Fuzzy Logic and Passivity Based Control applied to ...

Fuzzy Logic and Passivity Based Control applied to Hybrid DC Power Source using Fuel Cell and Battery M. Mohammedi, M. Becherif, A. Aboubou, O. Kraa, M.Y. Ayad and M. Bahri Abstract— Fuel cell systems offer clean and efficient energy production and are currently under intensive development by several manufacturers for both stationary and mobile applications. Therefore, their slow dynamics and difficulty to respond to the abrupt changes of load leads to prefer the association of a FC with one or more power sources of high dynamics, such as supercapacitors or batteries. In this paper, a hybrid DC power sources system using FC as a main source, a DC link and battery as an auxiliary power source is studied. The main purpose of this work is the control of the hybrid DC source by Pssivity-Based Control (PBC) where the equilibrium points are computed as function of the desired battery current. PBC is a very powerful nonlinear technique dealing with important system information like the system’s total energy. The second aim is to determine the desired battery current by a Fuzzy Logic Control (FLC) as function of battery state of charge SoC and the hydrogen quantity QH2 in order to ensure the desired behaviour of the system. Stability proof and simulation results are given. Index Terms— Fuel cell, battery, Passivity-Based Control, Interconnection and damping assignment, Port Controlled Hamiltonian System, Fuzzy Logic System, Hybrid System.

I. I NTRODUCTION The fuel cell (FC), because of its cleanness, high efficiency and high reliability, is emerging as an attractive power supply source for applications such as distributed generation power systems and electrical vehicles [1]. In the case of electrical vehicles, a fuel cell system alone, integrated into a vehicular power train, is not always sufficient to supply propulsion power for a vehicle, because fuel cell systems have some deficiencies, such as high cost, slow response and no regenerative energy recovery during braking [3]. Hybridization of a fuel cell system with energy storage devices can be a solution to these drawbacks. One motivation for hybridization is that energy storage devices can provide instant peak power during transient conditions of vehicle operation and improve fuel economy by taking advantage of regenerative braking power; at the same time, hybridization can also be helpful in decreasing the cost of the vehicle because of the possibility to use a smaller fuel cell. The fuel cells advantages such as high efficiency, self-contained unit, modularity capability, being clean make them attractive to be used as autonomous DC power supplies in some This work was not supported by any organization M. Mohammedi is with MSE Laboratory, University of Biskra, 07000, Algeria mohammedi [email protected] M. Becherif is with FCLab/FEMTO-ST, UMR CNRS 6174, UTBM, France [email protected] A. Aboubou is with MSE Laboratory, University of Biskra, 07000, Algeria aboubou [email protected] M.Y. Ayad is with Industrial Hybrid Vehicle Applications, France

[email protected]

applications. However, their slow dynamics (mainly due to their auxiliaries) and difficulty to cold start leads to prefer the association of a fuel cell with one or more power sources of high dynamics, such as supercapacitors or batteries [4]. In this paper, a state-space model has been developed and used by authors for modelling the electrical behaviour of electrical power source for hybrid vehicle applications. This hybrid DC link is equipped with a fuel cell system as a main source and a battery as an auxiliary power source as well. The state space dynamic model is useful for developing an effective energy management strategy based on Passivity Based Control (PBC) using Fuzzy Logic Control, which is capable to determine the desired current of battery according to the battery state of charge (SoC) and the remaining hydrogen quantity (QH2 ). Passivity based-control (PBC) is a generic name given to a family of controller design techniques that achieves system stabilization via the route of passivation, that is, rendering the closedloop system passive with a desired storage function (that usually qualifies as a Lyapunov function for the stability analysis) [5], [6]. If the passivity property turns out to be output strict, with an output signal with respect to which the system is detectable, then asymptotic stability is ensured [6]. The objective of the PBC is to bring back as quickly as possible the state in the vicinity of the equilibrium state, the solution sought here corresponding well with a compromise between the previous goal and the means implemented to reach that point [2]. II. M ODELLING OF H YBRID DC L INK Fig. 1 shows the scheme of the studied hybrid sources system. This system is composed of two sources, the first one is a FC and the second one is a battery that is connected to the DC link through an unidirectional DC/DC and a current bidirectional DC/DC converters respectively. For reducing FC’s hydrogen quantity and improving the dynamic behaviour of studied hybrid system, the FC is supplying the mean energy to the load, whereas the storage device (auxiliary source) is used to supply the load in the transient and steady states too, contrary to the hybrid system in [8], [9], [10] where the battery has been used to supply the load only in the transient state. The control design of our system is shown in the same figure where the FC and the battery supply the load and recover energy to charge the storage device. Hence, the desired battery current I B is determined by fuzzy logic control as function of battery state of charge SoC and hydrogen quantity QH2 .

battery current according to the battery SoC and the hydrogen quantity QH2 of FC, in order to ensure the desired behaviour of the system and to reduce the hydrogen consumption. In the transient and steady states the load is supplied by the FCbattery hybrid sources and the controller has to maintain the DC link voltage to a constant value and the battery current has to track its reference. A. Passivity based control Fig. 1.

