Economic Dispatch Using Lambda Itteration Method Neglecting Generation Limits & Transmission Losses Using Matlab

a)    Calculation Using Lambda Iteration Method

Lambda Iteration method Uses an Initial Assumption of Lagrange Multiplier and uses Continuous iterations to find correct value of Lambda. This correct value is found at Satisfaction of Demand.

Steps of Calculation Include

Ø  1^st Step to Solve a Dispatch Problem is to write cost characteristic equations of committed units for Meeting Demand .(Refer To Equal Incremental Cost Criteria)

Ø  Assume a Value of Lagrange Multiplier and Calculate Generation Sharing (Generation Allocation) at that value of Lambda 

Ø  Calculate Difference in Value of Power demand and Total Generation.(Error)

MATLAB Program

clc

clear all

Pd=input('Total Power Demand = ');

Lambda=input('Assumed Value of Lagrange Multiplier = ');

n=input('No of Units Commited to Satisfy Demand = ');

x=Pd*0.10:10:Pd;

for i=1:n;

    a=input('No Load Speed Cost of Unit($/h) = ');

    b=input('Slope of linear line of unit ($/MWh) = ');

    c=input('Coefficient of 2nd order term of Unit(($/h)/MW^2) = ');

    A(i,:)=[a b c];

    y=[2*c*x+b];

    plot(x,y)

    hold on

end

a=A(:,1);

b=A(:,2);

c=A(:,3);

DelP=5;

iter=0;

while abs(DelP) >=0.01

    iter=iter+1;

    P=(Lambda-b)./(2*c);

    DelP=Pd-sum(P);

    X=sum(1./(2*c));

    Del_Lambda=DelP/X;

Economic Dispatch Of thermal generating units Using Equal Incremental Cost Criteria Neglecting Generation Limits & Transmission Losses

Objective

                            In Power Systems Generation Allocation (Economic Dispatch) is Major Problem, Therefore we will try to Allocate Generation to Committed Units so as to Satisfy Demand With Minimum Expenses(Minimum $/h) . We will Use Lagrange Multiplier Techniques in this Session.i.e;

a)      Incremental Cost Criteria

b)      Lambda(LaGrange Multiplier) Iteration Technique

Explanation

                              Incremental Cost Criteria is stated as

“Allocate Generation so that total Demand is Satisfied & All Incremental Costs are equal in Economic Dispatch Problem”

It is Most Widely Used Technique for Solution of Economic Dispatch Problems.

However it is Constrained Continuous Problem Therefore it is Difficult to Solve system keeping in mind the Inequality constraints. Transmission losses are also to be neglected for simplicity. Therefore Neglecting

         b)    Calculation Using Lambda Iteration Method

Lambda Iteration method Uses an Initial Assumption of Lagrange Multiplier and uses Continuous iterations to find correct value of Lambda. This correct value is found at Satisfaction of Demand.

Steps of Calculation Include

Ø  1^st Step to Solve a Dispatch Problem is to write cost characteristic equations of committed units for Meeting Demand Using equation (1).

Ø  Assume a Value of Lagrange Multiplier and Calculate Generation Sharing (Generation Allocation) at that value of Lambda using equation (3).

Ø  Calculate Difference in Value of Power demand and Total Generation.(Error)

Matlab Program & Results

Equal Incremental Cost Criteria

Code

function [Lambda,P,Total_Cost]=IncrementalCC(Pd,A)

Pd=input('Total Power Demand = ');

n=input('Number of Generating Stations = ');

x=Pd*0.10:10:Pd;

for i=1:n;

    a=input('No Load Speed Cost of Unit($/h) = ');

    b=input('Slope of linear line of unit ($/MWh) = ');

    c=input('Coefficient of 2nd order term of Unit(($/h)/MW^2) = ');

    A(i,:)=[a b c];

    y=[2*c*x+b];

    plot(x,y)

    hold on

end

xlabel('Power(MW)');

ylabel('Incremental Cost ($/MWh)');

Title('Incremental Cost(Economic Dispatch Problem Solution)');

hold off

a=A(:,1);

b=A(:,2);

c=A(:,3);

Lambda=(Pd+sum(b./(2*c)))/sum(1./(2*c))

P=(Lambda-b)./(2*c)

Total_Cost=[sum(a)+sum(b.*P)+sum(c.*(P.^2))]

Results

Conclusion

                                Economic Dispatch is Online Function Carried on real Time (Minimum Cost of Production meeting Constraints). Economic Dispatch is an Important for Power System Operation because Economy is a Major factor which Influence Structure and Complexity of System. Even Unit Commitment Involve Economic Dispatch Solution as 1^st step. In this Session we Solved Problem Neglecting T/L Losses and Generation Limits. However When we take into Account T/L Losses.
Equal-Incremental Criteria is most widely used technique for Solution of ED Problem; 

Analysis & Plot of Convex and Non –Convex heat rate characteristics of thermal generating units Using MATLAB.

