AIM:
To understand the fundamentals of economic dispatch and solve the problem using
classical method with and without line losses.
SOFTWARE REQUIRED:
MATLAB 5.3
THEORY:
Mathematical Model for Economic Dispatch of Thermal Units
Without Transmission Loss:
Statement of Economic Dispatch Problem
In a power system, with negligible transmission loss and with N number of spinning thermal
generating units the total system load PD at a particular interval can be
met by different sets of
generation schedules
{PG1^(k) , PG2^(k) , ………………PGN^(K) }; k = 1,2,……..NS
Out of these NS set of generation schedules, the system operator has to choose the set of schedules,
which minimize the system operating cost, which is essentially the sum of the production cost of
all the generating units. This economic dispatch problem is mathematically stated as an
optimization problem.
Given: The number of available generating units N, their production cost functions, their
operating limits and the system load PD,
To determine: The set of generation schedules,
Necessary conditions for the existence of solution to ED problem
The ED problem given by the equations (1) to (4). By omitting the inequality constraints
(4) tentatively, the reduce ED problem (1),(2) and (3) may be restated as an unconstrained
optimization problem by augmenting the objective function (1) with the constraint "Phi" multiplied by
LaGrange multiplier, "Lamda" to obtained the LaGrange function, L as
The solution to ED problem can be obtained by solving simultaneously the necessary conditions
(7) and (8) which state that the economic generation schedules not only satisfy the system power
balance equation (8) but also demand that the incremental cost rates of all the units be equal be
equal to "Lamda" which can be interpreted as “incremental cost of received power”.
When the inequality constraints(4) are included in the ED problem the necessary condition (7) gets
modified as
PROCEDURE:
1. Enter the command window of the MATLAB.
2. Create a new M – file by selecting File - New – M – File
3. Type and save the program.
4. Execute the program by either pressing Tools – Run.
5. View the results.
EXERCISE-1:
The fuel cost functions for three thermal plants in $/h are given by
C1 = 500 + 5.3 P1 + 0.004 P1^2 ; P1 in MW
C2 = 400 + 5.5 P2 + 0.006 P2^2 ; P2 in MW
C3 = 200 +5.8 P3 + 0.009 P3^2 ; P3 in MW
The total load , PD is 800MW.Neglecting line losses and generator limits, find the optimal
dispatch and the total cost in $/h by analytical method. Verify the result using MATLAB
program.
PROGRAM:
alpha = [500; 400; 200];
beta = [5.3; 5.5; 5.8]; gamma = [0.004; 0.006; 0.009];
PD = 800;
DelP = 10;
lamda = input('Enter estimated value of Lamda = ');
fprintf(' ')
disp(['Lamda P1 P2 P3 DP'...
' grad Delamda'])
iter = 0;
while abs(DelP) >= 0.001
iter = iter + 1;
P = (lamda - beta)./(2*gamma);
DelP = PD - sum(P);
J = sum(ones(length(gamma),1)./(2*gamma));
Delamda = DelP/J;
disp([lamda,P(1),P(2),P(3),DelP,J,Delamda])
lamda = lamda + Delamda;
end
totalcost = sum(alpha + beta.*P + gamma.*P.^2)
EXERCISE-2:
The fuel cost functions for three thermal plants in $/h are given by
C1 = 500 + 5.3 P1 + 0.004 P1^2 ; P1 in MW
C2 = 400 + 5.5 P2 + 0.006 P2^2 ; P2 in MW
C3 = 200 +5.8 P3 + 0.009 P3^2 ; P3 in MW
The total load , PD is 975MW.
Generation limits:
200 £ P1 £ 450 MW
150 £ P2 £ 350 MW
100 £ P3 £ 225 MW
Find the optimal dispatch and the total cost in $/h by analytical method. Verify the result
using MATLAB program.
PROGRAM:
cost = [500 5.3 0.004
400 5.5 0.006
200 5.8 0.009];
mwlimits = [200 450
150 350
100 225];
Pdt = 975;
dispatch
gencost
RESULT:
Thus the fundamentals of economic dispatch and solve the problem using
classical method with and without line losses understood.
