Travaux dans le cadre du PEPS Gestion Echantillonnée des Systèmes Energétiques, dirigé par D. Normand-Cyrot

Transcription

1 Travaux dans le cadre du PEPS Gestion Echantillonnée des Systèmes Energétiques, dirigé par D. Normand-Cyrot Participants: D. Normand Cyrot, R. Ortega (L2S), O. BETHOUX, M. HILAIRET (LGEP), M. GHANES, J.P. BARBOT (ECS-Lab) Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 1 / 22

2 Commande aux perturbations singulières : Plan 1 Introduction Problem statement Objectives 2 3 Fuel cell FC boost converter Super-Capacitors boost converter DC bus and load Two time scales complete model 4 PI Controllers Singular Perturbation Control 5 Experimental set-up Experimental results without CL Experimental results with CL Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 2 / 22

3 Problem statement Objectives The problem of the power management of an hydrogen Fuel Cell (FC) system associated to a reversible impulse energy source (the supercapacitors) is considered (parallel power architecture management [Thounthong et al ]. The FC must deliver a slowly varying current, not more than 4A/s for a 0.5kW/12.5V [Thounthong et al. (2009) ] or 10A/s for a 20kW/48V [Corbo et al. (2009)]. That is why the FC needs to be associated with other sources (here the supercapacitors) which supply short pulse energy demanded by the load and fill the temporary failure of the FC. SCs FC DC/DC i fc v fc DC/DC i sc v sc C i l v b Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 3 / 22

4 Objectives Problem statement Objectives To respect of FC dynamics (mainly limited by the time response of the air compressor. To control of the storage device (SCs) state of charge. To control the power response (positive or negative) required by the load. Background: FC system controls PI [Azib et al. 2004], [Thounthong et al. 2009], RST [Caux et al. 2005], Predective [Vahidi et al. 2006], Flat [Payman et al. 2008], [Hissel et al. 2008], State feedback [Becherif et al. 2006], [Hissel et al. 2008], Optimal [Rodatz et al. 2005], Empirical [Paladini et al. 2007], Frequency decoupling [Duvat et al. 2009] Passivity [Hilairet, Bethoux, Ortega] and... (non exhaustive). Drawbacks : load knowledge, closed-loop system stability with FC dynamics constraints. PEPS Project : Passivity [Bethoux, Hilairet, Ortega], Digital [D.N. Cyrot], SP [Ghanes et al. 2011]. Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 4 / 22

5 anti-windup anti-windup Introduction Outer loop: Controller based on singular perturbation approach [P. Kokotovic 1986 and H. Khalil, 1996] is proposed to respect the slower FC dynamics, control of the storage device (SCs) state of charge and the power response required by the load with a DC bus regulation. This approach is well adapted to the problem of FC-SCs control where the FC and SCs currents must be slow and fast respectively. A stability proof of the system is given. Inner loop: 2 PI Controllers Problem statement Objectives SCs PaC DC/DC ifc vfc DC/DC isc vsc + IP controlleur L fc L sc i * fc IP controlleur + i * sc Boucle de courant ifc C il vb Charge isc Boucle de courant vb v0 vfc i * fc i * sc R0 i * fc Commande par Pert. Sing i * sc Pertes Hacheurs id v * b vb v * sc vsc Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 5 / 22

6 Principle of singular perturbation control To remove the short time effects t ǫ which are asymptotically stable or oscillating, in order to have only the long time effects t 1 (0 < ǫ << 1). Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 6 / 22

7 For 0 < ǫ << 1 : The following approximation Σ ǫ : { ẋ = f(x,z,ǫ) ǫż = g(x,z,ǫ) Σ 0 : { ẋ = f(x,z,0) 0 = g(x,z,0) (1) is justified if : g(x,z,0)=0 has a solution z = φ(x), where the Jacobian { g(x,φ(x),0) z } is regular in the considered state space (implicit function theorem). Thikhonov theorem : the Jacobian { g(x,φ(x)) z } is Hurwitz. The reduced system is then ẋ = f(x,φ(x),0) which has a unique solution for t [0,T], 0 < T <. Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 7 / 22

