Optimal Power Distribution Control for a Network of Fuel Cell Stacks

0.47 MB·

by  Resmi Suresh M P

Optimal PowerDistributionControlfor a NetworkofFuel
CellStacks
Resmi SureshM Pa, GaneshSankaranb,SreeramJoopudib,
ShankarNarasimhana, SumanRoy Choudhuryc, RaghunathanRengaswamya
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1 aIndianInstitute of Technology, Madras, Chennai, TamilNadu, India600036.
0
2 bGyanData Private Limited, Taramani,Chennai, India

n cNaval MaterialsResearch Laboratory, Ambernath, India
a
J

7
2
Abstract

]
Y In power networks where multiple fuel cell stacks are employed to deliver the required power, optimal sharing of the power
S demand between different stacks is an important problem. This is because the total current collectively produced by all
the stacks is directly proportional to the fuel utilization, through stoichiometry. As a result, one would like to produce the
.
s required power while minimizing the total current produced. In this paper, an optimization formulation is proposed for this
c
power distribution control problem. An algorithm that identifies the globally optimal solution for this problem is developed.
[
Throughan analysisoftheKKTconditions,thesolution totheoptimization problem isdecomposed intooff-lineandon-line

1 computations. The on-line computations reduce to simple equation solving. For an application with a specific v-i function
v derivedfrom data,weshowthatanalyticalsolutionsexistforon-linecomputations.Wealsodiscussthewiderapplicabilityof
5 theproposed approach for similar problems in otherdomains.
7
2
7 Keywords: Power sharing; Optimization;Fuelcell; Analytical solution
0
.
1
0 1 Introduction degradationcould be due to many factors such as elec-
6
trolyte degradation, catalyst poisoning, water flooding
1
etc. [2,3]. Such degradation can result in reduced cur-
: Fuelcellshavegainedconsiderableattentioninthefield
v rentbeingdrawnfromthestackatthesamevoltage.As
of power conversion, especially in the past few decades
i a result, the overall system performance could be com-
X [9,2,10].Fuelcellsarecapableofextractingenergyfrom
promised, and in the worst case of complete stack fail-
the fuel more efficiently than internal combustion en-
r ure,thewholesystemcouldfail[3].Thisissueisusually
a gines [2]. Due to this higher fuel-efficiency, less harmful
addressedbydesigningareliableparallelconnectedfuel
emissions,lowoperationalnoise andabsence ofmoving
cellarchitecturewithanadditionalpowerbusasshown
parts, fuel cells have been proposed for a wide variety
in Fig. 1 [8]. Another approachto improve reliability is
ofapplications[7,2].Thepowerobtainablefromasingle
the use of multiple power sources in the same network
fuelcellisratherlowincomparisontothepowerrequire-
[4,5]. While considerable research has been devoted to
ment for most of the applications. Hence, to meet real-
concerns related to output voltage, state of charge, re-
istic power requirements, active surface area has to be
liabilityandnetworkconfiguration,verylittle workhas
increased.Inviewofthis,multiplefuelcellsarestacked
focused on developing operational strategies that min-
or connected in series and/or parallel arrangements to
imize the fuel consumption through online control. We
enhancethe output voltageandpower[3,1].
addressthis controlprobleminthis paper.
Whilefuelcelltechnologyhasseveraladvantages,there
are also problems related to long term durability and 2 ProblemStatement
graduallossofperformanceovertime.Theperformance
Consider a fuel cell network with N branches and N
f,i
Emailaddress: [email protected] (Raghunathan stacksintheithbranch(asshowninFig.2).Thecontrol
Rengaswamy). problemisoneofminimizingthetotalcurrentdrawn(a
Preprintsubmitted toAutomatica 28 January 2016
branchisthedecisionvariable.Subscriptslbandubde-
notethe lowerandupper bounds respectively.
Assumptions:
1 The potentialacrosseachfuel cellis assumedto be a
staticmapf :
ij
V =f (I ) i=1:N; j =1:N (2)
ij ij i f,i
∀
2 Numberofstacksineachbranchofthenetworkunder
consideration,is assumedto be one (N =1 i=
f,i
∀
1:N).
