Parallel interconnection of buck converters revisited

Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress The International Federation of Automatic Control Proceedings of the 20th9-14, World Toulouse, France, July 2017 The International Federation of Congress Automatic Control Available online at www.sciencedirect.com Toulouse, France, July 9-14, 2017 The International Federation of Automatic Control Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017

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IFAC PapersOnLine 50-1 (2017) 15792–15797 Parallel interconnection of buck Parallel interconnection of buck Parallel interconnection of buck Parallel interconnection of buck revisited revisited revisited revisited ∗

converters converters converters converters

Jean-François Trégouët ∗ Romain Delpoux ∗∗ Jean-François Jean-François Trégouët Trégouët ∗ Romain Romain Delpoux Delpoux ∗ Jean-François Trégouët ∗ Romain Delpoux ∗ ∗ ∗ Laboratoire Ampère CNRS UMR 5005, Université de Lyon, INSA ∗ Laboratoire Ampère CNRS UMR 5005, Université de Lyon, INSA Laboratoire Ampère UMR 5005, Université de Lyon,(e-mail: INSA 25 avenue JeanCNRS Capelle, 69621 Villeurbanne, France ∗Lyon, Laboratoire Ampère UMR 5005, Université de Lyon,(e-mail: INSA Lyon, 25 Jean Capelle, 69621 Villeurbanne, France Lyon, 25 avenue avenue JeanCNRS Capelle, 69621 Villeurbanne, France (e-mail: [email protected]). Lyon, 25 [email protected]). Jean Capelle, 69621 Villeurbanne, France (e-mail: [email protected]). [email protected]). Abstract: This paper addresses the problem of current-sharing for interconnected power Abstract: This paper the of for power Abstract: It This paperonaddresses addresses the problem problem of current-sharing current-sharing for interconnected interconnected power converters. focuses heterogeneous set of buck converters, feeding an unknown resistive Abstract: This paper addresses the problem of current-sharing for interconnected power converters. It focuses on heterogeneous set of buck converters, feeding an unknown resistive converters. It focusesDC on bus. heterogeneous of buck converters, feeding an unknown resistive load via a common As a mainsetcontribution, it proposes a novel framework which converters. It focuses on heterogeneous set of buck converters, feeding an unknown resistive load via a common DC bus. As a main contribution, it proposes a novel framework which load via a common DC bus. As a main contribution, it proposes a novel framework which completely separate voltage regulation from current distribution dynamics, hence preventing load via a common DC bus. As a main contribution, it proposes a novel framework which completely separate voltage regulation from current distribution dynamics, hence preventing completely separate voltage regulation from current distribution dynamics, preventing undesirable interaction between them. Such a reformulation is performed on thehence open-loop model completely separate voltage regulation from current distribution dynamics, hence preventing undesirable interaction between them. Such a reformulation is performed on the open-loop model undesirable interaction betweenare them. a reformulation is performed the open-loop so that control law candidates not Such restricted to some particular class.onArbitrary largemodel set of undesirable interaction between them. a reformulation isseparation performed onArbitrary the open-loop model so law are not restricted some class. large set so that that control control law candidates candidates are not Such restricted tofrequency some particular particular class. Arbitrary large set of of heterogeneous converters can be handled and noto assumption is required, so that control law candidates are not restricted to some particular class. Arbitrary large of heterogeneous converters can be handled and no frequency separation assumption is required, heterogeneous converters can be handled and no frequency separation assumption is required, hence offering tractability without sacrificing performance. Remarkably, voltage regulationset boils heterogeneous converters can be handled and no frequency separation assumption is required, hence offering tractability without sacrificing performance. Remarkably, voltage regulation boils hence to offering tractability without sacrificing performance. Remarkably, down the control of a single virtual buck converter. Controller design voltage exampleregulation exploitingboils the hence offering tractability without sacrificing performance. Remarkably, down to the of virtual buck Controller design example exploiting the downstructure to the control control of a a single single virtual buck converter. converter. Controller design voltage exampleregulation exploitingboils the new is provided. down to the control of a single virtual buck converter. Controller design example exploiting the new is new structure structure is provided. provided. new structure is provided.Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. © 2017, IFAC (International Keywords: Power-system control, Parallel power converters, Buck converters, Current-sharing, Keywords:structure. Power-system control, control, Parallel Parallel power power converters, converters, Buck Buck converters, converters, Current-sharing, Current-sharing, Keywords: Power-system Cascaded Keywords: Power-system control, Parallel power converters, Buck converters, Current-sharing, Cascaded structure. structure. Cascaded Cascaded structure. avoidably lower performance. Relying on models with com1. INTRODUCTION avoidably lower performance. performance. Relying on on models have with comcom1. INTRODUCTION INTRODUCTION avoidably lower Relying models with 1. plex impedances, interesting research directions been avoidably lower performance. Relying on models with com1. INTRODUCTION plex impedances, interesting research directions have been plex impedances, interesting researchseparation: directions have been opened to go beyond this frequency In (Thotplex impedances, interesting research directions have been Substituting a single high-capacity centralized electrical opened opened to go beyond this frequency separation: In (Thotto go beyond this frequency (Thottuvelil and Verghese, 1997) arbitraryseparation: number ofInidentical Substituting single high-capacity centralized electrical Substituting aa single high-capacity centralized electrical opened to go beyond this frequency separation: In (Thotpower converter by multiple distributed converters con- tuvelil tuvelil and Verghese, 1997) arbitrary