Management of fuel cell power and supercapacitor state-of-charge for electric vehicles

Management of fuel cell power and supercapacitor state-of-charge for electric vehicles

Electric Power Systems Research 160 (2018) 89–98 Contents lists available at ScienceDirect Electric Power Systems Research journal homepage: www.els...

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Electric Power Systems Research 160 (2018) 89–98

Contents lists available at ScienceDirect

Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr

Management of fuel cell power and supercapacitor state-of-charge for electric vehicles A. Tahri a,∗ , H. El Fadil a , F.Z. Belhaj a , K. Gaouzi a , A. Rachid a , F. Giri b , F.Z. Chaoui c a b c

ESIT Team, LGS Laboratory ENSA, Ibn Tofail University, 14000 Kénitra, Morocco Laboratoire d’Automatique de Caen, Université de Caen, Bd Marechal Juin, B.P 8156, 14032, Caen, France ENSET, Mohammed V University, Rabat, 10000, Morocco

a r t i c l e

i n f o

Article history: Received 15 November 2016 Received in revised form 6 March 2017 Accepted 6 February 2018 Keywords: Fuel cell Supercapacitor Power converters Power management Nonlinear controller

a b s t r a c t In this paper a power management system (PMS) is designed to achieve, for automotive applications, a control strategy aiming to split the load power between a fuel cell and a supercapacitor accounting for the fuel cell limited dynamics, its rated power and bounded supercapacitor voltage. The power sources are connected to a DC bus through boost and buck-boost converters. The converters are controlled to regulate the Dc bus voltage and the supercapacitor current must track a reference provided by the PMS unit. The fuel cell is the main source and the supercapacitor is the auxiliary one, which recovers power at a braking or a decelerating mode. The supercapacitor current is also controlled in order to keep the state-of-charge (SOC) within accepted bounds. Thus, the fuel cell charges the supercapacitor when the SOC is too low, and, the supercapacitor feeds the power-train, whenever it is overcharged. Meanwhile, the fuel cell dynamics is perfectly controlled during algorithm commutations. Theoretical analysis and results, for a practically validated high-fidelity simulation model, show that the proposed controller and the power management system meet all the objectives. © 2018 Elsevier B.V. All rights reserved.

1. Introduction Scientists agree on the greenhouse effects of fossil fuels and their depletion is inevitable, that have encouraged researchers and industry to seek clean and sustainable energy sources. Among the promising electric sources is fuel cell because it consumes hydrogen and its byproduct is merely water and heat [1,2]. Even though, a fuel cell (FC) is a source that has relatively low power level. Furthermore, it cannot neither provide power to fast changing loads nor recover braking energy. Therefore, an energy storage system (ESS) is necessary to ensure better performance in hybrid electric vehicles [3]. An ESS can be implemented by a battery or a supercapacitor (SC). In confronting these two storage devices, the supercapacitor charging time is advantageous because it can reach 1–10 s, compared with the new fast lithium-ion battery which can be charged at 70% in few minutes [4]. In addition, the supercapacitor can provide with better performances peak powers, it have a long lifecycle

∗ Corresponding author. E-mail addresses: [email protected] (A. Tahri), [email protected] (H. El Fadil), [email protected] (F.Z. Belhaj), [email protected] (K. Gaouzi), [email protected] (A. Rachid), [email protected] (F. Giri), [email protected] (F.Z. Chaoui). https://doi.org/10.1016/j.epsr.2018.02.003 0378-7796/© 2018 Elsevier B.V. All rights reserved.

and it’s virtually free of maintenance. Therefore, in this work, we consider a supercapacitor bank as an energy storage system. The use of this technology combining the two sources of energy has an undeniable asset for the following reasons: - The fuel cell will be employed to meet the average and permanent power demand of the vehicle, - The supercapacitor, meanwhile, will be used to meet the peak and transient power demand. It also allows energy recovery during braking and deceleration phases. - As the fuel cell is not used to support rapid load changes, it would avoid the problem of ‘fuel starvation’ which would cause permanent damage to the proton exchange membrane of the cell [4]. This combination gives an efficient fuel cell hybrid power system because the fuel cell has relatively lower efficiency at low and high output power [5]. Moreover, this hybridization can downsizing the fuel cell then reducing the power system cost because the FC is the most expensive component [6]. In this work, we develop a new power management system based on a multi-loop nonlinear controller for a boost and buckboost converters connected respectively, to a FC, as a main source, and to a SC as an auxiliary one. The whole control unit is designed

