asme2e - peimannm.sdsu.edu · dscc 2015 october 28-30, 2015, columbus, usa dscc2015-9605 design...

Proceedings of the ASME 2015 Dynamic Systems and Control ConferenceDSCC 2015

October 28-30, 2015, Columbus, USA

DSCC2015-9605

DESIGN OPTIMIZATION OF SOLENOID ACTUATED BUTTERFLY VALVESDYNAMICALLY COUPLED IN SERIES

Peiman NaseradinmousaviAssistant Professor

Department of Mechanical EngineeringSan Diego State University (SDSU)

San Diego, California 92115Email: [email protected]

C. NatarajEndowed Professor of Engineered Systems

Department of Mechanical EngineeringVillanova University

Villanova, Pennsylvania 19085Email: [email protected]

ABSTRACTIn this effort, we present novel nonlinear modeling of two

solenoid actuated butterfly valves operating in series and thendevelop an optimal configuration in the presence of highly cou-pled nonlinear dynamics. The valves are used in the so-called“Smart Systems” to be employed in a wide range of applicationsincluding bioengineering, medicine, and engineering fields. Typ-ically, tens of the actuated valves are instantaneously operatingto regulate the amount of flow and also to avoid probable catas-trophic disasters which have been observed in the practice.Wefocus on minimizing the amount of energy used in the system asone of the most critical design criteria to yield an efficientop-eration. We optimize the actuation subsystems interactingwiththe highly nonlinear flow loads in order to minimize a lumpedamount of energy consumed. The contribution of this work is toinclude coupled nonlinearities of electromechanical valve sys-tems to optimize the actuation units. Stochastic, heuristic, andgradient based algorithms are utilized in seeking the optimal de-sign of two sets. The results indicate that substantial amount ofenergy can be saved by an intelligent design that helps selectparameters carefully but also uses flow torques to augment theclosing efforts.

1 IntroductionGenerally “Smart Systems” have received much attention

for a wide range of applications to be operated efficiently with

respect to the amount of energy used in the actuator units. TheUS Navy has particularly focused on developing reliable anden-ergy efficient systems to minimize the cost of operation and alsoto increase crew safety through a stable performance.

Automation systems typically consist of actuators, sensors,controllers, valves, piping, electrical cabling and communicationwiring. Many types of actuator-valve systems are in use [1, 2].One of the most critical systems to be utilized in cooling pur-poses is the so-called “Smart Valve” system. The main objec-tive of the smart valves is to shut down automatically in caseofbreakage and to reroute the flow as needed.

These sets include many interdisciplinary components inter-acting with each other through highly coupled nonlinear dynam-ics. We have previously analyzed a solenoid actuated butterflyvalve dealing with electromagnetics and fluid mechanics [3–6].High fidelity mathematical models were developed for both thesingle and coupled actuated valves. For the single set [3], ourfocus was on developing a nonlinear model to analyze the com-plicated physics of the system to be used in dynamic analysis[4]and optimization. [5]

We have captured transient chaotic and crisis dynamics ofthe single valve actuator for some critical parameters helping todefine the safe domain of operation. Determining the safe op-erational domain through the stability map helped us define thelower and upper bounds of the optimization tasks [5] and opera-tion, which reduced the amount of energy used in the single set.The first phase was to optimize the system design, particularly

1 Copyright c© 2015 by ASME

the actuation unit coupled with the mechanical and fluid parts.We then optimized the valve operation to be closed in an efficientfashion yielding the minimum energy consumption. In both theoptimization schemes, the roles of flow torques are important tohelp close the valve with minimal energy.

It is of great interest to emphasize that the smart valve sys-tems contain many of the actuated sets and hence, they wouldnot be independent of each other. These dependencies have beenobserved in the practice and probable malfunction of each setmay expectedly result in the catastrophic behavior of the wholesystem. Therefore, we have developed the coupled dynamicmodel of two actuated valves operating in series [6]. A peri-odic noise was applied on the upstream valve to evaluate its ef-fects on the downstream set of the valve and actuator. A power-ful tool of the nonlinear dynamic analysis (power spectrum)wasthen employed to present the same oscillatory response of thedownstream set with that of the upstream one; as expected, thedownstream set revealed the same frequencies of response witha smaller amplitude. Any slight dynamic change of the upstreamset, on the other hand, was shown to be effective for the down-stream set through the media trapped between two valves.

