Control of a direct internal reforming molten carbonate fuel cell system using wavelet network-based Hammerstein models

Journal of Process Control 22 (2012) 653–658

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Short communication

Control of a direct internal reforming molten carbonate fuel cell system using wavelet network-based Hammerstein models Wei Wu a,∗ , Da-Wei Jhao b a b

Department of Chemical Engineering, National Cheng Kung University, Tainan 70101, Taiwan, ROC Department of Chemical and Materials Engineering, National Yunlin University of Science and Technology, Douliou, Yunlin 64002, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 24 August 2011 Received in revised form 8 December 2011 Accepted 19 January 2012 Available online 14 February 2012 Keywords: Hammerstein model Wavelet network Molten carbonate fuel cell Control design

a b s t r a c t Inspired by a direct internal reforming molten carbonate fuel cell (DIR-MCFC) coupled with complicated nonlinear dynamics, the identification and control design of the Hammerstein model is presented. Through the sequential identification procedure, the static nonlinearity block is considered as the wavelet network which is trained and validated by the on-line learning algorithm, and the linear dynamic block is described by the state-space model in which parameters are estimated by the recursive least square algorithm. Using the numerical interpolation technique to approximate the implicit nonlinear function, we present a composite control framework consists of a nonlinear inversion and linear control. Through the closed-loop simulation tests, the nonlinear inversion design for the nonlinearity cancellation of a class of nonlinear systems is validated. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction Fuel cells are an attractive option for power generation directly from fuels, avoiding the inefficient combustion. Regarding the flexible consideration of a fuel cell with respect to the type of fuel, the high operating temperature can enable internal reforming designs such as the molten carbonate fuel cell (MCFC) connected to the internal reforming design [1]. Especially, the direct internal reforming molten carbonate fuel cell (DIR-MCFC) system usually has quite complicated mathematical model due to integrated chemical/electrical reactions [2,3], such that its modeling and dynamic simulation become a challenging task [4]. To explore the identification of fuel cell systems, Li et al. [5] presented the wavelet network dynamic model for the direct internal reforming solid oxide fuel cell (DIR-SOFC). Since the wavelet network possesses both of the characteristics of wavelets and neural networks, it can effectively identify complicated coupling systems [6]. Recently, Hammerstein/Wiener types of model structures coupled with wavelet networks were used to model highly nonlinear system with acceptable accuracy [7]. Huo et al. [8] developed the Hammerstein model, which consisted of a radial basis function neural network in series with autoregressive with exogenous input model, could precisely identify a nonlinear dynamic model of SOFC system. Recently, Yang et al. [9] showed that a complex nonlinear MCFC stack had been identified by a MIMO Takagi-Sugeno fuzzy

model, and Jurado [10] used a fuzzy Hammerstein model to create a predictive control strategy for a SOFC system. Regarding the control strategies for the MCFC systems, Yang et al. [11] proposed a model-based variable structure control (VSC) controller to effectively regulate the stack temperature, Shen et al. [12] presented an adaptive fuzzy control strategy based on the neural network identification to regulate the temperature of MCFC systems, and Sheng et al. [13] used a cascade control scheme with two PID controllers to regulate the spatial temperature difference of a two-dimensional MCFC model. This paper shows that the DIR-MCFC system has a strongly nonlinear behavior. For the purpose of control design, the accurate and simple Hammerstein model structure is first constructed. In our approach, the wavelet network is trained and validated via an online learning algorithm to represent the static nonlinearity block, and the parameters of the state-space model are directly estimated by the recursive algorithm to represent a linear dynamic block. Second, a composite control framework, which consists of a nonlinear inversion and conventional linear controller, is employed. Since the inversion technique is directly derived from the numerical interpolation function which is denoted as the approximation of the input–output map of the static nonlinearity. Furthermore, the performance of the composite control strategy is verified through simulations. 2. Wavelet network-based Hammerstein model

∗ Corresponding author. E-mail address: [email protected] (W. Wu). 0959-1524/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2012.01.011

The structure of the wavelet network-based Hammerstein model, depicted in Fig. 1, consists of a wavelet network block

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W. Wu, D.-W. Jhao / Journal of Process Control 22 (2012) 653–658

linear dynamic G is described by the discrete state-space model with matrices A, B, C, and D,