8 System with feedback: { χ = f(χ,ζ) ζ = g(χ,ζ)+ β(χ)u (2) with χ R m, ζ R n, u R n and β regular for all χ. u = 1 ǫ β(χ) 1 (ζ α 0 (χ)) the dynamics (2) becomes : { χ = f(χ,ζ) ǫ ζ = ǫ g(χ,ζ) (ζ α 0 (χ)) (3) Dynamics (3) are similar to the dynamics (1), thus, it is possible to use the Thikhonov theorem and the slow dynamic of (3) is equal to χ = f(χ,α 0 (χ)) (4) where φ(χ,ǫ) = i=0 α i(χ) ǫi i! is computed iteratively [Vasileva1963]. Here the first approximation of the slow manifold i.e. φ(χ,ǫ) α 0 (χ) is used. Nevertheless, for example, when the system behavior is too closed to a fold it is necessary to do a better φ s approximation. Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 8 / 22

9 Fuel cell FC boost converter Super-Capacitors boost converter DC bus and load Two time scales complete model Fuel cell modeling : [Pukrushpan et al., 2004] FC voltage v fc is computed according to the current stack i fc by a 5 th order polynomial function of the stack current i fc to improve the simulation time and use a simple equation without loss of accuracy. fuel cell voltage v fc 40 V i fc (A) 45 Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 9 / 22

10 Fuel cell FC boost converter Super-Capacitors boost converter DC bus and load Two time scales complete model FC boost converter : In an electric power system, the FC must be connected to a fixed DC bus voltage. The FC voltage must be increased; it is often less than the DC bus voltage. The boost converter is controlled by binary input w 1 (t). v fc i fc L fc v b i fc w 1 (t) Defining α 1 as the duty cycle of w 1 (t), this subsystem can be represented by its average model (here, the switches are ideal): di fc 1 dt = dv b dt = 1 C ( ) L (1 α1 fc )v b +v fc ( ) (5) (1 α1 )i fc i l where v b : DC link voltage, i l : DC current delivered to the load. The product between control input α 1 and state variable v b shows a non-linear behavior of the converter. Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 10 / 22

11 Fuel cell FC boost converter Super-Capacitors boost converter DC bus and load Two time scales complete model ControlDC in abus Fuel Cell andsystem load model GT CSE-GDR PACS, 19/01/12 11 / 22 SCs boost converter: SCs can be charged or discharged; the storage elements are connected to the DC bus through a reversible power converter. The SCs have a constant capacity (C sc ) and negligible losses and associated with an inductance (L sc ) and a boost converter. Two types of operations are possible: a buck operating mode when SCs receive energy from the DC bus, and a boost operating mode when SCs supply energy to the DC bus. Csc i sc L sc v sc w 2 (t) We define α 2 as the duty cycle of control variable w 2 (t). The second sub-system is represented by an average model as follows: di sc ( ) dt = 1 L (1 α2 sc )v b +v sc dv sc dt T 1 T 2 v b = isc C sc (6)

12 Fuel cell FC boost converter Super-Capacitors boost converter DC bus and load Two time scales complete model DC bus and load model: The load is modeled by a (R l,l l ) circuit. R l varies according to the power required by the load. The average model is: dv b dt = 1 C ( (1 α1 )i fc +(1 α 2 )i sc i l ) ( ) (7) di l dt = 1 L Rl l i l +v b Csc DC/DC i fc L fc v fc DC/DC u 1(t) i sc L sc v sc u 2(t) i sc T 1 T 2 Define α 2 as the duty cycle of w 2 (t), the second average model is: di sc ( ) dt = 1 L (1 α2 sc )v b +v sc (8) dv sc dt = isc C sc C i l v b L l R l Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 12 / 22