number of identical and Verghese, 1997) arbitrary number of identical converters are considered whereas the case of two different Substituting a single electrical power converter converter byis multiple multiple distributed converters conpower by distributed converters contuvelil andare Verghese, 1997) arbitrary number of identical nected in parallel ahigh-capacity strategy thatcentralized becomes more and converters converters are considered whereas the case of two two different considered of different is treated in whereas (Sun et the al., case 2005). Theoretical power converter by multiple distributed converters connected in parallel is a strategy that becomes more and nected in parallel is aparalleling strategy that becomes more and converters converters are considered whereas the case of two different more popular. Indeed, converters offers several is treated in (Sun et al., 2005). Theoretical is treatedof in et system al., 2005). Theoretical certificates the(Sun overall are not formally nected in parallel aparalleling strategyreliability that becomes more and stability more popular. popular. Indeed, paralleling converters offers several more Indeed, converters offers several converters is treated in (Sun et al., 2005). Theoretical advantages such asis increased due to redunstability certificates of the overall system are not formally stability of the overallcontrol systemschemes are not formally provided,certificates though. Furthermore, (the somore popular. Indeed, paralleling converters offers several advantages such as increased reliability due to redunadvantages such as increased reliability due to ease redunstability certificates of the overall system are not formally dancy and distribution of stresses of components, of provided, provided, though. Furthermore, control schemes (the imsothough. Furthermore, control schemes (the socalled “master/slave” and “democratic”) are a priori advantages such as increased reliability due to redundancy and distribution of stresses of components, ease of dancy and distribution stresses ofthermal components, ease of called provided, though. Furthermore, control schemes (the imsomaintenance and repair,of improved management called even “master/slave” and “democratic”) areconservatisms a priori priori im“master/slave” and “democratic”) are a posed if discussion about their intrinsic dancy and distribution of improved stresses ofthermal components, of called “master/slave” and “democratic”) are a priori immaintenance and repair, repair, improved thermal management maintenance and management and reduced output ripple by interleaving phase ofease Pulse posed even if discussion about their intrinsic conservatisms about their to intrinsic conservatisms seems even hardiftodiscussion handle. In addition that, generalization maintenance and repair, thermalphase management and reduced reduced output rippleimproved by(Thottuvelil interleaving phase of Pulse Pulse posed and output ripple by interleaving of posed even discussion about their to intrinsic conservatisms Width Modulation (PWM) and Verghese, seems hardifto tonumber handle.ofIn In addition to that, generalization seems hard handle. addition that, generalization to arbitrary non identical converters leads to and reduced output ripple by interleaving phase of Pulse Width Modulation (PWM) (Thottuvelil and Verghese, Width Modulation (PWM) (Thottuvelil and 2011). Verghese, seems hard to handle. In addition to that, generalization 1997; Huang and Tse, 2007; Cid-Pastor et al., An- to to arbitrary number of non identical converters leads to to arbitrary number of non identical converters leads calculus burden. Width Modulation (PWM) (Thottuvelil and Verghese, 1997; Huang and Tse, 2007; Cid-Pastor et al., 2011). An1997; Huang and for Tse,considering 2007; Cid-Pastor et al., 2011). An- calculus to arbitrary number of non identical converters leads to other motivation such a system originates calculus burden. burden. 1997; Huang and for Tse, 2007; Cid-Pastor et al., 2011). other its motivation for considering suchwhich system originates other motivation considering such aa system originates believe that limitations of those existing results are calculus burden. from similarity with Microgrid, consists inAnan We other motivation considering suchwhich a system We believe believe that limitations of those thosewhich existing results are from its its similarity with Microgrid, which consists in an an We that limitations of existing results from similarity with Microgrid, consists in inherent to the retained methodology heavily relyare on interconnection offor distributed energy sources andoriginates storages We believe that limitations of those existing results are from its similarity with Microgrid, which consists in an inherent to the retained methodology which heavily rely on interconnection of distributed energy sources and storages to the retained which heavily on interconnection of distributed energy et sources and storages inherent interconnection of SISOmethodology transfer functions. In this rely paper, through power converters (Guerrero al., 2011). inherent to the retained methodology which heavily rely on interconnection of distributed energy sources and storages interconnection of SISO transfer functions. In this paper, through power converters (Guerrero et al., 2011). interconnection of SISO transfer functions. In this paper, through power converters (Guerrero et al., 2011). we seek a different strategy which rather adopts multiinterconnection of SISO transfer functions. In this paper, An essential feature offered by parallel interconnection of we through power converters (Guerrero et al., 2011). we seek a different strategy which rather adopts multiseek view-point, a different allowing strategy for which rather decomposition adopts multigeometric An essential essentialisfeature feature offered by bytoparallel parallel interconnection of variable An offered interconnection of we seek state a different strategy which rather decomposition adopts multiconverters the possibility distribute load current. variable view-point, allowing for geometric decomposition variable view-point, allowing for geometric of both and input spaces. An essential feature offered by parallel interconnection of converters is the possibility to distribute load current. converters is the possibility to distribute load current. variable view-point, allowing for geometric decomposition Indeed, if regulation of output voltage imposes overall of of both both state state and input spaces. and