3 Minimum possible power from the fuel cell network
isachievedwhenallstacksareoperatingattheircor-
responding I . Pathological cases where P <
lb ij|Ii,ub
Fig. 1. Schematic of a parallel connected fuel cell network P forjthstackinith,whichmayarisedepending
with n branches ij|Ii,lb
onthe voltage-currentprofile,areneglected.
surrogate for total fuel utilized) while meeting a fixed
power requirement. This can be posed as the following Using equation (2), voltage constraints can be trans-
optimizationproblem. formedtocurrentconstraints.Also,anetworkwithmul-
tiple stacks in a branch can be converted to an equiva-
lent networkwith a single stack in eachbranch, as cur-
rent flow in each stack of a branch is equal. Power in
each branch of the resulting equivalent network can be
writtenas
P =P =φ V I =φ f (I )I (3)
i eff,i i i i i i i i
where V , P and φ are the voltage, power deliverable
i i i
andefficiencyofthesinglefuelcellstackintheithbranch
ofthe equivalentcircuit.
Usingalltheseassumptions,thegeneralconstrainedop-
timizationproblemdescribedin(P1)canbereducedto
the following minimization problem (P2) for a network
asshowninFig.1.
Fig. 2. Schematic of a parallel connected fuel cell network N
with N branchesand Nf,i fuelcell stacks intheith branch. min I = I (4a)
net i
Ii i=1
X
N N
N
subjectto P = φ f (I )I =P (4b)
min I = I (1a) i i i i i req
net i
Ii i=1 Xi=1 Xi=1
X I I 0 i=1:N (4c)
N Nf,i i,lb− i ≤ ∀
I I 0 i=1:N (4d)
subjectto Pij =Preq (1b) i− i,ub ≤ ∀
i=1 j=1
XX Theoptimizationproblem(P2)inequation(4)involves
V V i=1:N; j =1:N
ij ≥ ij,lb ∀ f,i a linear objective function with one nonlinear equality
(1c)
constraint and 2N linear inequality constraints. Due to
Vij Vij,ub i=1:N; j =1:Nf,i non-linearequalityconstraintpresent,this optimization
≤ ∀
(1d) problembecomesnon-convex.
whereI isthetotalcurrentfromallbranches,I rep-
net i
resentsthecurrentdrawnfromtheithbranchofthefuel 3 Theoretical AnalysisusingKKTConditions
cell network. V , P (=φ V I ) and φ are voltage,
ij ij ij ij ij ij
power produced and efficiency of the jth stack in the Theorem1 Whenafuelcellnetworkisusedtoachieve
ith branch respectively. Current (I ) drawn from each apowerofP ,optimumissuchthatstacksoperatingat
i req
2
acurrentvaluewithinallowablelimits(I <I <I ) Using dL =0,
aremaintained at equal dPi. i,lb i i,ub dIi
dIi
dP
i
1 λ µ +γ =0 i=1:N (10)
Proof: For any V-I characteristics, Karush-Kuhn- − dI − i i ∀
(cid:18) i(cid:19)
Tucker (KKT) conditions [6] can be used to prove the
above theorem. Lagrangian of the minimization prob- Forthe givencase,KKTconditionscanbe reduced
lemdescribedin (4)canbe writtenasfollows. to
N N N 0 fori=1:M
L= Ii+λ Preq Pi + µi(Ii,lb Ii) µi =
− ! − ( 0 fori=1+M :N
i=1 i=1 i=1 ≥
X X X
N
0 fori=1:K
+ γ (I I ) (5) γ = ≥
i i i,ub i
i=1 − ( 0 fori=1+K :N
X
There are 2N inequality equations out of which first N Substituting inequation(10),
arelowerboundsonI andthelastNareupperbounds
i
on I . Necessary conditions for optimality as described
i dP
i
byKKTconditions forall iǫ 1,...,N are 1 λ + γi =0 i=1:K (11)
{ } − (cid:18)dIi(cid:19)Ii,ub ∀
µi ≥0 (6) 1 λ dPi =0 i=1+K :M (12)
γi ≥0 (7) − (cid:18)dIi(cid:19)Ii,opt ∀
µi(Ii,lb−Ii)=0 (8) 1 λ dPi µ =0 i=1+M :N (13)
γi(Ii−Ii,ub)=0 (9) − (cid:18)dIi(cid:19)Ii,lb − i ∀
DifferentcasesinKKTconditionsareobtainedbyvary-
I is the optimum current drawn from ith
ing the number of active and inactive constrains, all of i,opt
stack.Fromequation(12),
whichfallsintoanyofthe following2casesfordifferent
valuesofMandK.
1
λ= i=K+1:M (14)
Case 1 dPi ∀
dIi Ii,opt
(cid:16) (cid:17)
For1 M N and1 K N,
≤ ≤ ≤ ≤
Constraints1:M areinactive, M +1:N areactive.