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Nomenclature ck , k = 1,2 Design parameters Cdc DC link capacitor [F] ı,  Tuning parameter of the power management system ek , k = 1,2 Error between the variable  i and its reference Hi , k = 1,2 Binary outputs of the hysteretic blocs ¯i1 Average value of the boost output current (A) ik , k = 1,2 Converters output currents (A) Ifcref Reference signal of the fuel cell output current (A) Iscref Reference signal of the supercapacitor current (A) im Input courant of the inverter (A) io Load current of the hybrid source (A) Fuel cell output current (A) ifc isc Supercapacitor output current (A) Iscref Reference signal of the supercapacitor current (A) Input inductance of the boost converter [H] Li L2 Input inductance of the buck-boost converter [H] Efficiency of the buck-boost converter in the boost o mode u Efficiency of the buck-boost converter in the buck mode Binary code of the buck-boost converter operating m mode k , k = 1,..3 Duty cycles of the binary inputs k , k = 1,..3 12 Average value of the common input variable of the buck-boost converter Pscref Reference signal of the buck-boost output power (W) Psc Output power of the buck-boost converter (W) Output power of the hybrid DC source (W) Po Pof Power management filter output (W) Pfc min , Pfc max Low and high limits values of the fuel cell output power (W) Design parameter of the high gain observer h Ri The equivalent series resistance (ESR) of the inductance Li [] Rsc Supercapacitor equivalent series resistance (ESR) [] xˆ Estimate value of the variable x x˜ Estimation error of the variable x˙ Time derivative of the variable x 1 Average value of supercapacitor current isc [A] 2 Average value of the DC link voltage vdc [V] Average value of the fuel cell current ifc [A] 3 u12 Common binary input variable of the buck-boost converter uk , k = 1,..3 Binary inputs variables of the converters DC link voltage [V] vdc Vdcref Reference value of the DC link voltage [V] vsc Supercacitor output voltage (V) Fuek cell output voltage (V) vfc Vcth1 High theoretical limit of the supercapacitor voltage (V) Low theoretical limit of the supercapacitor voltage Vcth2 (V) Vc min , Vc max Low and high limits values of the supercapacitor voltage (V) Vi , k = 1,2 Intermediate variables of the hysteretic blocs (V) Vi , k = 1,2 Tuning variables of the supercapacitor voltage nominal (V)

V Vcnom vc ωn , Z

Lyapunov functions Nominal value of the supercapacitor voltage (V) Supercapacitor voltage defined by vc = vsc + Rsc isc Natural frequency and damping ratio of the power management filter

to meet the four main objectives: (i) a tight regulation of the DC bus voltage; (ii) the SC current must track its reference; (iii). The load power must be split between the two sources accounting for the FC rated power and its low power slope; and (iv) the recovery power at a braking or a decelerating mode must be stored in the supercapacitor whilst the SC voltage must be kept in the feasible bounds. There is many works which dealing with the subject of energy management in FC/SC hybrid power systems as an optimization problem which can be carried out off-line for a specific driving cycle. A cost function is minimized, satisfying some constraints, aiming, mainly, minimizing hydrogen consumption [1,4,7–9]. The optimization problem can also be solved with a Model Predictive Control (MPC) [10]. In Ref. [11] a linear control strategy is designed based on a sharing of load power between the two sources taking into account the slow dynamics of the fuel cell. In Refs. [12,3] a flatness control strategy accounts for fuel cell output rated power and its slow dynamics, it also allows limiting the current of an SC during charging and discharging processes. The work in Ref. [13] proposes an effective energy management strategy based on the Passivity Based Control (PBC) using Fuzzy Logic estimation. In Ref. [14–16] many controllers are designed based on the Lyapunov approach without considering the dynamics limits of the fuel cell and the supercapacitor SOC constraints. In this work we consider an on line power management system. It achieves a control strategy aiming the power split between the sources accounting for, the fuel cell slow dynamics, its weak rated power and bounded SC state-of-charge (SOC). The power management algorithm is designed based on a controller developed according the Lyapunov stability tools [17]. The PMS generates a single output which is the SC current reference, whereas the FC current reference is utilized for regulating the DC bus voltage. Therefore, SC delivers the transient load power and recovers braking or decelerating energy in a way that the FC gives power in steady states with limited dynamics and rated power. Furthermore, the control algorithm maintains the SC voltage within accepted bounds. When the supercapacitor is overloaded, it provides some of the energy required by the load in the steady state. In addition, when the SC state- of-charge becomes critical, it receives energy from the fuel cell. In both latter SOC cases the PMS allows bring the voltage of the supercapacitor to its nominal value, and the dynamics of the FC is perfectly controlled during algorithm commutations. Compared to the existing literature, the present contribution contains several novelties, among which the following. a) A power management system is designed based on Lyapunov controllers for the first time, whereas in the existing literature using this kind of controllers, the fuel cell dynamics is not accounted for Refs. [14–16]. b) The SC voltage limitation algorithm accounts for FC slow dynamics in order to avoid discontinuities and fast changes in the fuel cell output power, even if the SC state-of-charge is out of bounds. That is, in the previous studies [3,11,12,17], the fuel cell constraints are only treated in the normal case of the supercapacitor SOC. c) Compared with the previous works [14–16] the closed loop system is of second order which leads to less complicated control lows.

