Control of N -parallel Connected Boost Converters

Feeding a Constant Power Load: an Automotive

Case Study

Jérémy Malaizé and Wissam Dib

Control and System Department

IFP New Energy

1-4 avenue de Bois-Préau

92 852 Rueil-Malmaison, FRANCE

Abstract�This paper concern is in the control of an arbitrarynumber of voltage sources feeding a constant power load,namely electric machines, through boost converters connected inparallel. The proposed control scheme achieves global asymptoticconvergence of the currents within the paralleled branches andthe voltage at the input ports of the load towards their respectivereference trajectories. Through an automotive case study, weassess the relevancy of this controller. We consider a full electricshuttle dedicated to inner-city trips, and featuring an hybridenergy storage system, composed of battery and ultra-capacitorcells. Some simulation results illustrate this method.

I. INTRODUCTION

To comply with stringent pollutants emissions constraints,

and to ful ll the growing consumer demand for more fuel-

ef cient cars, manufacturers lean towards more complex pow-

ertrains featuring green components, such as electric machines.

However, the side effect to the advent of more electric cars is

a dramatic increase in complexity of the onboard electric grid

architecture. This trend is accompanied by a crucial need for

dedicated control solutions to ensure the stability of the grid

and the desired performances. This paper is more speci cally

concerned with automotive applications featuring an associ-

ation of several voltage sources feeding electric machines,

viewed as constant power loads. This situation occurs as soon

as several battery modules are required to meet the overall

energy needs for a given vehicle, see [1]. This may also arise

within Hybrid Energy Storage Systems (HESS), see [2], made

up of a combination of voltage sources of different natures,

for instance battery and ultracapacitor (UC) cells. We consider

an active coupling between the voltage sources and the load

with the use of additional DC-DC power converters, and this

paper deals with the control of these systems.

The current and voltage control of standalone DC-DC units

is widely addressed in literature, with solutions ranging from

backstepping-based controllers [3], to passivity-based schemes

[4], [5], [6], to synergetic control designs [7]. However, great

care must be taken when connecting a DC-DC converter

to a constant power load. This arrangement may lead to

unstable operating modes [8], [7], and additional re nements

are required from a control point of view to arrive at the

desired performances.

When it comes to the association of several DC-DC convert-

ers in parallel ( gure 1), the degrees of freedom offered by this

architecture may be used to provide extra features. Through

this redundancy, system designers may intend to improve reli-

ability and ef ciency. There is a wide eld of research related

to this context, the underlying idea consisting in equaling the

currents within the different paralleled branches. Appealing as

this may seem, this leads to very challenging control issues

[9], and the system under study may even be showed to be

chaotic [10] if not controlled properly. Extensive work has

been done to perform the so-called current sharing between the

different converters. Several nonlinear control techniques have

been reported, namely sliding mode control [11], synergetic

control [12] or robust control schemes [13].

For automotive applications, equaling currents within the

different voltage sources may not be relevant. Instead, one

may attempt to conduct the power shunt within the sources

so as to minimize the size or the stress of the most valuable

components, such as the battery cells. This is the case in [14]

and [15], while in [16], the optimization of the losses within

the different sources is at stake.

This paper is concerned with the control of an arrangement

of an arbitrary number of boost converters in parallel, connect-

ing a load featuring a constant behavior to voltage sources. We

propose a globally stabilizing controller for:

• the currents within the voltage sources to track some

prescribed trajectories, obtained via [16],

• the dc-link voltage at the input ports of the load to rally

a desired value.

The suggested control law is then applied to an automotive

case study, namely the control of a UC - battery combination

for a full electric shuttle. Realistic simulation results prove the

relevancy of the approach.

This paper is organized as follows. In section II, the control

objectives are clearly stated, and a complete modeling of the

system to control is given in section III. The proposed control

scheme is reported in section IV. Section V is devoted to the

detailed decription of the main components of the powertrain.

In section VI, we provide some details on the implementation

of the controller as well as some simulation results showing

978-1-61284-972-0/11/$26.00 ©2011 IEEE 528

the ability of the proposed scheme to perform as expected in

a vehicular application.

II. CONTROL OBJECTIVES

I0 = P0/V0

V0

I2

I1

V2

V1

r0 C0

IN

VN

Iout

1

Iout

2

Iout

N

Fig. 1. Schematic view of an N -paralleled arrangement of boost convertersconnected to a constant power load.

