Constrained Explicit Predictive Control Strategies for PEM Fuel Cell ...

Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec. 12-14, 2007

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Constrained Explicit Predictive Control Strategies for PEM Fuel Cell Systems A. Arce, D.R. Ram´ırez, A.J. del Real and C. Bordons

Abstract— This paper presents the development of an explicit predictive control strategy for a stand-alone PEM (Polymer Electrolyte Membrane) fuel cell. This fuel cell can be considered as a good benchmark since it is representative of the state of the art of PEM technology and is used by many research groups. The experiments are performed on a detailed nonlinear simulator of a fuel cell which has been validated experimentally. In order to achieve real-time implementation of the control strategy, the predictive control algorithm must be computed in a explicit way because of the sampling time of this system, which is in the order of milliseconds. The work shows the development and simulation results of the constrained explicit predictive controller that reduces the computational effort needed.

I. INTRODUCTION Fuel cells have been developed considerably in the recent years. Although they were invented more than a century ago, they have received a great deal of attention in the last decades as good candidates for clean electricity generation both in stationary and automotive applications. There are many unresolved issues regarding materials, manufacturing and maintenance, automatic control being one of the most important. There are many types of fuel cells, this work being focussed on PEM (Polymer Electrolyte Membrane) cells, which run at low temperature and show fast dynamical response, which make them suitable for mobile applications. Good performance of these devices is closely related to the kind of control that is used, so a study of different control alternatives is justified. Fuel cells can operate stand-alone or integrated with other devices like DC/DC or DC/AC converters which provide the power source that supplies electricity to a load or to the grid, or batteries and ultracapacitors which supply current peaks to improve the transient states. Therefore, the control design depends on the system topology. Many control strategies have been proposed in literature, ranging from feedforward control [5],LQR [5],[7], Neural Networks [8], [9] or Model Predictive Control [11]. However, most manufacturers use simple control strategies such as static feedforward or PI controllers. The exact details of these control strategies usually are not publicized by the manufacturers. This work is focused on a fuel cell operating stand-alone, analyzing the operation of the devices which supply the reactant flows in order to regulate the fuel cell system, specifically addressing the air pump voltage control while Work partially supported by the Spanish Ministry of Science and Technology under Grant DPI 2004-07444-c04-01 and European Commission under grant 511368 (Hycon Network of Excellence). All the authors are with Escuela Superior de Ingenieros, Departamento de Sistemas y Automática, University of Seville, 41092 Camino de los Descubrimientos, Spain. Corresponding Author: Alicia Arce Email address: [email protected]

1-4244-1498-9/07/$25.00 ©2007 IEEE.

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the fuel cell works in dead-end mode with a proportionallycontrolled hydrogen feeding. The hydrogen valve control is used to manipulate anode pressure according to cathode pressure. In this fuel cell, power and nominal temperature are not very high, so an air cool fan can be used to refrigerate the fuel cell. The purge valve is controlled to avoid the flooding phenomenon which decreases the stack voltage. Model Predictive Control (MPC) seems to be a good candidate to control fuel cells and its use has been proposed and tested on simulation in in [1] and [3]. However, the real-time implementation of the basic GPC (Generalized Predictive Control) formulation [4] presents some problems associated to the computational burden needed to solve the optimization problem on-line. The main contribution of this paper is the development of a constrained explicit predictive controller [6] to regulate the air flow acting on the air pump voltage. The constrained explicit predictive control algorithm reduces the computational on-line effort moving it to previous off-line computation, reducing this way the execution time. As a result of that, it is feasible for this kind of system with fast characteristic time. Furthermore, different control structures are presented for the three control strategies analyzed: Maximum efficiency, starvation prevention and voltage tracking. The paper is organized as follows: the fuel cell model and the control model are described in sections II and III. Section IV analyzes the control strategies. The constrained explicit predictive controller is presented in section V. Section VI shows the results obtained with the controller. Finally, the main conclusions are drawn in section VII. II. N ONLINEAR M ODEL In this work a nonlinear model of a real PEM fuel cell is used as a simulator. This model combines theoretical equations and experimental relations, resulting in a semiempirical formulation. It describes the following areas: fluid dynamics in the gas flow fields and gas diffusion layers (oxygen, nitrogen, liquid water and vapor); thermal dynamics and temperature effects; and a novel algorithm to compute an empirical polarization curve. As a result, this model can predict both steady and transient states due to variable loads (such as flooding and anode purges), as well as the system start-up. The model is extensively described in [2] and a Simulink model is available on request to the authors. In the following a brief description of the model components and results is given. The model parameters have been tuned to mimic a 1.2 kW Ballard PEM fuel cell (Nexa Power Module,see figure 1), which can be considered as a benchmark since it is widely