Capturing the coupled dynamics of two actuated valveswould help us optimize the design of both the actuation unitsin which an interesting connection can be distinguished betweenthe currents. The currents are subject to the interconnected flowtorques and pressure drops of the valves. Note that, for the sin-gle set, we neglected the interactions among the actuators/valvesoperating in series although the magnetic parts are remarkablyaffected by the dynamics of the neighbor sets [6].

Important nonlinear phenomena in electromechanical sys-tems have also received considerable attention. Belatoet al.[7] analyzed chaotic vibrations of an electromechanical systemwhich includes a nonlinear dynamic system consisting of a sim-ple pendulum whose support point is vibrated along a horizontalguide by a two bar linkage driven by a DC motor with limitedpower. Nonlinear dynamic analysis of a micro electromechani-cal system (MEMS) has been carried out by Xieet al. [8] basedon an invariant manifold method proposed by Boivinet al. [9].Ge and Lin [10] have studied dynamical analysis of electrome-chanical gyrostat system subjected to an external disturbance.

Optimization of solenoid actuators has recently receivedsome attention. Ju and Woong [11] have focused on the optimaldesign of solenoid actuators using a non-magnetic ring. Electro-magnetic actuator-current development has been carried out byHameyer and Nienhaus [12], and Sunget al. [13] studied devel-opment of a design process for on-off type of solenoid actuators.Kajima [14] has considered a dynamic model of the plunger typeof solenoids. Karr and Scott [15] utilized the genetic algorithmto optimize an anti-resonant electromechanical controller oper-ating in frequency domain. Mahdi [16] carried out optimizationof the PID controller parameters to operate nonlinear electrome-chanical actuator efficiently. A coupled electromechanical opti-

mization of the cost of high speed railway overheads has beencarried out by Jimenez-Octavioet al. [17]. Nowak [18] has fo-cused on presenting an algorithm of the optimization of the dy-namic parameters of an electromagnetic linear actuator operatingin error-actuated control system. Other contributions in designoptimization of electromechanical actuators include [19–25].

This paper begins with a brief nonlinear dynamic model ofthe actuators/valves operating in series [6], provided here forcompleteness. Then, the optimal design process is formulatedto help select the efficient actuator parameters coupled with theelectromagnetical, mechanical, and fluid parts in order to yieldan energy efficient system. Most electromechanical systemsusedin the flow control lines have been studied by neglecting the in-terconnections imposed by other sets. From another aspect,thelinearization method, as one of the simplest practices, is widelybeing utilized in many of analytical investigations particularlyfor the systems with a higher level of complexity and coupling.The results of both the isolated- and linearized analyses mayexpectedly be valid within a narrow domain of operation lead-ing typically to significant inaccuracy and unreliability of theresults. The contribution of this work is to optimize both theactuated valves dynamically coupled in different aspects whileour previous effort [5] was on optimizing a single unit by ne-glecting its dynamic coupling with another set. A considerableamount of energy saving was obtained via the isolated analysisbut would obviously could be affected by the dynamic interac-tions among sets. In this effort, a lumped cost function willbeminimized, with respect to the stability and physical constraints,using global optimization tools in order to obtain efficientdesignand operation of the actuation units and valves, respectively. Thiswould provide an interesting opportunity to utilize the coupledoptimization scheme, which is being developed here, for otherlarge-scale networks including oil and gas fields, municipal pip-ing systems, petrochemical plants, and aerospace. The needforoptimization clearly exists for such networks in order to improveefficiency.

2 Mathematical ModelingThe system being optimized consists of two solenoid actu-

ated butterfly valves operating in series as shown in Fig. 1(a).The actuators are connected to the valves’ stems through therack and pinion arrangements. Applying electric voltages (ACor DC), the magnetic forces move the plungers and consequentlyrotate both the valves to desirable angles. Note that we utilize areturn spring to open the valves; this is a common practice amongmanufacturers.