We consider a set of N revertible step-up converters, as

represented in gure 1. They are connected to a load exhibiting

a constant, or at least slowly-varying, power behavior with

value P0. The voltage V0 at the input ports of the load, and

the current I0 entering it are actually related by I0 = P0

/V0.

The N boost converters are used to supply the load from Nvoltage sources, each of them having a magnitude Vn and an

output current In, for 1 ≤ n ≤ N .

The remainder of this paper is devoted to the design of a

controller meeting the following requirements:

• voltage stability: the voltage V0 converges towards a

given and slowly-varying voltage reference trajectory V ∗

0 .

• power ow: for n ≥ 2, the current In drained from the

n-th source converges towards a given and slowly-varyingcurrent reference trajectory I∗n.

• measurements: for n ≥ 0, the voltages Vn as well as the

currents In are assumed to be known.

It is not possible to maintain a desired voltage at the load

input ports and, at the same time, control the currents !owing

out of the N sources. The design of the currents and voltage

references is beyond the scope of this paper. Finally, the

control actions consist in driving the duty cycles of each

converters so as to achieve the desired tracking performances.

III. SYSTEM MODELING

Each revertible step-up converter admits a lumped-

parameter electrical equivalent circuit as depicted in gure 2

with:

• un is the duty cycle of the n-th converter, and T is the

switching period common to each converter,

• the power switches K1,n and K2,n respectively remain

closed and opened during unT , and conversely opened

and closed for a time window of length (1− un)T ,

Vn

Ln Rn

Cn rn

In

V0

Ioutn

K2,n

K1,n

Fig. 2. Electrical architecture of a boost converter.

• Ln and Cn are the current smoothing inductor and

voltage smoothing capacitor of the n-th converter respec-tively, with respective parasitic and leakage resistors Rn

and rn.

One may cast an averaged modeling over the period T of the

circuit represented in gure 2:

LnIn = −RnIn + Vn − unV0

Ioutn = unIn − CnV0 − V0

/rn

(1)

for any n such that 1 ≤ n ≤ N . In the end, one easily derives

the full state-space representation including the dynamics in

V0 by writing down the Kirchoff�s law at the input ports of

the constant power load:

P0

V0=

N∑

n= 1

Ioutn − C0V0 − V0

/r0.

Combining the previous equation with the second line in (1),

the state-space realization of the whole system represented in

gure 1 eventually reads:

CV0 = −V0

r−

P0

V0+

N∑

n= 1

unIn

LnIn = −RnIn + Vn − unV0, 1 ≤ n ≤ N

(2)

where 1/r =

N∑

n= 0

(1/rn)and C =

N∑

n= 0

Cn.

Recall that the current references I∗n for n ≥ 2 and voltage

reference V ∗

0 are set by an autonomous trajectory generation

unit. They are now assumed to be strictly constant, and to

solve the equations (2) at equilibrium, namely:

0 = −V ∗

0

r−

P0

V ∗

0

+

N∑

n= 1

u∗

nI∗

n

0 = −RnI∗

n + Vn − u∗

nV∗

0 , 1 ≤ n ≤ N,

(3)

where u∗

n is the duty cycle of the n-th converters at equi-

librium. Let us de ne the tracking errors between the actual

values of the currents and voltage and their respective trajec-

tories:

V0 = V0 − V ∗

0 , In = In − I∗n. (4)

529

IV. CONTROLLER DESIGN

The main contribution of this paper is stated in the fol-

lowing proposition where the suggested control law ensures

the asymptotic convergence of V0 and In to their respective

desired values V ∗

0 and I∗n.

Proposition 1:Let un be the state feedback de ned as

follows:

un = u∗

n + un − knyn, 1 ≤ n ≤ N, (5)

with u∗

n de ned by (3), yn given by:

yn = V0I∗

n − InV∗

0 , (6)

and un de ned as:

un = −γynV0

V ∗

0

, γ =P0V0

(V ∗

0 + V0

) N∑

n= 1

y2n

. (7)

kn are some positive scalars. The control law (5) achieves

global asymptotic convergence of the system (2) to its desired

references.

Proof:The proof proceeds in three steps:

(i) We rst introduce two algebraic equations guiding us for

the remainder of the proof.

(ii) We then show that applying the state feedback (5) makes

the outputs yn converge to zero.

(iii) We eventually prove the zero-state detectability, namely

that the errors V0 and In vanish to zero when yn converge

to zero.

