I. INTRODUCTION

Computer Simulations ofOptimum Boost andBuck-Boost Converters

S. RAHMAN, Member, IEEE

F.C. LEE, Member, IEEEVirginia Polytechnic Institute and State University

The developmeiit of mathematical models suitable for minimum

weight boost and buck-boost converter designs are presented. The

facility of an augumented Lagrangian (ALAG) multiplier-based

nonlinear programming technique is demonstrated for minimum

weight design optimizations of boost and buck-boost power con-

verters. ALAG-based computer simulation results for those two

minimum weight designs are discussed. Certain important features of

ALAG are presented in the framework of a comprehensive design ex-

ample for boost and buck-boost power converter design optimization.

The study provides refreshing design insight of power converters and

presents such information as weight and loss profiles of various

semiconductor components and magnetics as a function of the switch-

ing frequency.

Manuscript received January 20, 1982.

This work was supported by Subcontract G82313 CH8M to VirginiaPolytechnic Institute and State University from TRW Defense andSpace System Groups under NASA Lewis Research Center PrimeContract NAS3-2105 1.

Authors' address: Electrical Engineering Department, VirginiaPolytechnic Institute and State University, Blacksburg, VA 24061.

0018-9251/82/0900-0598 $00.75 1982 IEEE

The advantages of a comprehensive power converterdesign approach were demonstrated previously on abuck converter and a half-bridge converter [1, 2]. Thedesign allows one to identify a set of power converterdesign parameters which satisfies all design re-quirements and concurrently minimizes the converterweight and/or loss. This paper presents an extension ofthe previous work to design optimizations of a boostand buck-boost converter. First, mathematical modelsare formulated for the design and operation of thesetwo power converters. Based on these mathematicalmodels, an effective nonlinear programming technique,augmented Lagrange (ALAG) penalty functionalgorithm, is selected to search for the optimal set ofconverter design parameters. The facility of ALAGnonlinear programming technique is demonstrated suc-cessfully for the boost and buck-boost design. In addi-tion, six variable scaling options are designed and im-plemented in the ALAG program to facilitate program-ming convergence and to expedite the rate of con-vergence for the complex and multidimensionalnonlinear optimization problems.

At the beginning the mathematical background ofthe ALAG-based nonlinear programming algorithm ispresented briefly. This is followed by a discussion ofboost and buck-boost converter power circuits. A moregeneral buck-boost converter circuit is examined indetail. Various design requirements and physicaloperating characteristics of this converter are summariz-ed in the form of equality and inequality constraints.The minimum weight design requirement is formulatedas the objective function. For a detailed study of a boostconverter see Lee et al. [3].

Following the presentation of the mathematicalmodel, the programming requirements for using theALAG-based algorithm are discussed. Specialcharacteristics of an exterior point based optimizationalgorithm (like ALAG) are discussed here. The in-fluence of scaling on the rate of convergence is alsodiscussed. Impacts of variable scaling versus constraintscaling are examined. This results in a compromiseselection of both scaling techniques in the algorithm.

Finally, detailed results are presented for two sampleproblems of minimum weight design optimizations of aboost and a buck-boost converter. The sample designsprovide detailed values for each component (e.g., R, L,C, cross-sectional area and number of turns of L, etc.)for various switching frequencies. These show the trade-offs between weight and power loss as a function ofswitching frequency.

A. Power Converter Optimization-MathematicalModel

The utility of a design optimization is to pinpoint thedetailed converter design which meets given performance

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-18, NO.5 SEPTEMBER 1982598

specifications and to achieve concurrently the minimiza-tion of a certain converter characteristic defined by thedesigner. The task may be represented as amathematical model as follows:

P1: minimize f(X, Y),

Subject to

XEE+

i= 1,2,...,p

hj(X, Y,Z)=0, j= 1,2,...,q.

Here X is an n-dimensional vector representingpower- and control-circuit parameters to be optimallyselected. The components of X are values of R, L, C,the switching frequency, and the design details of themagnetic components such as core area, mean corelength, permeability, wire size, number of turns, andturns ratio of multiple-winding magnetics. The E+ is thepositive orthant of n-dimensional Euclidean space E".

The YE El represents the vector of constants relatedto component characteristics. These constants areknown to designers. Examples include winding and coredensities, winding resistivity, window-fill factor of thecore, winding-pitch factor (i.e., the ratio of the meanlength of one-turn winding to the core circumference),transistor and diode conduction and switchingcharacteristics, core-loss parameters, intended max-imum operating flux of given magnetics, and equivalentseries resistance (ESR) as well as energy-storage char-acteristics of filter capacitors.

The Z E E+ represents the vector of performance re-quirements to be met by the optimum design. Control-independent requirements include input-outputvoltages, output power, maximum weight, minimumefficiency, source EMI, and maximum output ripple.Control-independent requirements include regulatorstability, minimum audiosusceptibility rejection, max-imum output impedance, transient response subjectedto a step change of the input voltage or the load. Furtherdetails of design optimization approach can be found inRahman and Lee [4].

II. ALAG-BASED OPTIMIZATION TECHNIQUE

The augmented Lagrangian penalty function (orALAG) technique is essentially a method for solving aconstrained optimization problem using an unconstrainedoptimization method with the help of some transforma-tion. In other words, this is a concept of minimizingf(x)(see P1) with an unconstrained optimization methodwhile maintaining implicit control over the constraintviolations by penalizing the augmented f(x) at pointswhere the constraints are violated.

A. Outline of a Transformation

For reasons of simplicity let us consider only the set ofdesign variables X and the inequality constraints from P1.