A. Tahri et al. / Electric Power Systems Research 160 (2018) 89–98

d) In contrast to Ref. [17] some elements of this original contribution are pointed out in the following • The supercapacitor voltage is controlled whilst it’s ignored in Ref. [17]. • When the supercapacitor is overcharged (i.e. SOC > socmax ) it participate in feeding the load. Moreover, when the supercapacitor is undercharged (i.e. SOC > socmax ), the FC must feed the load and charge the supercapacitor. In these two cases, our proposed approach guarantees that the variation slope of the FC power is limited. Whilst in Ref. [17] the PMS accounts for FC slow dynamics only when the supercapacitor SOC is normal i.e. socmin ≤ SOC ≤ socmax . • The PMS allows bringing the voltage of the supercapacitor to its nominal value. Whilst in Ref. [17] it’s assumed that the SC voltage is between lower and upper bounds. • In this paper, the simulation is conducted using the MATLAB Sympower Toolbox, which is high fidelity professional software and which provides quite accurate models of all components (see Fig. 5). Whilst, in the conference paper the simulation is only based on the model (3a)–(3d) using MATLAB SIMULINK.

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2.2. Modeling of the DC source subsystem Applying Kirchhoff laws on the circuit shown by Fig. 1 one gets the following bilinear switching model of DC source subsystem [14,15]. 1 disc = (vsc − vdc u12 − R1 isc ) L1 dt

(1a)

1 − u3 dvdc = i − i1 Cdc fc dt

(1b)

difc dt

=−

vfc R2 1 − u3 i − vdc + L2 fc L2 L2

(1c)

u12 is the common control input variable of the buck-boost converter defined as follows u12 = m(1 − u1 ) + (1 − m)u2

(2a)

where m being a binary variable defined as follows



m=

1 if

iscref ≥ 0

(Boost mode)

0

iscref < 0

(Buck mode)

if

(2b)

The theoretical study proves that the nonlinear controller satisfies all servo objectives and the power management system achieves a suitable load power split between the main source and the auxiliary one. The results are validated by like real simulation model using MATLAB Simpower System toolbox, Version 7.8.0.347. The paper is organized as follows: in section II, we described and modeled the system. In section III, we designed the controller of the whole system. In section IV, a power management system is presented. In section V the controller and the power management system performances are illustrated through simulations using MATLAB/SIMULINK software. A conclusion, a reference list and an Appendix A end the paper.

The switched model (1a)–(1c) is beneficial to make a precise simulation of the studied system, but unfortunately it is not suitable for the synthesis of a control law as it involves binary inputs. For control purpose it is more convenient to use a model with continuous inputs. This model is obtained by using the so called averaging technique which uses averaged variables along the switching period [18]. It follows, from (1a) to (1c) and (2a) to (2b), that the averaged model [17] is as follows

2. System overview and modeling 2.1. System overview The studied hybrid electric storage system (HESS) of an electric vehicle is depicted by Fig. 1. It is composed by a Proton Exchange Membrane (PEM) fuel cell connected to a boost converter (BC) and a supercapacitor linked to a buck boost converter (BBC). The converters are connected in parallel on the DC bus and their control is performed according to the PWM principle. Note that, we do not study here the control of the inverter (INV) and the induction motor.

vsc 12 R1 ˙ 1 = − 2 − 1 + L1 L1 L1

(3a)

¯i1 (1 − 3 ) ˙ 2 = 3 − Cdc Cdc

(3b)

R2 (1 − 3 ) 1 ˙ 3 = − 3 − 2 + v L2 L2 L2 fc

(3c)

¯i1 = i0 − 12 1

(3d)

12 = m(1 − 1 ) + (1 − m)2

(4)

where the variables are defined in Table 1 and the signals 1 , 2 and 3 are averaged variables, performed over switching period, of the dc–dc power converters inputs u1 , u2 , and u3 . They are continues input variables since the converters are controlled according PWM principle.