To understand the fundamentals of economic dispatch and solve the problem using
classical method with and without line losses.
SOFTWARE REQUIRED:
MATLAB 5.3
THEORY:
Mathematical Model for Economic Dispatch of Thermal Units
Without Transmission Loss:
Statement of Economic Dispatch Problem
In a power system, with negligible transmission loss and with N number of spinning thermal
generating units the total system load PD at a particular interval can be
met by different sets of
generation schedules
{PG1^(k) , PG2^(k) , ………………PGN^(K) }; k = 1,2,……..NS
Out of these NS set of generation schedules, the system operator has to choose the set of schedules,
which minimize the system operating cost, which is essentially the sum of the production cost of
all the generating units. This economic dispatch problem is mathematically stated as an
optimization problem.
Given: The number of available generating units N, their production cost functions, their
operating limits and the system load PD,
To determine: The set of generation schedules,
The ED problem given by the equations (1) to (4). By omitting the inequality constraints
(4) tentatively, the reduce ED problem (1),(2) and (3) may be restated as an unconstrained
optimization problem by augmenting the objective function (1) with the constraint "Phi" multiplied by
LaGrange multiplier, "Lamda" to obtained the LaGrange function, L as
(7) and (8) which state that the economic generation schedules not only satisfy the system power
balance equation (8) but also demand that the incremental cost rates of all the units be equal be
equal to "Lamda" which can be interpreted as “incremental cost of received power”.
When the inequality constraints(4) are included in the ED problem the necessary condition (7) gets
modified as
PROCEDURE:
1. Enter the command window of the MATLAB.
2. Create a new M – file by selecting File - New – M – File
3. Type and save the program.
4. Execute the program by either pressing Tools – Run.
5. View the results.
EXERCISE-1:
The fuel cost functions for three thermal plants in $/h are given by
C1 = 500 + 5.3 P1 + 0.004 P1^2 ; P1 in MW
C2 = 400 + 5.5 P2 + 0.006 P2^2 ; P2 in MW
C3 = 200 +5.8 P3 + 0.009 P3^2 ; P3 in MW
The total load , PD is 800MW.Neglecting line losses and generator limits, find the optimal
dispatch and the total cost in $/h by analytical method. Verify the result using MATLAB
program.
PROGRAM:
alpha = [500; 400; 200];
beta = [5.3; 5.5; 5.8]; gamma = [0.004; 0.006; 0.009];
PD = 800;
DelP = 10;
lamda = input('Enter estimated value of Lamda = ');
fprintf(' ')
disp(['Lamda P1 P2 P3 DP'...
' grad Delamda'])
iter = 0;
while abs(DelP) >= 0.001
iter = iter + 1;
P = (lamda - beta)./(2*gamma);
DelP = PD - sum(P);
J = sum(ones(length(gamma),1)./(2*gamma));
Delamda = DelP/J;
disp([lamda,P(1),P(2),P(3),DelP,J,Delamda])
lamda = lamda + Delamda;
end
totalcost = sum(alpha + beta.*P + gamma.*P.^2)
EXERCISE-2:
The fuel cost functions for three thermal plants in $/h are given by
C1 = 500 + 5.3 P1 + 0.004 P1^2 ; P1 in MW
C2 = 400 + 5.5 P2 + 0.006 P2^2 ; P2 in MW
C3 = 200 +5.8 P3 + 0.009 P3^2 ; P3 in MW
The total load , PD is 975MW.
Generation limits:
200 £ P1 £ 450 MW
150 £ P2 £ 350 MW
100 £ P3 £ 225 MW
Find the optimal dispatch and the total cost in $/h by analytical method. Verify the result
using MATLAB program.
PROGRAM:
cost = [500 5.3 0.004
400 5.5 0.006
200 5.8 0.009];
mwlimits = [200 450
150 350
100 225];
Pdt = 975;
dispatch
gencost
RESULT:
Thus the fundamentals of economic dispatch and solve the problem using
classical method with and without line losses understood.
1 comments:
Write commentscan u teach how to run the coding for exercise 2?
ReplyEmoticonEmoticon