Comissionamento GAMS

8/10/2019 Comissionamento GAMS

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Learning GAMS for

Power Generation Operation and ControlLuigi Vanfretti

[email protected]

Rensselaer Polytechnic Institute

April 4, 2006.

Luigi Vanfretti, RPI p.1/30


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Outlook of this talk

What are we going to talk about?

About GAMS

What is GAMS?Using GAMS for POG&C

Downloading GAMS

Some Features of GAMSParts of a GAMS Model (Program)

Some problems of POG&C that can be solved with

GAMSEconomic Dispatch

Unit Commitment



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Downloading GAMS

A free demo of the current distribution, as well as pastdistributions, is available at http://www.gams.com/

Without a license, the GAMS distribution has thefollowing limitations:

Model limits:Up to 200 constraints

Up to 300 variables

Up to 2000 nonzero elements (1000 nonlinear).

Up to 30 discrete variables.

Solver (Global) limits:Up to 10 constranints.

Up to 10 variables.

Note that this limitations are not a problem foreducational purposes since we are not solving large

scale networks.

http://www.gams.com/http://www.gams.com/


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GAMS Features

The most important features of GAMS are:

Capability to solve from small scale to large scaleproblems with small code by means of the use of indexto write blocks of similar constraints from only oneconstraint.

The model is independent from the solution method, and

it can be solved by different solutions methos by onlychanging the solver.

The translation from the mathematical model to GAMS is

almost transparent since GAMS has been built toresemble mathematical programming models.

GAMS also uses common english words, thus is easy to

understand its statements.



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Parts of a GAMS Model

The parts of a GAMS Model or Program are defined in thetable below:

COMMAND PURPOSE

SET Used to indicate the name of the indices

SCALAR Used to declare scalars

PARAMETER Used to declare vectors

TABLE Declare and assign the values of an array

VARIABLE Declare variables, assign type and boundsEQUATION Function to be optimized and its constraints

MODEL Give a name to the model, and list the related constraints

SOLVE Indicate what solver to use

DISPLAY Indicate what you would like to display as output

Table 1: Main Commands of GAMS

We will now define each of this parts with the use of a eco-

nomic dispatch example presented in [1].



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Power System Example 1 - Economic Dispatch



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Sets

Sets are use to declare indices and its possible values. An index or more should

be declared along with its possible values. Here we present the sets of the

Economic Dispatch Problem.

Example:

SETS

G index of generators /G1*G2/

N index of buses /N1*N3/

MAP(G,N) indicates in what bus are the generators connected /G1.N1,G2.N2/;

The two indices are G and N. The text that follows is a string (used to make

comments and are ignored by the compiler). The index values are specified by

two slash(/) symbols.

GAMS uses an to define sets in a compact way, the sets could also be defined

as/N1, N2, N3/.

The sets here are dynamic, they have been mapped through MAPG(G, N), where

the links between the sets have been defined.



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Scalars

Scalars indicate a constant value as, for example, the fuelcost of a generating unit in $ / MBtu.

Example

Scalar fcost fuel cost in dollars_MBtu /1.1/

The scalar fcostis declared with a common value of1.1. Slash symbols are used to declare the value of thescalar.

Since the fuel cost of the generating units can beincluded in the cost function of each generating unit, wewill not use this scalar in our example, neverless, notethat one scalar could be defined for each generating unit

fuel cost.



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Parameters

Parameters are structures used to define data arrays of one or more dimensions,

they are vectors of data.

Example

PARAMETER

LOAD(N) load at bus N

/ N3 0.85 /

We have declared the parameter LOAD, which is defined by the set N. The

parameter assigment is defined only for Bus 3 by / N3 0.85 /(since is the

only bus which has a load).

GAMS will automatically check that the index values are valid set values foreach index, displaying errors when invalid set values are entered. For

example,/ N4 0.9 /, will yield in an error since there is no Bus 4 declared

in the Set N.



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Tables - I

Matrices of data can be defined with tables ortwo-dimensional parameters (that depend on more thanone set). Tables are structures used to define data matrices

of data of more than one dimension.Example:

TABLE GDATA(G,*) generator input data

PMIN PMAX COST

* (kW) (kW) (dollar/kWh)

G1 0.15 0.6 6

G2 0.10 0.4 7;

The table GDATA(G,*)is defined by the index G.

For every pair of set elements, as G1 and PMIN, a tablevalue value is specified, in this case 0.15.



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Tables - II

There can also be Tables that use more than one set, these emerge from the use

of dynamic sets.