Structure of the studied hybrid source and control system design.

In order to develop an energy management algorithm based on PBC of hybrid source used in the whole vehicle cycle, a dynamical modelling based on differential equations is used to describe the studied hybrid DC link. The sixth order overall state space model is written by the following expressions:

1) Port controlled hamiltonian system: PCH systems were introduced by Van der Schaft and Maschke in the early nineties and had ever since drawn much attention in electrical, mechanical and electromechanical systems. Some of the advantages of expressing systems in the PCH form are the fact that they cover a large set of physical systems and capture important structural properties [12]. Consider the non-linear system given by: x˙ = f (x) + g(x)u

                          

dVS dt dIFC dt dVDL dt dIDL dt dIB dt dIL dt

=

1 CS [(1 −UFC )IFC − IDL ]

where x ∈ Rn is the state vector, f (x) and g(x) are locally Lipschitz functions and u ∈ Rm is the control input. A PCH form of the system (1) is given by:

1

=

LFC [−(1 −UFC )VS +VFC ]

=

1 CDL [IDL + (1 −UB )IB − IL ]

=

1 LDL [VS −VDL ]

=

1 LB [−(1 −UB )VDL − rB IB + eB ]

=

1 LL [VDL − RL IL − EL ]

(1)

x˙ = [ℑ − ℜ]∇Hd + g(x)u

(5)

The desired closed loop energy function are given by the following expression:

Let’s have UFC the control of the FC boost converter and UB the control of the battery buck-boost converter. Then the control vector is: The control vector µ is presented by: µ = [µ1 , µ2 ]T = [(1 −UFC ), (1 −UB )]T

(4)

(2)

with VFC = f (iFC ), PEMFC static model is given as follow [11]: VFC = E0 − A. log( iFCi0−in ) n o (3) −in ) − Rm (iFC − in ) + B. log(1 − iFCiLim Hence VFC = f (iFC ). E is the reversible no loss voltage of the fuel cell, E0 is the measured open circuit voltage, iFC is the delivered current, i0 is the exchange current, A is the slope of the Tafel line, iLim is the limiting current, B is the constant in the mass transfer, in is the internal current and Rm is the membrane and contact resistances. In the sequel, VFC will be considered as a measured disturbance, and from physical consideration, it comes that VFC ∈ [0,Vd ], where Vd is the desired DC link voltage. III. E NERGY M ANAGEMENT BASED ON PBC In the proposed hybrid DC source, the fuel cell and the battery must cooperate to supply the load. The control of the hybrid DC source is effectuated by a PBC where the equilibrium points are computed as function of the desired battery current. A FLC is used to estimate the desired

1 x Hd = xeT Qe 2

(6)

2) Reconstruction of the state model: It consists of determining the equilibrium values of state vector and building a system, whose inputs are u and Y and whose exit is a vector x˜ , estimated from the state vector x of the source process. After some simple calculations, the equilibrium values of states x and control vector µ can be expressed as:  L T x = [Vd , I FC ,Vd , I DL , I B , VdR−E ]   L   µ 1 = V1 (VFC − LFC x˙ 2 ) d     µ 2 = V1 (eB − rB x5 − LB x˙ 5 )

(7)

d

where µ = [µ 1 , µ 2 ]T = [(1 −U FC ), (1 −U B )]T

(8)

V B = eB − rB x5

(9)

and

where VB is battery voltage, eB is electromotive force of battery, rB is battery’s internal resistance. After dynamic’s error where x˜ = x − x, the new state-space model is:

             

˙1 xe

=

˙2 xe

=

˙3 xe

=

˙4  xe      ˙5  xe      ˙6  xe

= = =

1 e2 − xe4 + (µ1 − µ 1 )x2 ] CS [µ1 x 1 e1 + (µ 1 − µ1 )x1 ] LFC [−µ1 x 1 x4 + µ2 xe5 − xe6 + (µ2 − µ 2 )x5 ] CDL [e

(10)

1

x1 − xe3 ] LDL [e 1 e3 − rB xe5 + (µ 2 − µ2 )x3 ] LB [−µ2 x 1 x3 − RL xe6 ] LL [e

where [ℑ(µ) − ℜ0 ] =  µ1 0 CS LFC − µ1 0  CS LFC   0 0    1 0  CS LDL   0 0    0 0