Fig. 1. Power stage circuit of the hybrid electric vehicle.

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e2 = 3 − Ifcref

Table 1 Averaged model variables. Physical variable

Average version

SC current isc Boost converter output voltage vdc Fuel cell current ifc Buck-boost converter binary input u12 Boost converter output current i1 Boost converter binary input u3

1 2 3 12 ¯i1 3



1 = m(1 − 12 ) 2 = (1 − m)12

In order to stabilize the system errors e = (e1 , e2 ), we propose the following control laws [17] 12

L1 = 2

3 = 1 −

Remark 1. Both actual inputs 1 and 2 of the buck-boost converter, are obtained from the combined control input 12 , according to (2b) and (4), as follows (5)

(9b)



(vsc − R1 1 ) c1 e1 + − I˙ scref L1

L2 2



c2 e2 +

(vfc − R2 3 ) L2



(10a)



− I˙ fcref

(10b)

Where c1 and c2 are positive design parameters. Indeed, replacing 21 of (8a) in (3a) and 3 of (8b) in (3c), the closed loop system dynamics is obtained e˙ 1 = −c1 e1

(11a)

e˙ 2 = −c2 e2

(11b)

Consider the following Lyapunov function candidate 3. Control design and analysis V= 3.1. Control objectives

 1 2 e1 + e22 2

(12a)

Its time-derivative along the trajectories (9a)–(9b) gives The control of the HESS is performed under the following objectives. a) Tight regulation of dc-bus voltage vdc b) Perfect tracking of SC current isc to its reference signal Iscref provided by the PMS bloc. c) The closed loop system must be asymptotically stable. 3.2. Controller design for DC–DC power converters The DC bus voltage vdc must be regulated to a given reference Vdcref = 475 V. But the boost converter has a non minimum phase feature when the input is 3 and the output is vdc [18,19]. To cope with this issue we resort to an indirect design strategy. Therefore the objective is converted to that the current ifc must track a given reference Ifcref which is obtained using the power conservation principle applied to the boost converter Ifcref =

Vdcref ¯i1 1 vfc

(7a)

Vdcref ˆi1 1 vfc

4. Power management system The task of the power management system (PMS) is generally to distribute the load power between the FC and the SC. The PMS is to generate the supercapacitor current reference signal Iscref , taking into account the following constraints - The electrical power of the fuel cell must be maintained within an interval [Pfc min , Pfc max ]. - supercapacitor voltage must be kept within an interval [Vc min , Vc max ], - The absolute value of the slope of the current of the fuel cell must be limited to a maximum value [3]. Let Psc the output power of the SC-buck-boost converter, and let Po the load power (related to load torque and vehicle speed), then the FC-Boost converter output power Pfc can be written (according to Fig. 1) as follows (13a)

Pfc = vdc ¯i1 ,

Psc = 2 vc isc ,

Po = vdc io

(13b)

and (8)

In light of these considerations and the objective of a better track of the supercapacitor current to its reference Iscref , we define the following errors e1 = 1 − Iscref

i) The closed loop system is GAS. ii) The error e1 vanishes implying perfect tracking of SC current isc to its reference Iscref iii) The error e2 vanishes implying tight dc bus voltage regulation. 䊐

with (7b)

Accordingly, (6) can be rewritten as follows. Ifcref =

As V˙ Is negative definite, the equilibrium point (0,0) for the error system e = (e1 , e2 ), is globally asymptotically stable (GAS) [21]. It follows that

Pfc = Po − Psc

where  h > 0 is a design parameter and ˜ 2 = 2 − ˆ 2 , with ˆi1 1 − 3 ˙ ˆ 2 = 3 − + 2h ˜ 2 Cdc Cdc

(12b)

(6)

where 1 ≤ 1 being an efficiency factor introduced to take into account all losses: switching losses in the converters and the losses in the inductances equivalent series resistance ESR R2 . FC-boost load averaged current ¯ı1 , introduced in (6), is accessible to measurement, which is generally the case. But measurement accuracy may deteriorate due to current sensor failure. Therefore, it is of practical interest to improve system reliability by implementing an online current observer (ˆi1 ) in parallel with, or instead of, the current sensor. In Refs. [16,20] a high gain observer is designed ˆi˙ 1 = − 2 Cdc ˜ 2 h