ConstraintsN+1:N+K areinactive,N+K+1:2N = dPk+1 = dPk+2 =...
areactive. ⇒ (cid:18)dIk+1(cid:19)Ik+1,opt (cid:18)dIk+2(cid:19)Ik+2,opt
dP
M
= (15)
(a) If K M, for i = M +1 : N, I = I = I
i i,lb i,ub dI
which≤is notfeasible. (cid:18) M (cid:19)IM,opt
(b) IfK >M,fori=K+1:N,I =I =I which
is notfeasible. i i,lb i,ub Hence, ddPIii of all stacks operating within their al-
lowablelimits areequal.Also,from(11)and(13),
Case 2
dPj
CFoorn1st≤raiMnts≤1N:Manadre1i≤naKctiv≤e,NM, +1:N areactive. γj =−1+ (cid:16)ddPIij(cid:17)Ij,ub ∀ j =1:K;
ConstraintsN +1:N +K are active,N +K+1:2N dIi Ii,opt
(cid:16) (cid:17)
areinactive. i=K+1:M (16)
(a) If K > M, For i = 1+M : K, I = I = I
i i,lb i,ub
whichis notfeasible. dPj
(b) If K ≤M, µj =1 (cid:16)dIj(cid:17)Ij,lb j =1+M :N;
− dPi ∀
Ii,ub for i=1:K dIi Ii,opt
I = (cid:16) (cid:17)
i i=K+1:M (17)
( Ii,lb fori=1+M :N
3
Itcanbeseenthatγ 0issatisfiedifandonlyif I , then according to the algorithm, any incremental
j N
≥
power demand dP should be met by drawing addi-
net
dP dP tionalcurrentfromthose ’l’suchthat,
j i
(18)
dI ≥ dI
(cid:18) j (cid:19)Ij,ub (cid:18) i(cid:19)Ii,opt dPnet dPnet
n ǫ N (25)
dI ≥ dI ∀
andµj 0canbe achievedonly if (cid:18) l (cid:19)Il (cid:18) n (cid:19)In
≥
ForapowerrequirementofP ,anyconfigurationthat
dP dP req
j i
(19) satisfies the above criterion will be a local optimum to
dI ≤ dI
(cid:18) j (cid:19)Ij,lb (cid:18) i(cid:19)Ii,opt the problem. Multiple solutions are possible depending
on the V-I profile. Global optimum can be obtained
Combining (18)and(19), based on the value of objective function for those so-
lutions which satisfy all these constraints.The solution
dP dP dP whichhas lowestvalue of objective function will be the
j i r
(20)
dI ≤ dI ≤ dI globaloptimum.Thisoptimumwillhavemaximumrise
(cid:18) j (cid:19)Ij,lb (cid:18) i(cid:19)Ii,opt (cid:18) r (cid:19)Ir,ub intotalpowerfromthenetworkforaspecificincreasein
j =M +1:N; i=K+1:M; r =1:K (21)
currentdrawn dPnet = N dPi .
dInet i=1 dIi
Thus, first order optimality conditions imply that the (cid:16)(cid:16) (cid:17) P (cid:17)
only feasible solution to the concerned optimization Remark1 Theoptimizationproblemdescribedin(4)is
problem (P2) is such that ’M’ stacks are operating at convexforanyV-Iprofile,V =f (I ),witharestriction
i i i
a current higher than their minimum allowable current thatcorrespondingpower(P =φ I f (I ))isaconcave
i i i i i
(Ii,lb) andout ofwhich’K’stacksare operatingatIi,ub function(orinotherwordsHessianofpowerisnegative
andthe conditions(15)and(20)aresatisfied. definite). For this convex subproblem, though multiple
solutionsareobtainedbysolvingequations (22),(23)and
(24), onlyone ofthem will satisfy theactual constraints
4 ProposedAlgorithm
of the optimization problem (P2), resulting in a unique
solution.Proof can befound in Appendix A.
