study, the weightings are determined via the following entropy-based approach; that is,(39)λi=[∑i=1n(1-ς×Vi)]-1,0<ς<1.Step 4.
Calculate the degree of association between the data to be measured in accordance with(40)K(xi)={-ρ(xi,Vi)|Vi|,-1≤ρ(xi,Vi)≤0orρ(xi,Vi)<-1ρ(xi,Vi)ρ(xi,Vt)-ρ(xi,Vi),otherwise,
where
(41)ρ(xi,Vi)=|xi-ai+bi2|-bi-ai2,ρ(xi,Vt)=|xi-ci+di2|-di-ci2.Step 5.
Calculate the relative values of the degree of association of the various grade sets using the normalization equation shown in (42) such that the degree of association of each grade set falls within 〈1,-1〉.
(42)K(q)*=2K(q)-K(q)min-K(q)maxK(q)max-K(q)min.Step 6.
Determine which grade set the matter element belongs to. IfK(q)*=1, the degree of correlation is of the evaluation result of type i. The probability of other set types is determined according to the degree of association. The outcome of this step is used to identify the gain value η which ensures that the output power of the FC tracks the maximum power point.


## 5. Simulation Results and Discussion

The performance of the proposed extension sliding mode (ESM) controller was evaluated by means of a series of Matlab/Simulink simulations based on the integrated model shown in Figure5. Note that in performing the simulations, the maximum output power of the FC was assumed to be 550 W and the simulation run time was specified as 10 s. In addition, the matter-element module was set as shown in (43).Figure 5
Integrated Simulink model of FC and ESM control system.The characteristic parameters of the simulated fuel cell system and DC-DC converter are shown in Table1.Table 1
Parameters of fuel cell system and DC-DC converter.
Parameter
Value
T (K)
318.15
P
H
2 (atm)
0.8
P
O
2 (atm)
1
Load
10 Ω
Inductance
800 mH
Capacitance
800μFFigure6 shows the simulation results obtained for the output power of the FC given an operating temperature of 25°C. Figures 7, 8, and 9 show the simulated response of the controlled FC given step changes in the temperature, hydrogen pressure, and membrane water content from 25°C to 60°C, 0.5 atm to 10 atm, and 10 to 20, respectively. The results confirm the ability of the proposed ESM control scheme to stabilize the output power of the FC at the point of maximum power despite perturbations in the FC operating conditions. Consider the following:
(43)R=(slidingS〈50,90〉〈20,50〉〈10,20〉〈5,10〉〈1,5〉〈0.5,1〉〈0.1,0.5〉〈0.05,0.1〉〈0,0.05〉).Figure 6
Evolution of FC output power under normal operating conditions.Figure 7
FC output power response given step change in operating temperature.Figure 8
FC output power response given step change in hydrogen pressure.Figure 9
FC output response given step change in membrane water content.Figure10 and Table 2 compare the transient and steady-state performance of the FC when controlled by three different schemes, namely, the ESM scheme proposed in the present study, the conventional SM scheme [21], and an extremum seeking (ES) scheme [15]. It is seen in Figure 10 that the ES method yields a slow transient response and a poor steady-state stability.  Furthermore, it is seen that while the ESM and SM methods both stabilize the output power at the maximum power point, the transient response of the proposed ESM method is faster than that of the traditional SM method.Table 2
Comparison of different control methods.
Stable time (sec)
Steady error
ΔP (W)
ΔV (V)
ΔI (A)
ESM
0.3
0.0
0.0
0.0
SM
0.7
0.0
0.0
0.0
ES
3.3
12.2
2.5
4.34Figure 10
Comparison of control performance of ESM, SM, and ES methods.


## 6. Conclusions

A sliding mode (SM) controller based on extension theory has been proposed for stabilizing the output power of a fuel cell (FC) at the maximum power point. In the proposed approach, the sliding mode is used to track the maximum power point, while extension theory is used to determine the appropriate value of the controller gain which ensures convergence of the controlled system to the sliding surface. The simulation results have shown that the proposed controller successfully stabilizes the output power of the FC at the maximum power point and has a faster transient response time than existing sliding mode (SM) or extremum seeking (ES) control schemes.



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*Source: 371064-2013-07-18.xml*