The mathematical modeling of such a coupled system obvi-ously needs some simplifying assumptions to avoid useless andcumbersome numerical calculations. The first one is to neglectthe magnetic diffusion. During the diffusion time, there isnopowerful magnetic force to move the plunger and the valve sub-


k1 c1

Plunger

Coil

Coil

Yoke

k2 c2

Plunger

Coil

Coil

Yoke

α1 α2Pin Pout

qin qout

qv

(a)

R1

Pin PoutP1 P2

R2

α1 α2

RL

L

(b)

FIGURE 1. (a) Two actuated butterfly valves in series; (b) A modelof two valves in series without actuation

sequently would not rotate in that time interval. Note that thediffusion time has an inverse relationship with the amount of cur-rent [3] such that a large value of the current yields a negligiblediffusion time and vice versa. We apply a current ofi = 13(A)for both the actuators yielding the diffusion time ofτd ≈ 6(ms)which can be easily neglected considering the nominal operationtime.

The second assumption is to utilize laminar flow for boththe valves. This is a common practice to avoid the tedious nu-merical calculations of a turbulent regime and also to develop ananalytical model to be used in the nonlinear dynamic analysis.The dynamic analysis needs to be done to capture the danger-ous responses of the system [4]. The validity of laminar flowassumption needs to be examined particularly with respect to theamounts of inlet velocity and pipe diameter given in Table 1.Us-ing these values, the Reynolds number indicates the existenceof the turbulent regime and questions the assumption of lami-nar flow. We hence carried out experimental work shown in Fig.2 to examine the validity of the assumption. Shown in Fig. 3

FIGURE 2. The experimental work carried out for a single set

TABLE 1. The system parameters

ρ 1000kgm3 v 3m

s

µ 0.5 Pin 256(kPa)

J1,2 0.104×10−1(kg.m2) bd1,d2 103 N.m.srad

N1,2 3300 C11,12 1.56×106(H−1)

gm1,m2 0.1(m) V1,2 24(Volt)

r1,2 0.05(m) Dv 8(in)

Ds1,s2 0.5(in) Pout 2(kPa)

k1,2 60(N.m) C21,22 6.32×108(H−1)

R1,2 1.8(Ω) L 3(m)

µ f 0.018 (Kg.m−1.s−1)

is the total torque, sum of both the hydrodynamic and bearingones, for the inlet velocity ofV0 ≈ 2.7

(ms

)and valve diameter of

Dv = 2 (inches) revealing an acceptable consistency among theexperimental data and the formula utilized in the analytical stud-ies based on the laminar flow. This also gives us the confidenceto use the analytically (and computationally) derived mathemat-ical expressions for the hydrodynamic and bearing torques.Wepreviously discussed the important roles of both the torques onthe dynamic response of a single actuated valve and subsequentlysuch effects are expected to be observed for the coupled sets[6].

We modeled the coupled system as two changing resistorsfor the opening/closing valves plus a constant one in the middleof the valves, shown in Fig. 1(b). The inlet and outlet pressuresare supposed to be known. Applying the assumption stated forthe dominant laminar flow, the Hagen-Poiseuille [26] formula


0 20 40 60 80−8

−6

−4

−2

0

2

4

6

α (degree)

Ttot(N

.m)

Expr1

Expr2

Expr3

Theory

FIGURE 3. The experimental and analytical total torques for the inletvelocity ofV0 ≈ 2.7

(ms

)and valve diameter ofDv = 2 (inches)

expresses the pressure drop between two valves (points 1 and2):

P1−P2 =128µ f L

πD4v

︸︷︷︸

RL

qv (1)

where,µ f is the fluid dynamic viscosity,Dv stands for the valvediameter,qv indicates the volumetric flow rate,L is the pipelength between two valves, andRL is the constant resistance. Thevalve’s “Resistance (R)” and “Coefficient (cv)” are the most im-portant parameters of the regulating valves including butterflyones. The valve’s resistance and coefficient are nonlinear func-tions of the valve rotation angle [27]:

Ri(αi) =891D4

v

c2vi(αi)

, i = 1,2 (2)

The pressure drop across the valve is stated as follows [28]:

