$title Winding Factor of Electrical Machines (WINDFAC,SEQ=224)

$onText
This model determines the optimal winding factor for electrical machines.

Michna, M, and Gdanska, P, Winding Factor of Electrical Machines, 1984.

Keywords: mixed integer nonlinear programming, electrical engineering, winding factor
$offText

Scalar
   ms 'number of phases'     / 3 /
   p  'number of pole pairs' / 2 /
   K  'harmonic order'       / 5 /
   ns 'coil span'            / 1 /;

Variable
   q     'number of slots per one phase and per one pole'
   Nz    'number of slots'
   alfae
   tauz  'slots pitch'
   s     'span';

Integer Variable q, Nz, s;

q.lo    =  1;
q.up    = 10;
Nz.lo   =  1;
s.lo    =  1;
alfae.l =  1.5;
tauz.l  =  1.0;

Variable
   kz1, kz3, kz5 'coil-group factor'
   ks1, ks3, ks5 'coil-span factor'
   kw , kw3, kw5 'winding factor'
   kw1           'winding factor for first harmonic';

kw1.lo = 0.8;

Equation
  def_Nz,  def_alfae, def_tauz, def_s,   def_kz1, def_ks1, def_kw1
  def_kz3, def_ks3,   def_kw3,  def_kz5, def_ks5, def_kw5, def_kw;

def_Nz..    Nz    =e=  2*ms*q*p;

def_alfae.. alfae =e= (2*pi*p)/Nz;

def_tauz..  tauz  =e=  Nz/(2*p);

def_s..     s     =e=  tauz - ns;

def_kz1..   (q*sin(alfae/2))*kz1 =e= sin(q*alfae/2);

def_ks1..   ks1 =e= sin((s*pi)/(tauz*2));

def_kw1..   kw1 =e= ks1*kz1;

def_kz3..   (q*sin(3*alfae/2))*kz3 =e= sin(3*q*alfae/2);

def_ks3..   ks3 =e= sin((3*s*pi)/(tauz*2));

def_kw3..   kw3 =e= ks3*kz3;

def_kz5..   (q*sin(5*alfae/2))*kz5 =e= sin(5*q*alfae/2);

def_ks5..   ks5 =e= sin((5*s*pi)/(tauz*2));

def_kw5..   kw5 =e= ks5*kz5;

def_kw..    kw  =e= kw3*kw3 + kw5*kw5;

Model wind / all /;

solve wind minimizing kw using minlp;