Example:

TABLE LDATA(N,N,*) line input data

SUS LIMIT

* (S) (kW)

N1.N2 2.5 0.3N1.N3 3.5 0.5

N2.N3 3.0 0.4;

TABLE LDATA(N,N,*)defines a 3D array that relates two sets (in this case

only the set N is used, but any other different set could be used) with two

different parameters.

For example: we are assigning a susceptance, SUS2.5, and a limit, LIMIT

0.3, to the line between Bus 1 and Bus 2, N1.N2.



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Variables - I

Variables are declared in GAMS in the following way:

Example:

VARIABLES

z objective function

p(G) output power for generator G

d(N) angle at bus N;

p.lo(G)=GDATA(G,PMIN);

p.up(G)=GDATA(G,PMAX);

d.fx(N3)=0;

The variable zrepresents the objective function and must be included always.

The index of the variables also has to be included.Optionally, upper and lower bounds can be declared using suffixes. We have

defined upper and lower bounds to the variable, p(G), through GAMS suffixes.

The lower bound usses the suffix lo,p.lo(G)defines the minimun power

output at node G.



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Variables - II

Variables can also be set to a constant value. For example, we have set

variable dto a fixed value of zero with: d.fx(N3)=0;. This actually sets

the angle at the reference bus at 0.

There are different types of variables: binary, free, integer, negative andpositive.; For example, we could have defined variable p(G)as a positive

variable and then set its bounds:

Example:

POSITIVE VARIABLE p(G);

p.lo(G)=GDATA(G,PMIN);

p.up(G)=GDATA(G,PMAX);

Variables can also have intial values, for example, we could indicate an initialpower at Bus 1 like this:

p.l(G1)=0.2;


E i I


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Equations - I

In the Equationscommand we include the objective function and the constraints of the problem.

Example:

EQUATIONS

COST objective function

MAXPOW(N,N) maximum line power limitMINPOW(N,N) minimum line power limit

LOADBAL(N) load balance equation;

ALIAS(N,NP);

COST.. z =e= SUM(G,GDATA(G,COST)*p(G));

MAXPOW(N,NP).. LDATA(N,NP,SUS)*(d(N)-d(NP))=l= LDATA(N,NP,LIMIT);

MINPOW(N,NP).. LDATA(N,NP,SUS)*(d(N)-d(NP))=g=-LDATA(N,NP,LIMIT);

LOADBAL(N).. SUM(G$ MAP(G,N),p(G))+SUM(NP,LDATA(N,NP,SU)*(d(N)-d(NP))+

LDATA(NP,N,SUS)*(d(N)-d(NP)))=e=LOAD(N);

From the example above we note that each constraint must be declared first, after they have been

declared, they are defined using two dots (..), to link the constraint name with the equation. A

semicolon has to be included after each constraint equation.

A summation,

ixij , is expressed as sum(i,x(i,j))in GAMS.

Additionally, there are equations symbols such as: =e=, which meansequal, and =l=, which means

less than or equal to. Please refer to [5,6] for details.


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Equations - II

The equivalent mathematical form of the objective and constraint equations is:

Objective function (Cost equation):

COST.. z =e= SUM(G,GDATA(G,COST)*p(G));

Is equivalent to:

z =n

i=1

{PiCi(Pi)}(1)

In this problem we consider power transfer bounds.The lower limit is represents the minimun power that each generator should

output; the upper limit represents the maximum power that the generator con

output due to termic limits.

This can be written as:

Pmini Pi Pmax

i(2)


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Equations - III

The power transmitted in a transmission line is limited by thermal and stability considerations (angle).

The power transmitted from Busi to Busj through the line i j is:

Bij(i j)(3)

Considering equations (??) and (??) we have the equation:

Pmin

i Bij(i j) Pmax

i(4)

Equation (??) translates in the GAMS constraint equations:

MAXPOW(N,NP).. LDATA(N,NP,SUS)*(d(N)-d(NP))=l= LDATA(N,NP,LIMIT);

MINPOW(N,NP).. LDATA(N,NP,SUS)*(d(N)-d(NP))=g=-LDATA(N,NP,LIMIT);

The Power Balance equation is given by:

Pi = Di

Bij(i j)(5)

Which translates to: LOADBAL(N).. SUM(G$

MAP(G,N),p(G))+SUM(NP,LDATA(N,NP,SUS)*(d(N)-d(NP))+

LDATA(NP,N,SUS)*(d(N)-d(NP)))=e=LOAD(N);

The term

Bij(i j)implicitly considers losses (but as a constrain, they are not optimized).


M d l


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Model

This command wil define the equations that should beincluded in the model to be solved. For the ED example wehave a cost equation, a maximum power inequality,

minimun power inequality and a load balance equation.Example:

MODEL ed /COST,MAXPOW,MINPOW,LOADBAL/;

This is actually an excellent feature of GAMS, since it willonly require us to elliminate one of the equations to solvea different model.