input spaces. converters is the possibility toamong distribute load current. Indeed, if if regulation ofcurrent output voltage imposes overall Parallel Indeed, regulation output voltage imposes overall interconnection arbitrary number of buck (stepof both state and input of spaces. current, distribution ofof converters remains Indeed, if regulation of output voltage imposes overall Parallel interconnection of arbitrary arbitrary numbercharacteristics of buck buck (step(stepcurrent, distribution of current among converters remains interconnection of number of current, of current amongforconverters remains down) DC/DC converters having distinct free. Thedistribution most wide-spread strategy dealing with this Parallel Parallel interconnection of arbitrary number of buck (stepcurrent, distribution of current among converters remains down) DC/DC converters having distinct characteristics free. The most wide-spread strategy for dealing with this DC/DC distinct characteristics free. The most wide-spread strategy for dealing this down) is considered in converters this paper.having Resulting electrical circuit is degree of freedom is the so-called balanced currentwith sharing down) DC/DC converters having distinct characteristics free. The most wide-spread strategy for dealing with this is considered in this paper. Resulting electrical circuit is degree of freedom is the so-called balanced current sharing considered in this degree freedom distributes is the so-called balanced current sharing is depicted by Fig. 1. paper. Resulting electrical circuit is which of uniformly currents among converters, is considered in this paper. Resulting electrical circuit is degree of freedom is the so-called balanced current sharing depicted by Fig. 1. which uniformly distributes currents among converters, depicted by Fig. 1. which uniformly distributes currents among converters, see e.g. (Thottuvelil and Verghese, 1997; Huang and Tse, contribution depicted by Fig. 1. of this paper consists of a novel framewhich distributes currents among converters, see e.g. e.g.uniformly (Thottuvelil and Verghese, Verghese, 1997; Huang and Tse, Tse, Main see (Thottuvelil and 1997; Huang and 2007). Main which contribution of this thisseparate paper consists consists ofregulation novel frameframeMain contribution of paper aa novel work completely voltageof from see e.g. (Thottuvelil and Verghese, 1997; Huang and Tse, 2007). 2007). Main contribution of this paper consists of a novel framework which completely separate voltage regulation from work which completely separate voltage regulation from Nevertheless, it is well known that control of both volt- current distribution dynamics, hence preventing undesir2007). work which completely separate voltage regulation from current distribution dynamics, hence preventing undesirNevertheless, it is is distribution well known known that that control of both both voltvoltdistribution dynamics, undesirNevertheless, it well of able interaction between them.hence Thispreventing suggests that the age and current may control badly interact and, current current distribution dynamics, hence preventing undesirNevertheless, it is well known that control of both voltable interaction between them. This suggests that thea age and current distribution may badly interact and, interaction between them. suggests that age and induce currentclosed-loop distributioninstabilities may badlyif interact and, able two related control objectives areThis not competing. Suchthe in turn, not properly able interaction between them. This suggests that theaa age and current distribution may badly interact and, two related control objectives are not competing. Such in turn, induce closed-loop instabilities if not properly related control objectives areopen-loop not competing. Such in turn, (Thottuvelil induce closed-loop instabilities not properly two reformulation is performed on the model so that treated and Verghese, 1997).if Traditionally, two related control objectives are not competing. Such a in turn, induce closed-loop instabilities if not properly reformulation is performed on the open-loop model so that treated (Thottuvelil and Verghese, 1997). Traditionally, reformulation is performed on the open-loop model so that treated (Thottuvelil and Verghese, 1997). Traditionally, this issue is addressed by resorting to frequency separation, control law candidates are not restricted to some particreformulation is performed on the open-loop model so that treated (Thottuvelil and Verghese, 1997). Traditionally, control law candidates are not restricted to some particthis issue is addressed by resorting to frequency separation, lawArbitrary candidates areset notofrestricted to some particthis issue is addressed by resorting to frequency separation, ular class. large heterogeneous converters so that bandwidth of current distribution is narrower than control control lawArbitrary candidates notof to some particthis issue is addressed by resorting to frequency separation, ular be class. Arbitrary large set ofrestricted heterogeneous converters so that that bandwidth of current current distribution is narrower narrower than ular class. set heterogeneous converters so bandwidth of distribution is than can handled and large no are frequency separation assumption that of voltage regulation. This allows hierarchical control ular class. Arbitrary large set of heterogeneous converters so that bandwidth of current distribution is narrower than can be handled and no frequency separation assumption that of voltage regulation. This allows hierarchical control be handled and no frequency separation that of voltage regulation. Thison allows hierarchical control can is required, hence offering tractability withoutassumption sacrificing structure (see e.g. standards Microgrid management can be handled and no frequency separation that of voltage regulation. Thison allows hierarchical control is required, required, hence offering tractability withoutassumption sacrificing structure (see e.g. e.g. standards on Microgrid management hence offering tractability without sacrificing structure (see standards Microgrid management performance. (ISA-95) introduced in (Guerrero et al., 2011)) but un- is is required, hence offering tractability without sacrificing structure (see e.g. standards on Microgrid management performance. (ISA-95) introduced in (Guerrero et al., 2011)) but un(ISA-95) introduced in (Guerrero et al., 2011)) but un- performance. (ISA-95) introduced in (Guerrero et al., 2011)) but un- performance.