V˙ = −c1 e12 − c2 e22

(9a)

vc = vsc + Rsc isc

(13c)

Consequently we can get Pfc = Po − 2 vc isc

(13d)

where 2 is a constant that takes into account the losses of the SC-buck-boost converter. The value of 2 depends on the converter mode (buck or boost mode). Recall that, these two modes depend

A. Tahri et al. / Electric Power Systems Research 160 (2018) 89–98

93

Fig. 2. Proposed power management system.

on the direction of energy flow between the supercapacitor and the charge or the fuel cell (charging and discharging of the supercapacitor). This constant is explicitly stated as follows



2 = o

if

Iscref ≥ 0

2 = u

if

Iscref < 0

vc-V1

H1

vc-V2

H2

(14)

In other words 2 = o during the boost mode and 2 = u during the buck mode. Inspired by (12a) and (12b), reference variables can be expressed as Pscref = Po − Pfcref

(15a)

Pscref = 2 vc Iscref

(15b)

1

(16a)

0

H2

This implies that Iscref =

Po − Pfcref 2 vc

and Pfcref = Po − 2 vc Iscref

Po − Pof 2 vc

(17a)

with Pfc min ≤ Pof ≤ Pfc max

Vcth1 V 1

(16b)

A supercapacitor is known for its fast dynamic. It is suitable for rapid load changes and can withstand a large number of chargedischarge cycles without degradation [8]. Moreover, a fuel cell is known for its slow dynamics. Its power should be limited in a given interval [Pfc min , Pfc max ]. In addition, the slope of the FC current or power must be limited to avoid the phenomenon of fuel starvation. For these reasons, a second order filter is used to limit the dynamics of the power supplied by the fuel cell [3,12], as illustrated in Fig. 2, (the design of the filter is explained in Appendix A). Hence, in the normal case of the the supercapacitor SOC, we have Pfcref = Pof . Which gives, in view of (16a) Iscref =

H1

(17b)

Furthermore, the Iscref reference signal should be controlled to maintain the voltage of the supercapacitor in an interval [Vcmin , Vcmax ]. It follows, then, that the role of the block “supercapacitor voltage Limitation” can be described as follows:

Vcnom

ΔV2 V2

Vcth2

vc

Fig. 3. Hysterical block to bring vc to vc nom.

- When vc < Vcmin the discharge of the supercapacitor should be disabled. It is then allowed to be charged by the fuel cell. - If vc > Vcmax , the supercapacitor charging must be disabled and it must contribute to the propulsion of the vehicle In addition, to bring the supercapacitor voltage close to the normal value Vcnom , we introduce a hysterical block as shown in Fig. 3, with V cth2 = V 2 + V 1 And V cth1 = V 1 − V 1

(18a)

Vcth1 and Vcth2 are the voltage thresholds of the supercapacitor. To keep the SC voltage in the recommended limits while ensuring a certain safety margin, these threshold voltages are chosen as follows V cmin < V cth1 and V cmax > V cth2 Remark 2. lows [22]. soc =

(18b)

The supercapacitor state-of-charge is defined as fol-

vc − VC min VC max − VC min

(19a)

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Fig. 4. Flow chart showing the decision tree of the rule-based control algorithm.

Consequently, in the normal case socmin ≤ SOC ≤ socmax with socmin =

vcth1 − VC min VC max − VC min

(19b)

and

Iscref =

socmax =

vcth2 − VC min VC max − VC min

.

(19c)

During both propulsion and braking the controller enforces upper and lower state-of-charge boundaries. The imposed upper bound socmax prevents against overcharging. The lower bound socmin is chosen for efficiency of operation [22]. 䊐 The detailed function of the “SuperC Voltage Controlling Functions” is explained in the algorithm shown in Fig. 4. In this algorithm one can see that the boundary conditions are taken into account. Especially when the supercapacitor voltage Vc is lower than Vcth1 (lower limit) or higher than Vcth2 (higher limit). Accordingly, and using (19a)–(19c), the SOC is always between SOCmin and SOCmax . Also, Fig. 4 shows how to toggle between supercapacitor charge and discharge by adequately imposing the reference current Iscref according to the fuel cell power limits Pfcmin and Pfcmax . Furthermore, in order to bring the SC voltage to its nominal value one can replace the first conditions of the algorithm of Fig. 4 by their corresponding logical conditions (combining H1 an H2 of Fig. 3) as follows