For example, we could solve ED without consideringmaximun and minimun powers by declaring:

MODEL ed /COST,LOADBAL/;


S l


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Solve

Solveis the command used to solve the optimizationproblem that we defined. For our problem the solvestatement is:

Example:

SOLVE ed USING lp MINIMIZING z;

This command tells GAMS to solve Economic Dispatch,ed, using a linear programing solver, lp, minimizing thevariable z; and thus, minimizing the cost function.

There are many solvers, options and suffix that can beused in the solution of an optimization problem; pleaserefer to [5,6].


Displa


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Display

Additionally to the information displayed in the standard gams output file, named

with a.lstextension, the information can be displayed using the display

command.

Example:DISPLAY p.l, d.l;

With this command GAMS will display the output power at the generators and

the angle at each bus. The suffix l, indicates GAMS to display the level of the

variable.

If we solve the model with this command the output will show:

---- 42 VARIABLE p.L output power for generator G

G1 0.565, G2 0.285

---- 42 VARIABLE d.L angle at bus N

N1 -0.143, N2 -0.117


P S t E l 2 U it C it t


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Power System Example 2 - Unit Commitment

This example is also taken from [1].

The system of the Economic Dispatch problem is used. A three hour unit commintment

is presented. The demands are: t1 = 150, t2 = 500and t3 = 400. The generators have

the following reserves: Unit 1 = 15, Unit 2 = 50 and Unit 3 = 40.

The data for each of the genetors follows:

Table 2: Data for the Unit Commintment ProblemGenerator # 1 2 3

Pmax 350 200 140

Pmin 50 80 40

Rampup limit 200 100 100

Rampdown limit 300 150 100Fixed Cost 5 7 6

Startup Cost 20 18 5

Shutdown Cost 0.5 0.3 1.0

Variable Cost 0.100 0.125 0.150


P S t E l 2 U it C it t


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SETS:

SETS

K index of periods of time /1*4/

J index of generators /1*3/

TABLES:TABLE GDATA(J,*) generator input data

PMIN PMAX T S A B C E

* Ramp Ramp Fixed Variable Start Shutdown

* Down Up Cost Cost UP Cost

* Limit Limit Cost

* (kW) (kW) (kW/h) (kW/h) (E) (E) (E) (E/kWh)

1 50 350 300 200 5 20 0.5 0.100

2 80 200 150 100 7 18 0.3 0.125

3 40 140 100 100 6 5 1.0 0.150;

TABLE PDATA(K,*) data per period

D R

* Load Reserve

* (kW) (kW)

2 150 153 500 50

4 400 40;


Power System Example 2 Unit Commitment


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VARIABLES:

VARIABLES

z objective function variable

p(J,K) output power of generator j at period k

v(J,K) is equal to 1 if generator j is committed in period ky(J,K) is equal to 1 if generator j is started-up at the beginning of period k

s(J,K) is equal to 1 if generator j is shut-down in period k;

VARIABLE INITIALIZATION AND TYPE:

POSITIVE VARIABLES p(J,K);

** Status decisions are modeled using binary variables.

BINARY VARIABLES v(J,K),y(J,K),s(J,K);

** Initial values are stated for some variables.

v.fx(J,1)=0;

p.fx(J,1)=0;




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EQUATIONS:

EQUATIONS

COST objective function

PMAXLIM(J,K) maximum output power equation

PMINLIM(J,K) minimum output power equation

LOAD(K) load balance equation

RESERVE(K) spinning reserve equation

LOGIC(J,K) start-up shut-down and running logic

RAMPUP(J,K) maximum up ramp rate limit

RAMPDOWN(J,K) maximum down ramp rate limit;

** The objective function is an equality equation. The remaining constraints are defined for all periods K,

** except for the initial one. To model this exception the $(ord(K) GT 1) condition is included.

COST.. z =e= SUM((K,J), GDATA(J,A)*v(J,K)+GDATA(J,B)*y(J,K)+ GDATA(J,C)*s(J,K)+GDATA(J,D)*p(J,K));

PMAXLIM(J,K)$(ord(K) GT 1).. p(J,K)=l=GDATA(J,PMAX)*v(J,K);

PMINLIM(J,K)$(ord(K) GT 1).. p(J,K)=g=GDATA(J,PMIN)*v(J,K);

LOAD(K)$(ord(K) GT 1).. SUM(J,p(J,K))=e=PDATA(K,D);