∆Pi(αi) = 0.5Ri(αi)ρv2 (3)

where,α indicates the valve rotation angle,v is the flow velocity,andρ stands for the density of the media. Rewriting Eq. 3 yields,

∆Pi(αi) =π2D4

vv2

16︸︷︷︸

q2v

8×Ri(αi)ρπ2D4

v︸︷︷︸

Rni(αi)

= Rni(αi)q2v (4)

We established that both the hydrodynamic (Th) and bearing (Tb)torques [28, 29] are too sensitive to the pressure drop obtainedvia Eq. 4 leading us to reformulate them to be stated as follows.

fi(αi) =16Tci(αi)

3π(

1− Ccci(αi)(1−sin(αi))2

)2 (5)

Thi =16Tci(αi)D3

v∆Pi

3π(

1− Ccci(αi)(1−sin(αi))2

)2 = fi(αi)D3v∆Pi (6)

Tbi = 0.5Ad∆PiµDs =Ci∆Pi (7)

where,µ is the friction coefficient of the bearing area,Ds indi-cates the stem diameter of the valve,Ci =

π8 µD2

vDs, andTci andCcci stand for the hydrodynamic torque and the sum of upper andlower contraction coefficients, respectively; they dependon thevalve rotation angle [3].

The nonlinear dynamic analysis carried out for a set of theactuator/valve [4] provides the criteria needed to determine thebounds of the optimization tasks; the stability analysis obviouslyneeds to be done using an analytical model. The same practicewe would utilize for the coupled system with the aid of fittingsuitable curves oncvi andRni in order to model the system ana-lytically. For our case study ofDv=8 (in), the valves’ coefficientsand resistances are formulated as follows.

cv(αi) = aα3i +bα2

i + cαi +d (8)

Rni(αi) =7.2×105

(aα3i +bα2

i + cαi +d)2(9)

where, a = 461.9, b = −405.4, c = −1831, andd = 2207.Clearly the mass continuity principle impliesqin = qout = qv.Rewriting Eq. 4 then yields,

Pin −P1

Rn1(α1)=

P2−Pout

Rn2(α2)(10)

Rn1P2+Rn2P1 = Rn2Pin +Rn1Pout (11)

One can easily derive the coupledP1 andP2 terms by combiningEqs. 1 and 11 as follows:

P1(Rn1,Rn2,RL) =Rn2Pin +Rn1Pout+Rn1RLqv

(Rn1+Rn2)(12)

P2(Rn1,Rn2,RL) =Rn2Pin +Rn1Pout−Rn2RLqv

(Rn1+Rn2)(13)


Eqs. 12 and 13 state the roles ofRn1, Rn2, andRL on the variationsof P1 andP2 with the given values ofPin, Pout, andqv, as observedin practice. Therefore, it is fairly straightforward to conclude thedynamic sensitivity of the downstream set to any slight changesof the upstream one. We then can rewrite both the hydrodynamicand bearing torques dependency on all the resistances as follows.

Thi = fi(αi)D3v∆Pi(Rn1,Rn2,RL) (14)

Tbi = Ci∆Pi(Rn1,Rn2,RL) (15)

Note thatfi is a function of many nonlinear terms which includethe changingTci andCcci in addition to the valves’ angles. For asystematic analysis of the whole system, the following functionsare fitted to theD3

v fi of each valve:

Th1 = (a1α1eb1α11.1− c1ed1α1)

︸︷︷︸

D3v f1

(Pin −P1)

= (a1α1eb1α11.1− c1ed1α1)×

1(aα3

1+bα21+cα1+d)2

∑2i=1

1(aα3

i +bα2i +cαi+d)2

(Pin −Pout−RLqv) (16)

Th2 = (a1α2eb1α1.12 − c1ed1α2)

︸︷︷︸

D3v f2

(P2−Pout)

= (a1α2eb1α21.1− c1ed1α2)×

1(aα3

2+bα22+cα2+d)2

∑2i=1

1(aα3

i +bα2i +cαi+d)2

(Pin −Pout−RLqv) (17)

where, a1 = 0.4249, b1 = −18.52, c1 = −7.823× 10−4, andd1 =−1.084. We also replace the sign function (sign(αi)), whichis being used in the bearing torque statement to present its resis-tance role, by the smooth function tanh(Kαi) for ease of analysis.Figs. 4 and 5 are results of the stability analysis of the coupledsystem; we will report this effort as another article.