,P(x, r) = f(x) + +[g(x), r]

where *( ) is the augmented f(x), r, in general, is a vectorof controlling parameters, and 4) is a real-valuedfunction which imposes the penalty. However, the actionof imposing the penalty is controlled by r. It may beobserved here that with a suitable choice of 4. and itscontrol r, one can use an efficient unconstrainedalgorithm (e.g., Broyden's quasi-Newton method, see[5]) in order to completely solve the constrainedproblem.

One approach to implementing the above is asfollows. Choose 4) and a sequence rk such that x(rk) isdetermined some way and x(rk) -> x* as k -- oo. x* is thevalue of the vector x for which f(x) is minimized.

B. Penalty Function Technique

As mentioned earlier, the penalty function trans-formation technique imposes an increasing penalty onthe augmented objective function as constraint violationincreases. Various penalty functions and their modifica-tions have been proposed in connection with the penaltyfunction based transformation. Penalty functions utilizedin the algorithms for our study are discussed in the fol-lowing.

Hestenes [6], Powell [7], and Haarhoff and Buys [8],at about the same time, proposed very similar modifica-tions to the traditional penalty function technique forequality constraints to alleviate some of the computa-tional difficulties. The modified penalty functions con-tained controlling parameters 0 E Ep+q and o E E+P+qwhereas the traditional penalty function included onlyone controlling parameter r E E+.. Powell [7] andHaarhoff and Buys [8] proposed algorithms in whichthe vector of parameters a is changed only when the rateof convergence is not satisfactory and e is changed everyiteration so as to enforce constraint satisfaction. The im-portant feature of this approach is that lI/ri is not re-quired to tend to zero for the convergence of the algo-rithm.

Rockafellar [9] proposed a suitable modification ofthe Powell-Hestenes augmented Lagrangian (ALAG)penalty function to solve the inequality constrainedproblem. Fletcher [10] proposed algorithms for solvingthe inequality constrained problem using the ALAGpenalty function technique. The augmented Lagrangianpenalty function for P1 is obtained by combining thePowell-Hestenes penalty function and the Rockafellarpenalty function.

tp(x, A, a) = f(x) - > (Ajhj - I hj

+ 2 z(ii) ? -

gi = min[(gi - A5/oi), 01, ai=AJ/,, VL

In the preceding function Ai/ori represents a penalizing

RAHMAN/LEE: COMPUTER SIMULATIONS OF OPTIMUM BOOST AND BUCK-BOOST CONVERTERS

gi(X, Y, Z) >.- O.

599

threshold for ith inequality constraint. Increases in :ri toenforce faster convergence reduce the penalty thresholdlevel and lead to closer constraint satisfaction. For in-equality constraints, when gi > 0, Ai (or Oi) is relaxedto zero. Otherwise it is changed so as to make the cor-responding constraint active at the current solution X.

The ALAG penalty function algorithms are based onthe following duality results. Let X (A) = X (A, cr) bethe unconstrained minimizer of (x, A, o-) for specified Aand Cr. Then the dual function at X(A) may be representedas d(A) = k[X(A), A, a]. The duality results may besummarized as follows:

ad/aAj = -hj, j = 1, 2, ..., q

ad/a Xi = -min(gi, Xi/o1), i = 1, 2, ... p

A2d [-NTGlN 0?]I3- I

where G is the Hessian of W with respect to X. S is adiagonal matrix with elements oi corresponding to in-active inequalities gi >, 0, and the columns of the matrixN are the gradients of equalities and violated ine-qualities gi < 0, at X(A). These duality results indicatethat A should be varied so as to maximize d(A). Fordetails regarding iterative methods for varying A and as-sociated penalty function algorithm see [11].

III. BOOST AND BUCK-BOOST POWERCONVERTERS

In this section the circuit schematics (see Figs. 1 and2) for boost and buck-boost converters are presentedalong with a detailed examination of a more generalbuck-boost converter circuit. The purpose is to definethis converter design problem in such a way that theapplicability of ALAG-based optimization techniquecan be demonstrated. The input-output relationship andderivations of the constraints are given in [12]. Theresults of design optimization are presented in a latersection.

A. Design Variables

There are 24 unknown variables representing thedetails of buck-boost converter magnetic design. Thiscircuit contains a two-stage input-filter, a two-windingenergy storage inductor, a power transistor, a diode,and an output filter. The design variables are

RL, R2, Rp dc winding resistances of inductors LI,L2, Lp, respectively; note: Np = N, isused in the design example

R, input filter damping resistorL1, L, input filter inductors

Lp primary inductance of two-winding in-ductor

C,, C4, C5 filter capacitorsA1, A2, AP core cross-sectional area of inductors

LI, L2, and Lp, respectivelyZA, Z,, Zp mean magnetic path length of induc-

tors L,, L,, and Lp, respectivelyN1, N,2, Np number of turns on inductors LI, L,,

and Lp, respectively; note: Np = N, isused in this design example

A,1 A,2, Ap, winding areas per turn for Ll, L,, andLp, respectively

F transistor switching frequencyeff overall operating efficiency.

It should be mentioned here that in order to separatethe resonant frequencies of the two input filter stages thefollowing relationship between LI and L2 is used: L1I/L2= PE2; see [13] for details.

In order to perform the tradeoff study betweenswitching frequency, and power loss and weight, thevariable F is held constant for each run of the optimiza-tion program. Therefore there is a net total of 22variables in the design optimization program.

B. Design Constants

These design constants are obtained either throughmanufacturer's specifications or the designers' own ex-periences; numerical values are given in the parenthesis,all units are in MKS system:

Fig. 1. Schematic of boost converter.

I n

Fig. 2. Schematic of buck-boost converter.