The quantities γ and un are actually related by the following

equations:

−V ∗

0

r+

N∑

n= 1

(u∗

n + un) I∗

n −P0

V ∗

0 + V0

= −γ

N∑

n= 1

ynIn

−RnI∗

n + Vn − (u∗

n + un)V∗

0 = γynV0.(8)

One may derive the explicit relations (8) by using the equilib-

rium (3), and solving (8) for un and γ.

We may now rewrite equations (2) in terms of the tracking

errors (4), and use the relations (8):

C˙V 0 = −

V0

r+

N∑

n= 1

((un − γyn) In − knynI

∗

n

)

Ln˙In = −RnIn − (un − γyn) V0 + knynV

∗

0 .

(9)

To prove the assertion reported in (ii), we consider a

Lyapunov candidate function de ned as follows:

V = C(V0

)2

+

N∑

n= 1

Ln

(In

)2

. (10)

We then compute its time derivative with the use of the error

dynamics (9):

V =−V 20

r−

N∑

n= 1

RnI2n

+ V0

N∑

n= 1

((un − γyn) In − knynI

∗

n

)

+

N∑

n= 1

In

(− (un − γyn) V0 + knynV

∗

0

).

The second order terms InV0 of the second and third lines

in the previous equation cancel each other out, and using the

negativity of the rst two terms, one eventually gets:

V ≤ −

N∑

n= 1

knyn

(V0I

∗

n − InV∗

0

)= −

N∑

n= 1

kny2n,

where the de nition (6) of yn has been used. Via Lasalle�s

invariance principle, this equation suf ces to conclude towards

the convergence of yn to zero for any n.

We now move on to the proof of the assertion reported in

(iii). If all the yn converge to zero, one may conclude that

there exists a unique scalar λ such that the following chained

equalities hold:

V0

V ∗

0

=I1I∗1

= . . . =INI∗N

= λ.

This latter relation shows that the vector of the tracking errors

is proportional to the equilibrium state. We may then rewrite

the set of equations (9) as:

λCV ∗

0 = −αV ∗

0

r+

N∑

n= 1

(λunI∗

n − λγynI∗

n − knynI∗

n)

λLnI∗

n = −λRnI∗

n − λunV∗

0 + λγynV∗

0 + knynV∗

0

If we multiply the rst line of this equation by V ∗

0 and the

following lines by I∗n, and then add up all these lines, one getsthe following differential equation for λ:

λ = −

(V ∗

0 )2 /

r +

N∑

n= 1

Rn (I∗

n)2

C (V ∗

0 )2+

N∑

n= 1

Ln (I∗

n)2

λ.

This allows to conclude that λ → 0, and that the tracking

errors vanish to zero when applying the state feedback (5).

Remark1:Note that the de nition of un is not jeopardized

even if yn tends to zero. This is actually due to the zero-state

detectability of the outputs yn that cause the voltage error to

decay to zero with the same rate as yn. For suf ciently small

values of the yn, the component un of the feedback (5) still

remain bounded, as it admits a continuous extension, and may

be proved to converge to zero.

530

V. BATTERY - UC COMBINATION IN AN AUTOMOTIVE

APPLICATION

A.Vehicle overview

From now on, we suggest to validate the control scheme (5)

through simulations of a vehicular application. We consider

a shuttle dedicated to inner city transportation and featuring

a full electric powertrain including two in-wheel hub motors

mounted on the rear axle. These motors are powered by a hy-

brid energy storage system (HESS) made up of an arrangement

of battery and ultra-capacitor (UC) cells, connected in parallel

via two boost converters. This architecture is consistent with

the one considered in this paper, namely with N = 2.Referring to gure 1, we shall hereafter consider that:

• the two in-wheel hub motors constitute the constant

power load, and P0 equals the total amount of electric

power required for traction,

• the battery pack plays the role of the rst voltage source

in gure 1, and V1 and I1 equal the battery pack voltageand the current within the battery respectively,

• the UC pack is connected to the second branch in gure

1, such that V2 and I2 equal the voltage at the input portsof the UC pack and the current within it respectively.

B.Vehicle chassis

We denote by ν the vehicle speed, M its mass subject to

variations according to the passengers getting on and off at

each stop and τ the wheel torque resulting from the combined

actions of the hub motors and the braking system. The vehicle

is considered as a point mass M with dynamics given by :

Mν =τ

r−

1

2ρSCxν |ν| −Mg (sin (α) + µν + fsign (ν)) ,

(11)

where the notations are as follows :

• r is the wheel radius and g the acceleration due to gravity,• ρ is the air density, S the reference area and Cx the drag

coef cients in order to model aerodynamic drag,

• µ and f respectively model viscous and dry rolling

resistance

• α is the slope along the shuttle trip.