Anewalgorithmisproposedtofindanoptimumsolution
totheminimizationproblem(P2)describedusingequa-
tion(4).Foranetworkof’N’brancheswithasinglestack 5 Implementation
ineachbranch,todeliverP ,sayI >I forfirst’m’
req i i,lb
stacks and among these, ’k’ stacks operate at I = I
i i,ub Theimplementationofproposedalgorithmisdescribed
at optimum. Stacks k+1 to m are then operating at a
in Fig. 3. Input to the algorithm will be the V-I model
current in between their bounds, I < I < I . At
i,lb i i,ub parameters, bounds on current and the required power
optimum,accordingto theorem1,
(P ).Usingthenewalgorithm,majorityofthecalcula-
req
tionsforestimating optimumcurrentcanbe performed
dP dP
i = j i,j =k+1:m (22) off-line, reducing the on-line computations. The major
(cid:18)dIi(cid:19)Ii,opt (cid:18)dIj (cid:19)Ij,opt ∀ stepintheoff-linemodeistoidentify observablepoints
(kink points which decide whether to add or remove
subjectto stacksfromthelistofstacks,forwhichcurrentneedsto
be optimized). Observable points are decided based on
m (dPj/dIj) of each stack evaluated at both Ij,lb and Ij,ub,
P =Peff (23a) wherejcorrespondstothebranchnumber.Eachvaluein
i req
i=Xk+1 the list of (dPj/dIj) corresponds to an observable point.
dP dP dP ForafuelcellnetworkwithNbranches,therewillbe2N
j i r
(23b) observable points. These observable points are then ar-
dI ≤ dI ≤ dI
(cid:18) j (cid:19)Ij,lb (cid:18) i(cid:19)Ii,opt (cid:18) r (cid:19)Ir,ub rangedsuchthatthepowercorrespondingtotheobserv-
ablepointdecreasesdownthelist.Correspondingtothe
suchthat j =m+1:N; i=k+1:m; r=1:k ith valueof(dP/dI)inthesortedlist,ifforanyj ǫ1:N,
and (dPj/dIj)Ij=Ij,lb ≤ (dP/dI)i, then jth stack should be op-
Preefqf =Preq− N Pj,min− k Pr,max (24) e(edrrPaa/ttiidnnIgg)iaa>ttI(jad,lPbcj./uAdrIrnjed)nIjft=orIijna,unbby,etjthwǫee1nen:jNthI,sitfa(cadknPjds/hdIoIju)lIdj=.bIOje,lpbotp>i--
j=m+1 r=1 j,lb j,ub
X X mum current to be drawn from these stacks can be ob-
Corresponding to a power requirement of Preq, let the tained by solving (dPj/dIj)Ij,opt = (dP/dI)i. Further, if
optimal current drawn from branches be I1, I2,..., and for any j ǫ 1 : N, (dPj/dIj)Ij=Ij,ub ≥ (dP/dI)i, then jth
4
Start on-linesuchthatequations(22),(23)and(24)aresatis-
fied.Theexactsolutiontothissetofequationshasbeen
f(Ii),Iub,Ilb obtained analytically for a specific function Vi = fi(Ii)
and is discussed in section 6. If multiple solutions are
CalculatePobt,minandPobt,max obtainedby solvingthese setofequationswhichsatisfy
the bounds on current, then global optimum is found
Arrangecellsindecreasingorderof(dP) based on the value of objective function. The solution
dI
calculatedforeachcellatIi,lbandIi,ub. with lowest value of net current is the global optimum
i = 1 : 2Nrepresentsindexofsortedlist
forthe problem.
ComputePobtateachobservablepoints
6 Analytical solution for a specific convex sub-
problem
Preq PrePqob≤t,mPionbt,≤max no Eprbroeowroe:rbRtcaeaqinnuneidoretd V−I characteristics of a fuel cell
26
Experimental Data
n=0 yes
Model Fit
n=n+1 24
Preq≤Pobt,n no Updaten ge (V) 22 (cid:9)MV o=d e2l7 O.3b8t7a3in −ed 0.7804 I0.5
a
yyeess Correspondingobservablepoint=n-1 Volt 20
FindcellsoperatingatIub(kcells)andIlb(N−mcells)
18
Findcellnumbersforwhichcurrent
needstobeoptimized(m kcells) 16
− 0 50 100 150 200
Current (A)
Calculate
Preefqf (24)
Fig. 4. Comparison of experimental data obtained from
Solve(dPi) = constant i = k+1 : m NMRLwith theassumed V-Icharacteristics form.
dIiand ki=1Pi ∀= Preefqf Voltage and current data (See Fig. 4) for a typical fuel
cell stack are obtained from Naval Materials Research
MultiplesoPlutions
Laboratory(NMRL). V-I characteristics of the form as
Currentincellsi = k + 1 : m described in (26) is assumed for which corresponding
power is a concave function. The assumed profile has
Current i=1:k&i=m+1:N
∀ beentestedonthedataanda99.87%goodfitisobtained
Allpossiblelocaloptima asshowninFig.4.