RL FC winding pitch factor = (mean length per turn)Eo (core circumference) (1.9)

Fw core window fill factor (0.4)p conductor resistivity (0. 172 x 10 -7 Q - m)D, core density (7800)DC conductor density (8900)

+ BI maximum operating flux density (0.4)RL DK weight per farad (DK,, DK4, DKS/210, 1100, 72)

LE VST transistor saturation voltage drop (0.25 V)VBE transistor emitter-to-base voltage drop (0.8 V)TSR transistor turn-on rise time (0.15 Ls)TSF transistor tum-off fall time (0.2 ,us)

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-18, NO. 5 SEPTEMBER 1982600

VD diode conduction voltage drop (0.9 V)TND diode turn-on rise time (0.03 us)TFD diode turn-off fall time (0.05 pis)TRE diode turn-off recovery time (0.03 ,us)KH heat sink weight density (15.4 W/kg)KS source weight density (30.8 W/kg)XN turns ratio (1.0).

C. Power Converter Performance Requirements

The following power converter performance re-quirements are specified. These performance re-quirements are employed in the next section to for-mulate design constraints.

El input voltage (28 V)E. output voltage (37.5 V)P0 output power (70 W)S frequency-dependent source conducted in-

terference (0.1 A); this specification limits themaximum percentage of the switching currentbeing reflected back to the source to ensure thatthe source is not disturbed much by the switchingaction downstream

VR output ripple factor (1 percent) which is definedas output ripple factor (percent) = (peak-to-peak output ripple voltage)/(nominal dc outputvoltage)

PEI input-filter resonant peaking limit (2); the inputfilter peaking at its resonant frequency should belimited in order not to degrade the stability andthe audiosusceptibility of the converter.

Objective Function: F = WI + WTW + WC + WS +WH where

WI (core weight) = DI(A1Z, + A2Z2 + ApZp)

WTW (winding weight) = 4FcDC(Ac1N1 Va+ AC2N2fA2 + 2Acj,,NP0VXp)

WC (capacitor weight) = DK3C3 + DK4C4 + DR5C,WS (source weight) = P0/(eff KJ)

WH (heat sink weight) = Po(l - eff)/(eff K.,)

D. Design Constraints

Highlights of the buck-boost converter char-acteristics include power loss, core window area, coreflux density, magnetic winding resistance, and someother converter properties. These are modeled in theform of constraints presented in the following:

1. Loss constraint C(l) = 0:

C(1) = PJ(1/eff) - 1]- PIF - PQ - PD - POF

where PIF = input filter copper loss

PIF = [P0/(effEl)J2 (RI + R2).

2. PQ = transistor saturation loss + base drive loss +transistor turn on loss + transistor turn off loss,

PQ = (POVST)/(eff El) + (0.1 POVBE)/(eff El)

+ (TsRF/6) {[(EO + VD)/nI + Es + 2VsT}

[(POI(eff El)) (Eo + nE1)/EO

- E1EO/(2Lp(Eo + nE1)F))]

+ (TSFF/6) {[(EO + VD)In] + Es + 2VST}

{(PO/(eff E))((Eo + nE,)/EO]

+ [E1EO/(2Lp(Eo + nE1)F)J}.

3. PD = diode conduction loss + turn on loss + turnoff loss,

PD = (POVD)/(effEO) + [nE1 + E0)/12]

{[Po(Eo + nE1)/(eff nE1E4)J

+ [EIEOI(2nLp(Eo + nE1)F)]}

- [(nE1 + Eo)(TsD + 3TRE)F/12]{[PO(EO

+ nE1)I(eff nE1Eo)]-[EjEOI(2nLp(Eo+nE,)F)]}.

4. POF = two-winding inductor (1) copper and corelosses,

POF = [Eo/(Eo + nE1)]{[PO(E0 + nE1)/(eff E1E0)]2

+ [E52Ej/(12L2(EO + nE,)2F2)1}Rp

+ [nE,/(Eo + nE)] {[EjE^/( 1 2n2L2(E0

+ nE,)2F2)] + (Eo + nE1)2P^/

(eff2n2E#E2)}n2Rp + [EIEo/((Eo + nEj)Np);- (80 Zp VT) (0.0022).

5. PCAP = output filter capacitor ESR loss,

PCAP = [nE1R5/(EO + nEi)1([E2Ej/(12n2Lp2 (Eo+ nE,)2F2)J {[(EO + nE1)Po(nE1E0 eff)]

- (pdlEO)}2 + Pj/(nE,EO)).6. Parasitic resistance for LI, L2, T, C(2) = C(3) =C(12) = 0,

RAHMAN/LEE: COMPUTER SIMULATIONS OF OPTIMUM BOOST AND BUCK-BOOST CONVERTERS 601

C(2), C(3), C(12) = RiAci - 4QFcNi Vi, i = 1,2,p.

7. Input filter peaking constraint C(4) = 0,

C(4) = (PE1)2 - [1 + (R3C3/L1)]/{(C4/C3)2

+ (R3C3/L1)[1 - (C4/C3) - (L2/L1)(C4/C3)]2}.

8. Operating flux density constraint C(5) = C(6) =C(9) = 0,

C(5), C(6) = NAi - L,Po/(eff EBsi) i= 1. 2

C(9) = NpAp - (L1/BSp){[Po(Eo + nE1)/(eff EIEo)]

+ E1Eo/(2Lp(Eo + nE1)F)}

9. Window area constraint C(7) = C(8) = C(10) = 0.

C(7), C(8) = (NiAci/Fw)0°5 - Z1/2Tr + \/A7/2;

i = 1, 2.

C(10) = (2N0Acp/Fw)05 - Z42T A2+ .

10. Output ripple factor constraint C(II) = 0,

C(11) = VR - [P0(E0 + nE1)/(effEIE6n)+ Ej1(2Lp(Eo + nE1)nF)]R5

- Po/(2EO(Eo + nE1)C5F).

