A core Course on Modeling Introduction to Modeling 0LAB0 0LBB0 0LCB0 0LDB0 S.20

Formalization phase phase: formalize relations

... such that energy costs are minimal energy costs are minimal no blinding no blinding enough visual contrast enough visual contrast... such that energy costs are minimal energy costs are minimal no blinding no blinding enough visual contrast enough visual contrast How to optimize road illumination

totP = p * nLanterns relationsdimensions assumptions [kWh] = [kWh/lntrn ]*[lntrn] ignore losses in wiring enCostpH = ppkWh * totP [Euro/h] = [Euro/kWh]*[kW]only electricity costs todo enCostpH blnd contrast ppkWh totP p nLanterns roadLength dL maxP maxInt minP minInt nLanterns =1+roadLength/dL [lntrn] = [m ]/[m/lntrn]equal distances blnd=max(maxP,maxInt)-maxP contrast=minP-min(minP,minInt) [kW/m 2 ] independent of color,... maxP =... driver visual capabilities minP =... driver visual capabilities roadLength =... from problem owner p = (choose) from the designer dL = (choose) from the designer [kW/m 2 ] [m] [kW] [m] ppkWh =... [Euro/kWh]from energy supplier

lantern height power road width surface reflectance traffic density car speed height driver visual capabilities rides on consists of operated by sees adjacent n 1 n n 1 1 authority expenses pays n 1 1 illuminatelocated on 1 How to deal with maxInt and minInt? Focus on the sees relation

lantern height power road width surface reflectance driver visual capabilities sees illuminate How to deal with maxInt and minInt?

EL r L r E r perceived intensity = f(L,E,r) perceived intensity = f 1 (L,r)*f 2 (r,E) How to deal with maxInt and minInt?

LE r L r E r perceived intensity = f(L,E,r) perceived intensity = f 1 (L,r)*f 2 (r,E) How to deal with maxInt and minInt?

perceived intensity = f 1 (L,r)*f 2 (r,E) L r E r How to deal with maxInt and minInt?

relation road - eye: intensity f 2 (r,E) c*B(r) (does hardly depend on E) perceived intensity = f 1 (L,r)*f 2 (r,E) L r E r How to deal with maxInt and minInt?

relation road - eye: intensity f 2 (r,E) c*B(r) (does hardly depend on E) perceived intensity = f 1 (L,r)*f 2 (r,E) L r E r How to deal with maxInt and minInt?

relation road - eye: intensity f 2 (r,E) c*B(r) (does hardly depend on E) relation lamp – road: B(r) = f 1 (L,r) = p/|L-r| 2 perceived intensity = f 1 (L,r)*f 2 (r,E) L r E r multiple lamps: B = B 1 + B 2 + B = n B n How to deal with maxInt and minInt? A slightly more accurate formula also takes into account that brightness is reduced when light strikes the road under a skew angle: B(r)=p cos /|L-r| 2, where cos = h/|L-r|

relation road - eye: intensity f 2 (r,E) c*B(r) (does hardly depend on E) relation lamp – road: B(r) = f 1 (L,r) = p/|L-r| 2 multiple lamps: B = B 1 + B 2 + B = n B n Therefore maxInt = max r road c*B(r) = max r road c*( n p/|L n -r| 2 ) = max r road c*( n p/|L n -r| 2 ) minInt = min r road c*B(r) = min r road c*( n p/|L n -r| 2 ) = min r road c*( n p/|L n -r| 2 ) How to deal with maxInt and minInt?

relation road - eye: intensity f 2 (r,E) c*B(r) (does hardly depend on E) relation lamp – road: B(r) = f 1 (L,r) = p/|L-r| 2 multiple lamps: B = B 1 + B 2 + B = n B n l - n dL w - Wh r Compute |L n -r| 2 using Pythagoras: |L n -r| 2 = h 2 +(w-W h ) 2 +(l-n dL) 2, where r = (l,w,0); W h = ½ width of the road; h = lantern height. h How to deal with maxInt and minInt?

Summary develop a model using the todo list, introduce quantities when necessary; develop a model using the todo list, introduce quantities when necessary; translate relations from conceptual model into functions in formal model; translate relations from conceptual model into functions in formal model; try expressions involving few as possible quantities (e.g., prefer f 1 (x,y)*f 2 (y,z) over f(x,y,z) ); try expressions involving few as possible quantities (e.g., prefer f 1 (x,y)*f 2 (y,z) over f(x,y,z) ); if possible, try to approximate f 1 (x,y) by a simpler f 2 (x) for relevant range of ys if possible, try to approximate f 1 (x,y) by a simpler f 2 (x) for relevant range of ys demo demo