The state variables are defined as follows.

[z1,z2,z3,z4,z5,z6 = α1, α1, i1,α2, α2, i2]

We previously developed the magnetic force and rate of currentterms [3] to be used in deriving the state space equations as fol-lows.

Fmi =C2iN2

i i2i2(C1i +C2i(gmi− xi))2 (18)

diidt

=(Vi −Ri i i)(C1i +C2i(gmi− xi))

N2i

−C2i i i xi

(C1i +C2i(gmi− xi))(19)

z1 = z2 (20)

z2 =1J1

[r1C21N2

1z23

2(C11+C21(gm1− r1z1))2 +(Pin−Pout−RLqv)×

1(az31+bz21+cz1+d)2

∑i=1,41

(az3i +bz2i +czi+d)2

[

(a1z1eb1z11.1− c1ed1z1)−C1×

tanh(Kz2)]−b1dz2− k1tz1] (21)

z3 =(V1−R1z3)(C11+C21(gm1− r1z1))

N21

−

r1C21z3z2

(C11+C21(gm1− r1z1))(22)

z4 = z5 (23)

z5 =1J2

[r2C22N2

2z26

2(C12+C22(gm2− r2z4))2 +(Pin−Pout−RLqv)×

1(az34+bz24+cz4+d)2

∑i=1,41

(az3i +bz2i +czi+d)2

[

(a1z4eb1z41.1− c1ed1z4)−C2×

tanh(Kz5)]−b2dz5− k2tz4] (24)

z6 =(V2−R2z6)(C12+C22(gm2− r2z4))

N22

−

r2C22z5z6

(C12+C22(gm2− r2z4))(25)

where,x indicates the plunger displacement,r is the radius of thepinion, Fm stands for the motive force,C1 andC2 are the reluc-tances of the magnetic path without airgap and airgap, respec-tively, N is the number of coils,i indicates the applied current,gm is the nominal airgap,J is the polar moment of inertia of thevalve’s disk,bd is the equivalent torsional damping,Kt indicatesthe equivalent torsional stiffness,V is the supply voltage, andR indicates the electrical resistance of coil. Note that K=1 re-sulted in a good approximation to the sign function. Eqs. 20-25constitute the sixth order dynamic model of the coupled actua-tors/valves.

3 Optimal DesignEfficient optimization schemes are needed to be utilized in

minimizing the amount of energy used by the whole systemwith respect to the stability criteria we developed earlier[4, 30].Neglecting the parameters’ constraints established through thestability analysis would result in the failure of the whole sys-tem shown in Fig. 4 revealing the chaotic dynamics of both thevalves/actuators for a set of the critical parameters; a positiveLyapunov exponent shown in Fig. 5 confirms the chaotic be-havior of the system. We selected critical values of the equiv-alent viscous damping and friction coefficient of the bearing area


(bdi = µi = 5×10−4), based on the stability analysis, to presentthe chaotic dynamics of both the valves and actuators. The de-tailed results and discussions will be presented as anotherarticle.The US Navy has experienced such a disaster by running the flowline within the critical parameters due to the military incident.

0 10 20 30 40 50−200

−150

−100

−50

0

50

100

150

200

αi (degree)

αi

(

rad

s

)

α1

α2

FIGURE 4. The chaotic dynamics of the valves/actoators

0 50 100 150 200

−30

−25

−20

−15

−10

−5

0

Time(s)

λi

λ1=−0.057139

λ2=−0.12185

λ3=−0.13351

λ4=−0.13048

λ5=−30.7429

λ6=0.26673

FIGURE 5. The Lyapunov exponents indicating the chaotic dynamicsof the system

The problem is one of constrained optimization with pos-sibly several local minima. Therefore, we need to utilize effi-cient optimization approaches to get the global minimum; theconstraints are the stability and physical ones. The cost functionwe wish to minimize is a lumped term of the energy used in boththe sets.