⎧ V > vc ⎪ ⎨ cth1

¯ 1 &H2 →H

Vcth2 < vc

¯2 → H1 &H

Vcth1 ≤ vc ≤ Vcth2

→ H1 &H2

⎪ ⎩

Po − Pof 2 vc

Po − Pfc min

(22a)

2 vc

and this until vc = Vcth2 − V1 − V2 c)Case 3: Supercapacitor operating out of bounds with vc < Vcth1 If vc < Vcth1 then the fuel cell must give full power to power the traction motor and allow charging the supercapacitor. In this case Pfcref = Pfc max , and Iscref =

Po − Pfc max

(22b)

2 vc

until vc = Vcth1 + V1 + V2 . In the latter two cases, the final values Vcth1 + V1 + V2 and Vcth2 − V1 − V2 must be located in the vicinity of Vcnom . Moreover, in the two cases of SC functioning out of range (vc > Vcth2 and vc < Vcth1 ), to avoid Pfc discontinuities that can be caused by algorithm commutations, we impose the reference current of the supercapacitor to take following values: If vc > Vcth2 (vc − (Vcth2 − V1 − V2 )) Po − Pfc min Vcth2 2 vc

Iscref =

(23a)

If vc < Vcth1 (20)

where the “&” symbol designates the logic AND. The target is now to determine the functions i (i = 1,2,3) in Fig. 4. We will do this by examining the following cases: a) Case 1: Normal operating of the supercapacitor (vc ∈ [Vcth1 , Vcth2 ]) In normal operating of the supercapacitor (that is to say vc ∈ [Vcth1 , Vcth2 ]), according to (17a), we can write

1 (t) =

When vc > Vcth2 the supercapacitor must contribute to the propulsion of the vehicle and Pfcref should be maintained at its minimum value Pfcmin . The reference current becomes

(21a)

b) Case 2: Supercapacitor operating out of bounds with vc > Vcth2

Iscref = ı

(−vc + (Vcth1 + V1 + V2 )) Po − Pfc max Vcth1 2 vc

(23b)

where ≥ 1 and ı ≥ 1 are tuning parameters designed to compensate for attenuation caused by the expressions (vc −Vcth2 +V1 +V2 ) (−vc +Vcth1 +V1 +V2 ) and introduced, respectively, V V cth2

cth1

in (22a) and (22b) in order to oblige Iscref to continuously vanishes, since isc = 0 in the steady state. Afterwards we put ˇ1 (t) =

(vc − Vcth2 + V1 + V2 ) Vcth2

(24a)

(−vc − Vcth1 + V1 + V2 ) Vcth1

(24b)

and ˇ2 (t) = ı

A. Tahri et al. / Electric Power Systems Research 160 (2018) 89–98

95

Fig. 5. Power management under Vcth 1 ≤ vc ≤ Vcth 2 and controllers behaviors.

We obtain then, given the equations (16b) and (23a)–(23b), both expressions of Pfc , in the steady state, corresponding to the values of Iscref in the two latest cases If vc > Vcth2 Pfc = Po − ˇ1 (t)(Po − Pfc min )

(25a)

If vc < Vcth1 Pfc = Po − ˇ2 (t)(Po − Pfc max )

(25b)

In the last two expressions, we replace PO by Pof (see Fig. 3) to prevent Po rapid changes to affect Pfc , so we get If Vcth2 < vc Pfcref = Pof − ˇ1 (t)(Pof − Pfc min )

(26a)

If vc < Vcth1 Pfcref = Pof − ˇ2 (t)(Pof − Pfc max )

(26b)

It follows, in view of (16a), the Iscref current is obtained as follows If Vcth2 < vc Iscref = (Po − (1 − ˇ1 (t))Pof − ˇ1 (t)Pfc min )/(2 vc (t))

(27b)

Thus, the terms of the functions 2 (t) and 3 (t) are written so

2 (t) = (Po − (1 − ˇ1 (t))Pof − ˇ1 (t)Pfc min )/(2 vc (t))

(28a)

and

3 (t) = (Po − (1 − ˇ2 (t))Pof − ˇ2 (t)Pfcmax )/(2 vc (t))

Element

Its characteristics

Fuel cell Inductances L1 and L2 Inductances ESR: R1 and R2 Supercapapcitor, Csc Supercapacitor ESR: Rsc PWM switching frequency

12.5 kW PEMFC, Vfcmax = 500 V, Ifcmax = 70 A 4.2 mH 4.2 m 9.95 F 0.144  10 kHz

Table 3 Controller parameters. Parameter

Value

c1 c2 h  ı Efficiency factors 1 , o, ␭u Filter natural frequency ωn Filter damping ratio z

100 2000 500 10 35 0.95; 0.95; 1.05 31.6 rad s−1 1.58

(27a)

If vc < Vcth1 Iscref = (Po − (1 − ˇ2 (t))Pof − ˇ2 (t)Pfc max )/(2 vc (t)

Table 2 System parameters.