11. Frequency dependent source EMI constraint C( 3): 0,

C(13) = (S/\/T+7F7203)T) (1/\7WT7+l2)

- [(L2C4/L1C3)(2rF I) (lID)

- (C4/C3)(2nF V7 3I )2]-f

where

C(14) = 0.97 - eff : 0C(15) = RT-R - R2 0C(16) = C3 - 1.0 X 10-6 > 0C(17) = C4 - 1.0 X 10-6 > 0.

In constraint C(15), the sum of R, and R2 are limited towithin a certain predetermined value RT (= 0.1 Q in thepresent examples).

IV. SPECIAL FEATURES OF ALAG

A. Exterior Point Minimization

ALAG penalty function based optimization is an ex-terior point unconstrained minimization technique. Thissequential method is characterized by its use of infeasi-ble points. Such an example is presented in Fig. 3. It canbe seen here that the initial starting point for x is outsidethe feasible region, and the algorithm forces con-vergence to a feasible point in the limit. The intuitivebasis can be seen as follows. The penalty term will bevery large if x stays too far from the feasible region.Therefore, in the interest of reducing (and ultimatelyeliminating) the penalty function, optimum x willeventually be inside the feasible region. Thus the advan-tage of an exterior point algorithm is that the initialestimates for the design variables need not be feasible.

FEASIBLEREGION

TRAJECTORY OFUNCONSTRAINEDMINIMA

-XIx2z '12

Fig. 3. Minimize -xx2 subject to x + x2 0, -xI + x2 + I : 0.

A = [2Po(Eo + nE1)I(Xeff E1E)] sin[ITET/(4 + nEf)]

B = [E1Eo/(irLp(Eo + nEi)F)](cos[ET4/(F4E + nE)]

- sin[rEo/(Eo + nE1)]/[rEJ,Q/(F + nE1)])

D = R3(C/L1 )°5.

12. Other inequality constraints C(14), C(15), C(16),C(17) > 0.

The following constraints are employed to confinethe design variables in certain reasonable ranges in orderto facilitate fast convergence:

B. Scaling

Penalty function methods can be computationallyineffective if certain constraints dominate others. Dueto the wide ranging values of design variables and thecomplicated nature of the constraints such computa-tional difficulties are not uncommon in the use ofALAG technique. Fortunately, the use of appropriatescaling can greatly alleviate this problem. Two suchscaling methods (variable and constraint) are discussedbelow.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-18, NO.5 SEPTEMBER 1982602

Variable Scaling. In a switching power converterdesign, the values of design variables are scattered overa wide range. For example, the capacitance may be onthe order of 10A and the switching frequency on theorder of 105. These widely scattered values of variablesare causes of convergence difficulties. Therefore thevariable scaling technique is provided in the computerprogram to scale all the variables. For example, for xl= 0.5 x 10-,x2 = 0.8 x 103, onecanuse VSCAL(1)= 10-7, VSCAL (2) = 102, SO that

xl/VSCAL (1) = 5.0

x2/VSCAL (2) = 8.0

where VSCAL (1), VSCAL (2) are scale factors for therespective variables. See Available Scaling Optionsbelow.

For a reasonably acceptable accuracy, the tolerancefor variable convergence is set around EPS = 10-6 to10-7. For the program to exit from the iterative com-putation through the variable convergence criterion, itmust satisfy the following requirement:

maxk k) - i e for all i.

That is, the largest difference between two values (fromconsecutive iterations) of any variable must be less thanthe tolerance required. It should also be mentioned thatthe program can also exit via a constraint convergencecriterion discussed in the following.

Constraint Scaling. It is very unlikely that the initialguess of starting point can satisfy all the constraints tothe extent that each equality constraint residual isrelatively smaller than the constraint tolerance and eachinequality constraint is also satisfied. If that is the case,then we have already solved the problem without usingthe computer. In reality, based on the set of initialguesses, the constraint values can vary over a widerange. It is desirable to scale each constraint by a factorso that the effect of violating a given constraint is of thesame order of magnitude as the effect of violating anyother constraint. Unfortunately, there are so far nouniversal guidelines for selecting the constraint scalingparameters. It has been observed that faster con-vergence can be achieved by the proper selection ofthese parameters. However, improper use of constraintscaling can cause divergence problems. Details on theavailable scaling options are given in the next section.

Whenever the maximum scaled constraint violationAKK(k) is less than AKMIN (constraint tolerance), programconvergence is reached.

The ALAG-based optimization program can be runby using variable scaling alone. However, judiciousselection of constraint scale factor has been found to aidrapid convergence.

Available Scaling Options. There are six differentuser-selectable scaling options programmed in the

START

Read input data|

Set EPS(l)compute circuit CONSTANTS

CALL SCALEScompute all the scale factors

Scale the input orStarting X o vector

CALL ALAGAUse Augmented Lagrangion Penalty

function algorthm to solve the problemFrom ALAGA

r nd values of varablesusing variable scale factors

Computeoptimal performancefactors for the circuit

PrintResults

Fig. 4. Flowchart showing major operations in algorithm.

optimization model and simulation package. In eachcase, with the exception of first two, the subroutineSCALES (see flow chart in Fig. 4) determines the scalefactors on the basis of the input information. The op-tions are as follows:

1) No scaling; the variable and function values areused unmodified in the main program.

2) All scale factors are set by the user.3) Scaled variable values lie within a user selectable

range. No scaling for constraint and the objective func-tions.

4) Variable scaling is same as in option 3. However,scaled values of constraints and objective functions liebetween 1.0 and -1.0.

5) Scaled variable values and scaled moduli of theconstraint and objective functions lie within a user-selectable range.

6) Variable and objective function scalings are sameas in option 5. However, the squares of the scaled con-straint functions lie within a user selectable range.