minE =2

∑i=1

∫ t f

0Vii dt (26)

subject to :z1 < 90, z4 < 90

We previously reported that the analytical model would notbe valid atz1 = 90 andz4 = 90 [3–6, 28]. The cost function istypically determined with respect to the scale and performanceof the network. Tens of such actuated valves are used in the USNavy fleet and a lower lumped amount of energy consumed inthe network is needed to reduce the cost of operation. This wouldlead us to select a lumped cost function to be minimized. Afterselecting some of the parameters as predetermined, the designvariables to be used in the optimization process are chosen asfollows: C11, C12, C21, andC22 are the magnetic reluctances,gm1

andgm2 indicate the airgaps, andN1 andN2 are the number ofcoils for both the actuators.

We hence define the design variables as the following:

θ = [C11,C12,C21,C22,N1,N2,gm1,gm2]T (27)

The state equations, as discussed earlier, need to be satis-fied at all times during the optimization process and the designvariables are subject to the following lower and upper bounds.

θmin = [0.8e6,0.8e6,4e8,4e8,2900,2900,0.08,0.08]T (28)

θmax = [2e6,2e6,9e8,9e8,3400,3400,0.11,0.11]T (29)

These bounds were established based on practical system con-siderations, stability analysis, and physical constraints. We em-ploy three global optimization tools including simulated anneal-ing, genetic, and gradient based (GlobalSearch) algorithms toprovide a clear map of optimization efforts with respect to thelocality/globality of the cost function minima.

The first method (SA) was independently developed byKirkpatrick et al. [31] and by Cerny [32]. The second one (GA)has been designed based on a heuristic search to mimic the pro-cess of natural selection [33].

One of the advantages of the simulated annealing procedureis to select a new point randomly. We hence need to set the initialguesses as random numbers. The algorithm covers all new pointsto reduce the objective. At the same time, with a certain prob-ability, points that increase the objective are also accepted. Thealgorithm avoids to be trapped in local minima by using points


that raise the objective and has the potential to search globallyfor more possible solutions.

The genetic algorithm is significantly more robust than otherconventional ones. It does not break easily in the presence ofslight changes of inputs, and noise. For a large state-space, the al-gorithm may potentially exhibit significantly better performancethan typical optimization techniques.

The design variables in practice are not of the same order,and caused serious numerical errors in our initial studies.Wesolved this issue by conditioning them using a normalizationscheme as follows.

Nni =Ni

103 ;C1in =C1i

106 ;C2in =C2i

108 ;gmin = 10gmi

Note that random initial guesses we also used in the opti-mization process (as required by simulated annealing) as follows.

θnr = θlb +(θub−θlb)× rand(0,1) (30)

where rand(0,1) is a random number between zero and one. Wedeveloped the algorithm in MATLAB and captured many inter-esting results.

4 ResultsThe predetermined parameters given in Table 1 were ob-

tained from the experimental work we have done for the singleset as shown in Fig. 2. Figs. 6-9 present the optimization pro-cess for the design variables utilizing the simulated annealing,genetic, and GlobalSearch algorithms. The SA, genetic, andGSalgorithms terminate after 23920, 1740, and 4563 iterations, re-spectively, satisfying the tolerances defined for both the variablesand the lumped cost function. All methods yield lower valuesofC11, C12, C21, C22, gm1, andgm2, and higher values for the num-ber of coils with respect to their corresponding nominal valueslisted in Table 2.

These optimal variables to be used in the actuation parts areconsiderably more efficient in that higher and lower values of theactuation forces and currents are obtained as shown in Figs.10and 11, respectively. It is fairly straightforward to conclude thatfor very narrow ranges of optimal design variables,C2i ’s revealmore dominant roles with respect to their scales thanC1i ’s andNi ’s. We are hence able to distinguish the effects of even aslightly lower value ofC22 of the upstream actuator, in compar-ison with the downstream one, on the amounts of both the cur-rents and magnetic forces. A smallerC22 yields a higher valueof the motive force, and the actuator subsequently has more free-dom to consume a lower level of the current/energy. Figs. 10 and11 confirm the effects of optimalC2i ’s used in the upstream anddownstream actuators revealing slightly higher and lower values