0 0

− CS L1DL 0

0 0

0

1 CDL LDL

µ2 CDL LB

− CDL1LDL

0

0

µ2 − CDL LB

0

d − rB +rV LB2

1 CDL LL

0

0

(11)

   − CDL1 LL    0    0     − RL2L L

The representation (10) in function of the gradient of the desired energy (6) and can be written as: ˙ = [ℑ(µ) − ℜ]∇Hd + A xe



0 0

(17) 

rB + rVd RL ℜ = diag 0; 0; 0; 0; ; 2 LB2 LL 0



= ℜ0T

(18)

with ∇Hd = [CS xe1 ; LFC xe2 ; CDL xe3 ; LDL xe4 ; LB xe5 ; LL xe6 ]T and [ℑ(µ) − ℜ] =  µ1 0 CS LFC µ − 1 0  CS LFC   0 0    1 0  CS LDL   0 0    0 0

0 0

− CS L1DL 0

0 0

0

1 CDL LDL

µ2 CDL LB

− CDL1LDL

0

0

µ2 − CDL LB

0

− LrB2 B

1 CDL LL

0

0

(12)



0 0

   1  − CDL LL   0    0    RL  − L2 L

(13)  1 CS [(µ1 − µ 1 )x2 ]  1 [(µ − µ1 )x1 ] 1  LFC   1    [(µ − µ )x ] 2 2 5   CDL 

A=     

    1  − µ )x ] [(µ 2 3 2 LB  0 0

(14)

Where ℑ(µ) = −ℑ(µ)T is a skew symmetric matrix defining the interconnection between the state space and ℜ = ℜT ≥ 0 is a symmetric positive semi-definite matrix defining the damping of the system. The following control laws are proposed [8], to introduce a parameter r in the damping matrix of the system:  UFC = U FC (15) UB = U B − re x5 with r is a positive design parameter. 3) Stability proof: Using the control laws precedents (8), (15), the system of equations (11) becames: ˙ = [ℑ(µ) − ℜ0 ]∇Hd + A0 xe

  0 0 0 0 0  0  0 0 0 0     rx xe    5 5  0 0 0 0  CDL   0   A =  0  = 0 0 0 0        0  0 0 0 0    0 0 0 0 0 

(16)

 0 0   0   ∇Hd = F∇Hd 0   0  0

0 0 rx5 CDL LB

0 0 0

(19) A0 = F∇Hd  0 0   0   [ℜ0 − F] = 0   0   0

(20)

0 0

0 0

0 0

0 0

0

0

rx5 0 − CDL LB

0

0

0

0

0

0

0

rB +rVd LB2

0

0

0

0

 0 0   0   0   0  RL 

(21)

LL2

The derivative of the desired energy function (6) along the trajectories of (15) is non positive if and only if the following matrix is non negative definite, [ℜ0 − F] ≥ 0 [13]. Proof: ˙ = [ℑ(µ) − ℜ0 ]∇Hd + F∇Hd = ℑ∇Hd − [ℜ0 − F]∇Hd xe ˙ = ∇H T ℑ∇Hd − ∇H T [ℜ0 − F]∇Hd H˙ d = ∇HdT xe d d T ˙ Hd = ∇Hd [F − ℜ0 ]∇Hd ≤ 0 [F − ℜ0 ] is negative semi definite if and only if all eigenvalues of [F − ℜ0 ] are negative. Mathematically, the eigenvalues of triangular matrices are the diagonal elements. In our case, [F −ℜ0 ] is an upper triand gular matrix so its eigenvalues are: 0; 0; 0; 0; − rB +rV ; − RL2L LB2 L 0 Then the eigenvalues of [F − ℜ ] are negative since r, Vd , rB , LB , RL and LL are positive constants. Hence, the derivative of the desired energy function (6) along the trajectories of (16) and the proposed control (15) is

TABLE I T HE D IFFERENT S IMULATION PARAMETERS

negative semi definite. Consequently, the origin of the closed loop dynamics (15) is stable.

RLE load, controller and DC link parameters eB (V ) VDL (V ) RL (Ω) LL (mH) E(V ) 12 42 10 10 10

IV. F UZZY L OGIC C ONTROL The proposed desired battery current estimation is based on a classical Fuzzy Logic control which defined under Matlab toolbox . A FLC needs to define both input and output membership functions, fuzzification method, scaling factor values, type of membership, rules, rule processing (Mamdani, Sugeno), inference mechanism, t-norms, s-norms and defuzzification method [14], [15]. A typical block diagram of a fuzzy logic control is detailed in [16], [17]. The FLC input parameters are the battery (SoC) and the remaining quantity of hydrogen (QH2 ), the fuzzy output is the desired battery current I B . The dynamic behaviour of FLC is defined by the rule base given in Table. II.