Nomenclature Acell Cps hj

Cp F h¯ in i

h¯ in j istack N˙ ain , N˙ cin Na , Nc ms mhj ˙ hj m ncell R Tamb in Thj out Thj UA

effective cell area, cm2 heat capacity of average stack, J/g/K

x(k + 1) = Ax(k) + Bv(k) y(k) = Cx(k) + Dv(k)

heat capacity of jacket, J/g/K Faraday constant (=96,485 C/mol) inlet specific enthalpy of species i into the anode, J/mol inlet specific enthalpy of species j into the cathode, J/mol current demand, A inlet molar flow rate at the anode/cathode, mol/s total moles at the anode/cathode, mol total stack mass, g mass of steam in the jacket, g mass flow rate of steam in the jacket, g/s number of cells in the stack universal gas constant (=8.314 J/mol/K) ambient temperature, K the inlet steam temperature

where y(k) is the output of the Hammerstein model at the kth sampling instant. Thetransformed input v(k) = f (u(k)). Sincethe set of parameters,  = wi , aij , bij , f¯j  i = 1, . . . , N; j = 1, . . . , m , in the wavelet network is undetermined, the identification algorithm is used to find the optimal parameters, i.e., the minimization of the following objective function should to be satisfied.

Vstack in xa,i

the outlet steam temperature product of the overall heat transfer coefficient and heat transfer area, W/K stack voltage, V inlet mole fraction of species i into the anode

in xc,j

inlet mole fraction of species j into the cathode

J1 () =

(3)

1 1 2 2 E(k) (yss (k) − v(k)) = 2 2 Nt

Nt

k=1

k=1

(4)

subject to the steady state model ss (k) = F(u(k)) yss (k) = h(ss (k)),

(5)

k = 1, 2, ...

where  ss (k) and yss (k) represent the steady-state value of the system state and output at the kth sampling instant, respectively. the learning procedure Regarding    for wavelet networks, the data set (u(k), yss (k)) k = 1, . . . , Nt is first used to train the wavelet







network, and the new data set (u(k), yss (k)) k = Nt + 1, . . . , Nt is used to verify the trained wavelet network. Moreover, the optimization algorithm is performed by a gradient-based method [14] such that the derivative of the objective function with respect to  is shown by

 ∂v ∂J1 =− E ∂ ∂ Nt

(6)

i=1

Hence the updating laws of parameters wi , aij , bij , and f¯j (i = 1, . . ., N ; j = 1, . . ., m) are derived as follows: (i) weights

∇ wi = −

∂E = Ei (u(k)) ∂wi

(7)

(ii) translations

∇ bij = − Fig. 1. Wavelet network-based Hammerstein model.

∂E w = E i i (u(k)) aij ∂bij

(8)

uj − bij ∂E = Ewi i (u(k)) ∂aij a2

(9)

(iii) dilations followed by a linear dynamic block. The identification of the wavelet network is assumed as the static nonlinear function f, i.e.

f (u(k)) =

N 

ij

(iv) bias wi i (u(k)) + f¯

(1)

i=1

where u(k) is the input of the Hammerstein model at the kth sampling instant, the additional parameter f¯ is denoted as a bias to deal with nonzero mean function on prescribed domain, and  represents the multi-dimensional wavelet by using the product of the number of m scalar wavelets

i (u(k)) =

∇ aij = −

  m  uj − bij j=1

aij

(2)

∇ f¯ = 1

(10)

where  is the learning rate. Moreover, the network parameters are on-line updated at each sampling instant shown by k+1 = k −

∂J1 ∂

(11)

Regarding the identification of linear dynamic G, another objective function J2 with respect to ϑ is described by

2 1  yp (k) − G(q)v(k) Nt Nt

J2 (ϑ) =

(12)

k=1

Notably, parameter bi and ai are the translation and dilation vectors, respectively. Regarding the wavelet network, the multidimensional wavelet  is the activation function and the parameter wi (i=1, . . ., N) is the weight of the hidden layer. Moreover, the

subject to the process model x(k + 1) = F(x(k), u(k)) yp (k) = h(x(k))

(13)

W. Wu, D.-W. Jhao / Journal of Process Control 22 (2012) 653–658

yp is the process output and the discrete-time transfer function of state-space model G(q) is written as G(q) = C(qI − A)−1 B + D =

˛(q) ˇ(q)

Moreover, the nonlinear inversion f−1 is approximated by the following function



(14) (v(k))

=

where the polynomials ˛(q) and ˇ(q) are ˛(q) = qn + ˛1 qn−1 + · · · + ˛n−1 q + ˛n ˇ(q) = qm + ˇ1 qm−1 + · · · + ˇm−1 q + ˇm

minimizing the objective function J2 . Therefore, an iterative algorithm is used to obtain a solution, i.e. ϑi+1

∂J2 = ϑi −  ∂ϑ

(16)

where  is a positive-definite matrix. To carry out an effective iteration procedure, the Levenberg–Marquardt algorithm in Matlab® is usually adopted.