Objectives

Discontinuity Of heat rate characteristics occurs when we consider wall operation in Thermal Generating Units. This is called Non-Convex Heat rate Characteristic. This multiwall or step wise curve is different from Single wall heat rate curve.

So our objective is to

ü  Analyze & Plot  Non-Convex Heat rate curves

ü  Compare Convex & Non-Convex Curves

Theory

Introduction

Most Power system Contain Multi-Machine, Multiwall operations systems for generation of power ,therefore their heat rate curves are not simply linear but are Stepwise curves At switching of different walls.

Non-Convex Characteristics equation and Convex Characteristic Equation is given by

Explanation

Where c, b and a are the fuel cost coefficients of the respective unit, e and f are the valve-point coefficients of the respective unit. The units of the above coefficients are ($/MW2 h), ($/MWh), ($/h), ($/h) and (1/MW), respectively. Pmin (in MW) is the minimum capacity limit of unit. The added sinusoidal term in the production cost function reflects the effect of valve-points. Hence this problem is non-convex and non-differentiable considering valve-point effects

MATLAB Program

Code

clc

clear all

disp('Analysis and Plot of Convex & Non-Convex Characteristics Of thermal Operating unit');

syms a b c e f

g=input('Generator Total Power Capacity (MW) = ');

Pmin=input('Lower Limit Of Operation Zone(Pmin) = ');

Pmax=input('Upper Limit Of Operation Zone(Pmax) = ');

ss=input('Enter Const Suitable step size = ')

x=Pmin:ss:Pmax;

c=input('Value of p1(2nd Degree Coefficient) =');

b=input('Value of p2(1st Degree Coefficient) =');

a=input('Value of p3(3rd Degree Coefficient) =');

disp('Enter Values of  valve-point coefficients of the unit')

e=input(' e =  ');

f=input(' f = ');

m=abs(e*sin(f*(Pmin-x)));

p=[c b a];

fx1=polyval(p,x);

fx=fx1+m;

plot(x,fx,'o')

hold on

plot(x,fx1,'---')

xlabel('P');

ylabel('f(x)=a*P^2 + b*P + c + |e*sin(f*Pmin-P)| ');

title('300W Generating Unit(Comparison of Convex & Non-Convex Characteristics)');

Matlab Interface

Resulting Curve

Plot Of Characteristics Curves for Generating Stations

Theory

Usually in case of Thermal Generating units’ Major concern is Fuel consumption per Kwh and it depicts Major part of station production costs. In defining the characteristics of steam turbine units, the following terms will be used

H = Btu per hour heat input to the unit (or MBtu/h)

F = Fuel cost times H is the p per hour (Jt/h) input to the unit for fuel

Generally, the minimum load at which a unit can operate is influenced more by the steam generator and the regenerative cycle than by the turbine.

There are three type curves used for study of characteristics of thermal generating station.

X-tics of Thermal Unit

1.     Heat Rate Curve (Input/output Curve)

The input to the unit is shown on the ordinate may be either in terms of heat energy requirements [millions of Btu per hour (MBtu/h)] or in terms of total cost per hour (Jt per hour). The output is normally the net electrical output of the unit in (MW).

In other Words, This curve shows relation of Heat consumption rate per Kwh to Load Supplied by Generating Station.

Where  

2.     Incremental Heat Rate Curve

Incremental heat rate characteristic is the slope (the derivative) of the input-output characteristic (∆H/∆P or ∆F/∆P). The data shown on this curve are in terms of Btu per kilowatt hour (or Rs per kilowatt hour) versus the net power output of the unit in megawatts. This characteristic is widely used in economic dispatching of the unit.

3.     Net Heat Rate Curve

This characteristic is H/P versus P (KW). It is proportional to the reciprocal of the usual efficiency characteristic developed for machinery.

MATLAB Program

a)    Input-Output Curve

Code

clc
clear all
disp('Plot of Characteristic Equations');
p=input('Generator Total Power Capacity (MW) = ');
x=p*[1 0.8 0.6 0.4 0.25];
Pmin=min(x);
Pmax=max(x);
P=Pmin:5:Pmax;
c=input('Value of p1(2nd Degree Coefficient) =');
b=input('Value of p2(1st Degree Coefficient) =');
a=input('Value of p3(3rd Degree Coefficient) =');
y=[c*P.^2+b*P+a];
y1=[(2*c*P)+b];
y2=[c*P+b+a*P.^-1];
subplot(1,2,1),plot(P,y,'r')
title('Input-Output Characteristics');
xlabel('Power');
ylabel('Heat rate (MBTU/hr) ');
subplot(1,2,2),plot(P,y1)
title('Incremental Heat rate Characteristics');
xlabel('Power');
ylabel('MBTU/Kwah ');
hold on
subplot(1,2,2),plot(P,y2,'g')
title('Input-Output Characteristics');
xlabel('Power');
ylabel('MBTU/Kwah');

Resulting Curves