13 Fuel cell FC boost converter Super-Capacitors boost converter DC bus and load Two time scales complete model FC-SCs complete model: ẋ 1 ẋ 2 = (1 α1)x4+(1 α2)x5 x3 i d C = x5 C sc ẋ 3 = R l x 3+x 1 L l ẋ 4 ẋ 5 = (1 α1)x1+z L fc = (1 α2)x1+x2 L sc (9) with: - State space x(t) = [x 1 ; x 2 ; x 3 ; x 4 ; x 5 ] t = [v b ; v sc ; i l ; i fc ; i sc ] t - Control inputs u(t) = [u 1 ; u 2 ] t = [1 α 1 ; 1 α 2 ] t - Measures y(t) = x and z(t) = v fc - Converter losses i d = i d1 +i d2 = (V0+R0 x4) x4 x 2 + (V0+R0 x5) x5 z Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 13 / 22

14 PI Controllers Singular Perturbation Control PI Controllers: Inner loop Two fast actuators are applied to the previous compete model (9). This fast actuators are two PI controllers which assign respectively I fc and I sc to a desired value I fc and I sc. Then I fc is considered as the input u 1 and I sc as the input u 2. Consequently, from the complete model (9), the reduced slow system is given by: Slow reduced model: ẋ 1 = 1 C (z3 x 1 u 1 + x2 x 1 u 2 x 3 i d ) ẋ 2 = u2 C sc (10) ẋ 3 = R l x 3+x 1 L l with - State : x = [x 1 ; x 2 ; x 3 ] t = [v b ; v sc ; i l ] t - Control inputs : u = [u 1 ; u 2 ] t = [i fc ; i sc ] t - Measures y = [v b ; v sc ; i l ] t and z 3 = v fc. - Converters losses i d = (V0+R0 u1) u1+(v0+r0 u2) u2 x 1. Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 14 / 22

15 PI Controllers Singular Perturbation Control Singular Perturbation Control: Outer loop The desired equilibrium point are x = [vb ; v sc; v b R l ], with vb and the DC bus and SCs desired voltages. v sc Slow controller: I fc : Its dynamic must be very slow (i.e. di fc dt < 4As 1 ). u 1 = i fc = x 1 z 3 I lm C cs T slow e sc (11) where - I lm : average of current load (filtered with a low past filter). - e sc = V sc Vsc - T slow : chosen so that e sc will be sufficiently slow. - e sc is not filtered (Vsc is a constant). - V sc is proportional to I sc integral and then implicitly filtered. Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 15 / 22

16 PI Controllers Singular Perturbation Control Singular Perturbation Control: Outer loop The desired equilibrium point are x = [vb ; v sc ; v b R l ], with vb and vsc the DC bus and SCs desired voltages. Fast controller: As u 1 = i fc is a slow input, the second input : u 2 = i sc can be faster than u 1 but slower that the current loops driven by both PI fast actuators. u 2 = x 1 x 2 [ C T fast e b z 3 x 1 ( x 1 z 3 I lm C cs T slow e sc )+x 3 +i d ] (12) with - e b = V b V b - T fast is chosen in order than the feedback on e b is slower that both PI fast actuators but faster that the loop on e sc. Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 16 / 22

17 PI Controllers Singular Perturbation Control Singular Perturbation Control: The desired equilibrium point are x = [vb ; v sc ; v b R l ], with vb and vsc the DC bus and SCs desired voltages. Proposition: The reduced model of the FC-SCs system (10) under the controls (11)-(12): and u 1 = i fc = x 1 z 3 I lm C cs T slow e sc u 2 = x 1 x 2 [ C T fast e b z 3 x 1 ( x 1 z 3 I lm C cs T slow e sc )+x 3 +i d ] is locally exponentially stable for Vb and V sc constants with a bias convergence of x 2 = v sc to x1i d x 2C cs. Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 17 / 22