The designer chooses the scaling option on the basisof input information. He must pay particular attentionto the diversity of variable and constraint functionvalues. Also, this option allows him to place varyingamounts of emphasis on different quantities (e.g.,variable, constraint, and objective functions). Fig. 4presents a flowchart summarizing the major operationswithin the ALAG-based optimization algorithm usedfor this research project. Note that the optimization iscarried out internal to the main subroutine ALAGAwhich in turn uses some other subroutines to performthe optimization.

RAHMAN/LEE: COMPUTER SIMULATIONS OF OPTIMUM BOOST AND BUCK-BOOST CONVERTERS 603

TABLE IOptimization Results for Buck-Boost Converter

Buck-Boost Converter Optimization Results

F 20K 30K 40K 50K 60K 70K 80K 90K 100K 110K 120K

Al 0.324x10 0.288x10 0.254x10 0.226x10 0.202x10 0.174x10 0.938xl0 0.942x10 0.954x10 0.934x10 0.911x10

Ni 50.613 44.098 41.895 40.892 40.050 43.077 61.340 57.774 53.123 51.336 48.837

ACI 0.543x10 0.444x10 0.398x10 0.366x10 0.335x10 0.323x10 0.321x10 0.309x10 0.284x10 0.272x10 0.251x10

Zl 0.473xlO-I 0.417xlO-1 0.387x10-1 0.366xlO- 0.346xlO-I 0.340x10oI 0.345x10-I 0.333xlO- 0.315xlO-1 0.306x10- 0.291x0_I

RI 0.699x10 0.698x10 0.695x10 0.694x10 0.702x10 0.729x10( 0.766x10 0.752x10 0.756xl0 0.754x10 0.768x10

LI 0.217x10 0.167x10 0.141x10 0.123x10 0.108x10 0.103x10 0.805x10 0.763x10 0.712x10 0.671x10 0.619x10

A2 0.203x10-4 0.166xl-U4 0.145x10-4 0.128x10-4 0.125x10-4 0.167x10-4 0.121x10-4 0.113x10-4 0.103x10-4 0.966x10-5 0.0964x10-5

27.lb7 25.535 24.420 24.121

0.534xlO6 0.452xlO6 0.400xlO 6 0.370xlO6

0.355xlO1 0.318xlO 1 0.295xlO 1 0.279x10

0.300xlO1 0.302xlO 1 0.304xlO1 0.305xlO

d.72,xlO 0.558xlO4 0.469xlO 4 0.409xlO

0.327xlO 0.248xlO 40.207xlO 0.180x10

68.957 63.842 59.057 55.911

0.172xlO 0.142xlO 0, 125xlO 0.115xlO

0.452xlO1I 0.395xlO 1 0.358xlO 1 0.335xl(I

21.543 14.872 15.952

0.336xlO 0.295xlO 0.310xlU

0.262xlO 0.246xlOI

0.234xlO

0.297xlO 0.271x1lOI 0.234xlO

0.358xlO 0.342xl() 0.268xlO 4

0.165xlO 0.14Ox10 0.160xlO

54 . 317 50.082 2 7 . 32 5

0.lllxlO 0.135xlO6 0.179xlO60 332xlO O.324xlO 0.301xlO

16.099 16.440 16.539 15.377

-6 0.284xl0 0.274x10-6 0.271x10-60.29 7xl0

0.228xlOI 0.222xlO I 0 217xlO 1 0.212xlO I

0.238xlO 0.243xlO 0.245xlO 0.231xlO

0.254xlO 0.237xlO 0.224xlO 0.206xlO

-4 -4 -4 -5

0.149xlO 0.146xlO4 0 131xlO 0.886xlO

27.883 29,712 30.358 44.22

0.175xlO 0.188xlO 0.181xlO 0.175xlO

0.297xlOl 0.307xl0 0.299xlO-1 0.314xlO-

RP 0.301 0.294 0.281 0.269 0.261 0.182 0.796xlO n.807xlO n.791x10 n.79hxlO 0.384xlO4 -4 --44 -44 087l-4 071l-4 07xl-4 -484l

LP 0.940xlO 0.692x10 0,545x10 0.461xlO 0.428xlO 0.326x(10 0.148xlO 0.156xlO 0.188xlO 0.172x10) 0.179xlO4

C3 0.103x10-3 0.779x10-4 0.621x10-4 0.519x10-4 0.453x10-4 0.389x10-4 0.438x10-4 0.353x10-4 0.128x10-4 0 115xl(-14 0.774x10-5R3 1.128 1.106 1.136 1.166 1.173 1.228 1.129 1.330 1.408 0.974 0.931

C4 0.162xlO-4 0.112x10-4 0.898x10-5 0.760xlU-5 0.672x10-5 0.563>;10- 5 0.836c10-5 0.788x1l)-5 0.720xlO-5 0.605x1)-5 0.399x10-5

C5 0.148x10 3 0.121x10 3 0.107xlO 0.985x10 0 907xO1 0.892xlO 0.112xlO 0.lUlxlO 0.881xlO 0.868xl() 0.819xlt)E 0.8208 0.8238 0.8271 018295 0.8319 0.0567 0.8751 0.8761 0.8783 0.8751 0.8691

PQ 1.4679 1.6866 1.9036 2 1187 2 3252 2.4924 2.8014 2.9771 3,1171 3.3447 3.5515

PD 2.0944 2.1174 2.1389 2.1650 2 1994 2.1445 1.9662 2.0273 2.1182 2.1492 2.2235

PCAp 0.1958 0.2275 0.2489 0.2637 0.2736 0.2761 0.3397 0 3188 0.2964 0.3016 0.3018

