```! LINGO model to represent or approximate (PieceLin2DL.lng) a function of two variables via piecewise linear interpolation, using a triangular tesselation of the 2D space, using a minimum number of binary variables, i.e., ~ log( number of cells); ! Keywords: Piecewise Linear, Interpolation, Hydro Electric, Triangulation, Non-convex Function, Linearization; SETS: HSET; ! The horizontal dimension; VSET; ! The vertical dimension; HXV( VSET, HSET): HV, VV, FV, WGT; ENDSETSDATA: HSET = 1..3; VSET = 1..3; ! Matrix of Data points for head(pressure), volume and power output(function value) for a hydro-electric generator; HV, VV, FV = 195 1800 20 ! Row 1; 195 3500 52 195 5100 69 217 1900 26 ! Row 2; 217 3600 61 217 5200 80 240 2000 30 ! Row 3; 240 4100 78 240 5600 93 ; ENDDATA ! Approach: the two dimensional space is partitioned via triangulation, i.e., partitioned into triangles. Use linear interpolation within a cell. Triangles are used, as opposed to say, quadrilaterals, because a point within a triangle is uniquely determined by weights on the three corner points. With quadrilaterals there are alternate weightings that give the same x,y values, but different function values. ! Parameters: HV(i,j) = horizontal or X or Head value at data point i,j, VV(i,j) = vertical or Y or flow volume value at data point i,j, FV(i,j) = function value or power at data point i,j, ! Variables: WGT(i,j) = weight given to data point i,j; ! The appropriate diagram with 3*3 = 9 points and 2*2*2 = 8 triangle cells: _______ | /|\ | | / | \ | i |/__|__\| |\ | /| | \ | / | |__\|/__| j ; ! YH = 1 if chosen cell is above middle horizontal line, YV = 1 if chosen cell is right of middle vertical line, YC = 1 if chosen cell is in center diamond. else 0. Observe that specifying values for all of YH, YV, YC will imply a unique cell out of the 8 possible triangles; ! Weights must form a convex combination; [CNVX] @SUM( HXV(i,j): WGT(i,j)) = 1; ! Compute associated horizontal or X value; [COMPX] @SUM( HXV(i,j): HV(i,j)* WGT(i,j)) = HA; ! Compute associated vertical or Y value; [COMPY] @SUM( HXV(i,j): VV(i,j)* WGT(i,j)) = VA; ! Compute associated function value; [COMPFV] @SUM( HXV(i,j): FV(i,j)* WGT(i,j)) = FA; ! Choice must be binary, all or nothing; @BIN(YH); @BIN(YV); @BIN(YC); ! Implications of setting YV; WGT(1,3) + WGT(2,3) + WGT(3,3) <= YV; WGT(1,1) + WGT(2,1) + WGT(3,1) <= 1-YV; ! Implications of setting YH; WGT(3,1) + WGT(3,2) + WGT(3,3) <= YH; WGT(1,1) + WGT(1,2) + WGT(1,3) <= 1-YH; ! Implications of setting YC; WGT(2,2) <= YC; WGT(1,1) + WGT(3,1) + WGT(1,3) + WGT(3,3) <= 1-YC; ! Define an arbitrary optimization problem to illustrate the method; ! An arbitrary tradeoff cost between head and volume; [ARBOBJ] Min = VA + 15*HA; ! We need this much power; [ARB1] FA >= 67; ! Arbitrarily put some constraints on Head; [ARB2] HA <= 229; [ARB3] HA >= 227; ```