18 PI Controllers Singular Perturbation Control Proof: Choosing the following Lyapunov candidate function V T = V 1 +V 2 +V 3 with V 1 = 0.5e i, V 2 = 0.5e 2 b and V 3 = 0.5e 2 sc, where e i = (x 3 i lm ) 2. By applying the controls (11)-(12) to the reduced model (10), the differentiation of V is: R V = e l e i+e b i L l e2 b T fast x 1 C x e sc ( 2 [ T fast e b z3 x 1 ( x1 z 3 i lm Ccs T slow e sc )+x 3 i d ] ) C cs By choosing T fast << 1, it follows that e b converges to zero independently of e i then e i also converges exponentially to zero and this uniformly with respect to e sc. Then V = R l L l e 2 i 1 T fast e 2 b z3 T slow x 2 e 2 sc + x1i d x 2C cs e sc. Which is locally exponentially stable for x 2 > 0 with a bias ( x1i d x 2C cs ) depending on i d. Moreover T slow must be chosen so that e sc dynamics is slower according to PI fast actuators. Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 18 / 22

19 Experimental set-up Experimental results without CL Experimental results with CL LGEP Experimental set-up : W, 46 A, and 26 V Nexa fuel cell. Ballard design, built with Proton Exchange Membrane technology. - Ultracapacitor bank of 26 F, 30 V, and 50 A designed. - Supply sources are interconnected to DC bus using two choppers. - DS1103 controller board is used for implementing the energy management control strategy, using Matlab/Simulink software. - The hybrid power source is connected to a programmable electronic load. Rated power of 1800 W (imax = 150A/Vmax = 60V ). Real time DS1103 card Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 19 / 22

20 Experimental set-up Experimental results without CL Experimental results with CL Results without converter losses - T fast = 50ms and T slow = 2s. - i L vary from 0 to 20 A. - Mauvaise régulation de la tension du bus DC. - Réponse douce de i fc ( d ifc dt < 10A/s). - Le contrôle de l état de charge est correctement réalisé, neanmoins une faible erreur subsiste comme on l a mentionné dans la preuve. i load (t) [V] I pac_max = 40A, Pente_Ipac_max = 30A/s T rapide = 50 ms, T lent = 2 s i FC (t) [A] i UC (t) [A] t (s) t (s) t (s) v BUS (t) [V] v FC (t) [V] v UC (t) [V] t (s) t (s) t (s) Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 20 / 22

21 Experimental set-up Experimental results without CL Experimental results with CL Results with converter losses - T fast = 10ms and T slow = 2s. V 0 = 3V, R = 0,170Ω. - i L vary from 0 to 20 A. - Bonne régulation de la tension du bus DC effectuée par la partie rapide du contrôle aux perturbations singulières quand la demande en puissance de la charge augmente. - Réponse douce de i fc ( d ifc dt < 10A/s). - le contrôle de l état de charge est correctement réalisé, neanmoins une faible erreur subsiste comme on l a mentionné dans la preuve. i Ipac_max = 45A, Pente_Ipac_max = 20A/s charge (t) [V] i PAC (t) [A] i SC (t) [A] v BUS (t) [V] t (s) t (s) v PAC (t) [V] t (s) v SC (t) [V] t (s) t (s) t (s) Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 21 / 22

22 Experimental set-up Experimental results without CL Experimental results with CL Electrical caracteristics of hybrid system Fuel Cell: Parameter Name Value Open circuit voltage 45V Rated voltage 26V Rated current 46A Ultracapacitors: Parameter Name Value Capacitance 26F Rated voltage 30V Rated current 50A Optimal Voltage (V UCref ) 50V Inductors and Capacities: Parameter Name Value Inductor L 1 200µH Inductor L 2 100µH Rated current L 1 50A Rated current L 2 50A Capacities C bus 14mF Optimal DC Bus Voltage (V BUSref ) 50V Control in a Fuel Cell System GT CSE-GDR PACS, 19/01/12 22 / 22