The wheel torque τ is split up into the mechanical breaking

torque τbrk and τelec the torque supplied by each motor:

τ = τbrk + 2τelec. (12)

C.Hubmotors

Among existing electric machine technologies, we chose the

!ux-concentrated type interior permanent magnet synchronous

motor as the best candidate for the considered in-wheel hub

motor. The motor is connected to the wheel via a reduction

gear of value γ. For a given driving cycle, one may derive

the maximum required torque at low speed, as well as the

maximum rotation speed and the required power at high

speed. Using an analytical approach to design the machine

parameters, see [17], we end up with a static map, plotted in

gure 3, that suf ces to model each hub motor together with its

voltage inverter throughout the remainder of this work. Let us

0

5000

10000

"200"100

0100

2000

2000

4000

6000

8000

ωmot [rpm]τmot [N m]

Pl[W

]

Fig. 3. Static mappings of the in-wheel hub motor losses.

denote ωm ot and τm ot as the rotation speed and the delivered

torque, gure 3 shows the motor losses power Pl (ωm ot, τm ot).More speci cally, this mapping Pl is used to compute the value

of the load power P0:

P0 = V0I0 = 2 (ωm otτm ot + Pl (ωm ot, τm ot)) . (13)

The total load power may be related to the vehicle speed and

the wheel torque viathe reduction gear. One may replace ωm ot

and τm ot in (13) by the following relations:

ωm ot = γν/r, τm ot = γ (τ − τbrk )/(2γ) . (14)

D.HESS

We now proceed to describing the energy storage sys-

tem,composed of a combination of battery and UC cells.

Regarding the battery, we consider an array of N(s)b × N

(p)b

identical elementary cells, in an arrangement of N(p)b branches

in parallel, each branch consisting of N(s)b cells in series. In

the end, the current I1 within the battery and voltage V1 at its

input ports are given by :

I1 = N(p)b Icellb , V1 = N

(s)b V cell

b , (15)

where Icellb and V cellb are the current and voltage of each cell

respectively. Using notations of the same kind, the UC pack

is made up of N (s)u × N (p)

u identical cells, and one de nes

the current within the pack, as well as the voltage at its input

ports as :

I2 = N (p)u Icellu , V2 = N (s)

u V cellu . (16)

1)Batterycellmodeling:Each battery cell is modeled as

a three-state lumped parameter dynamical system, see [18],

where the considered states are the cell state of charge xb and

two voltages(V

(1)b , V

(2)b

)associated with internal capacitors.

The complete modeling reads:

˙SoCb =−I1

N(p)b Qb

V(1)b =

I1

N(p)b C

(1)b

−V

(1)b

R(1)b C

(1)b

V(2)b =

I1

N(p)b C

(2)b

−V

(2)b

R(2)b C

(2)b

,

(17)

where :

531

• the functions R(i)b and C

(i)b depend on the battery SoC

and temperature Tb,

• Qb is the battery cell nominal storage capacity, mb the

cell mass and cb its speci c heat capacity,

In the end, the battery voltage V1 may be expressed as :

V1 = N(s)b

(Eb − V

(1)b − V

(2)b

)−

N(s)b

N(p)b

R(0)b I1, (18)

where Eb is the no-load cell voltage depending on xb and Tb.

2)Ultracapacitor cellmodeling:To derive an appropriate

modeling for each UC cell, the lumped parameter approach

in [19] is used. An internal voltage VC , related to a capacitor

Cu depending on VC , together with the cell temperature Tu,

is considered as the only state variable. The associated state-

space realization may be given by :

VC = −Icellu

N(p)u Cu

−VC

Rleaku Cu

(19)

where Rleaku accounts for leakages within the UC. In the same

vein as in (18), one easily derives the expression for the UC

pack voltage :

V2 = N (s)u VC −

N(s)u

N(p)u

R(0)u I2. (20)

E.Driver

The driver�s objective is to get the vehicle following a

desired path, which is commonly addressed as making the

vehicle speed ν track a velocity pro le ν∗. When operating

the acceleration and braking pedals, the driver sets the wheel

torque τ in (11). In the following, τ is given as the output

of PI controller with additional feedforward terms to enhance

velocity tracking :

τ = kp (ν∗ − ν) + ki

∫ t

0

(ν∗(s)− ν(s)) ds

+1

2ρSCxν

∗ |ν∗|+Mg (µν∗ + fsign (ν∗)) ,

(21)

where kp and ki are two scalars, and the other notations are

the same as in (11).