Minimumtotalcurrent
Optimum
Current:Iopt V(Ii)=fi(Ii)=ai+bi Ii (26a)
suchthat a 0 (26b)
Stop i ≥ p
Onlinemode b 0 i=1:N (26c)
i
≤ ∀
Fig. 3. Proposed algorithm: A fuel cell network with N
As power corresponding to this V-I profile is concave,
branches is considered such that, to deliver the required
corresponding optimization problem is convex as de-
power(Preq)mcellsshouldbeoperatedatacurrenthigher scribedinAppendixAandthuswillhaveauniqueglobal
thanI , outof which k cells operateat I .
lb ub
solution. Analytical solution is derived for this specific
stack should be operating at I . Hence, correspond- V-Icharacteristics.
j,ub
ing to each observable point, current in each stack is
pre-calculatedoff-line. This informationcan be used to Consider ’n1’ cells operating at I at optimum such
j,opt
evaluatethe powerateveryobservablepoint. that I < I < I . The following equations can
j,lb j,opt j,ub
beobtainedusingTheorem1andareusedtodetermine
When there is a specific power requirement, the stacks optimum current drawn from those ’n1’ cells as shown
to be operated at current values between their bounds inFig.3.
areidentifiedbasedonthepre-calculatedpowerateach P1+P2+...+Pn1 =Preefqf (27)
observablepoint.Upperandlowerboundconstraintson
currentareinactive for these stacks.The optimumcur- dP1 dP2 dPn1
= =...= (28)
rent to be drawnfrom eachof these stacks is evaluated dI1 dI2 dIn1
5
Usingequation(3)and(26),andsubstitutingx = I , multiple stacks in a branch. Finally, efficiency of the
j j
equation(27)and(28)reducesto algorithmiscomparedwiththatofstandardoptimizers.
p
n1 7.1 Working ofalgorithm
(φ a +φ b x )x2 =Peff (29)
j j j j j j req
j=1
X Consideranexampleofafuelcellnetworkconsistingof
3 fuel cell stacks that are connected in parallel using a
3 3
φ1a1+ φ1b1x1 = φjaj + φjbjxj j =2:n1 powerbus.Parametersusedforsimulationaredescribed
(cid:18) 2 (cid:19) (cid:18) 2 (cid:19) ∀ in Table 1. The variation of dPi/dIi with √Ii for all the
(30) stacks are shown in Fig. 5. It can be clearly seen that
Forj =1:n1,rearrangingequation(30), stack1hasthehighestdPi/dIiatitslowerboundandsoit
should start operating first according to the algorithm.
xj =gjx1+hj (31) The observable points as explainedin section 4 are cal-
x2j =gj2x21+2gjhjx1+h2j (32) culated and tabulated in Table 2 and are also marked
in Fig. 5. Power obtainable from the network when all
x3j =gj3x31+x21 3gj2hj +x1 3gjh2j +h3j (33)
Table1
where (cid:0) (cid:1) (cid:0) (cid:1) Parameters for afuel cellnetwork with 3 fuelcell stacks
Stack Lowerbound Upperbound φ a b
g = φ1b1 and h = (φ1a1−φjaj) (34) Number oncurrent(A) oncurrent(A)
j j
φ b 1.5φ b
j j j j
1 2.103 106.8127 1 47.655 -1.297
Substituting in equation (29) and simplifying, we get a 2 0 325.6562 1 39.895 -0.557
cubic equation. 3 6.646 236.4155 1 33.847 -0.5976
stacksareoperatingattheircorrespondinglowerbounds
n1 n1
x3 φ b g3 +x2 3φ b g2h +φ a g2 is 310.976 W. For each observable point, stacks which
1 j j j 1 j j j j j j j  areto be operatedat their correspondinglowerbounds
j=1 j=1
X X(cid:0) (cid:1) andupperboundscanbepre-calculatedandaregivenin
 n1    Table2.ItcanbeseenfrombothTable2andFig.5that
+x1 3φjbjgjh2j +2φjajgjhj =Preefqf (35) stack1startsoperatingfirstonincreasingthepowerre-
Xj=1(cid:0) (cid:1) quirement from Pmin (= 310.976W). When the power
  requiredreachesthepowercorrespondingtothe second
Current drawn from all stacks can be evaluated using observable point (corresponding to dP2/dI2 evaluated at
equations(31)and(36)aftersolvingforx1 (usingequa- I2,lb), then stack 2 also starts to operate. Each stack
tion(35)). starts to operate one by one and for the stacks oper-
I =x2 (36) ating,optimumcurrentsarecalculatedusinganalytical
opt,j j
solution described in section 6. If observable point cor-
Astheanalyticalsolutionisobtainedbysolvingacubic
respondingtoupperboundcurrentofstackjisreached,
equation (Equation (35)), it will result in three sets of
thenfromthatpoweronwards’jth’stackshouldbeoper-
solutions.Butastheproblemisconvexandhasaunique
atingatitsupperboundcurrent.Thiscanbeseenstart-
minimum, it is guaranteed that there can only be one
ing from observable point 4, where stacks reaches their
solutionsetamongthosethreewhichwillsatisfyallcon-
corresponding upper bounds one by one. The process
straintsofthe actualoptimizationproblem(P2).