Copyright © 2017, 2017 IFAC 16362 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 16362 Copyright 2017 responsibility IFAC 16362 Peer review©under of International Federation of Automatic Control. Copyright © 2017 IFAC 16362 10.1016/j.ifacol.2017.08.2317

Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Jean-François Trégouët et al. / IFAC PapersOnLine 50-1 (2017) 15792–15797

L1

set

i1

d1

σ

E1 C

R

v

im

DC BUS

Buck Converter 1

Lm

15793

Load

dm Em

Buck Converter m

Fig. 1. Electrical schematic. The new framework possesses remarkable features. (i) Adopting the form of a cascaded structure, it allows for standard related tools to be used and, in turn, formal stability proofs to be more easily derived. (ii) Arbitrary and load-dependent current-distribution can be considered, making this scheme suitable for heterogeneous set of converters for which balanced current sharing is not optimal with respect to power losses (Trégouët et al., 2016). (iii) Circuit theory interpretation shows that voltage regulation boils down to classical output voltage control of a single buck converter whereas current distribution assignment is nothing but a set of independent regulation of current flowing through a single coil connected to a controlled voltage source. The paper is structured as follows. In Section 2, we formalize our problem statement and introduce suitable preliminary assumption for the problem to be well-posed. In Section 3, detailed model description is provided. In Section 4, open-loop equations are manipulated to arrive at our main contribution, which significance is discussed. In Section 5, an example of control design, which structure is naturally suggested by the new framework, is provided. Finally, in Section 6 we discuss relevance of the new framework to cop with untreated difficulties like constantpower load and input saturation. Notation: The symbol Im stands for the identity matrix of dimensions m × m. The null matrix of size m × n is denoted by 0m×n . The vector (column matrix) of size m for which every entry is 1 (0) is denoted by 1m (0m ). The notation xk refers to the k-th element of the vector x, with 1 being the index of the first element. The operator ’diag’ builds diagonal matrix from entries of the input vector argument. 2. PROBLEM STATEMENT The electrical circuit represented by Fig. 1 is considered. It corresponds to parallel interconnection of m buck converters sharing a single capacitor and connected to a common resistive load R. Magnitude of the load is supposed to be constant and to belong to the compact and connected set L ⊂ R>0 . In addition to that, it is emphasized that R is unknown, as in most of practical situations. Converters are controlled via PWM and dk refers to duty cycle of k-th converter where k belongs to the following

K := {1, . . . m}. Voltage of DC bus is denoted by v and current in k-th inductor Lk is referred to as ik . As shown by next section, state vector x of the related model gathers those signals:  Rm+1  x := [i v] . Magnitude of voltage sources Ek are supposed to be known and constant. Capacitor C is connected in parallel to the load. Bus voltage regulation to a given value vr ∈ R>0 represents the foremost control goal. 1 Nonetheless v only depends on the total current, that is the sum σ of each ik . Thus, additional degrees of freedom remain in the way σ is distributed among converters. For this reason, this paper considers current control distribution as an additional control objective. This gives a way to control power-flow through the interconnection. Note that if voltage reference vr is typically constant, current vector reference ir might depend on R in order to allow for load-depend power-flow. Satisfying this last requirement is often none trivial as load R is unknown. Let us now formulate the problem addressed in this paper. Problem 1. Given a reference profile xr (·) := (ir (·), vr ) : L → Rm × R>0 , design (load independent) state-feedback control law x → d such that, for all R ∈ L, closedloop system globally and asymptotically converges to equilibrium xr (R). For this problem to admit solutions, it is clear that prescribed xr (R) must be a reachable equilibrium, that is there exists an input trajectory d(t) that transfers x(t) from the origin to xr (R) and constraints x(t) to remain at this point. Remarking that constant equilibria are such that the sum of current of each branch equals v/R and that 1m i − v/R = 0 ⇔ [R1m −1] x = 0, (1) it comes out that xr (R) must satisfied the following assumption for the problem to admit solutions. Hypothesis 2. It holds xr (R) = [ir (R), vr ] ∈ Ker {[R1m −1]} , for all R ∈ L. 3. MODEL DESCRIPTION Throughout this paper, it is assumed that (i) frequency of the commutation fs is sufficiently large for the dynamics to be approximated by an average (ripple-free) continuoustime model, (ii) converters remain in continuous conduction mode and (iii) electrical components are ideals, i.e. parasitic resistances can be neglected. By virtue of Kirchhoff’s circuit laws, under previous assumptions, dynamics of the circuit depicted by Fig. 1 is governed by dik = −v + Ek dk , dt dv = σ − v/R, C dt

∀k ∈ K, Lk

(2a) (2b)

Note that buck converter is a device that can only reduce input voltage so that each Ek must be larger than voltage reference vr .