(28b)

Remark 3. Assuming vdc and io are measurable the power Po = vdc io is instantaneously captured. The sign of this power indicate whether the engine is in the traction mode (Po > 0) or in the regenerative mode (Po < 0). The value of the power Po is then sent to our proposed algorithm (shown in Fig. 4) to determine an adequate Iscref allowing the charge (Iscref < 0) or the discharge (Iscref > 0) of the supercapacitor. 5. Simulation results The performances of the proposed power management and the nonlinear controller are illustrated through simulations with the Matlab Simpower System toolbox (SPS). The binary inputs u1 and u2 are generated from 12 according to (5) [20]. The parameter

characteristics of the system are listed in Table 2. The controller parameters are fixed as in Table 3. The design parameters c1 , c2 and  h have been selected using a ‘trial-and-error’ search method and proved to be suitable. In this simulation we choose V1 = V2 = 4 V. Fig. 5 shows that the FC provides the average power required by the load while the SC provides power during transient phases. Indeed, the sharing of roles between the FC and the SC can be explained as follows: - The maximum power of the FC being fixed at 13 kW and the startup power demand is of 29 kW, we find that between 0 and 3 s, the power management system controls the SC to provide the power difference which is 16 kW. In presence of a step change of the demanded output power Po at 3 s, the FC power is constant and the supercapacitor gives the extra power to the load. - At time 4 s, a quick deceleration mode occurs, resulting in a recovery of energy in the SC between 4 s and 5 s. Here the FC provides the power for the load with a slope inferior to the given maximum value, thanks to the filtering bloc (see Fig. 2). - At time 6 s, a braking mode (the torque is negative) is operated in order to maintain the constant speed in the case of a downhill vehicle, for example. The recovered power is then stored in the SC While the FC gives the minimum power to feed these accessories

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Fig. 6. Power management and SOC control when vc < Vcth 1 .

Fig. 7. Power management and control of SC when vc > Vcth 2 .

and to maintain constant the DC bus voltage. It should be noted that in this mode, the voltage of the supercapacitor is maintained between presumed limits, namely Vcth1 = 196 V and Vcth2 = 404 V. This figure also shows the controller performances, namely a tight regulation of the DC bus voltage to its desired value and a good tracking of the SC current to its reference signal provided by the PMS system. Fig. 6 illustrates the case of a deep discharge of the SC between 0.8 s and 6 s which causes the voltage vc being below Vcth1 (at 6 s vc = 180 V while Vcth1 = 196). One can see from this figure that, when Po < Pfc max (at 8 s Po = 2.85 kW and Pfc max = 13 kW), the FC feeds the traction motor and charges the SC until vc = Vcth1 + V1 + V2 = 204 V at time 14 s. In this case the SC current vanishes (isc = 0 A). Moreover, we note that, in spite of the rapid changes in load power Po (especially at t = 8 s), the FC is not overstressed preventing the phenomenon of “fuel starvation”. This is, indeed, made possible thanks to the good tracking of the current isc to the reference signal 3 (t) generated by the PMS. That is, at the end of the mode vc < Vcth1 (t = 14 s) Iscref = 0, guaranting that Iscref and Pfc vary smoothly at the commutation instants (from vc < Vcth1 to Vcth1 ≤ vc ≤ Vcth2 ). Fig. 7 illustrates the case of a long recovery mode. As vc becomes greater than Vcth2 (at 4 s vc = 408 V while Vcth2 = 404). We see during this mode (vc > Vcth2 ) that the SC contributes to the traction motor power (isc > 0) until vc = Vcth2 − V1 − V2 = 396 V (at 14 s vc = 396 V