46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007

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used by many research groups worldwide and is representative of state–of–the–art PEM technology. The benchmark is equipped with a controller which overtakes the control and safety tasks. The stack is composed of 46 cells with a 110cm2 membrane each one. The system is auto-humidified and air-cooled by a small fan. As for the hydrogen feeding of the fuel cell, a dead-end mode with flushes was adopted. Also, a PC was used for the acquisition of the measured values and, in order to simulate a variable power demand, the energy produced was delivered to an electronic load.

1 Experimental data Fuel cell voltage (V) 0.9

0.8

0.7

0.6

0.5

0.4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

−2

Current density (A cm

Fig. 3.

Fig. 1.

)

Polarization curve and experimental data

The simulated results obtained with the model have been compared to experimental data from the Ballard stack demonstrating the accuracy of the proposed model methodology. Figure 4 shows the comparison of the non-linear model and the experimental data, both electrical variables (voltage) and thermal variables (stack temperature). As it can be seen, the model fits the dynamic quite correctly. Therefore the complete model can be used to design and test controllers before being implemented on a real fuel cell (this is specially relevant if it is taken into account that commercial fuel cells cannot usually be manipulated as they are sold, thus their control strategies cannot be changed).

1.2 kW Nexa power module

50

Stack Voltage (V)

40

20 10 0

Simulated Data Experimental Data 0

500

1000

0

500

1000 time(s)

320 315 310 305 300 Simulated Data Experimental Data

295 290

Fig. 4.

Fig. 2.

1500

325

Stack Temperature (K)

Fuel stacks are formed by numerous fuel cells connected in series, but in order to supply energy, several auxiliary devices are required. These devices vary according to the application they are designed for and fuel cell stack size. The non-linear model includes not only the stack itself but also the models of each auxiliary device. Figure 2 presents the fuel cell system scheme with the auxiliary equipment such as air pumps, fuel valves and cooling fans.

30

1500

Validation results of the non-linear model

III. C ONTROL MODEL The non-linear model used as the complete simulator is too complex to act as a control model. Thus, a simplified linear discrete model is needed to design the controller. This model will be obtained by linearization of the nonlinear model around an operation point. The model structure (i.e. the input and output variables) must be selected before the linearization. Particulary, the system has two modes of operation, depending on the control strategy (see figure 5).

Fuel Cell system scheme

The key to model the fuel cell is the polarization curve which is represented in figure 3. It is calculated from experimental data obtained from the fuel cell benchmark.

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Fig. 5.

Block diagram of the two cases studied


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WO2 ,in WO2 ,reacted

(1)

where WO2 ,in is the input steam of oxygen and WO2 ,reacted is the oxygen which has reacted inside the stack. The critical value for λO2 is set to 2. The hydrogen valve has not been considered as an input since it is assumed to be controlled by a proportional controller. The target of this controller is to maintain the anode pressure constant. Furthermore, the influence of the air supply controller is more noticeable than the hydrogen supply controller. Regardless of the case, the control system is affected by a measurable disturbance which is the required stack current. This measurable disturbance is considered as a non manipulable input. In the second operation mode (see figure 5), the system has the air pump voltage as an input and the stack voltage as an output. Also, the required stack current appears as another input, acting again as a measurable disturbance. Also, a suitable operation point must be selected before the linearization. After the study described in section IV, the optimal oxygen excess was set above 4 because it is the value nearest to the optimal point for all the feasible current range. The operating current is 15 A as it is the mean value of the whole range. This point is attained at an air pump voltage Vcp of 69.91%. Taking all these considerations into account, two linear models, one for each operation mode, have been obtained. The continuous linear model for the first case is: 0.012179s2 + 0.018025s + 0.7242 s3 + 3.45s2 + 7.324s + 5.745 −0.26586s − 0.001432 GIst ,λO = 2 s + 0.005318

GVcp ,λO = 2

40

Non−linear Discrete linear

35

30 20

30

40

50

60 Time (s)

70

80

90

100

10 Non−linear Discrete Linear

8 6 4 2 0

0

20

40

60

80

100

Time (s)

Fig. 6.