Through above training and validation algorithms, if the identification of the wavelet network-based Hammerstein model is complete, then the following nonlinear inversion is expected to cancel the static nonlinearity u(k) = f −1 (v(k))

(17)

such that the linearized model is obtained. If the external input v is denoted as the digital PID control,

v(k) = PID (e(k), e(k − 1), . . .) = Kc ⎣e(k) +

k t 

I

e(j) +

⎤

D (e(k) − e(k − 1))⎦ t

(18)

j=1

Then the composite control, u(k) = f−1 ◦ PID (e(k), e(k − 1), . . .), is obtained, where (Kc , I , d ) are tuning parameters to ensure the stable output regulation. t is the sampling period, and e(k) is the feedback error at the kth sampling instant. In our approach, the controller tuning will follow the continuous cycling method shown in [15]. Remark 1. In continuous time setting of PID controller, the continuous cycling method for PID controller tuning is easily determined empirically to verify closed-loop responses that have a 1/4 decay ratio. For a small sampling period, the suitable controller settings obtained for a continuous controller can be directly utilized in a digital PID controller. Although the static nonlinearity represented by the wavelet network framework is vague, the input–output map between input u and v can be directly evaluated by simulation tests. To address the simple and explicit nonlinear function, the numerical interpolation technique is used to approximate the nonlinear mapping. In our approach, the cubic spline technique is adopted to fit the input–output profile from the wavelet network. The spline function is constructed by connecting third-order polynomials for each interval between knots as represented by

v(k) = ai u(k)3 + bi u(k)2 + ci u(k) + di , i = 1, 2, ...n

3

 3

 i +

i +

v(k) 2ai

v(k) 2ai

+

 −





i +

i +

v(k)

2

2ai

v(k) 2ai

2

+ (ri − p2i )

3

,

(20)

3

+ (ri − p2i ) + pi

where vi ≤ v(k) < vi+1 , i=1, 2, . . ., n, and pi = −

bi 3ai

i = p3i + ri =

ci 3ai

bi ci − 3ai di 6a2i

(21)

When the discrete PID controller is added, the composite control law with the aid of the above nonlinear inversion is synthesized. 4. Control of a non-isothermal fuel cell model

3. Controller design

⎡

+

(15)

the set parameters, ϑ= Similarly,    of ˛i , ˇj  i = 1, . . . , n; j = 1, . . . , m , need to be determined by

655

ui ≤ u(k) < ui+1 , (19)

Notably, n data points with n − 1 intervals induce 4(n − 1) unknown coefficients in Eq. (16). 4(n −   Under specified 1) conditions, all unknown constants, ai , bi , ci , di  i = 1, 2, ..., n , can be evaluated.

For the control of a non-isothermal nonlinear process, a kWclass of a DIR-MCFC system is illustrated. Referring to the previous work [16], the mathematical model of a DIR-MCFC system is described by Eqs. (A1)–(A11) in Appendix A. Moreover, the corresponding state space representation is shown by ˙ = f (, u) y = h()

(22)

where  = [xa,H2 , xa,CH4 , xa,CO , xa,CO2 , xa,H2 O , xc,CO2 , xc,O2 , T, Thw,out ]T is the state variable, the air flow rate at the cathode N˙ cin is treated as the manipulated input u, and the controlled variable y is the stack temperature. Obviously, the nonisothermal DIR-MCFC system is a highly nonlinear process. To address the wavelet networkbased Hammerstein model and its control implementation, the wavelet network-based Hammerstein model is first constructed under the prescribed inlet conditions, e.g. 3.01 g/s ≤ Ncin ≤ 4.3 g/s, istack = 200 A, and Nain = 1 m/s. When the numbers of sampling, Nt = 926, are used to identify and new samplings, Nt = 185, are applied for validation, the identification algorithm and control design are stated as follows. (i) Regarding the static nonlinearity part, the input–output map of u(k) and v(k) is depicted in Fig. 2(a) in which weights of wavelet networks are estimated by an iteration algorithm. Since this input–output map consists of 77 data points with 76 intervals, the cubic spline function is used to precisely fit the curve. By Fig. 2(a), the effect of approximation is quite good. Using the approximation of nonlinear inversion by Eqs. (20) and (21), the inversion of the input–output map is depicted in Fig. 2(b). (ii) Regarding the linear dynamic part, the Matlab toolbox is utilized to find the order and the parameters of the discrete-time state-space model. Moreover, the discrete-time transfer function is shown by 0.02694q2 + 2.126 × 10−5 q + 1.644 × 10−8 q3