and isc = 0). The switch to the normal mode thereby smoothly through the PMS which then delivers the signal 2 (t). The result is a smooth trajectory of the power Pfc of the FC, thus avoiding the problem of “fuel starvation”. Finally, Fig. 8 shows the controller and power management system performances in the presence of an engine speed profile (depending on the NEDC between 700 s and 800 s) and a trajectory of the load torque. We note that, in addition to the controller of the HESS and the power management system we used also the regulator obtained in Ref. [15], that is to control the inverter asynchronous motor combination. We notice that the supercapacitor provides the energy to the load at startup during the interval [0.6 s] and recovers energy during deceleration phases (interval [6 s,7 s]) and braking state (interval [12 s,14 s]). 6. Conclusion In this study, we dealt with the problem of power optimization of a hybrid energy storage system consisting of a fuel cell (main source) and a supercapacitor (auxiliary source). The study consists of two different parts: control of power stage and the power management system. The first part of the paper is devoted to the control of two power converters (BC and BBC). The control is addressed based on the Lyapunov approach, the control objectives are: (i) a good regulation of the DC Link voltage; (ii) a perfect tracking of SC current to its

A. Tahri et al. / Electric Power Systems Research 160 (2018) 89–98

97

Fig. 8. Power management under a part of NEDC profile combined with a load torque profile.

reference provided by the PMS. It is formally demonstrated, using a rigorous test performed on a higher fidelity simulation model, that the proposed PMS and the regulator have achieved all the objectives for which they were designed The second part is devoted to the power management system (PMS) that allows an appropriate distribution of the power demanded by the load between the two sources. The objectives of the PMS are the following: - Taking into account a slow dynamic of the fuel cell, in order to avoid the phenomenon “fuel starvation” of the FC. - Recovering energy during braking or deceleration phases. - Maintaining the SC voltage between two levels, fixed a priori.

where ωn is the natural frequency and z is the damping ratio. One can find the solution of the differential equation derived from (29a), in over damped case (z > 1) with Pof (0) = 0 and P˙ of (0) = 0 when the input is a unit step, PoL (s) = 1s , Pof (t) = 1 −

Acknowledgments



P˙ of (t) = −p1

−z +

2zs ωn

+1

z2 − 1

−1

ep2 t



and p2 = ωn −z +

(29b)



z2 − 1 .

z2 − 1

z2 − 1

2

ep1 t + p2

−z −





2

z2 − 1

ep2 t

(30a)

z2 − 1

2

z 2 −1

2

demonstrate for all t. P˙ of (t) < p2

−z −

z 2 −1



z2 − 1



ep2 t

(30b)

z2 − 1

2

Through calculation one finds P˙ of (t) <

ωn



2

ep2 t

(30c)

z2 − 1

˙ can be bounded as follows For all time, the derivative s(t) P˙ of (t) <

ωn



2

(31)

z2 − 1

We study the response of the Pfc filter (see Fig. 3) when the input is a PoL step whose final value is Pfc max . Using (31) we obtain the

< Pfc max

dPof dt

ωn



2

(32)

z2 − 1

Since Pfcref = Pof , we choose ωn and z in order to satisfy the dynamics of the fuel cell. Thus we obtain

According to Fig. 2, one has +

2



z2 − 1

z2

p1 and p2 are negative, then√for all t Since √ 2 2 √ z −1 ep2 t > 0 and −p1 −z+ √ z −1 ep1 t < 0. Therefore we p2 −z−

dt

PoL (s)





+





dPof

1

−z −

Applying the derivative property of the exponential yields

Appendix A.

s2 ωn2

e

p1 t

−1



upper bound of

Pof (s) =

z2 − 1

z2

2

This work is supported by the Moroccan Ministry of Higher Education (MESRSFC) and the CNRST under the grant PPR/2015/36.

Design of the Pfc filer



With p1 = ωn −z −

Indeed, theoretical approach and simulation results demonstrate; - The fuel cell power is always kept between a low bound Pfc min and a high bound Pfc max . - The variation slope of the fuel cell power is limited even during steps changes of the load power. - The fuel cell constraints are accounted for when it charges the supercapacitor when it has a low state of charge. - When the supercapacitor is over charged it feeds the load accounting for the fuel cell slow dynamics. - In both cases, when the supercapactor is overcharged or it’s undercharged, the power management system can bring the supercapcitor state of charge to its nominal value. - The control of the supercapacitor state of charge allows to store all energy recovered in a decelerating and a breaking phases.

−z +

 (29a)

dPfc dt



= Pfc max max

ωn



2

z2 − 1

(33)

98

A. Tahri et al. / Electric Power Systems Research 160 (2018) 89–98

This equation shows that, given Pfcmax ,

dPfc dt

and choosing max

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