economic load dispatch using pso methode

7/28/2019 economic load dispatch using PSO methode

1/53

CHAPTER 1

INTRODUCTION

LITERATURE SURVEYORGANISATION OF THE

THESIS

1


2/53

INTRODUCTION

Since an engineer is always concerned with the cost of products and services, the efficient

optimum economic operation and planning of electric power generation system have always

occupied an important position in the elctric power industry. With large interconnection of the

electric networks , the energy crisis in the world and continous rise in prices , it is very essential

to reduce the running charges of electric energy. A saving in the operation of the system of a

small percent represents a significant reduction in operating cost as well as in the quantities of

fuel consumed. The classic problem is the economic load dispatch of generating systems to

achieve minimum operating cost.

1.1 LITERATURE SURVEY :

In order to have better understanding about the project the literature survey has been done.

As part of literature survey, several books, magazines, journals, websites and technical papers

are studied.

The economic dispatch problem is one of the fundemental issues in power systems. ELD

problem using Lambda iteration method is discussed in detail by D.P.Kothari and J.S.Dhillon[1].

The optimal value of lambda for a given power demand is calculated which is then used to

compute the economic generations iteratively and few overall computations are involved in this

approach. And an IEEE paper by Dr.Maheswarapu Sydulu has been referred to solve economic

load dispatch problem neglecting losses [2]. And for the method of solving economic load

dispatch problem by Particle Swarm Optimization method is discribed in detailed by

M.Sudhakaran, et,al. [3].

2


3/53

1.2 ORGANISATION OF THE THESIS:

In chapter 2 review of economic load dispatch has been discussed. Different traditional methods

applied to find out solution to economic load dispatch problem have been discussed.

In chapter 3 Economic Operation of Power system, Optimum load dispatch, Cost functions and

System constraints are discussed in detail.

In chapter 4 the previous approaches available for solving the economic load dispatch problem

were discussed and brief introduction was given for the available methods.

In chapter 5 Lagrange method has been explained while considering as well as neglecting losses.

The problem formulation, algorithms and flow charts for the Lagrange method have been

presented.

In chapter 6 A Very Fast and Effective Non- Iterative Lambda Based Method has been discussed

and the algorithm and flow chart are presented.

In chapter 7 we have clearly explained the concept of Particle Swarm Optimization method.

Basic parameters of PSO are explained. And the, algorithm and flowchart have been represented.

In chapter 8 the results obtained by solving the ELD problem for the 3 generator system and six

generator system for different load demands are presented and duly discussed.

In chapter 9 the conclusions deduced for the results obtained are presented along with the future

scope of the project.

3


4/53

CHAPTER 2

ECONOMIC LOAD DISPATCH

INTRODUCTION

4


5/53

ECONOMIC LOAD DISPATCH

2.1 INTRODUCTION :

The economic load dispatch is an important function in modern power system like unitcommitment, load forecasting, available transfer capability calculation, security analysis,

scheduling of fuel purchase etc. ELD is solved traditionally using conventional optimisation

technique such as Lambda iteration, non iterative lambda based approach, gradient method.

Lately, ELD is also addressed by intelligent methods like genetic algorithm(GA)[6],[7]

evolutionary programming(EP)[8],[9]. Dynamic programming(DP)[10], tabu search[11], Particle

Swarm Optimisation(PSO)[4] etc. for calculation simplicity , existing methods use second order

fuel cost functions which involve approximations and constraints are handled seperately ,

although sometimes valve-point effects are considered.

Lamda iteration , gradient method[12],[13],[14] can solve simple ELD calculations and

they are not sufficient for non linear optimization problems with several constraints, There are

several intelligent methods, among them genetic algorithm is proved to be more effective in

solving complex non linear problems. Lately, it has beed proved that Particle Swarm

Optimization method yields better optimal solution compared to GA [6] in solving non linear

optimization problems. For a practical problem like ELD, the intelligent methods must be

modified accordingly so that they are suitable to solve the problem with more accurate multiplefuel cost functions and constraints and they can reduce randomness. Intelligent methods are

iterative techniques that can search optimal solution depending on the problem domain and

execution time limit.They are general purpose searching techniques based on principles inspiredfrom the genetic and evolution mechanisms observed in natural systems andpopulations ofliving beings. These methods have the advantage of searching the solution space more

thoroughly . The main difficulty is their sensitivity to the choice of parameters. Among

intelligent methods, PSO is simple and promising. It requires less computation time and memory.It has also standard values for its parameter.