+ 0.008824q2 + 2.01 × 10−5 q + 7.254 × 10−9

(23)

Notably, the wavelet network-based Hammerstein model is stable in regard to the prescribed operating conditions and constraints. Consequently, the wavelet network-based Hammerstein model is depicted in Fig. 3(a), when the input pattern of air flow rate is shown   in Fig. 3(b).  (iii) When the constants, ai , bi , ci , di  i = 1, 2, ..., 77 , are directly evaluated from the cubic spline function which have been

656

W. Wu, D.-W. Jhao / Journal of Process Control 22 (2012) 653–658

(a) 475

950 our work PID

I/O map

470

set point

cubic spline

930

Temperature(K)

465

v

460 455

910

890

450 445

0.65

0.7

0.75

0.8

u

0.85

0.9

870 950

1000

1050

1100

1150

Time(s)

0.9

(b)

Fig. 4. Responses of stack temperature using the composite control (our work) and the PID control for the setpoint tracking.

0.85

u

0.8

0.7

(a)

900

Temperature(K)

0.75

895

our work PID

0.65 445

450

455

460

465

470

475

v Fig. 2. (a) The input–output map of wavelet network vs. the numerical approximation; (b) nonlinear inversion. 940

(a)

890

Hammerstein DIR-MCFC

Temperature(K)

920

885 0.995

900

880

1

2

3

4

5

Time(s)

6

7

8

9

4

1 0.95 0.9

air flow rate (g/s)

1.01

1.015 4

x 10

PID our work

10

x 10

Temperature(K)

(b)

0

1.005

Time(s)

(b) 900

860

840

1

(a)

0.85

895

890

0.8 0.75 0.7

885 0.995

0.65 0

1

2

3

4

5

Time(s)

6

7

8

9

1

1.005

Time(s)

1.01

1.015 4

x 10

10 4

x 10

Fig. 3. A wavelet network-based Hammerstein model: (a) output responses of plant and model; (b) corresponding input pattern of air flow rate at the cathode.

Fig. 5. Responses of stack temperature using the composite control (our work) and the PID control for disturbance rejection while (a) +20% istack , (b) −20% istack .

W. Wu, D.-W. Jhao / Journal of Process Control 22 (2012) 653–658

shown in Fig. 2(a), and the approximation for the nonlinear inversion by Eq. (20), shown in Fig. 2(b), is constructed. Therefore, a composite control consists of the static nonlinear control law, u(k) = (v(k)) and a discrete PID control, v(k) = PID (e(k), e(k − 1), . . .). (iv) Referring the open-loop test in Fig. 3, it implies that the DIR-MCFC system is an open-loop stable system. Using the continuous cycling tuning procedure with Ziegler-Nichols settings, three PID controller parameters of the closed-loop system are easily determined a priori. Based on fixed tuning parameters, Kc = 0.2328, I = 1, and

d = 0.25, Fig. 4 shows a comparison of the setpoint tracking while the composite control, u(k) = (PID (e(k), e(k − 1), . . .)), and the traditional PID control, u(k) = PID (e(k), e(k − 1), . . .), are employed, respectively. The tracking performance by use of the composite control is better than the PID control. It implies that the design of nonlinear inversion can reduce the effect of system nonlinearities. Moreover, Fig. 5 shows that the composite control can ensure the better output regulation than the PID control while ±20% changes in the electrical load are considered. Notably, the traditional PID control with conservative tuning parameters would cause undesired oscillation, however the composite control with the same tuning parameters can carry out the nonlinearity cancellation to improve the control performance.

657

and the mole balance equations for the electrochemical half reactions at the cathode are









in x˙ c,j = (Nc )−1 N˙ cin xc,j − xc,j + 1.5xc,j re + Rc,j ,

j ∈ Sc ≡



CO2 , O2



(A2)

where Ra = [3rR + rW − re , − rR , rR − rW , rW + re , − rR − rW ] and Rc = [−re , −0.5re ]. The Faraday’s law of electrolysis re = istack Acell ncell /2F, and the rate equations rR and rw are shown as, respectively 3.71 × 1017

rR =

g(xa,i , T )2

e(−240.1/RT ) ×



xa,CH4 xa,H2 O 2.5 xa,H

P

−0.5

rW =



−



5.43 g(xa,i , T )2

e

(−67.13/RT )



2

30.42− 2.7×104 /T

1.03 × 1010 e

2



0.5 xa,H xa,CO P 1.5

xa,CO xa,H2 O xa,H2

P−



(A3)



xa,CO2 P 3

e[−3.80+(4.2×10

/T )]