continuesinasimilarmannertillallstacksareoperating
at their upper bound. Fig. 5 shows how various stacks
shouldbeoperatedatoptimaforvariouspowerrequire-
7 Resultsand Discussion
ment regimes. Fig. 6 shows the optimum current cal-
culatedusing the proposedalgorithmforvariouspower
The algorithm proposed in section 4 has been imple- requirements.
mentedonvariousfuelcellnetworks.Allfuelcellstacks
are assumed to follow V-I characteristics described in For any power requirement, we get three solutions by
equation (26). Parameters for various stacks are ob- solving equation (35). For example, when P = 8000,
req
tained by perturbing the parameters calculated for a threesolutionsobtainedaregiveninTable3.Butasex-
typical fuel cell stack V-I profile shown in Fig. 4. The plainedinsection6,itcanbeseenthatonlyonesolution
results obtainedfor three different case studies are pre- satisfiesupperandlowerboundsforcurrent.Itisanin-
sented here. The working of algorithm is shown using terestinginferencethatonlyasinglesolutionwillsatisfy
thefirstcasestudywhilethesecondoneaimsatdemon- alltheconstraintsoutofallthesolutionsobtainedusing
stratingtheapplicationofalgorithmforanetworkwith algorithmfor aconvexproblem.
6
Fig. 5. dPi/dIi variation for different stacks in the network for which parameters are described in Table 1. Variation is shown
with respect to √I suchthat I variesfrom I toI .Each horizontal line representsan observablepoint.
lb ub
Table2
Observablepoints calculated for thefuelcell network with 4 stackswhose parameters aredescribed in Table 1
Sl.no Corresponding Corresponding Stacksoperating Stacks Stacksoperating
totalpower dP/dI atImin tobeoptimized atImax
1 310.976 44.834 1,2,3 - -
2 890.577 39.895 2,3 1 -
3 6183.777 31.536 3 1,2 -
4 12037.033 27.548 - 2,3 1
5 16200.563 24.818 - 3 1,2
6 19206.708 20.064 - - 1,2,3
Table3 ble4.Forapplyingthealgorithm,anequivalentnetwork
Three solutions obtained by solving equation for the fuel
with single stack in each branch is constructed for this
cell network with parameters shown in Table 1 for a power
network such that total power from each branch of the
requirementof 8000 W.
originalnetworkisthesameasthepowerfromthatsin-
Stack1 Stack2 Stack3 TotalCurrent gle stack in the corresponding branch of the equivalent
network.
Solution1 1140.37A 4808.83A 3350.97A 9300.17A
Solution2 81.02A 136.23A 17.07A 234.32A
Solution3 1.93A 36.6A 153.407A 191.94A
Let P be the power acquired from ith branch in the
i
7.2 Equivalent network formation for a network with equivalentnetwork,then
multiplestacks in each branch
Nf,i Nf,i
Consider a network of 30 fuel cell stacks unequally dis- P = P = φ V I (37)
i ij ij ij i
tributedin15brancheswithparametersasgiveninTa- j=1 j=1
X X
7
Table5
350 Comparisionoftheresultsobtainedforunequaldistribution
Cell 1 of fuel cell stacks in different rows using a standard opti-
300 Cell 2 mization routine and the proposed algorithm. Preq=75000
Cell 3 W, φ=0.8 Numberof fuel cell stacks = 30. Parameters and
250 optimumcurrent drawn aredescribed in Table 4.
A) Optimizer ProposedAlgorithm
nt (200
e
urr150 TotalCurrent(A) 828.88 828.88
C
TotalPower(W) 75000 75000
100
Timetakenforoptimizing(sec) 0.104 0.0024
50
sets of solutions obtained from the algorithm satisfied
0
0 0.5 1 1.5 2 upper and lower bound constraints. It is surprising to
Power (W) ×104 seethatacubicequationasgivenin(35)willalwaysre-
sult in one and only one solution that satisfies all the
Fig. 6. Optimum currents obtained for stacks for various constraints for a network with all stacks having a con-
powerrequirement cavepowerprofile.Fortheconvexsubproblem,aunique
optimumisguaranteedfromthe algorithm.