CHAPTER 3

5


6/53

ECONOMIC OPERATION OF POWER SYSTEM

INTRODUCTION

OPTIMUM LOAD DISPATCH

COST FUNCTIONSYSTEM CONSTRAINTS

ECONOMIC OPERATION OF POWER SYSTEM

6


7/53

3.1 INTRODUCTION:

The engineers have been very successful in increasing the efficiency of boilers, turbines and

generators so continously that each new added generating plants of a system operates more

efficiently than any older unit on the system. In operating the system for any load condition thecontribution from each plant and from each unit within a plant must be determined so that the

cost of delivered power is minimum.

Any plant may contain different units such as hydro, thermal, gas etc. These plants have

different characteristic which gives different generating cost of any load. So that there should be

a proper scheduling of plants for minimisation of cost of operation . The cost characteristic of

each generating unit is non-linear. So the problem of achieving the minimum cost becomes a

non-linear problem and also difficult.

3.1.1 OPTIMUM LOAD DISPATCH :

The optimum load dispatch problem involves the solution of two different problems. The

first of these is unit commitment or pre dispatch problem wherein it is required to select

optimally out of available genrating souces to operate to meet the expected load and provide

specified margin of operating reserve over a specified period of time. The second aspect ofeconomic dispatch is the online economic dispatch wherein it is required to distribute load

among the generating units actually paralleled with the system in such manner as to minimise the

total cost of supplying the minute to minute requirements of the system. The objective of this

work is to solve a nonlinear economic load dispatch problem.

3.2 MATHEMATICAL MODELLING:

7


8/53

The problem of Economic Dispatch can be mathematically stated in the form of an

optimization Problem with minimizing the total cost of the system as the objective function with

the system limits as the constraints. The cost and the system limits can be mathematically

modeled as follows.

3.2.1 COST FUNCTION :

Let C(i) mean the cost expressed in $/hr , producing energy in generator unit i, the total

controllable system production cost therfore will be

$/hr

The genrated real power PG(i) accounts for major influence on c(i). The individual real

generation are raised by increasing the prime mover torques and this requires an increased

expenditure of fuel. The reactive generation QG(i) donot have any measurable influence on c(i)

because they are controlled by controlling the field current.

The individual production cost c(i) of generatora unit i is therefore for all practical

purposes a function only of PG(i) , and for the overall controllable production cost ,thus

When the cost function C can be written as sum of terms where each term depends only

upon independent variable.

3.2.2 SYSTEM CONSTRAINTS :

Broadly speaking there are two types of constraints

i) Equality constraints

ii) Inequality constraints

The inequality constraints are of two types (i) hard type and (ii) soft type. The hard type are

those which are definite and specific like tapping range of an on-load tap changing transformer

where as soft type are those which have some flexibility associated with them like the nodal

8


9/53

voltages and phase angles between the nodalvoltages etc. soft inequality constraints have beenvery efficiently handled by penalty function methods.

i) EQUALITY CONSTRAINTS :

From observation we can conclude that cost function is not affected by the reactive power

demand. So the full attention is given to real power balance in the system. Power balance

requires that the controlled generation variables PG(i) obey the constraint equation

ii) INEQUALITY CONSTRAINTS :

i) Generator constraints :

The KVA loading in a generator is given by P2 +Q2 and this should not exceed a specified

value of power because of the temperature rise conditions.

The maximum active power generation of a source is limited again by thermal consideration and

also minimum power generation is limited by the flame instability of a boiler. If the power

output of the generator for optimum operation of the system is less than a specified value Pmin,

the unit is not put on the busbar because it is not possible to generate that low value of powerfrom the unit. Hence the generator power P cannot be outside the range stated by the inequality

Pmin P Pmax

Similarly the maximum and minimum reactive power generation of a source is limited. The

maximum reactive power is limited because of overheating of rotorand the minimum is limitedbecause of the stability limit of the machine . Hence the powers of the generators Pp cannot be

outside the range stated by the inequali

Qmin Q Qmax

ii) Voltage constraints :

9


10/53

It is essential that the voltage magnitudes and phase angles at various nodes should vary

within certain limits . The normal operating angle of transmission lies between 30 to 45

degrees for transient stability reasons. A lower limit of delta assures proper utilisation of

transmission capacity.

iii) Running spare capacity constraints :

These constraints are required to meet

a) the forced outages of one or more alternators on the system and

b) the unexpected load on the system

The total generation should be such that in addition to meeting the load demand and losses aminimum spare capacity should be available i.e,

G Pp+Pso

Where G is the total generation and Pso is some pre-specified power. A well planned system is

one in which this spare capacity Pso is minimum.

iv) Transmission line constraints :

