Problem-Solving Example #1
A.C. Motor Design
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Figure 1 is a case study of re-engineering -- a simulation program for A.C. motor design that was reapplied using our Fortran Calculus language/compiler as an optimization program and fully tested in about an hour. The original program had been used for trial and error design calculations by a west coast manufacturer. Restructuring this mature engineering model into a design optimization program, only required introducing a FIND statement that defined a constrained optimization problem with 12 unknown parameters and 7 constraints. A FIND statement invokes a solution technique or solver that consists of automatic differentiation with a numerical algorithm. The solver involved in this case is a nonlinear programming algorithm called THOR. Figure 2 (part of output) is a report automatically generated by the solver, THOR, summarizing the iteration steps. This partial summary shows initial guesses of the parameters and the final results.
This case study illustrates the economic benefits of AD based software for reapplying existing application software in a higher-productivity mode. The existing engineering simulation model was used "as is". It was automatically elevated to optimization by the hidden differential arithmetic. There was no mathematical analysis or algorithm design required. Because this was a re-engineering of an existing program, no debugging was necessary, and since the optimization solvers are interchangeable, different algorithms could be substituted to verify solution correctness.
Figure 1. Calculus Code ... A.C. Motor (Optimal) Design
global all
problem acmotor
dimension cns(7)
dynamic botm, bnd
call input
call design
print *, '---------------initial design---------------'
call output
find coiltrns, ! number of turns per coil
& EPDIAM, ! Separating diameter
& STASLOTW, ! Stator slot opening width
& ROTSLOTW, ! Rotor slot opening width
& AIRGAP, ! Air gap
& STATOOTW, ! Stator tooth width
& STABAKIR, ! Stator back iron
& ROTTOOTW, ! Rotor tooth width
& ROTBAKIR, ! Rotor back iron
& STASLOTO, ! Stator slot opening depth
& ROTSLOTO, ! Rotor slot opening depth
& SLIP ! Slip
& IN DESIGN; BY THOR(TCON);
& WITH BOUNDS BND; AND LOWERS BOTM;
& HOLDING CNS; TO MAXIMIZE EFF
print *, '----------------optimized design-----------------'
call output
end
model design
f1 = 231258.0438 / coiltrns
q1 = sepdiam * pi / 56 : q2=sepdiam * pi / 69 : q3=sepdiam * pi / 28
c1 =q1/((q1-staslotw)+(staslotw *(.7 -(.036 * staslotw/ airgap))))
c2 =q2/((q2-rotslotw)+(rotslotw *(.7-(.036 * rotslotw/airgap))))
crnol = f1 * airgap * c1 * c2 / (6.96 * coiltrns * q3)
q4 = (statorod - stabakir - stabakir - sepdiam - .1) / 2 - .01
q5= (sepdiam+q4+.1) * pi / 56 - statootw - .016 : a3=q4 * q5
z1 = 7000 * a3
q6 = coiltrns * 2 / z1 : z3= - 2 : z4=32+z3 : q7=1.26**z3 * 162
q8=64 / 1.26**z3 : z5=stacklen+stacklen+pi * statorod / 14
rdc =z5 * coiltrns * 28 * q7 / 12000
v1 = (sepdiam - .06 - rotorid - rotbakir - rotbakir) / 2
v2 = (sepdiam - .06 - v1) * pi / 69 - rottootw
a4 = v1 * v2 : r2 = (.0133 / a4) * (stacklen / 12000) * u1
rrot= (56 * coiltrns)**2 * r2 / 138 : r4 = rrot
g3 = 3.2 * (q4 / (3 * q5)+.03 / (staslotw+q5)+stasloto / staslotw)
g4 = 3.2 * (v1 / (3 * v2)+.03 / (rotslotw+v2)+rotsloto / rotslotw)
g5 = (1 / c1+1 / c2 - 1)**2 * .26 / airgap
g6= (g3+g5 * q1) / 56+(g4+g5 * q2) / 69
xleak = xc * pi * coiltrns**2 * (.05+g6 * stacklen) ! leakage reactants
crstal = 115 / (sqrt(((rdc+r4)**2)+xleak**2)) ! stall current
crfl= 115 / (sqrt(((rdc+r4 / slip)**2)+xleak**2)) ! full load current
storq = 1.5792 * (crstal**2) * r4 ! stall torque
rtorq = 1.5792 * (crfl**2) * r4 / slip ! running torque
p8=rtorq * (1 - slip) * 1.264944649 : p6= (crstal**2) * (rdc+r4)
p7 = (crfl**2) * r4 / slip
flux = (1.4 * f1 / (statootw * stacklen) / 6450 ! flux density
b7 = (sqrt(crfl**2+crnol**2) * flux / crnol) / 1.4 : b2 =.13 * b7**1.9
b3 = (((statorod**2 - sepdiam**2) * pi / 4) - 56 * a3) * 0.28 * stacklen
feloss = b3 * b2 ! iron losses
culsta = (b7 * crnol) / flux)**2 * rdc ! stator copper losses
eff = .5 * p8 / (p7+feloss / 2+culsta) ! efficiency - maximize this
rac =13225 / feloss ! a.c. resistance
xmag =115 / crnol ! magnetizing reactants
culrot = crfl * crfl * rrot / staslotw ! rotor copper losses
cns(1) = statorod - .1 - sepdiam ! stator o.d. > = separation diam+.1
cns(2) = sepdiam - rotorid+.1 ! separation diam > = rotor i.d. - .1
cns(3) = ((sepdiam * pi / 69) - .035) - rottootw ! rotor tooth geometry
cns(4)=.5*(sepdiam - rotorid) -.025 -rotbakir ! rotor back iron geometry
cns(5) = storq - 60 ! stall torque > = 60
cns(6) = 19 - flux ! flux density < = 19
cns(7) =.05 - slip ! slip < = 5 percent
end
As this example illustrates, the benefits of multiple - parameter optimization in practical engineering calculation can often be dramatic. Because of the large number of parameters, something on the order of a billion to a trillion computer cases would have been necessary to achieve this result if using the original program. The bottom line benefit is dramatically reduced cost, higher engineering productivity, and immediate results to meet tight schedules.
Figure 2. (Partial) Output Report ... A.C. Motor (Optimal) Design
---- Thor Summary, Invoked At Acmotor[29] For Model Design
---- Convergence Condition After 35 Iterations
o
o
o
Loop Number...... [Initial] 35
Unknowns
coiltrns 2.900000e+01 3.214302e+01
sepdiam 4.000000e+00 5.010000e+00
staslotw 6.000000e-02 6.570807e-02
rotslotw 3.500000e-02 3.500000e-02
airgap 8.000000e-03 8.000000e-03
statootw 1.350000e-01 1.687119e-01
stabakir 1.500000e-01 3.955060e-01
rottootw 1.100000e-01 8.885508e-02
rotbakir 4.700000e-01 9.700004e-01
stasloto 3.500000e-02 2.500000e-02
rotsloto 1.500000e-02 0.000000e+00
slip 3.000000e-02 5.000000e-02
Objective
Eff 4.020854e-01 7.312523e-01
Inequality Constraints
cns ( 1) 1.010000e+00 0.000000e+00
cns ( 2) 1.225000e+00 2.235000e+00
cns ( 3) 3.712132e-02 1.042519e-01
cns ( 4) 6.750000e-02 7.249963e-02
cns ( 5) -7.555936e+00 6.415480e-04
cns ( 6) 1.258933e+01 1.437190e+01
cns ( 7) 2.000000e-02 0.000000e+00
--- End Of Loop Summary