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/* Power plant LP scheduler, example data with 25hrs for daylightsavings */
/* Implemented, inspected, written and converted to GNU MathProg by NASZVADI, Peter, 199x-2017 <vuk@cs.elte.hu> */
/* Fast electric power plant scheduler implementation based on new results in author's Thesis.
The base problem is: * given some power plants * a short time scale partitioned to equidistant intervals * the task is to yielding the cheapest schedule for the plants * the daily demand forecast is usually accurate and part of the input
The power plants has technical limitations: * upper and lower bounds of produced energy * and also a gradient barrier in both directions (can depend on time, but this GMPL implementation is simplified) * Also units with same properties (technical and price) should be scheduled together usually with near same performance values * Assumed a simplified network topology, which is contractive, so keeping Kirchhoff's laws is a necessary constraint too * All solutions must be integer
The LP relaxation is equivalent with the MIP problem due to the model's matrix interesting property: it is Totally Unimodular (proven in 2004 by author) and also a Network Matrix (2006, presented at OTDK 2016, Szeged, Hungary) so: * it is strictly polynomial if it is solved by most simplex algs * all base solutions become integer if the RHS vector is integer (it is in real life, so this is an acceptable assumption) * The transposed matrix is NOT a Network Matrix in most cases!
However, adding several other constraints easily turns the problem to be NP-hard, which is also pinpointed and discussed in the Thesis.
See more about electric power plants' scheduling in the author's Thesis (in Hungarian): http://www.cs.elte.hu/matdiploma/vuk.pdf
It is possible to run with custom parameters, what is needed to define is: * TIME set (daylightsaving cases or other than hour intervals) * PLANTS set (the 'Demand' is mandatory and usually negative) * PRICE parameter (can be negative if energy is sold to a consumer) * BOUND parameter (technical bounds) * MAXGRAD parameter (technical bounds)
Then generate a pretty-printed solution by typing: glpsol --math powpl25h.mod*/
set TIME, default { '00:00', '01:00', '02:00', '03:00', '04:00', '05:00', '06:00', '07:00', '08:00', '09:00', '10:00', '11:00', '12:00', '13:00', '14:00', '15:00', '16:00', '17:00', '18:00', '19:00', '20:00', '21:00', '22:00', '23:00', '24:00'};/* Time labels, assumed natural ordering. daylightsaving's bias can be inserted p.ex. in Central Europe like: ... '01:00', '02:00', '02:00b', '03:00', ... */
set TADJ := setof{(r, s) in TIME cross TIME: r < s}(r, s) diff setof{(t, u, v) in TIME cross TIME cross TIME: t < u and u < v}(t, v);/* Tricky adjacent time label generator because GMPL lacks order determination of set elements (except intervals composed of equidistant numbers) */
set PLANTS, default {'Demand'};/* Demand is a promoted, mandatory one, usually filled with negative MW values in data section */
set DIRECTION, default {'Up', 'Down'};/* All possible directions of gradients, do not touch */
param MAXINT, default 10000;/* A "macro" for bounding absolute value of all used numbers and used as default value */
param PRICE{PLANTS}, default MAXINT;/* Should be specified in data section, self-explanatory. can be negative if there are energy buyers */