The flow of active and reactive power through the transmission line is limited by thermal

capability of the circuit expressed as Cp Cp max

Where Cp max is the maximum loading capacity of the Pth line.

v) Transformer taps settings:

If an auto transformer is used , the minimum tap setting could be zero and the maximum one i.e,

0 t 1.0

Similarly for the two winding transformer if the tappings are provided on the secondary side,

0 t n

10


11/53

CHAPTER 4

LAGRANGE METHOD INTRODUCTION

ECONOMIC LOAD DISPATCH PROBLEM

(WITHOUT AND WITH LOSSES)

PROBLEM FORMULATION

ALGORITHM

FLOW CHART

11


12/53

LAGRANGE METHOD

4.1 INTRODUCTION :In this chapter Lagrange method has been discussed and thealgorithms are developed to solve the ELD Problem with and without losses [1]. The problem

formulation for economic load dispatch problem using lambda iterative method has been given.

And the algorithm and flowchart to corresponding problem are presented.

The economic load dispatch problem deals with the minimization of cost of generating thepower at any load demand. The study of this economic dispatch can be classified into two

different ways. One is economic load dispatch without the transmission line losses and other is

economic load dispatch with transmission line losses. In Lambda iteration method lambda is the

variable introduced in solving constraint optimization problem and is called Lagrange multiplier.

Since all the inequality constraints to be satisfied in each trial the equations are solved by the

iterative method

4.2 ECONOMIC LOAD DISPATCH PROBLEM NEGLECTING

LOSSES:

4.2.1 PROBLEM FORMULATION:

Economic dispatch (ED) is the scheduling of generators to minimise the total operating

cost depending on the various constraints.

This optimization problem can be mathematically stated as

Fi(Pi) = ai + biPi + ciPi2 $/hr ------------------------- (1)

subjected to constraints

Pgi = Pd + PL --------------------------- (2)

Pgmin Pg(i) Pgmax ----------------------------- (3)

Where,

i=1,2,3.....ng

12


13/53

aibi ci are cost coefficients

Pd = Given load demand

Pgi= Real power generation

ng = number of generation buses

Pl = Transmission power loss

The system losses are obtained by Krons loss formula

PL = Pgi * Bij * Pgj ----------------------------------- (4)Where,i=j=1,2ngB loss coefficient matrix.

The following formulae are used to solve the ELD problem using the Lambda iterative method

--------------------------------------- (5)

and Pgi is given by

---------------------------------------- (6)

and P = Pd + PL - Pgi ----------------------------------------- (7)

Equation describing Pgi including Loss coefficients is given by

Pgi = ------------------------------------- (8)

13


14/53

4.2.2 ALGORITHM FOR SOLVING ELD BY

LAGRANGE METHOD NEGLECTING LOSSES :

STEP 1: Read data,namely cost coefficients, ai,bi, ci

(i=1,2,.NG); convergence tolerance,

; step size ; and maximum allowed iterations,ITMAX,etc.

STEP 2: Compute the initial values of Pgi (i=1,2,NG) and by assuming

that PL =0.Then the problem can be started by Eqs (1) and

(2) and the values of and Pgi (i=1,2,,NG) can be obtained

Directly using Eqs.(5) and (6),respectively.

STEP 3: Assume no generator has been fixed at either lower limit or at upper limit.

STEP 4: Set iteration counter,IT =1.

STEP 5: Compute Pgi (i=1,2,R) of generators which are not fixed at either upper

or lower limits, using Eq (8),wher R is the number of participating

generators.

STEP 6: Compute P=PD- Pgi .

STEP 7: Check P, if yes then go to STEP 11.(it means the program moves

Forward without obtaining required convergence.)

STEP 8: Modify new ==P,where is the step-size used to increase or decrease

the value of in order to meet the STEP 7.

STEP 9: IT=IT+1, =new and GOTO STEP 5 and repeat.

STEP10: Check the limits of generator,if no more violations then GOTO STEP13,

14


15/53

Else fix as following.

If Pgi < Pgimin then Pgi =Pgimin

If Pgi>Pgimax

then Pgi =Pgimax

STEP11: GOTO STEP 4.