F.Vehicle controlunit

Given the overall vehicle architecture, the vehicle control

unit needs to perform two main tasks, that will delibarately be

only mentioned without any additional details:

• the traction supervision, namely to split up the total wheel

torque τ required by the driver into a mechanical braking

torque, and a electric hub motor torque, so as to ful ll

(12),

• the HESS supervision, namely for that particular case to

compute the reference trajectories V ∗

0 and I∗2 .

VI. SIMULATION RESULTS

A.Simulation setup

From this point on, we consider a high frequency modeling

of the boost converters. The power switches are assumed to

operate at T = 1ms, in an ideal and lossless manner. To

get rid of the high frequency components inherent to the

switchings, the available measurements pass through a low-

pass lter whose bandwidth is set to 10/T = 100Hz. Toimplement the control law (5), it is necessary to compute the

control inputs at equilibrium u∗

1 and u∗

2, as well as the current

within the battery at rest, namely I∗1 . Remember that V∗

0 and

I∗2 are set by the vehicle control unit, and I∗1 is constrained

as result of this choice. To solve (3) for the u∗

i and I∗1 , anumerical descent technique is used, and it of course allows

for the real-time operation of the controller. Finally, the boost

converters control unit is triggered at the same sampling time

T = 1ms as the switching, and the control inputs are kept

constant during intervals of length T .

B.Results

The vehicle follows the driving cycle reported in gure 4(a),

which is typical of an urban-like trip featuring several stops.

The total amount of electrical power P0 for these driving

conditions is given in gure 4(b). As exhibited in gure 4(d),

the desired voltage V ∗

0 at the input ports of the in-wheel

inverters is set to 650V , and the proposed scheme performs

well in maintaining the actual value of V0 to that value. The

current-sharing autonomous control unit aims at minimizing

the battery stress by asking the UC pack to deliver the high

frequency component of the total line current I0 = P0/V0.

In return, the battery supplies a rather constant current. The

evolutions of I1 for the battery and I2 for the UC are given

in gure 4(e). As given by the equations (18) and (20),

when current is drained from the voltage sources, their output

voltages change. The battery voltage exhibits a long term drift,

as can be seen in gure 4(c), this behavior is related to its

discharge. The UC pack voltage undergoes signi cant voltages

variations since larger currents are required, and it discharges

more rapidly than the battery.

The controller performance may be evaluated in gures 4(f),

4(g) and 4(h). Recall that this control scheme is rst intended

to deal with constant voltages V1 and V2, and for constant

power load P0. We have already showed that the power request

signi cantly changes along the cycle, so do the voltages. The

proposed scheme proves to be robust to these time-varying

conditions, while it is primarily not designed for that. In gures

4(g) and 4(h), one may notice some minor deviations around

the desired voltage, and the actual value of the UC current

tracks its time-varying reference.

VII. CONCLUSION

A new control scheme for the control of N boost converters

connected in parallel is proposed in this paper. Theoretical

proof as well as practical implementation results are provided.

The control scheme allows a tight control of the voltage at the

load ports, and of the currents split within the different sources.

532

100 200 300 400 5000

10

20

30

40

50

time (s)

ν(k

m/h

)

(a) Vehicle speed ν.

100 200 300 400 500"100

"50

0

50

100

time (s)

P0

(kW

)(b) Total electric power P0.

0 200 4000

200

400

600

time (s)

(V1,V

2)

(V)

V1

V2

(c) UC and battery voltages.

0 100 200 300 4000

200

400

600

800

time (s)

V0

(V)

(d) Load voltage.

0 200 400

"200

"100

0

100

time (s)

(I1,I 2

)(A

)

I2

I1

(e) UC and battery currents.

0 200 4000.5

0.6

0.7

0.8

0.9

1

time (s)

(u1,u

2)

(0-1

)

u1

u2

(f) Duty cycles (u1,u2).

100 110 120 130 140 150640

645

650

655

660

time (s)

V0

(V)

V0

V ∗

0

(g) Zoom in on the tracking of V ∗

0.

100 120 140"200

"100

0

100

200

time (s)

I 2(A

)

I2

I∗2

(h) Zoom in on the tracking of I2∗.

Fig. 4. Simulation results for a typical urban-like driving cycle.

Future work will consist in applying the proposed technique

to an experimental testbed.

ACKNOWLEDGEMENTS

This work is partially supported by a grant from the French

Agency for Environment and Energy Management, ADEME.

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