Comparison of the non-linear model with discrete-time linear one

IV. C ONTROL STRATEGIES

(2) (3)

Note that there is a transfer function from each input (air pump voltage and demande stack current) to the output (oxygen excess ratio). For the second case, the linear model obtained is: GVcp ,Vst =

Figure 6 compares the non-linear model with the discrete linear model for both operation modes. Note that the control models perform qualitatively close to the simulation models. The discrepancies between the non-linear and control models can be assumed for control purposes. Also these discrepancies reflect the real ones that should be faced by model based controllers applied to the fuel cell benchmark.

Stack Voltage (V)

λO2 =

be completed in a small amount of time. This makes very difficult the task of implementing certain advanced control techniques such as constrained predictive control.

Oxygen Excess

The first case has air pump voltage as an input and oxygen excess as an output. The variable called oxygen excess is related to the amount of oxygen inside the fuel cell stack and it is relevant because of the starvation. Starvation is an uncontrollable phenomenon that can destroy the stack if it is ignited, and is linked to the excess oxygen. Starvation appears if this variable decreases down a critical value. The oxygen excess is equal to:

2.896 · 10−7 s4 + 0.02302s3 + 0.4146s2 + 2.42s + 4.24 s5 + 15.73s4 + 56.27s3 + 118.4s2 + 118.6s + 37.7 (4) −0.2805s − 0.379 (5) GIst ,Vst = s − 1.004

After the linearization a discrete time model has been obtained. The fastest characteristic time of the transfer functions presented in (2) - (5) is 0.0886 seconds. Applying the rule of thumb of choosing a sampling time ten times lower than the fastest dynamic the sampling time was chosen to be 0.008 seconds. Note that the sampling time demands that all the computations required to obtain the control signal must

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This section presents a study of the steady state curves of the fuel cell benchmark where the trajectories of the different control targets are included. The steady state curves represent the variation of the net power that is the stack power minus the air pump power consumed against the variation of oxygen excess. This study takes in account three different control strategies: 1) Starvation prevention, 2) Maximum efficiency, and 3) Voltage control. The first one, starvation prevention, controls the level of the oxygen excess over a critical value set to 2. The second one, maximum efficiency, has the objective of supplying the maximum net power in each current demand.The last one, voltage control, tracks a voltage reference. Figure 7 shows the steady-state net power curves. They have the same shape as the efficiency curves, although their values are scaled, as every curve has to be divided by the value of the hydrogen energy. This value is the same for each curve. The different steady-state control trajectories are ploted in the same figure so that they can be compared.


FrC09.1 V. F ORMULATION OF THE EXPLICIT PREDICTIVE

1200

CONTROLLER 1000

Pneta (W)

800

In this section the formulation for the explicit predictive controller is presented. It is based on the GPC formulation (Generalized Predictive Control) ([4]). The objective of any predictive controller is to compute the future control sequence uk , uk+1 , . . . , ut+N−1 in such a way that the optimal j-step ahead predictions yk+ j|k are driven close to the set point sequence wk , wk+1 , . . . , wt+N−1 for the prediction horizon. The j-step ahead predictions are computed using a CARIMA prediction model with an extra input for a measured disturbance v:

Air Pump Saturation 100%

600

400

200

0

1

2

3

4

5 λO

6

7

8

9

2

Fig. 7.