0 0.5 1 1.5 2 2.5

x 104

4

4.2

4.4

4.6

4.8

5x 10

8

Iteration

C2i

(a)

0 500 1000 1500 20004

5

6

7

8

9x 10

8

Iteration

C2i

(b)

0 1000 2000 3000 4000 50004

5

6

7

8

9x 10

8

Iteration

C2i

(c)

FIGURE 6. The optimizedC2i ; red and blue squares stand for theupstream and downstream sets, respectively: (a) SA; (b) GA;(c) GS


0 0.5 1 1.5 2 2.5

x 104

2900

3000

3100

3200

3300

3400

Iteration

Ni

(a)

0 500 1000 1500 20002900

3000

3100

3200

3300

3400

Iteration

Ni

(b)

0 1000 2000 3000 4000 50002900

3000

3100

3200

3300

3400

Iteration

Ni

(c)

FIGURE 7. The optimizedN: (a) SA; (b) GA; (c) GS

TABLE 2. The nominal and optimal variables

Nominal SA GA GSC11106 (H

−1) 1.56 0.9001 0.9047 0.9065C21108 (H

−1) 6.32 4.002 4.001 4.001C12106 (H

−1) 1.56 0.8003 0.8036 0.8091C22108 (H

−1) 6.32 4.005 4.009 4.005

N1 3300 3400 3398 3400

N2 3300 3400 3400 3400

gm1(m) 0.1 0.08 0.08 0.08

gm2(m) 0.1 0.08 0.08 0.08

Etot 24643 13516 13521 13528

of the force and current, respectively, for the upstream setthanthose of the downstream one, even ifC11 is partially higher thanC12. The optimal number of coils shown in Fig. 7 would alsopartially help increase and reduce the motive forces and currentsof both the sets, respectively (Fmagi ∝ N2

i ); the same conclusionscan be made for the optimalC1i ’s andgmi’s presented in Figs. 8and 9.

We previously established the important roles of both thehydrodynamic and bearing torques on the valve motions. Thehydrodynamic torque is a helping load to push the valve to beclosed and is typically effective for when the valve angle islowerthan 60 [6]. Note that the bearing torque is a resistance load andremarkably becomes effective for the valve’s angle higher thanthat of 60. Hence, it would be interesting to evaluate the effectsof the optimization process on the efficient operations of boththe valves with respect to the modified behaviors of the hydrody-namic and bearing torques.

Fig. 12 reveals the optimal total torques acting on both thevalves with respect to the nominal ones. The hydrodynamictorques of both the optimal and nominal sets play drastic rolestypically for the range of 0≤ αi ≤ 60 to help close the valvesas shown in Fig. 12; we validated the effective range experimen-tally [6] which indicates the helping nature of the hydrodynamictorque by presenting positive values. Also considerably lowervalues of the bearing torques can be observed for the dominantrange ofαi ≥ 60 and the valves consequently would be subjectto smaller resistances. The circled area shown in Fig. 12 con-firms the reduced bearing torques to help consume lower valuesof the currents and lumped energy. Significantly smaller amountsof currents are used in the optimal sets, as shown by the circledarea in Fig. 11; this can be related to the decreased resistancetorques (the bearing ones) in addition to the increased magneticforces. Consequently, we would be able to apply a lower levelof


0 0.5 1 1.5 2 2.5

x 104

0.8

1

1.2

1.4

1.6

1.8

2x 10

6

Iteration

C1i

(a)

0 500 1000 1500 20000.8

1

1.2

1.4

1.6

1.8

2x 10

6

Iteration

C1i

(b)

0 1000 2000 3000 4000 50000.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2x 10

6

Iteration

C1i

(c)

FIGURE 8. The optimizedC1i : (a) SA; (b) GA; (c) GS

0 0.5 1 1.5 2 2.5

x 104

0.08

0.085

0.09

0.095

0.1

0.105

0.11

Iteration

gmi

(a)

0 500 1000 1500 20000.08

0.085

0.09

0.095

0.1

0.105

0.11

Iteration

gmi

(b)

0 1000 2000 3000 4000 50000.08

0.085

0.09

0.095

0.1

0.105

0.11

Iteration

gmi

(c)