CDL (mF) 15

TABLE II T HE RULE BASE S YSTEM

QH2 (Low) QH2 (Avg) QH2 (High)

SoC(Low) 0 Negative NegativeBig

SoC(Avg) PositiveBig Positive 0

SoC(High) PositiveBig Positive 0

VI. C ONCLUSION

xi (i = 1, 2, ..., n) are the FLC input parameters and the fuzzy output variables is denoted as z. The membership functions µSoC j (xi ) and µQH j (xi ) are represented as the input 2i

linguistic term Aij . B j is the linguistic term for the fuzzy output. V. S IMULATION R ESULTS OF THE H YBRID DC S OURCE C ONTROL The different simulation parameters are shown in TABLE I. We impose a reference voltage Vd that the DC link voltage must follow it. Fig. 2 presents the system response to changes in the DC bus voltage reference Vd , and load current IL . The DC bus voltage tracks well the reference, a low overshoot about (4%) and no steady state error are observed. Figs. 3 and 4 show the fuel cell VFC , IFC and the battery VB , IB responses successively. When the hydrogen quantity is decreasing, the battery supplies with the fuel cell power to the load in transient and steady state, if it is charged, by cons in the work [8], the storage device supplies the load just in transient state. The battery current IB tracks well its reference which is considered as the steady state of IB in PBC. Control signals of the DC/DC converters of the FC and battery storage are presented in Fig. 5. UFC and UB are in the set [0, 1]. Fig. 6 shows the variations of QH2 and SoC according to the FC and battery powers provided to the load. At 6 s the load resistance makes change at 12 Ω and then to 10 Ω at 8 s as shown in Fig. 7. Fig. 8 shows the powers transfer of the system and the sum of the battery and FC powers gives the load power. The hybrid DC power source with the proposed controller is robust towards load resistance and DC bus voltage reference changes.

45

VDL & Vd (V)

(22)

In this paper, a modelling and control principles of a hybrid DC power source with a FC as the main source, a battery storage as the secondary source and DC/DC converters have been presented. Passivity-Based Controller has been designed to control the interfacing DC/DC converters. In this paper, a modelling and the control principles of a DC hybrid source system, composed of a FC source and a battery source, have been presented. This source uses the fuel cell as mean power source and the battery as auxiliary power source which supplies the load in transient and steady state in function of SoC and QH2 . PCH structure of the overall system is given exhibiting important physical properties in terms of variable interconnection and damping of the system. The problem of the DC Bus Voltage control is solved using simple linear controllers based on an IDA-PBC approach. An intelligent Fuzzy Logic Control is adopted to determine the desired battery current used in PBC. Global Stability proofs are given and encouraging simulation results has been obtained. Many benefits can be expected from the proposed structure such that supplying the power picks in the transient and the steady state by using battery.

40

35

VDL Vd

30

0

1

2

3

4

5

6

7

8

9

5

6

7

8

9

time (s) 4 3 IL (A)

R j : IF x1 is A1j and x2 is A2j and . . . and xn is Anj T HEN z is B j

i

r 0.01

2 1 0

0

1

2

3

4 time (s)

Fig. 2.

DC Link voltage and its reference, Load current.

0.65 0.6

20 UFC

VFC (V)

25

15

0

1

2

3

4

5

6

7

8

0.55 0.5

9 0.45

time (s) 10

0.4 IFC (A)

5

0

1

2

3

4

5

6

7

8

9

5

6

7

8

9

time (s)

0

0.75

−5 0

1

2

3

4

5

6

7

8

9 UB

time (s)

Fig. 3.

0.7

FC Voltage and FC current. 0.65

0

1

2

3

4 time (s)

12.1

VB (V)

12

Fig. 5.

11.9

FC boost control and Battery converter control.

11.8 100

11.7

80 1

2

3

4

5

6

7

8

9 QH2 (%)

0

time (s) 8

40

IB

6 IB & IB (A)

60

20

IB

4

0

0

1

2

3

4

6

7

8

9

5

6

7

8

9

100

0

90 1

2

3

4

5

6

7

8

9

time (s)

Fig. 4.

SoC (%)

−2 0

5 time (s)

2

Battery Voltage, Reference and battery currents.

80 70 60 50

0

1

2

3

4 time (s)

R EFERENCES Fig. 6.

Fuel Cell QH2 and Battery SoC.

Fig. 7.

Load resistance change.

160 140 120 100 Powers (W)

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80 60 40 20 PB PFC

0

PLoad −20 0

1

2

3

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5 time (s)

Fig. 8.

Powers transfer.

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