param BOUND{(p, t, d) in PLANTS cross TIME cross DIRECTION}, default if t = '00:00' then if d = 'Down' then BOUND[p, t, 'Up'] else 0 else if p <> 'Demand' or d = 'Up' then sum{(u, v) in TADJ: v = t} BOUND[p, u, d] else BOUND[p, t, 'Up'];/* Obvious, technical bounds of each power plant unit (real or virtual like 'Demand'). If some parts are not given in data section, calculated from preceeding values. Also for time '00:00', its 'Down' values by default are the same as denoted with 'Up' */
param MAXGRAD{(p, d) in PLANTS cross DIRECTION}, default MAXINT;/* Usually nonnegative integer, might differ in distinct directions per unit in the cited thesis, it is allowed to gradient bounds to depend on time, but this is a simplified model */
var x{(t, p) in TIME cross PLANTS}, <= BOUND[p, t, 'Up'], >= BOUND[p, t, 'Down'];/* The schedule, dimension is MW */
s.t. kirchhoff{t in TIME: t <> '00:00'}: sum{p in PLANTS} x[t, p] = 0;/* Conservative property */
s.t. gradient{(p, t, u) in PLANTS cross TADJ}: -MAXGRAD[p, 'Down'] <= x[t, p] - x[u, p] <= MAXGRAD[p, 'Up'];/* Technical limitations, each unit usually cannot change performance arbitrarily in a short time, so limited in both directions per time unit*/
minimize obj: sum{(t, p) in TIME cross PLANTS}(x[t, p] * PRICE[p]);/* The objective is the cost of the schedule */
solve;
/* Pretty print solution in table */
printf '+--------+';for{p in PLANTS}{ printf '-% 6s-+', '------';}printf '\n';printf '|%7s |', ' ';for{p in PLANTS}{ printf ' % 6s |', p;}printf '\n';printf '+--------+';for{p in PLANTS}{ printf '-% 6s-+', '------';}printf '\n';for{t in TIME}{ printf '|%7s |', t; for{p in PLANTS}{ printf ' % 6s |', x[t, p].val; } printf '\n';}printf '+--------+';for{p in PLANTS}{ printf '-% 6s-+', '------';}printf '\n';
data;
set TIME := '00:00', '01:00', '02:00', '02:00b', '03:00', '04:00', '05:00', '06:00', '07:00', '08:00', '09:00', '10:00', '11:00', '12:00', '13:00', '14:00', '15:00', '16:00', '17:00', '18:00', '19:00', '20:00', '21:00', '22:00', '23:00', '24:00';
/* Generated random default values and names, the demand is the sum of 2 sinewaves. Also specified a treshold for nuclear plants from 15:00 till 19:00 The sun is shining only morning and in the afternoon: 07:00-18:00, so solar plant cannot produce electric energy after sunset.
Only touch this section, or export it to a data file!*/
set PLANTS 'Demand', 'Atom1', 'Atom2', 'Coal', 'Gas1', 'Gas2', 'Green', 'Oil', 'Solar', 'Dam';
param PRICE := 'Demand' 0 'Atom1' 2 'Atom2' 2 'Coal' 15.6 'Gas1' 12 'Gas2' 11.5 'Green' 8.8 'Oil' 23.3 'Solar' 7.6 'Dam' 3;/* price per MW */
param BOUND := [*, *, 'Up'] (tr): 'Atom1' 'Atom2' 'Coal' 'Gas1' 'Gas2' 'Green' 'Oil' 'Solar' 'Dam' := '00:00' 240 240 100 150 150 20 90 0 20 '01:00' 240 240 155 192 208 35 230 0 20 [*, *, 'Up'] (tr): 'Atom1' 'Atom2' := '15:00' 200 200 '19:00' 235 235 [*, *, 'Up'] (tr): 'Solar' := '07:00' 20 '18:00' 0 [*, *, 'Down'] (tr): 'Atom1' 'Atom2' 'Coal' 'Gas1' 'Gas2' 'Green' 'Oil' 'Solar' 'Dam' := '01:00' 100 100 50 62 68 0 75 0 20 [*, *, 'Up'] : '01:00' '02:00' '02:00b' '03:00' '04:00' '05:00' '06:00' '07:00' '08:00' := 'Demand' -868 -851 -842 -837 -791 -887 -912 -1046 -1155 [*, *, 'Up'] : '09:00' '10:00' '11:00' '12:00' '13:00' '14:00' '15:00' '16:00' := 'Demand' -945 -873 -797 -990 -1241 -1134 -815 -782 [*, *, 'Up'] : '17:00' '18:00' '19:00' '20:00' '21:00' '22:00' '23:00' '24:00' := 'Demand' -772 -827 -931 -1105 -1215 -1249 -1183 -952;
param MAXGRAD (tr) : 'Atom1' 'Atom2' 'Coal' 'Gas1' 'Gas2' 'Green' 'Oil' 'Solar' 'Dam' := 'Up' 30 30 35 89 95 5 56 2 4 'Down' 30 30 45 96 102 5 56 2 4;
end;
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