PMAG 11.5281 10.9421 10.3432 9.8383 9.3489 6.7959 4.8831 4.5803 4.1676 4 1959 4.4609

PT 15.2862 14.9736 14.6346 14.3857 14.1471 11.7030 9.9904 9.9035 9 6993 9.9915 10.5377

WS 2.7689 2.7589 2 7479 2.7398 2.7320 2 6529 2.5971 2 5943 2 5876 2.5971 2.6149

WHI 0.9925 0.9723 0 9503 0.9341 0.9186 0.7603 0.6487 0.6431 0 6298 0.6488 0.6843

WI 0.02992 0.02114 0 01678 0.01394 0.01214 0,01138 0.847xlO 0 793x10 0.762xlO 0.691x10 0.583xlO

WW 0.02419 0.01639 0.01275 0.01068 0.911xlO 2 0.857xlO 2 0.790x10 2 0.735xlO 2 O705xlO 2 0.652xlO 2 0.650xlO 2

WC 0.0501 0.03744 0.03065 0.02634 0.02342 0.1>2073 0.02646 0.02336 0.01694 0.0153 0.0119

WMAG. 0.0534 0.03753 0.02952 0.2462 0.02126 0(.01995 0.01638 0.01528 0.01467 0.01344 0.01234

WT 3.8650 3.8062 3.7583 3.7249 3.6954 3.4!3 3.2886 3.2759 3.2491 3.2747 3. 3234

V. OPTIMIZATION RESULTS OF BOOST AND To facilitate comparison of the optimal converterBUCK-BOOST CONVERTERS designs between the boost converter and the buck-boost

By treating the switching frequency as a constant in converter, the same input-output rquirments, design con-

stants, and converter performance specifications areeach optimization run, a set of converter parameter data used. ponviemre designmincsights,cthoss and

is obtained. This set of data represents the optimum weighT breakdo are inststhewloss and

converter design under the specified switching frequen- uwenght breakdowns are plottedagarnst the switching fre-

cy. A number of runs are executed by varying the fre- an boos.thconvereresquency between 20 kHz to 130 kHz in 10 kHz steps. adebintionseutiedlDetailed optimization results for buck-boost and boost breakdowns are shown below:converters are presented in Tables I and II, respectively.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-18, NO.5 SEPTEMBER 1982

N2

AC2

Z2

R2

L2

AP

NP

ACP

ZP

604

TABLE IIOptimization Results for Boost Converter

Boost Converter Optimization Design ResultsF 20K 30K 40K 50K 60K 70K 80K 90K 100K 110K 120K

Ai 0.469x10 0.438x10 0.436x10 0.358x10 0.459x10 0.279x10 0.229x10 0.236x10 0.208x10 0.175x10 0.174x10

N1 94.297 79.345 79.098 69.350 27.441 26.907 27.109 19.767 22.550 22.798 22.771

ACI 0.334x106 0.322x10 60.319x106 0.243x10 0.119xlO 6 0.856x1O 70.783x10 7 0.604x10 7 0.634x10 7 0.556x10 70.555x10 7

Zl 0.405xlO1 0.349xlO1 0.347x101 0.289x101 0.169xlO1 0.137xlO1 0.129x101 0.111X10- 1.112xlO1 0.105xlO1 0.104xlO1

RI 0.697xl10 0.676x10- I0.678x1O 1 0.708x1O 1 0.645x10 I 0.6"8:. 10 10.688xlO 1 0.659xl10 0.673x10 0.711x10 0.710x10

Li 0.665x10-4 0.527xI0O4 0.523xI0 0.376xlO14 0.189xlO 4 0.IlJxlU 40.931xI0 5 0.673x10 5 0.672x10 5 0.564x10 O.562x10

A2 0.306x10 5 0.250x10 50.247x10 5 0.186x10 5 0.506x10 5 0.247xi0 50.203x10 5 0.135x10 5 0.127x10 S 0.103x10 50.102xl0 5

N2 48.282 46.461 46.648 44.602 8.288 10.119 10.216 11.584 12.280 12.934 12.945

AC2 0.369xlO 6 0.351x106 0.349x10 6 0.278x1O 6 0.135x10 6 0.102x10 60.952x10 7 0.726x10 7 0.709x10 7 0.642x10 70.642x10Z2 0.291xlO 1 0.276x10 I0.275x1O 1 0.240x10 1 0.I30x10 0.106x10 10.100xlo 1 0.878x102 0.878x10 2 0.829x10 20.829x10R2 0.300x10 I 0.274x0l I0.274x1O 1 0.286x1O 1 0.181x10 I 0.204x1O 1 0.200xI10 0.243xlO 1 0.256x10 I 0.267x10 I0.267xl0L2 0.222x10 0.176x10 0.174xlO4 0.125x10 4 0.631x10 5 0.375x10 50.310x10 5 0.224x10 5 0.224x10 5 0.188x10 50.187x10 5

AS 0.178x10 0.127x10-4 0.108x10 0.902xl0 0.886x10-5 0.64-xlO 0.592x10 5 0.433x10 0.440x10 5 0.331x10 0.318x10

N5 34.332 37.577 34.494 35.774 35.111 45.280 45.681 74.055 70.519 76.611 76.534

AC5 0.398x10 6 0.329x106 0.295x106 0.259xIO 6 0.101xlO 6 0.912x10 70.816x10 7 0.396x10 7 0.347x10 7 0.291x10 0.285x10

Z5 0.340xlO 0.309xlO1 0.282xlO1 0.265x10 0.199xlO -.194x10 0.185xlO1 0.161xlO1 0.154x10 0.141x10 0.139xlO

R5 0.477xI0 0.531x10 0.504x10 0.541x10 0.1353 0.1648 0.1786 0.510 0.588 0.626 0.626

L5 0.255xl0 0.273x10 0.229x10 0.220x10 0.245x10 0.246x10 0.239x10 0.320x10 0.317xlO 0.244x10 0.239x104 -5 5 5 5 5~5 - -59x1-C3 0.133xlO 0.322xlO 0.318x10l 0.201x10. 0.200x105 0.200x10 0.200x10- 0.200x10 0 205x -5 0.200xl