1

16363

Proceedings of the 20th IFAC World Congress 15794 Toulouse, France, July 9-14, 2017 Jean-François Trégouët et al. / IFAC PapersOnLine 50-1 (2017) 15792–15797

E1 d1

+ −

(2a) Em dm



  −1 1 Γ 1 (Γ Γ ) 0 m m m m m T = . m 0m−1 0 1

i1

+

+

σ

(2b)

v

im

−

Fig. 2. Open-loop model. where σ :=



ik = 1m i,

(2c)

k∈K

refers to the total current. Eq. (2a) describes dynamics of output currents produced by each converter whereas (2b) corresponds to output voltage dynamics. The corresponding matrix formulation of previous dynamical equations is as follows:        diag {L} 0 d i 0 −1m i =  0 C dt v 1m −1/R v   diag {E} + d. (3) 0m

Fig. 2 gives graphical representation of the system dynamics and exhibits a natural feedback of voltage v into dynamics of i which is governed by input d. This diagram clearly shows that control of v can only by achieved “indirectly” through total current σ, driven by duty cycle d. 4. NEW OPEN-LOOP CASCADED FORMULATION This section presents the main result of this paper. Reformulation of open-loop model (2) is first obtained before being interpreted from circuit theory view-point. Then, discussions about relevance of this result for control purpose ends the section. 4.1 A new basis

As exhibited by (2b), voltage dynamics does not depend on each current ik individually but rather on total current σ. In order to better highlight this dependency, let us introduce the new coordinates (δ, σ, v) ∈ Rm−1 × R × R where σ explicitly appears:     δ −1 i σ =T , (4) v v with    Γm 0m−1  −1 T := 1m 0 (5) ∈ R(m+1)×(m+1) ,  0m 1 and where Γm ∈ Rm×(m−1) is defined as follows   1 0 ··· 0  .  −1 1 . . . ..         Im−1 0   . − m−1 . (6) Γm :=  0 −1 . . 0  =  0m−1 Im−1    . . .   .. . . . . 1  0 · · · 0 −1 Note that T is indeed invertible as Γm is a matrix basis of the orthogonal complement of Im {1m } in Rm . It can be easily verified that T reads

(7)

It is worth mentioning that the new coordinate δ ∈ Rm−1 admits a simple physical interpretation: From (4), this signal is related to i in the following way  δ = Γm i = [(i1 − i2 ) (i2 − i3 ) · · · (im−1 − im )] , which shows that δ reflects current distribution. This is nothing but the component of i which is missing in σ, as expected from invertibility of T . In the rest of the paper, we shall use the operator ∆ defined as follows  Rp  y → ∆y := Γp y = [(y1 − y2 ) · · · (yp−1 − yp )] (8) so that δ could alternatively be written as ∆i. In this new basis, matrix equation (3) becomes      −1  d  δ   0 0 −Γm diag {L}−1 1m   δ  σ = 0 0 −1m diag {L} 1m σ dt v v −1/(RC) 0 1/C    −1 Γm diag {L} diag {E} +  1m diag {L}−1 diag {E}  d, 0m

(9)

since it holds  −1   0m×m −1m −1 diag {L} 0m T T C 1m −1/R 0m     Γm 0m−1 −1 0 0 −diag {L} 1  m m = 1m 0 −1/(RC) 0m−1 1/C 0m 1   −1 0 0 −Γm diag {L} 1m −1 = 0 0 −1m diag {L} 1m  , 0 1/C −1/(RC)

and Hyp. 2 can be written as   [δr (R) σr (R) vr ] := T −1 [ir (R) vr ]  ∈ Ker  {[R1m −1] T }  = Ker 0m−1 R −1 , for all R ∈ L. Note that this set membership is nothing but the equilibrium condition of voltage dynamics (2b). Remark 3. (Constant current distribution). For some particular applications, power flow does not depend on load. A classical example of such a requirement is the socalled “balanced current sharing” for which each converter equally contributes the overall current σ. In such a case, it holds δr (R) ≡ 0 so that δ does not depend on R, unlike σr (R) which is equal to vr /R. 4.2 Block decomposition State matrix (9) admits a block triangular structure which allows to interpret the system as a cascade of two dynamical blocs: An upper-subsystem, governing coupled dynamics of (σ, v), feeds a lower-one, describing δ dynamics, with v. This cascaded structure originates from the independence of dynamics of δ from (σ, v), so that the upper-subsystem impacts the lower-one but there is no signal in the other way around.