Steady-State curves and trajectory of the control strategies

The magenta plot represents the steady state trajectory for the voltage control criterion. The stack voltage range, without breaking the critical value, is small. The trajectory plot corresponds to reference voltage of 32 V . For low required stack currents, the oxygen excesses are quite near to the critical point. With this control target, suitable values of λO2 cannot be assured for the entire current range. The black curve tracks the starvation prevention trajectory. Notice that the oxygen excess is in a safe position in steady-state, so the starvation phenomenon is effectively prevented. The red plot is the maximum efficiency trajectory. This control strategy assures at the same time that it supplies the maximum net power and stays in the safe starvation area, as can be seen in figure 7. All the strategies can be implemented with a predictive controller, although with some variations in the control loop structure. The required schemes are shown in figure 8. The maximum efficiency criteria requires a trajectory generator which interpolates the curve shown in figure 7. The only difference between the other two are the controller output. In the case of starvation prevention, the output is the oxygen excess and in voltage control it is the stack voltage.

1 A(z−1 )yk = B(z−1 )uk−1 + D(z−1 )vk + C(z−1 )ek Δ

(6)

where Δ = 1 − z−1 . The polynomials for the oxygen excess control case are obtained from the discrete transfer functions obtained from equations (2) and (5). The system output is the oxygen excess (λO2 )1 , the system input is the air pump voltage in percent units (Vcp ) and the measurable disturbance corresponds to the stack demand current (Ist ). The way in which the system will approach the desired trajectories will be indicated by a function J which depends on present and future control signals and disturbances: N2

Nu

j=N1

j=1

J = ∑ (yk+ j|k − wk+ j )2 + λ ∑ (Δuk+ j−1 )2 where: N1 and N2 define the beginning and end of the cost horizon, Nu is the control horizon and yk+ j|k is the output prediction for time k + j made at time k. The control signal is assumed to be constant after the control horizon and the future measured disturbances are supposed to be constant and equal to the last measured value. The can be grouped in a vector j-step ahead predictions y = yk+N1 |k · · · yk+N2 |k that can be computed from the following prediction equation (see [4]): y = Gu + Fx x

(7)

where G ∈ ℜN×Nu , u = [Δuk · · · Δuk+Nu −1 ] , Fx ∈ ℜN×dimx , N = N2 − N1 and x ∈ ℜdimx . The first term Gu represents the forced response and the second Fx x the free response (see [4]). The parameter vector x contains present and past values of yk and past values of Δuk and Δvk from which the free response of the system depends. In the context of a CARIMA model vector x represents the process state. For the fuel cell control model this vector has the form: x = [yk · · · yk−4 Δuk−1 · · · Δuk−3 Δvk−1 · · · Δvk−4 ]

(8)

With prediction equation (7) the cost function can be rewritten as: J(u, w, x) = (Gu + Fx x − w) (Gu + Fx x − w) + λ u u

Fig. 8. Top to bottom: Starvation Prevention, Maximum efficiency and Voltage tracking control schemes.

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(9)

1 The formulation of the controller for the stack voltage control is the same as that for the oxygen excess control. In this paper there are not simulation results of voltage control because this controller is not able to control the fuel cell stack while keeping the oxygen excess on the safe side for a wide range of the current demand ([1], [3]).


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where w = wk+N1 · · · wk+N2 . The optimal control sequence will be computed by solving at each sampling time the following QP problem: u∗ = arg min J(u, w, x) u∈U

(10)

where U {∀u : Ru ≤ b + Sx x} with R ∈ ℜq×Nu , b ∈ ℜq×1 , Sx ∈ ℜq×dimx is the convex set of all the feasible control sequences. Note that constraints on the output, control moves and control signal amplitude values can be easily written in this form ([4]). Moreover, the constraints on the amplitude of the control signal imply that the definition of U must be updated at each sampling time (specifically uk−1 must be included in the computation of b, see [4]). Problem (10) can be rewritten as an equivalent multiparametric Quadratic Programming (mpQP) problem ([6]): min s.t.

1 u Qu + θ C u 2 Ru ≤ b + Sθ θ

where θ is an augmented vector of parameters: θ = x wk+N1 · · · wk+N2 uk−1

f (θ ) = F i · θ + gi

i f H i · θ ≤ ki , i = 1, . . . , Nmpc

(20)

where the polyhedral sets {H i · θ ≤ ki }, i = 1, . . . , Nmpc are a partition of the given set of parameters θ . The simplest way to implement the piecewise affine feedback law (20) is to store the polyhedral cells {H i · θ ≤ ki } and perform an on-line linear search through them to locate the one which contains θ . When the number of regions is too high it should be better to use more efficient ways of implementing the search such as in [14], [13].