R3 0.163xlO1 0.162x10 0.162x10 0.162x10 0.524x10 0.524x10 0.524x10 0.524x10 0.523x10 2 0.523x10 0.522x10

C4 0.655x10 5 0.161x10 50.159x10 5 0.IlOlxlO 5 0.10OxlO 5 0.lOQxlO 50.lOOxlO S 0.10Ox10 5 0.996x10 6 0.103x10 50.100x10 5

C6 0.109x10 3 0.768x10 0.678x10 0.595x10 0.511x10 0.468x10 0.442x10 0.390x10 4 0.376x10 0.389x10 0.378x10

E 0.9397 0.9465 0.9467 0.9462 0.9389 0.9355 0.9336 0.8991 0.8929 ( 8R21 n( RRli

PQ 0.3883 0.4427 0.5106 0.5729 0.6319 0.6945 0.7581 0.3362 0.9056 0.9916 1.0570

PD 1.7682 1.7725 1.7776 1.7904 1.8243 1.8457 1.8622 1.9635 1.9918 2.020 2.0354

PCAP I.30x10 0.175x10 0.198x10 0.226x10 0.277x10 0.310x10 0.333x10 0.479x10 0.518x10 0.538x10 0.556x10

PMAG2.3232 1.7262 1.6362 1.5969 2.0648 2.2546 2.3244 5.009 5.4482 6.2944 6.2469

PT 4.4928 3.9589 3.9442 3.9829 4.5489 4.8259 4.9780 7.8568 8.3979 9.30nn 9.3050

WS 2.4186 2.4013 2.4007 2.4020 2.4204 2.4;294 2.4343 2.5278 2.5454 2.5766 2.5773

WH 0.2917 0.2571 0.2561 0.2586 0.2954 0.3133 0.3232 0.5101 0.5453 0.6078 0.6101

WI 0.691x10 2 0.479x10 20.409x10- 2 0.302x10 2 0.249x10 2 0.148x10 20.124x10 0.843x10 0.799x10 0.573x10 0.554x10 3

WW 0.113x10 0.834x10 0.757x10 0.518x10 0.136x10 0.108xlO 20.925x10 3 0.603x10 0.554x10 3.445x1O 0.434x10

WC 0.179xlO 1 0.798x10 0.730x10 0.581x102 O.519x10 2 0.489x102 U.470xj0 0.4J2x12 0.422xl2 0 2436xlO 0.424xlO

WMAG 0.182x10 0.131x10 0.117x10 0.821x10 0.385x10 0.256x10 0.217x10 0.145x10 0.135x10 0.102x10 2.989x10WT 2.7465 2.6794 2.6759 2.6747 2.7249 2.7502 2.7644 3.0437 3.0962 3.1898 3.1930

PQ total power dissipation in the transistor WC capacitor weightPD power dissipation in the diode WW winding (copper conductor) weightPCAP power dissipation in the output filter capacitor WMAG WW + WIPMAG total magnetic loss (core loss + winding loss) WT total weight.PT total lossWS source weight The difference in simulation results between the boostWH packaging weight and buck-boost converter designs are summarized in theWI magnetic core weight following:

RAHMAN/LEE: COMPUTER SIMULATIONS OF OPTIMUM BOOST AND BUCK-BOOST CONVERTERS 605

10 '

8.

F-

6

cn

U)

0 4-J4

2

20K 40K 60K 80K lOOK 120 K

F(KHZ)

Fig. 5. Loss breakdown for optimal buck-boost converter design.

F(KHZ)

Fig. 7. Loss breakdown for optimal boost converter design.

WT

3

Ws

2.5

3.5 WT

25

<[ 2.0

D-i|_1.5_

1-0WH

0.5

'~-

cr

(9

0

(9

"I

V.

WMAG

/WC

20K 40K 60K 80K IOOK 120KF(KHZ)

Fig. 6. Weight breakdown for optimal buck-boost converter design.

1) The buck-boost converter is heavier than theboost converter. In order to have the minimum weightdesign the buck-boost converter has to operate at a

higher frequency than the boost power converter.2) Switching losses of semiconductor devices are

higher for the buck-boost converter. This is logical sincethe switching current amplitude is considerably higherthan that of the boost converter due the same input andoutput voltage and the same power level.

3) Magnetic components for the buck-boost con-verter are generally larger in size and heavier in weight.

2.0 _

1.5Z

1.0 IrWH

05

WMAG

20K 40K 60K 80K IOOK 120K

F(KHZ)

Fig. 8. Weight breakdown for optimal boost converter design.

4) Magnetic losses PMAG (core loss + windingloss) for the buck-boost converter are dominated by thewinding loss in low frequencies. The PMAG losscharacteristic falls rapidly as the switching frequency in-creases. The high magnetic losses in low frequenciescause severe weight penalty. It is clearly demonstrated inFigs. 5 and 6 that in order to minimize the converterweight/loss, it is desirable to operate the converter atfrequencies in the range of about 80 kHz to 100 kHz.For the minimum weight/loss boost converter design,however, the optimal frequency lies in the range of 40kHz to 60 kHz as shown in Figs. 7 and 8.

VI. CONCLUSIONS

Nonlinear programming techniques have been suc-cessfully employed to generate computer simulations ofthe minimum weight designs of switching power con-verters. For power converter optimization, the ALAG

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-18, NO.5 SEPTEMBER 1982

- 10U)

8U)U)0

PT

606

package has been found to be quite effective in terms ofthe computation time, ease of coding, and rate of con-vergence.