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Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Jean-François Trégouët et al. / IFAC PapersOnLine 50-1 (2017) 15792–15797

Input d acts on both subsystems. To better understand this coupling, let us proceed to the change of input coordinates   1 −1 d = diag {E} diag {L} Γm (Γm Γm )−1 1m m       −1  λ diag ∆ (L ) diag {∆ E} 0 (10) 0 Eeq /Leq µ which decomposes d into λ ∈ Rm−1 and µ ∈ R. In this relationship, the following constants have been introduced: R  1/Leq := 1m diag {L} R  Eeq := min Ek .

−1

1m =



1/Lk

k

k

The operator ∆ derives from ∆, defined by (8), as follows  (∆y)k , (yk = yk+1 ) (11) Rp  y → (∆ y)k := 1, (otherwise) 

and, with slight abuse of notation, the inverse of any vector y ∈ Rp for which yk = 0, (k ∈ {1, . . . , p}) corresponds to component-wise inversion, i.e. −1

−1



y := diag {y} 1p = [1/y1 · · · 1/yp ] . (12) ensures that Note that the use of ∆ insteadof ∆ in (10)  d → (λ, µ) is a bijection as diag ∆ (L−1 ) diag {∆ E} is always invertible. 2 The resulting dynamics of the open-loop system reduces to      −1 δ δ 0 0 −∆(L ) d    σ = 0 0 −1/Leq   σ  dt v v 1/C −1/(RC)  0    −1   diag ∆ (L ) diag {∆ E} 0 λ  + , 0 Eeq /Leq µ 0 0

so that the upper-subsystem Σv governs voltage dynamics through total current σ controlled by input µ        d σ 0 −1/Leq σ E /L Σv : = + eq eq µ, (13) 1/C −1/(RC) v 0 dt v and drives the lower-subsystem Σδ corresponding to dynamics of δ driven by control signal λ   d Σδ : δ = −∆(L−1 )v + diag ∆ (L−1 ) diag {∆ E} λ dt (14) or, equivalently,    (∆L)k d δk = −v + (∆ E)k λk , (Lk = Lk+1 ) k dt Σδ : d   δk = (∆ E)k λk , (otherwise), dt for all k ∈ {1, . . . , m − 1}. The last relationship has been obtained by left-multiplying (14) by diag{∆ (L−1 )}−1 . Fig. 3 illustrates this structure. Example 4. Consider the case of m = 3 converters where L1 = L2 = L3 and E1 = E2 = E3 so that it holds   ∆ (L−1 ) = [(1/L1 − 1/L2 ) 1] , ∆ E = [1 (E2 − E3 )] . Obviously, this assertion holds provided that the problem is physically meaningful in the sense that both Ek and Lk are nonzero for all k ∈ K.

2

µ∈R

Σv

v∈R λ ∈ Rm−1

15795

Σδ

δ ∈ Rm−1

Fig. 3. New cascaded open-loop model. Subsystem Σδ reads (L1 − L2 )δ˙1 + v = λ1 δ˙2 = (E2 − E3 )λ2 , and (10) gives E2 1 1 E1 d1 − d2 = ( − )λ1 L1 L2 L1 L2 E2 E3 d2 − d3 = (E2 − E3 )λ2 L2 L3  1 E2 E3 E1 d1 + d2 + d3 = (min Ek )( )µ k L1 L2 L3 Lk k

so that λ1,2 are related to differences of duty cycles (and drives δ = ∆i), whereas µ reflects the sum d1 + d2 + d3 by which σ = i1 + i2 + i3 is controlled. 4.3 Circuit theory interpretation From its dynamical equation (13), upper-subsystem Σv can be physical interpreted as the single buck converter illustrated by Fig. 4 (a). Duty cycle of this virtual device corresponds to µ and current flowing through it is nothing but σ. It would be connected to the same load R and capacitor C as that of the global system. Its inductance and voltage input equal Leq and Eeq , respectively. Note also that Leq is nothing but the equivalent inductor resulting from parallel interconnection of every coil Lk . As far as Σδ is concerned, its dynamics can be interpreted as m − 1 instances of electrical circuit Σkδ depicted by Fig. 4 (b) and where (∆ E)k λk acts as controlled voltage source. Noticeably, neighbor coils sharing the same value make dynamics of δk independent of v since ∆(L−1 )k = 0 (whereas (∆ (L−1 ))k = 1). This means that bus voltage does not affect current distribution δk between neighbor branches having the same inductance value, i.e. Lk = Lk+1 . 4.4 Significance of the new formulation for control purpose Geometric decompositions i → (δ, σ) and d → (λ, µ) are the core of the reasoning underlining the construction of the control framework depicted by Fig 3. We believe that this reformulation facilitates control design for reasons that are now exposed. 1) From previous subsection, it comes out that λ parametrizes the part of d which is invisible by σ, whereas µ is the remaining part by which σ can be affected. As a result, regulation of v boils down to the design of the single input controller Cv : (σ, v) → µ, which is actually nothing but voltage regulation problem for a single buck converter. 2) The new formulation allows for modular design. Indeed, both voltage v (via σ) and current distribution δ can be controlled independently by way of µ and λ, respectively.