VI. S IMULATION RESULTS

(11) (12)

(13)

The last parameter uk−1 is included in θ to allow constraints on the amplitude of the control signal. Note that the dimension of θ can be kept as low as possible if the value of the set point is assumed to be constant over the prediction horizon. This assumption is followed in this work and thus: (14) θ ∈ ℜ(dimx+2)×1 = x wk+N1 uk−1

In this section are shown some simulation results obtained with the predictive controller in explicit form. The computation of the controller has been made using the Matlab functions mpqp and getcontroller that can be found in the Hybrid Toolbox for Matlab [12]. The default solver (quadprog) was used. The simulated results have been obtained for the starvation prevention and maximum efficiency strategies. The parameter space for which the explicit solution has been computed is bounded by:

Under this assumption the matrices in (11) and (12) can be computed as Q = 2 · (G · G + λ · IN×Nu ) and C = 2 · G · Fx [−1 · · · − 1]T1×N [0 · · · 0]T1×N

•

The system studied has some physical constraints that we have to take in account during the optimization. The input amplitude and output are bounded by minimum and maximum values and these constraints can be written as: T · u ≤ Umax · 1Nu ×1 + 0Nu ×dimx −1Nu ×1 · θ (15) −T · u ≤ −Umin · 1Nu ×1 + 0Nu ×dimx 1Nu ×1 · θ (16) (17) G · u ≤ ymax · 1N×1 + −Fx 0N×2 · θ N×1 N×2 ·θ + Fx 0 (18) −G · u ≤ −ymin · 1

•

where T ∈ ℜNu ×Nu is a lower triangular matrix of ones and 0 and 1 are vector and matrices of zeroes and ones respectively. Equations (15) and (16) show the constraints regarding the inputs physical limitations. Equations (17) and (18) refer to output range limits. All the constraints can be grouped in the form of (12). The control law defined by the optimization problem presented in equations (11) and (12) is continuous and piecewise affine ([6]). The solution of the mpQP problem yields an explicit description of the control law in the box θmin ≤ θ ≤ θmax , such that the optimal value of Δuk can be obtained as: (19) Δuk = f (θ )

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• • • •

−0.5 ≤ yk−i ≤ 10.5, i = 0, · · · , 4. −100 ≤ Δuk−i ≤ 100, i = 1, · · · , 3. −40 ≤ Δuk−i ≤ 40, i = 1, · · · , 4. −0.5 ≤ wk−1 ≤ 10.5. −0.5 ≤ uk−1 ≤ 100.5. −0.5 ≤ vk−1 ≤ 40.5.

These limits arise from the physical characteristics and parameters of the real fuel cell. The constraints are for the air pump voltage from 0 to 100% and for the oxygen excess from 0 to 10. On the other hand, the range of the suitable current demand is from 0 to 40 A. The prediction and control horizons were chosen equal to 3 and the weighting factor λ is 0.01. These values are the same used in [1] and [3] and have proved to be a god choice in terms of controller performance. The system is simulated by the non-linear model introduced in section II. The controller obtained has 8 different regions. Note that the number of regions is small enough so that the on-line evaluation can be done by performing a simple sequential search over the parameter space partition. A significantly greater number of regions would require more efficient ways of implementing the search ([14], [13]). Figure 9 represents the evolution of the required current during the simulation for each control strategy. Figure 10 shows the air pump voltage and the oxygen excess for this current demand with the maximum efficiency criteria. Figure 11 presents the air pump voltage fluctuations and the oxygen excess during the same simulating with the Starvation prevention criteria.


FrC09.1 TABLE I M AXIMUM AND AVERAGE EXECUTION TIME TABLE

40

35

IStack (A)

30

Controller Constrained GPC Constrained Explicit GPC

25

20

Maximum (ms) 33 7.7

Average (ms) 10 0.3

15

10

0

50

100

150

VII. C ONCLUSIONS In this paper the use of predictive control strategies for PEM fuel cells based on explicit descriptions of control law is presented. The traditional way of implementing the control law by means of the online solution of a QP problem is not appropiate for PEM fuel cells as the sampling times are in the order of milliseconds. This formulation reduces the execution time and allows the implementation of predictive controllers in real time. On the other hand there are some open questions. One of the most interesting is the possibility of expanding the controller so that the operation point is included as another parameter. This would allow the use of this controller in an adaptive control scheme with a very low computational burden.

