# Production of Electricity

## Contents

## Problem description

Power generators of four different types are available to satisfy the daily electricity demands (in megawatts) summarized in the following table. We consider a sliding time horizon: the period 10pm-12am of day d is followed by the period 0am-6am of day d + 1.

Daily electricity demands (in MW)

+-------+-------+-------+--------+--------+-------+--------+---------+ |Period |0am-6am|6am-9am|9am-12pm|12pm-2pm|2pm-6pm|6pm-10pm|10pm-12am| +-------+-------+-------+--------+--------+-------+--------+---------+ |Demand | 12000| 32000| 25000| 36000| 25000| 30000| 18000| +-------+-------+-------+--------+--------+-------+--------+---------+

The power generators of the same type have a maximum capacity and may be connected to the network starting from a certain minimal power output. They have a start-up cost, a fixed hourly cost for working at minimal power, and an hourly cost per additional megawatt for anything beyond the minimal output. These data are given in the following table.

Description of power generators

+---------+-----------+------+--------+--------+------------+--------+ |Available|Min. output|Max. |capacity|Fix cost|Add. MW cost|Start-up| |number |in MW |in MW |$/h | |$/h |cost | +---------+-----------+------+--------+--------+------------+--------+ |Type 1 | 10 | 750 | 1750 | 2250 | 2.7 | 5000 | |Type 2 | 4 | 1000 | 1500 | 1800 | 2.2 | 1600 | |Type 3 | 8 | 1200 | 2000 | 3750 | 1.8 | 2400 | |Type 4 | 3 | 1800 | 3500 | 4800 | 3.8 | 1200 | +---------+-----------+------+--------+--------+------------+--------+

A power generator can only be started or stopped at the beginning of a time period. As opposed to the start, stopping a power plant does not cost anything. At any moment, the working power generators must be able to cope with an increase by 20% of the demand forecast. Which power generators should be used in every period in order to minimize the total daily cost?

## Variables

demand MW needed each period available Generators of each type available mincap Minimal output of generator maxcap Maximal output of generator fixcost Fix cost per hour runningcost Cost per hour and MW above mincap startcost Cost for starting a generator periodlengths Hours in each period

## Reference

Applications of optimization... Gueret, Prins, Seveaux

% Marcus Edvall, Tomlab Optimization Inc, E-mail: tomlab@tomopt.com % Copyright (c) 2005-2009 by Tomlab Optimization Inc., $Release: 7.2.0$ % Written Oct 7, 2005. Last modified Apr 8, 2009.

## Problem setup

demand = [12;32;25;36;25;30;18]*1000; available = [10;4;8;3]; mincap = [750;1000;1200;1800]; maxcap = [1750;1500;2000;3500]; fixcost = [2250;1800;3750;4800]; runningcost = [2.7;2.2;1.8;3.8]; startcost = [5000;1600;2400;1200]; periodlengths = [6;3;3;2;4;4;2]; n1 = length(demand); n2 = length(available); start = tom('start',n1,n2,'int'); work = tom('work',n1,n2,'int'); padd = tom('padd',n1,n2); % Bounds bnds1 = {[start;padd] >= 0}; bnds2 = {work <= repmat(available',n1,1)}; % Relationship between work and padd addcap = repmat((maxcap-mincap)',n1,1); con1 = {padd <= addcap.*work}; % Demand in each period constraint con2 = {sum(repmat(mincap',n1,1).*work+padd,2) >= demand}; % 20% additional production possible con3 = {sum(repmat(maxcap',n1,1).*work,2) >= 1.2*demand}; % Relationship between start_pt and work_pt con4 = {start(1,:) >= work(1,:) - work(end,:)}; con5 = {start(2:end,:) >= work(2:end,:) - work(1:end-1,:)}; % Objective objective = sum(sum(repmat(startcost',n1,1).*start + ... repmat(periodlengths,1,n2).*(repmat(fixcost',n1,1).*work + ... repmat(runningcost',n1,1).*padd))); constraints = {bnds1, bnds2, con1, con2, con3, con4, con5}; options = struct; options.solver = 'cplex'; options.name = 'Production of Electricity'; sol = ezsolve(objective,constraints,[],options); PriLev = 1; if PriLev > 0 intervals = ['00-06';'06-09';'09-12';'12-14';'14-18';'18-22';'22-24']; starts = sol.start; running = sol.work; extra = sol.padd; [reacs, times] = size(starts); for time = 1:times, disp([' At ' intervals(time,:) '...']) for reac = 1:reacs, if running(reac,time) > 0, disp([ ' there are ' num2str(running(reac,time)) ... ' reactors of type ' num2str(reac) ' running' ]) if starts(reac,time) > 0, disp([' (we started ' num2str( starts(reac,time))... ' additional reactors)']) end if extra(reac,time) > 0, disp([' (we also produce ' num2str( extra(reac,time))... ' MW more than the minimal level)']) end end end end end % MODIFICATION LOG % % 051010 med Created. % 060109 per Added documentation. % 060125 per Moved disp to end % 090308 med Converted to tomSym

Problem type appears to be: mip ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2010-02-05 ===================================================================================== Problem: --- 1: Production of Electricity f_k 1465809.999999999500000000 f(x_0) 0.000000000000000000 Solver: CPLEX. EXIT=0. INFORM=102. CPLEX Branch-and-Cut MIP solver Optimal sol. within epgap or epagap tolerance found FuncEv 81 Iter 17 CPU time: 0.015625 sec. Elapsed time: 0.015000 sec. At 00-06... there are 3 reactors of type 1 running there are 4 reactors of type 2 running (we started 1 additional reactors) (we also produce 1600 MW more than the minimal level) there are 4 reactors of type 3 running there are 7 reactors of type 4 running (we started 3 additional reactors) (we also produce 3350 MW more than the minimal level) there are 3 reactors of type 5 running there are 3 reactors of type 6 running (we also produce 350 MW more than the minimal level) there are 3 reactors of type 7 running At 06-09... there are 4 reactors of type 1 running (we also produce 1750 MW more than the minimal level) there are 4 reactors of type 2 running (we also produce 2000 MW more than the minimal level) there are 4 reactors of type 3 running (we also produce 200 MW more than the minimal level) there are 4 reactors of type 4 running (we also produce 2000 MW more than the minimal level) there are 4 reactors of type 5 running (we also produce 950 MW more than the minimal level) there are 4 reactors of type 6 running (we also produce 2000 MW more than the minimal level) there are 4 reactors of type 7 running (we also produce 1950 MW more than the minimal level) At 09-12... there are 2 reactors of type 1 running (we also produce 1600 MW more than the minimal level) there are 8 reactors of type 2 running (we started 6 additional reactors) (we also produce 6400 MW more than the minimal level) there are 8 reactors of type 3 running (we also produce 6400 MW more than the minimal level) there are 8 reactors of type 4 running (we also produce 6400 MW more than the minimal level) there are 8 reactors of type 5 running (we also produce 6400 MW more than the minimal level) there are 8 reactors of type 6 running (we also produce 6400 MW more than the minimal level) there are 4 reactors of type 7 running (we also produce 3200 MW more than the minimal level) At 12-14... there are 3 reactors of type 2 running (we started 3 additional reactors) there are 1 reactors of type 3 running there are 3 reactors of type 4 running (we started 2 additional reactors) there are 1 reactors of type 5 running there are 3 reactors of type 6 running (we started 2 additional reactors) there are 1 reactors of type 7 running