RESERVE(K)$(ord(K) GT 1).. SUM(J,GDATA(J,PMAX)*v(J,K))=g=PDATA(K,D)

+PDATA(K,R);

LOGIC(J,K)$(ord(K) GT 1).. y(J,K)-s(J,K)=e=v(J,K)-v(J,K-1);

RAMPUP(J,K)$(ord(K) GT 1).. p(J,K)-p(J,K-1)=l=GDATA(J,S);

RAMPDOWN(J,K)$(ord(K) GT 1)..p(J,K-1)-p(J,K)=l=GDATA(J,T);




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MODEL:

The model declaration /ALL/will allow us to solve the UC problem

with all the constraints:

MODEL uc /ALL/;

MODEL:

If we are only interested in solving the UC problem without the reserve

contrarint, ramup and rampdown limits, we can declare the model as:

MODEL uc /COST,PMAXLIM,PMINLIM,LOAD,LOGIC/;

SOLVER:

GAMS will use a mixed-integer solver to find the solution of the

problem above, with the following command.

SOLVE uc USING mip MINIMIZING z;




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GAMS OUTPUT:The following is an abstract of the output from GAMS when all the constraints areconsidered:

---- 88 VARIABLE p.L output power of generator j at period k

2 3 4

1 150.000 350.000 260.000

2 100.000

3 50.000 140.000

---- 88 VARIABLE z.L = 77.300 objective function va

riable

GAMS OUTPUT:Solution without the reserve contraint, ramup and rampdown limits:

---- 87 VARIABLE p.L output power of generator j at period k

2 3 4

1 150.000 350.000 320.000

2 150.000 80.000

---- 87 VARIABLE z.L = 67.000 objective function va

riable



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Thank you for your attention!

The files used in this presentation are online at:

http://www.rpi.edu/~vanfrl/pogc.html


References
http://www.rpi.edu/~vanfrl/pogc.htmlhttp://www.rpi.edu/~vanfrl/pogc.html


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References

[1] E. Castillo, A. J. Conejo, P. Pedregal, R. Garca and N. Alguacil,Building and Solving Mathematical Programming Models inEngineering and Science,John Wiley & Sons, 2002. Spanish

version availablefreeat:

http://departamentos.unican.es/macc/personal/

profesores/castillo/Libro.htm

[2] J. Restrepo,Power Applied GAMS Tutorial, Power Engineering

Research Laboratory, McGill University.[3] J. Restrepo,Introduction to GAMS, Power Engineering Research

Laboratory, McGill University.

[4] D. Chattopadhyay, "Application of General Algebraic Modeling

System to Power System Optimization," IEEE Transactions on

Power Systems, vol. 14, no. 1, February 1999, pp. 15 - 22.


References
http://departamentos.unican.es/macc/personal/http://localhost/var/www/apps/conversion/tmp/scratch_5/profesores/castillo/Libro.htmhttp://localhost/var/www/apps/conversion/tmp/scratch_5/profesores/castillo/Libro.htmhttp://departamentos.unican.es/macc/personal/


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References

[5] R. E. Rosenthal.GAMS - A Users Guide,GAMS DevelopmentCorporation, Washington, DC, USA, 2006.http://www.gams.com/docs/gams/GAMSUsersGuide.pdf

[6] R. E. Rosenthal,A GAMS Tutorial,Naval Postgraduate School.

Monterey, California, USA.

http://www.gams.com/docs/gams/Tutorial.pdf

[7] A. Wood and B. F. Wollenberg,Power Generation, Operation and

Control,John Wiley & Sons, Inc. New York, 1996.[8] A. Wood,dispatch.gms : Economic Load Dispatch Including

Transmission Losses.

http://www.gams.com/modlib/libhtml/dispatch.htm

A. Wood,hydro.gms : Hydrothermal Scheduling Problem.

http://www.gams.com/modlib/libhtml/hydro.htm

A. Wood,fuel.gms : Fuel Scheduling and Unit Commitment Problem.

http://www.gams.com/modlib/libhtml/fuel.htm

http://www.gams.com/docs/gams/GAMSUsersGuide.pdfhttp://www.gams.com/docs/gams/Tutorial.pdfhttp://www.gams.com/modlib/libhtml/dispatch.htmhttp://www.gams.com/modlib/libhtml/hydro.htmhttp://www.gams.com/modlib/libhtml/fuel.htmhttp://www.gams.com/modlib/libhtml/fuel.htmhttp://www.gams.com/modlib/libhtml/hydro.htmhttp://www.gams.com/modlib/libhtml/dispatch.htmhttp://www.gams.com/docs/gams/Tutorial.pdfhttp://www.gams.com/docs/gams/GAMSUsersGuide.pdf

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