FIGURE 9. The optimizedgm: (a) SA; (b) GA; (b) GS


0 10 20 30 400

200

400

600

800

1000

1200

1400

1600

1800

Time (s)

Fmag i(N

)

Upstream SetDownstream SetNominal Sets

FIGURE 10. The optimal and nominal magnetic forces

0 10 20 30 400

2

4

6

8

10

12

14

Time (s)

i(A)

Downstream SetUpstream SetNominal Sets

FIGURE 11. The optimal and nominal applied currents

the lumped energy to carry out the closing efforts. From anotheraspect, the higher and lower values of the magnetic forces andbearing torques, respectively, lead to the valves’ higher rotationangles shown in Fig. 13;α1op = 85 andα2op = 83.

Consequently, reduced amounts of energies are consumedas shown in Figs. 14(a), 14(b), and 14(c). Shown in Figs. 14(a),14(b), and 14(c) reveal upward of 45.15%, 45.13%, and 45.10%energy savings obtained through the SA, genetic, and GS algo-rithms, respectively, for the whole optimal systems in compari-

10−2

10−1

100

101

−200

0

200

400

600

800

1000

1200

1400

Time (s)

Ttot(N.m

)

101

−40

−20

Upstream SetDownstream SetNominal Sets

FIGURE 12. The total torques acting on both the valves

0 20 40 60 800

0.5

1

1.5

α(degree)

α

(

rad

s

)

Upstream Valve OptimizedDownstream Valve OptimizedNominal Valves’ Angles

FIGURE 13. The optimal and nominal valves’ rotation angles

son with that of the nominal one.

∆ESA =Enominal−Eoptimal

Enomial×100≈ 45.15% (31)

∆EGA =Enominal−Eoptimal

Enomial×100≈ 45.13% (32)

∆EGS =Enominal−Eoptimal

Enomial×100≈ 45.10% (33)

We repeatedly examined the optimization schemes to avoid to


0 0.5 1 1.5 2 2.5

x 104

1.4

1.6

1.8

2

2.2

2.4x 10

4

Iteration

Energy

(a)

0 500 1000 1500 20001.2

1.4

1.6

1.8

2

2.2

2.4

2.6x 10

4

Iteration

Energy

(b)

0 1000 2000 3000 4000 5000

1.4

1.6

1.8

2

2.2

2.4

2.6x 10

4

Iteration

Energy

(c)

FIGURE 14. The optimized lumped amount of energy: (a) SA(Eopm= 13516); (b) GA (Eopm= 13521); (b) GS (Eopm= 13528)

be trapped in local minima. The negligible difference (lessthan0.15%) among the SA, genetic, and GS algorithms would poten-tially confirm the global minimum value which looks meaningfulwith respect to the physics of the system. The amount of energysaved is promising in that we typically run tens of valve-actuatorsets in a flow line and using such optimal configurations wouldhelp reduce the amount of energy consumption and subsequentlythe cost of operation for the whole network.

5 ConclusionsThis paper presented a novel nonlinear model of two actu-

ators and valves operated in series. We discussed the effects ofmutual interactions between the valves’ dynamics in correlationswith the flow nonlinear torques including both the hydrodynamicand bearing ones. These dependencies among different compo-nents were formalized to yield a sixth order dynamic model ofthe whole system. We used simulated annealing, genetic, andgradient based algorithms to carry out optimization and obtainedthe global minimum of the lumped cost function defined as thesum of energy consumed in each set.

The principal results of this paper can be summarized as fol-lows.

• Energy can be saved by a remarkable amount (as much as45%) by implementing optimal design.

• The optimal flow torques help consume a minimum level ofthe lumped energy.

• Lower values of the currents and subsequently instantaneousenergies (by plottingEins= vi i i vs. αi) are consumed partic-ularly for higher rotation angles.

• Higher values of the motive forces are obtained.• Better optimal performances would be obtained using opti-

mal design.

We currently focus on developing a comprehensive modelfor n valves and actuators to be operated optimally in series.

ACKNOWLEDGMENTThe experimental work of this research was supported by

Office of Naval Research Grant (N00014/2008/1/0435).

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