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Leq

5. EXAMPLE OF CONTROL DESIGN

σ

5.1 Stability conditions of the overall system Eeq µ

R

C

v

In the case where (possibly dynamical) sub-controllers Cv and Cδ are linear, it is clear that the overall closed-loop system is stable if closed-loop subsystems are such that

(a) (∆L)k

1

δk

⋆

where P1  P2 refers to the interconnection of P1 and P2 .

⋆

v

(∆ E)k λk

• Cv  Σv is stable; • Cδ  Σδ is stable for any constant input v,

δk

(∆ E)k λk

(Lk = Lk+1 )

(otherwise)

(b) Fig. 4. Circuit theory interpretation of (a) Σv and (b) Σkδ in the two cases.

v Σv

Cδ

µ

Fig. 5. Proposed control scheme. If this assertion clearly holds for Σv , effect of v on Σδ can also be canceled out even when Lk = Lk+1 by simply substituting λ by λ + v/(∆ E)k via an inner feedback loop. In such a case, one ends up with two disconnected subsystems, as suggested by the dashed line of Fig. 3. Nevertheless, apart from the special case where δr (·) is constant, it should be notice that load have to be estimated by controller of Σδ in order to properly define the desired reference. As a result, some communication from Σv to Σδ have to be maintained for this estimation to be possible since Σδ does not depend on R. For this reason, the map Cδ :

5.2 Control of individual subsystems In accordance with previous discussion, let us design some Cv and Cδ ensuring closed-loop stability of the subsystems.

Σδ

δ

σ

Cv

λ

In the more general framework where non-linear control laws are implemented, classical tools for non-linear cascade can be used. In particular, if (i) Cv  Σv is GAS and LES, (ii) Cδ  Σδ is 0-GAS (that is GAS when signals from Σv to Σδ are at the steady-state) and (iii) system trajectories are globally bounded, then the overall system is GAS (Sontag, 1989).

(δ, σ, v) → λ,

is considered as the general form of controller for Σδ . 3) Ordering of the cascade preserves hierarchy of control objectives. Indeed, in general, lower subsystem of cascaded system converges after the upper one has reached its equilibrium. As a result, leading control goals have preferably to be fulfill by highest subsystem. The proposed control scheme complies with this guideline as voltage regulation, performed by Σv , dominates power-flow control, related to Σδ , in the control objective hierarchy. Still, v cannot be controlled directly by control input, but only through σ. However, open-loop inertia of the total current σ is related to Leq which is typically very small, as parallel interconnection of coils ends up with a reduced equivalent inductance. Fig 5 summarizes this discussion by depicting the proposed control scheme.

Control of Σv (voltage regulation) As Σv can be interpreted as a virtual buck converter, every technique which aims regulating this system are applicable, see e.g. (SiraRamírez and Silva-Ortigoza, 2006; Olalla et al., 2009). In particular, let us use state-feedback with output integrator  of the form −kp ε − kd σ − ki ε where R  ε := vr − v refers to voltage deviation, so that Cv adopts the following formulation  z˙ = ε (15) Cv : µ = −ki z − kp ε − kd σ. Stability of the equilibrium satisfying v = vr is achieved if the augmented upper-subsystem dynamics, described by the following equation, is stable for all R ∈ L            σ E /L σ −kp −k eq eq d d  v + 0 vr v = A(R) + 0 kp dt z z 0 1 −ki (16) where   0 −1/Leq 0 3×3  A(R) := 1/C −1/(RC) 0 . R 0 −1 0 Quadratic stability gives a tractable sufficient condition for this to be satisfied: If there exist positive definite W = W  ∈ R3×3 and Y ∈ R1×3 such that 

W A (R) + A(R)W + [Eeq /Leq 0 0] Y + (17) Y  [Eeq /Leq 0 0] ≺ 0 holds for R equal to the maximum and the minimum of L, then W is invertible and the gains computed via (18) [−kd kp −ki ] = Y W −1 ensure stability of (16) for all R ∈ L (Boyd et al., 1994). Control of Σδ (current distribution assignment) Since current distribution to be achieved is load dependent, it is necessary to estimate R. Remarking that equilibrium