(A4) where g(xa,i , T ) = 1 + 8.23 × 10−10 e(89.2/RT ) xa,CO P + 6.12 × 10−14 e(70.65/RT ) xa,H2 P + 6.65 × 10−9 e(38.28/RT ) xa,CH4 P

5. Conclusions

+ 1.77 × 105 e(−88.68/RT )

This paper presents the identification of the Hammerstein model for a DIR-MCFC system where the static nonlinearity block is considered as the wavelet network and the linear dynamic block is described by the state-space model. In our approach, the wavelet network is trained and validated via an on-line learning algorithm to represent the static nonlinearity block, and the parameters of the state-space model are directly estimated by the recursive algorithm to represent a linear dynamic block. For the purpose of control design, a composite control is composed of a nonlinear inversion and conventional linear controller. The nonlinear inversion technique is directly replaced by the numerical approximation of the input–output map of the static nonlinearity and a discrete PID control is treated as the linear controller. Through the closed-loop simulation tests of the setpoint tracking and disturbance rejection, the nonlinear inversion design can improve the control performance due to the effect of the nonlinearity cancellation.

(ms Cps + N˙ ain dT = dt



xa,i h¯ i + N˙ cin

i ∈ Sa

⎣N˙ ain





+N˙ cin

The authors would like to thank the National Science Council of the Republic of China for financially supporting this research under contract no. NSC 99-2221-E-006-256. out dThj

dt

=

xc,j h¯ j )

j ∈ Sa



in ¯ in xa,i hi − xa,i h¯ i −

i ∈ Sa

−



in ¯ in hj − xc,j h¯ j xc,j

xa,i h¯ i



i ∈ Sa

i ∈ Sa

−

xc,j h¯ j

  ⎤

j ∈ Sc







i ∈ Sa ≡





H2 , CH4 , CO, CO2 , H2 O



˙ hj m mhj



in out Thj − Thj −

Rc,j

(A6)

Qheat

(A7)

hj

mhj Cp

in and T out represent the inlet and outlet steam temperawhere Thj hj ture, respectively. The heat capacity of steam is formulated by



out out Cp = 7.256 + 2.298 × 10−3 Thj + 2.83 × 10−9 Thj

2

(A8)

the specific enthalpies of species at the anode and cathode are written by



T

ref h¯ k = h¯ k +

(A1)



T − Tamb − Vistack + Qheat ⎦ Rt



in x˙ a,i = (Na )−1 N˙ ain xa,i − xa,i − xa,i (2rR + re ) + Ra,i ,

Ra,i

i ∈ Sc

hj

(i) the mole balance equations for the electrochemical half reaction at the anode are written as

(A5)

P

T

⎡

Acknowledgment

Referring the nonisothermal dynamics for a kW-class of a DIRMCFC is composed of:

xa,H2

(i) the thermal models for the stack and the heating jacket are described as

j ∈ Sc

Appendix A.

xa,H2 O

Tref





ak + bk + ck 2 d ,

k∈



Sa , Sc



(A9)

658

W. Wu, D.-W. Jhao / Journal of Process Control 22 (2012) 653–658

Table 1 Heat capacity coefficients for species from 298 K to 1500 K at constant pressure. Species

h¯ ref (kcal mol−1 )

a (cal K−1 mol−1 )

b × 10−3 (cal K−1 mol−1 )

c × 10−8 (cal K−1 mol−1 )

H2 O2 H2 O CO2 CO CH4

0.000 0.000 −57.800 −94.054 −26.417 −17.895

6.947 6.148 7.256 6.214 6.420 3.381

−0.200 3.102 2.298 10.396 1.665 18.044

0.481 −0.923 0.283 −3.545 −0.196 −4.300

and the heating heat transfer rate between the stack and heating jacket is shown by Qheat = UA

in − T ) − (T out − T ) (Thj hj

(A10)

in − T )/(T out − T )) ln((Thj hj





Notably, h¯ k , k ∈ Sa , Sc , can be evaluated from Table 1. Furthermore, the stack voltage under a specified current demand is expressed by Vstack =

2.44 × 105 − (472.45 + 0.04T ) T 2F +

0.5 xa,H2 xc,CO2 xc,O RT 2 0.5 ln P 2F xa,CO2 xa,H2 O



− istack 0.5 × 10−4 e[3016((1/T )−(1/923))] −0.42 −0.17 −1 +2.27 × 10−9 e(6435/T ) xa,H xa,CO xa,H 2

2O

2

P −1.59

−0.43 −0.09 −0.52 +7.51 × 10−10 e(9298/T ) xc,O xc,CO P 2

2

 (A11)

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