Time (s)

Fig. 9.

Stack current demand during a simulation

100

(%)

80

air,pump

60

V

40 20 0

0

50

100

150

Time (s) 8 Set Point Output System

λO

2

6 4 2 0

0

50

100

R EFERENCES

150

Time (s)

Fig. 10. The control and the output signals during the simulation of the maximum efficiency strategy

100

air,pump

80 60

V

40 20 0

0

50

100

150

Time (s) 6 Output System Set Point

λO

2

4

2

0

0

50

100

150

Time (s)

Fig. 11. The control and the output signals during the simulation of the Starvation prevention strategy

Table I shows the maximum and average computation times of the explicit controller versus a conventional implementation of the constrained GPC. As expected, the explicit implementation takes significantly less time to compute the optimal control signal for both the average and worst case scenario. Moreover, the explicit implementation meets the demands of a sampling time of 8 ms even in the worst case, whereas the conventional implementation cannot be applied with such time constraints. The computer used for the simulations was a 2.80 GHz P4 CPU with 512 Mb RAM running Windows XP and Simulink software.

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[1] C. Bordons, A. Arce and A.J. del Real, ”Constrained Predictive Control Strategies for PEM fuel cells”, IEEE proceedings of 2006 American Control Conference,2006. [2] A.J. del Real and A. Arce and C. Bordons, ”Development and Experimental Validation of a PEM Fuel Cell Dynamic Model”, Journal of Power Sources (in revision, available on request to the authors), 2007. [3] A. Arce, A.J. del Real and C. Bordons, ”Application of Constrained Predictive Control strategies to a PEM Fuel Cell Benchmark”, IEEE proceedings of 2007 European Control Conference, Kos, Greece, 2007. [4] E.F. Camacho and C. Bordons, ”Model Predictive Control, Second Edition”,2004. Springer-Verlag,London [5] J.T. Prukushpan, A.G. Stefanopoulou and H. Peng, ”Control of Fuel Cell Power Systems: Principles, Modelling and Analysis and Feedback Design”, series Advances in Industrial Control, 2004, Springer. [6] A. Bemporad, M. Morari, V. Dua and E.N. Pistikopoulous, ”The explicit linear quadratic regulator for constrained systems”. Automatica 38 (2002),pag 3-20. [7] P. Rodatz, G. Paganelli and L. Guzella, ”Optimization air supply control of a PEM Fuel Cell system”,IEEE Proceeding of American Control Conference,2003 [8] P.E.M. Almeida and M. Godoy Simões,”Neural optimal control of PEM fuel cells with parametric CMAC networks”,IEEE Transactions on Industry Applications [9] M.Y. El-Sharkh, A. Rahman,M.S. Alam, ”Neural networks-based control of active and reactive power of a satnd-alone PEM fuel cell plant”,Journal of Power Sources 135 (2004) 88-94. [10] J. Golber and D.R. Lewin. ”Model-Based control of Fuel Cells Regulatory Control”. Journal of Power Sources 135 (2004) 135-151. [11] A. Vahidi, A.G Stefanopoulou and H. Peng, ”Model Predictive Control for Starvation Prevention in a Hybrid Fuel Cell System”, in IEEE proceedings of 2004 American Control Conference,2004 [12] A. Bemporad (2004.Jan) Hybrid Toolbox-User’s guide [online]. Available: http://www.dii.unisi.it/hybrid/toolbox.(URL) [13] T. Johansen and A. Grancharova (2003). Approximate explicit constrained linear model predictive control via orthogonal search tree. IEEE Transactions on Automatic Control 48(5), 810–815. [14] P. Tøndel, T. A. Johansen and A. Bemporad (2003). Evaluation of piecewise affine control via binary search tree. Automatica 39(5), 945– 950.