ForaV-I profilegiveninequation(26),
Nf,i Forageneralproblem,ifrequiredpowerisfeasiblefrom
P = φ (a +b I )I (38) thefuelcellnetwork,thenallpossiblelocalminima can
i ij ij ij i i
beobtainedfromtheproposedalgorithmunlikeconven-
j=1
X p tional optimizers where a single local optimum is ob-
Nf,i
tained based on the initial guess. Also, all these pos-
= (φ a I +φ b I1.5) (39)
ij ij i ij ij i sible local optima are estimated in a very short time.
Xj=1 Globaloptimumisguaranteedusingtheproposedalgo-
=(a +b I )I (40) rithmunlikestandardoptimizersasitcanbeestimated
i i i i
easily using all possible local optima. If power is a con-
p
where a = Nf,i(φ a ) and b = Nf,i(φ b ) are cavefunctionofcurrentforallcellsinthenetwork,then
i j=1 ij ij i j=1 ij ij a unique local optimum is obtained from the algorithm
parametersinvoltageprofileforthestackinithbranchof
P P and this local optimum will be the global optimum to
theequivalentnetwork.Usingtheequivalentnetworkof
theproblem.Optimumsolutionandcomputationaltime
theabovementionednetwork,optimumcurrentineach
arehighly dependent onthe initialguess fedby the op-
branchisestimatedforapowerrequirementof75000W
eratorincaseofanyoptimizationroutine.Whereas,the
andis giveninTable 4.
algorithmproposeddoesnotrequireaninitialguessand
ismuchmorecomputationallyefficient.
7.3 Comparison of proposed algorithm with conven-
tional optimizers
Thoughthisproblemcanbesolvedusingstandardopti-
mizersinMATLAB orPython,itisnotpossibletoem-
The optimumcurrentestimated in eachbranchofvari- bed this optimizer in a chip due to various limitations
ousfuel cellnetworksusing proposedalgorithmis com- givenbelow.
pared with that of standard optimizers. The optimum
currents obtained using both approaches are found to
(a) Itcannotdeterminewhetherasolutiontotheproblem
bematchingwellforallthenetworks.Table4showsthe
isfeasible ornot.
comparison of optimum current estimated using stan-
(b) It is sensitive to the initial guess passed to the opti-
dard optimizer and proposed algorithm in Python en-
mizer.
vironment for one of the networks. The time taken for
(c) Robustnesscannotbe ensured.
optimization and the value of objective function using
(d) Increasedcomputationalloadonthe hardware.
bothapproacheswhenimplementedinPythonaregiven
(e) It requires large computational time and storage
in Table 5. It can be seen that the proposed algorithm
space.
ismuchfasterthanthe standardoptimizer.
(f) Itdoesnotguaranteeglobalsolution.
Multipletrialswererunfordifferentpowerrequirements
andforvariousfuelcellnetworkswhereallstacksareas- All the above limitations can be overruled by the new
sumedtohaveaconcavepowerprofile..Itwasobserved algorithmandthuscanbeembeddedintoachipforon-
thatinalltheseruns,onlyonesolutionamongthethree line applications.
8
Table4
Optimum current values obtained in case of unequal distribution of fuel cell stacks in different rows. Preq=75000 W, φ=0.8,
Numberof FuelCell Stacks= 30, 0.1 I inf
≤ ≤
Stack Row a b I (A) I (A)
opt opt
No. No. values values (Optimizationroutine) (ProposedAlgorithm)