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points in the state-space must verify v = Rσ, computation of v/σ can serve as a simple load-estimator which asymptotically converges to R provided that dynamics of (σ, v) is asymptotically stable. Controller Cδ has to ensure 0-stability of δr (R), that is stability when Σv has reached its equilibrium (σ, v) = (vr /R, vr ). Hence, the following static map is a suitable candidate  vr , (Lk = Lk+1 ) (19) Cδ : λ = −K(δr (vr /σ) − δ) + 0, (otherwise),

for all Hurwitz matrix K ∈ R(m−1)×(m−1) . Indeed, when Σv is at the equilibrium, closed-loop of (14) and (19) reads δ˙ = K(δ − δr (R)) which asymptotically converges to the equilibrium δ = δr (R). Stability of the whole system Load estimation via vr /σ introduces non-linearity in the controller Cδ . Yet, every conditions stated in Sec. 5.1 are satisfied so that GAS is ensured. 3 The following proposition sums up discussion of this subsection. Proposition 5. The map (i, v) → d described by (10), with µ delivered by (15) and λ defined as in (19), solves Prob. 1 if K is Hurwitz and (kp , ki , kd ) are computed via (18) where W  0 and Y verify (17) for all for R ∈ ∂L. 6. PERSPECTIVES 6.1 Constant power load In the case where the load cannot be modeled as resistive component but rather as a constant power load (CPL), voltage dynamical relationship (2b) becomes C v˙ = σ − p0 /v, where term v/R has to be substitued by p0 /v with p0 ∈ R>0 , see e.g. (Belkhayat et al., 1995). This introduces strong non-linearity that may cause system instability if not properly treated. Nevertheless, the proposed cascaded reformulation allows to isolate this non-linearity to Σv and let Σδ unchanged. Specifically, lower-right term of state matrix of (13) becomes −p0 /(Cv 2 ).

As a result, only Cv delivering µ has to be modified to cope with this difficulty. Note that the rough load estimator vr /σ now reads vr σ and asymptotically converges to p0 , provided that stability of equilibrium satisfying vr = v is ensured by Cv . 6.2 Saturation of d

So far, it has been implicitly assumed that λ and µ can be freely assigned. In practice, this is obviously not the case as duty cycle vector d belongs to [0, 1]m . By inverting (10), this constraint can be pre-computed by constructing sets Λ and Φ in which λ and µ has to, respectively, belongs to in order not to saturate d. Obviously, Λ depends on Formally, boundedness of trajectories requires δr (vr /σ) to remain bounded. As xr (R) has been assumed to be reachable and asymptotic value of σ, that is vr /R, is strictly positive, this can only happen if σ approaches 0 during transient. This issue should be simply addressed by saturate the quantity 1/σ into Cδ and prevent division by zero. 3

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instantaneous value of µ and, conversely, Φ takes λ as an input. However, considering that regulation of v is the major control goal, it is reasonable to modify trajectory of λ in order to avoid saturation of µ by driving boundaries of Φ(λ) away from µ. Once (σ, v) is sufficiently close to the desired equilibrium (or µ is sufficiently far from the boundaries of Φ(λ)), λ can operate into the “normal” mode. Implementation of more advanced anti-windup techniques are ongoing researches. REFERENCES Belkhayat, M., Cooley, R., and Witulski, A. (1995). Large signal stability criteria for distributed systems with constant power loads. In Power Electronics Specialists Conference, 1995. PESC’95 Record., 26th Annual IEEE, volume 2, 1333–1338. IEEE. Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. Cid-Pastor, A., Giral, R., Calvente, J., Utkin, V., and Martinez-Salamero, L. (2011). Interleaved converters based on sliding-mode control in a ring configuration. IEEE Transactions on Circuits and Systems I: Regular Papers, 58(10), 2566–2577. Guerrero, J., Vasquez, J., Matas, J., de Vicuna, L., and Castilla, M. (2011). Hierarchical control of droopcontrolled ac and dc microgrids ; a general approach toward standardization. IEEE Transactions on Industrial Electronics, 58(1), 158–172. Huang, Y. and Tse, C.K. (2007). Circuit theoretic classification of parallel connected dc-dc converters. IEEE Transactions on Circuits and Systems I: Regular Papers, 54(5), 1099–1108. Olalla, C., Leyva, R., El Aroudi, A., and Queinnec, I. (2009). Robust lqr control for pwm converters: an lmi approach. IEEE Transactions on industrial electronics, 56(7), 2548–2558. Sira-Ramírez, H. and Silva-Ortigoza, R. (2006). Control design techniques in power electronics devices. Springer Science & Business Media. Sontag, E.D. (1989). Remarks on stabilization and inputto-state stability. In Decision and Control, 1989., Proceedings of the 28th IEEE Conference on, 1376–1378 vol.2. Sun, J., Qiu, Y., Lu, B., Xu, M., Lee, F.C., and Tipton, W.C. (2005). Dynamic performance analysis of outerloop current sharing control for paralleled dc-dc converters. In Twentieth Annual IEEE Applied Power Electronics Conference and Exposition, 2005. APEC 2005., volume 2, 1346–1352. IEEE. Thottuvelil, V.J. and Verghese, G.C. (1997). Analysis and control design of paralleled dc/dc converters with current sharing. In Applied Power Electronics Conference and Exposition, 1997. APEC’97 Conference Proceedings 1997., Twelfth Annual, volume 2, 638–646. IEEE. Trégouët, J.F., Delpoux, R., and Gaulthier, J.Y. (2016). Optimal secondary control for dc microgrids. In 2016 IEEE 25th International Symposium on Industrial Electronics (ISIE), 510–515. Santa Clara, CA.

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