# electric-NCP.mod
#
# NCP formulation of EPEC derived from MPECs defined in AllFirms.mod
#
# MPEC of example due to Fukushima and Pang, "Quasi-Variational 
# Inequalities, Nash-Equilibria, and Multi-Leader-Follower Games".
# 
# The generator (firm) is the Stackelberg leader, and the ISO, 
# arbitrager, and market clearing are the followers in this game.
# Another possibility is to have the ISO as a Leader as well.
#
# ampl coding by S. Leyffer, Argonne National Laboratory, Jan. 2005.
#
# Change log:
# 
#######################################################################

### set definitions
set N;				# set of nodes in network
set F;				# set of firms (generators)
set ARCS in N cross N;

### parameters & constants
param c{F,N} default 0;		# cost per unit generation at node n by firm f
param P0{N};			# price intercept of sales function at node n
param Q0{N};			# quatity intercept of sales function at node n
param e{ARCS};			# ISO's unit cost of shipping along arcs
param CAP{F,N};			# production capacity at node n for firm f

### variables
var s{F,ARCS} >= 0;		# amount produced by f at node n1, sold at n2
var y{ARCS}   >= 0;		# amount of shipment from n1 to n2
var S{N}      >= 0;		# total sales at node n
var ss{ARCS}  >= 0;		# slacks for easier complementarity
var a{ARCS}   >= 0;		# amount bought by arbitrager at n1, sold at n2
var w{ARCS} default 1;		# unit charge of shipment received by ISO

### multipliers (each firm has different multipliers)
var l_cap{F,N}   >= 0;	        # capacity constraint	
var l_sal{F,N};			# total sales 
var l_slk{F,ARCS};              # definition of slacks
var l_arb{F,ARCS} >= 0;         # arbitrager
var l_mar{F,ARCS};              # market clearing
var l_ISO{F,ARCS} >= 0;         # ISO

minimize dummy: 0;

subject to

   ### first order equations wrt s[f,i,j]
   FO_s {f in F, (i,j) in ARCS}: 
       0 <= - (P0[j] - P0[j]/Q0[j]*S[j]) + w[i,j] + c[f,i]
            + l_cap[f,i] - l_sal[f,j] - l_mar[f,i,j]
       complements   s[f,i,j] >= 0;

   ### first order equations wrt y[i,j] for every leader f
   FO_y {f in F, (i,j) in ARCS}: 
      0 <= l_mar[f,i,j] + l_ISO[f,i,j]*(-w[i,j]+e[i,j])
      complements y[i,j] >= 0;

   ### first order equations wrt S[i] for every leader f
   FO_S {f in F, i in N}: 
       0 <=   P0[i]/Q0[i]*(sum{j in N: (j,i) in ARCS}(s[f,j,i]+a[j,i]-a[i,j]))
            + l_sal[f,i]
            + sum{j in N:(j,i) in ARCS} l_slk[f,j,i]*(P0[i]/Q0[i]) 
            - sum{j in N:(i,j) in ARCS} l_slk[f,i,j]*(P0[i]/Q0[i])
       complements   S[i] >= 0;

   ### first order equations wrt ss[i,j] for every leader f
   FO_ss{f in F, (i,j) in ARCS}:
       0 <= l_slk[f,i,j] + l_arb[f,i,j]*a[i,j] complements ss[i,j] >= 0;

   ### first order equations wrt a[i,j] for every leader f
   FO_a {f in F, (i,j) in ARCS}:
       0 <= - (P0[j]-P0[j]/Q0[j]*S[j]) + (P0[i]-P0[i]/Q0[i]*S[i]) + w[i,j]
            - l_sal[f,j] + l_sal[f,i] + l_arb[f,i,j]*ss[i,j] - l_mar[f,i,j]
       complements   a[i,j] >= 0;

   ### first order equations wrt w[i,j] for every leader f
   FO_w {f in F, (i,j) in ARCS}:
       0 <= (a[i,j] + s[f,i,j]) - l_slk[f,i,j] - l_ISO[f,i,j]*y[i,j]
       complements w[i,j] <= e[i,j];
    
   ### capacity constraint
   cap{f in F,i in N}: CAP[f,i] >= sum{j in N:(i,j) in ARCS} s[f,i,j]
	               complements l_cap[f,i] >= 0;

   ### total sales at node i
   sales{i in N}: 0 = - S[i] 
                      + sum{f in F, j in N:(j,i) in ARCS} s[f,j,i] 
                      + sum{j in N:(i,j) in ARCS}( a[j,i] - a[i,j] );

   ### define slacks for complementarity
   slacks{(i,j) in ARCS}: 0 = - ss[i,j] 
                              - (P0[j] - P0[j]/Q0[j]*S[j]) 
                              + (P0[i] - P0[i]/Q0[i]*S[i]) 
                              + w[i,j];

   ### market clearing condition
   market{(i,j) in ARCS}: y[i,j] = sum{f in F} s[f,i,j] + a[i,j]
	                 complements w[i,j];

   ### arbitrager's optimality conditions (follower)
   arbitrager{f in F, (i,j) in ARCS}: 
	0 <= -a[i,j]*ss[i,j] complements l_arb[f,i,j] >= 0;

   ### ISO's optimality conditions
   ISO{f in F, (i,j) in ARCS}: 
	0 <= -y[i,j]*(-w[i,j]+e[i,j]) complements l_ISO[f,i,j]>= 0;