1 1 49.25 -0.25
9.5299 9.5355
2 1 49.302 -0.302
3 2 49.353 -0.353 0.1 0.1
4 3 49.405 -0.405
5 3 49.457 -0.457 648.8999 648.9024
6 3 49.509 -0.509
7 4 49.56 -0.56
3.2599 3.262
8 4 49.612 -0.612
9 5 49.664 -0.664
2.6699 2.674
10 5 49.716 -0.716
11 6 49.767 -0.767
2.2799 2.2766
12 6 49.819 -0.819
13 7 49.871 -0.871
1.99 1.993
14 7 49.922 -0.922
15 8 49.974 -0.974 0.1 0.1
16 9 50.026 -1.026
1.6899 1.6946
17 9 50.078 -1.078
18 10 50.129 -1.129
1.55 1.55
19 10 50.181 -1.181
20 11 50.233 -1.233
21 11 50.284 -1.284 90.0699 90.0715
22 11 50.336 -1.336
23 12 50.388 -1.388
1.3 1.2976
24 12 50.44 -1.44
25 13 50.491 -1.491
26 13 50.543 -1.543 64.1899 64.1877
27 13 50.595 -1.595
28 14 50.647 -1.647
1.13999 1.1365
29 14 50.698 -1.698
30 15 50.75 -1.75 0.1 0.1
9
8 Conclusions [7] James Larminie, Andrew Dicks, and Maurice S McDonald.
Fuelcellsystemsexplained,volume2.WileyNewYork,2003.
Anoptimizationproblemisformulatedfortheoptimum [8] Steven C Michalske and Bradley L Spare. Parallel fuel
powerdistributioncontrolforafuelcellnetwork.Anew cell stack architecture. April 15 2010. US Patent App.
12/761,284.
and computationally efficient algorithmis proposed for
thisoptimizationproblemwhichcaneasilybeembedded [9] B Rohland, J Nitsch, and H Wendt. Hydrogen and fuel
in a chip for online applications. New algorithm is pro- cellsthe clean energy system. Journal of power sources,
37(1):271–277, 1992.
posed based on the observations when KKT conditions
areappliedtothegivenoptimizationproblem.Thealgo- [10]ColleenSpiegel.PEMfuelcellmodelingandsimulationusing
rithm proposed here guarantees global optimum to the MATLAB. AcademicPress,2011.
optimization problem concerned unlike local optimum
obtained using the conventional method of using stan-
dardoptimizersandinamuchshortertime.Thiscanbe A Relaxed OptimizationProblem
veryuseful inoptimizingthe currentandimplementing
control strategies in a real fuel cell network. Analytical
Consider the optimization problem (P3) where the
solutionfor a specific convexsubproblemis alsoformu-
equalityconstraintis relaxedto inequalityconstraint.
latedinthiswork.Thoughtheanalyticalsolutionisap-
plicableonlyforaspecificvoltagecurrentprofile,theal-
gorithmcanbeusedforanyotherV-Icharacteristicsas N
well, irrespective of whether the resulting optimization min Inet = Ii (A.1a)
problemis convexornot. Ii i=1
X
N
subjectto P P (A.1b)
Theproposedalgorithmcanbefurtherextendedtoother i req
≥
power distribution systems including pumps and com- i=1
X
pressors,generators, batteries and solar cells, though a Ii,lb Ii 0 i=1:N
− ≤ ∀
fuel cell network is considered through out this paper. (A.1c)
Forexample,inamultiple pumpsystemusedforwater I I 0 i=1:N
i i,ub
transport with varying demand, the algorithm can de- − ≤ ∀ (A.1d)
terminewhichpumpshouldoperateatwhatflowrateto
satisfythe requiredwaterdemandatanypointoftime. Lagrangianforthis problemis
Here, flow rate is analogous to current while head or
pressure drop is comparable to voltage. Also, electrical
N N N
systemsusedinaparallelnetworkconfigurationissimi-
L= I +λ P P + µ (I I )
i req i i i,lb i
lartothefuelcellnetworkanditsoptimumcurrentdis- − ! −
i=1 i=1 i=1
tributionamongconnectedsources,toachieveadesired X X X
N
power,canbeidentifiedusingthealgorithm.Itcanalso
+ γ (I I ) (A.2)
be applicable for cases where combinations of different i i− i,ub
systemsareconnectedtogethertoformanetwork. Xi=1
There are a total of 2N+1 inequality constraints out
of which first N are lower bounds on current, next N
References
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laxed power inequalty constraint. Necessary conditions
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i
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γ 0 (A.4)
Albuquerque, NM,DOE/NETL-2004/1206, 2004. i ≥
µ (I I )=0 (A.5)
[4] Zhenhua Jiang and Roger A Dougal. Strategy for active i i,lb− i
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−
advanced technology batteries. In Power Electronics λ 0 (A.7)
Specialist Conference, 2003. PESC’03. 2003 IEEE 34th ≥
Annual, volume 1,pages 81–87. IEEE, 2003. N
λ(P P )=0 (A.8)
[5] Zhenhua Jiang, Lijun Gao, and Roger A Dougal. Adaptive req − i
control strategy for active power sharing in hybrid fuel Xi=1
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10