Adopting the ALAG routine, a cost effectivecomputer-aided design approach is presented whichprovides a minimum-weight converter design down tothe details of component level while meeting all power-circuit performance requirements. This computer-aideddesign approach provides important design insightswhich help to assess the following important design con-cerns:

1) tradeoffs between weight and loss as the switch-ing frequency is increased

2) optimum converter design down to the details ofcomponent levels

3) optimum component designs as a function of theswitching frequency and their relationships to theoverall system optimization

4) significance of the U-shape curves representingtotal-weight and total-loss versus frequency are ob-served in Figs. 5 through 8; this allows the designer toeasily identify the optimum switching frequency or arange of frequencies over which the total weight/loss isminimum in the practical sense

5) impact of various critical component char-acteristics (viz. magnetic losses and switching losses ofsemiconductor devices) on the overall system

6) optimal converter topology for a given application.

Employing the nonlinear program based optimiza-tion technique, the power converter designer can con-ceive the overall optimum system design. This is done bytaking into consideration the power-circuit related per-formance requirements with the design objective ofeither minimizing weight, loss, or any other physicallyrealizable quantity. It thus sets the stage for a morescientific design approach instead of relying on subjec-tive brute-force, trial and error, piecemeal design.

RAHMAN/LEE: COMPUTER SIMULATIONS OF OPTIMUM BOOST AND BUCK-BOOST CONVERTERS 607

REFERENCES

[1] Wu, C.J., Lee, F.C., Balachandran, S., and Goin, H.L. (1980)Design optimization for half-bridge dc-dc converter.IEEE Power Electronics Specialists Conference Record,1980, Atlanta, Ga.

[2] Yu, Y., Lee, F.C., and Triner, J.E. (1979)Power converter design optimization.IEEE Transactions on Aerospace and Electronic Systems,May 1979, AES-15, 344-355.

[3] Lee, F.C., Rahman, S., Wu, C.J., and Kolecki, J. (1981)A new approach to the minimum weight/loss design ofswitching power converters.Proceedings ofPOWERCON, Apr. 1981, Dallas, Tex.

[4] Rahman, S., and Lee, F.C. (1981)Nonlinear program based optimization of boost and buck-boost converter designs.IEEE Power Electronics Specialists Conference Record,1981, 180-191.

[5] Broyden, C.G. (1972)Quasi-Newton methods.In W. Murray (Ed.), Numerical Methodsfor Unconstrained Op-timization.New York: Academic Press, 1972, 87-106.

[6] Hestenes, M.R. (1969)Multiplier and gradient methods.In Zadeh et al. (Eds.), Computing Methods in OptimizationProblems.New York: Academic Press, 1969, 143-164.

[7] Powell, M.J.D. (1969)A method for nonlinear constraints in minimizationproblems.In R. Fletcher (Ed.), Optimization.New York: Academic Press, 1969, 283-293.

[81 Haarhoff, P.C., and Buys, J.D. (1970)A new method for the optimization of a nonlinear functionsubject to nonlinear constraints.Computer Journal, May 1970, 13, 178-184.

[91 Rockafeller, R.T. (1973)A dual approach to solving nonlinear programming prob-lems by unconstrained optimization.Mathematical Programming, 1973, 5, 354-373.

[10] Fletcher, R. (1974)Methods related to Lagrangian functions.In P.E. Gill and W. Murray (Eds.), Numerical MethodsforConstrained Optimization.New York: Academic Press, 1974, 219-240.

[11] Balachandran, S., and Lee, F.C. (1981)Algorithm for power converter design optimization.IEEE Transactions on Aerospace and Electronic Systems,May 1981, AES-17, 422432.

[12] Lee, F.C., Rahman, S., Carter, R.A., Wu, C.J., Yu, Y., andChen, R. (1980)

Modeling and analysis of power processing-phase III.Prepared by TRW Defense and Space Systems Groups and theVirginia Polytechnic Institute and State University, NASA Report,NASA CR-16558, NASA LeRC, Dec. 1980.

[13] Yu, Y., Buchmann, M., Lee, F.C., and Triner, J.E. (1976)Formulation of methodology for power circuit design op-timization.IEEE Power Electronics Specialists Conference Record,1976.

Saifur Rahman (S'74-M'78) was born in Dacca, Bangladesh. He graduated fromthe Bangladesh University of Engineering and Technology in 1973 with a B.Sc.degree in electrical engineering. He obtained the M.S. degree in electrical sciencesfrom the State University of New York at Stony Brook in 1975 and the Ph.D. degreein electrical engineering from the Virginia Polytechnic Institute and State Universityin 1978.

He has taught in the Department of Electrical Engineering, Bangladesh Universi-ty of Engineering and Technology, State University of New York at Stony Brook,Texas A&M University, and Virginia Polytechnic Institute and State University,where he is currently an Assistant Professor. His industrial experience includes workat Brookhaven National Laboratory and the Carolina Power and Light Company.His areas of interest are large scale optimization, power system analysis, and alter-native energy systems.

Dr. Rahman is a member of the American Section of the International SolarEnergy Society.

Fred C.Y. Lee received the B.S. degree in electrical engineering in 1968 from Cheng-Kung University, Taiwan, Republic of China, and the M.S. and Ph.D. degrees inelectrical engineering, both from Duke University, in 1971 and 1974, respectively.

From 1974 to 1977, he was employed as a member of the technical staff of theControl and Power Processing Department, TRW Systems, where he was engaged indesign, simulation, and analysis for spacecraft power processing equipment. In 1977he joined the Virginia Polytechnic Institute and State University as an Assistant Pro-fessor in the Department of Electrical Engineering. He is presently an Associate Pro-fessor in that department. His research interests include power electronics, electricvehicle propulsion, nonlinear modeling and analysis, and design optimization ofpower processing components and systems.

Dr. Lee has published over 60 articles in the areas of power processing modeling,analysis, and design.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-18, NO.5 SEPTEMBER 1982608