STEP12: Compute the optimal total cost from Eq.(1)

STEP13:Stop

15


16/53

4.2.3 FLOW CHART FOR ELD USING LAGRANGE METHOD

NEGLECTING LOSSES

FIG 4.1 : FLOWCHART FOR ELD USING LAGRANGE METHOD(NEGLECTINGLOSSES)

16

Start

Read the values ofa

i, b

i, c

i, ,

,ITMAX,Pd

Compute initial values of Pg(i), using

eqs 5&6 , take Pl=0

Set

Compute P given by eq 7

If

|P|

Pgmax


17/53

4.2.4 ALGORITHM FOR ECONOMIC LOAD DISPATCH BY

LAGRANGE METHOD INCLUDING LOSSES AND CONSIDERING

LIMITS:

STEP 1: Read data,namely cost coefficients, ai,bi, ci , B- coefficients,Bij, Bi0, B00 (i=1,2,.NG; j=1,2,.NG);convergence

tolerance, ;step size ; and maximum allowed iterations,ITMAX,etc.

STEP 2: Compute the initial values of Pgi (i=1,2,NG) and by assuming

that PL =0.Then the problem can be started by Eqs.(1) and

(2) and the values of and Pgi (i=1,2,,NG) can be obtained

Directly using Eqs.(5) and (6) respectively.

STEP 3: Assume no generator has been fixed at either lower limit or at upper limit.

STEP 4: Set iteration counter,IT =1.

STEP 5: Compute Pgi (i=1,2,R) of generators which are not fixed at either upper

or lower limits, using Eq (8),wher R is the number of participating

generators.

STEP 6: Compute the transmission loss using Eq(3.14)

STEP 7: Compute P=PD+PL- Pgi . i=1,2,3..NG

STEP 8: Check P, if yes then go to STEP 11.(it means the program moves

Forward without obtaining required convergence.)

STEP 9: Modify new ==P,where is the step-size used to increase or decrease

the value of in order to meet the STEP 7.

17


18/53

STEP 10: IT=IT+1, =new and GOTO STEP 5 and repeat.

STEP11: Check the limits of generator,if no more violations then GOTO STEP13,

Else fix as following.

If Pgi < Pgimin then Pgi =Pgimin

If Pgi>Pgi max then Pgi =Pgimax

STEP12: GOTO STEP 4.

STEP13:Compute the optimal total cost from (1) and transmission loss from (4)

STEP14:Stop

4.2.5 FLOWCHART FOR SOLVING ELD USING LAGRANGE METHOD

( INCLUDING LOSSES)

18


19/53

19

Start

Read the values of ai,

bi, c

i, ,,ITMAX,P

d

Set IT=1

Compute initial value of

using eq 6

Compute Pgi , PL given by eq

Compute P = Pd + PL - Pgi

If |P|

&

IT>ITM

AX

new + *P

IT=IT+1, =

if

PgiP

g

max

Pgi= PgmaxPgi= Pgmin


20/53

FIG 4.2 : FLOWCHART FOR ELD USING LAGRANGE METHOD(INCLUDING LOSSES)

CHAPTER 5

20

Pgi

Compute the optimum

cost using eq 1

Stop


21/53

A VERY FAST AND EFFECTIVE NON-ITERATIVE

-LOGIC BASED METHOD

INTRODUCTION

PROBLEM FORMULATION

ALGORITHM

FLOW CHART

A VERY FAST AND EFFECTIVE NON-ITERATIVE -LOGIC

BASED METHOD

21


22/53

5.1 INTRODUCTION : In this chapter we have solved for the problem of economic load

dispatch neglecting losses by non-iterative lambda based method. The advantage of using this

method is been clearly explained and the problem formulation and algorithm and flowchart have

been represented.

This is a direct or non iterative method it does not demand any initial guess value of for

ED of units for a given P demand . Many of the existing conventional methods fail to impose P-

limits at voilating units. Improper selection of initial value of may cause slow convergence or

at atimes leads to divergence for conventional algorithms. Further , each specified P D the

problem is to be attempted afresh with new guess value of . If the load varies from PD min to PD

max on the plant having Kunits ,the solution time becomes significantly large for higher values of

K. Unlike to this, this proposed method always offers the solution in an non-iterative mode withvery low solution time as it is computationally very fast[2].

The proposed -logic method is a new contribution in the area of economic dispatch. It has two

stages at first , preprepared power demand(PPD) is to be calculated. At the second stage solution

vector (P1,P2,P3PK) is calculated with very less computation as is directly calculated for

specified PD without any iterative approach .

5.2 ELD BY NON-ITERATIVE METHOD

5.2.1 PROBLEM FORMULATION :

This method for solving ED consists of two stages i.e, (i) Pre Prepared Power Demand

Data (ii) Calculation of Solution P1,P2..Pk for specified power demand Pd.

The ED condition is given as

dFl/dPI = dFz/dPZ = ... ....= dFk/dPk

Fi(Pi) = ai + biPi + ciPi2 $/hr. .... --------------------------- (1)

dFi(Pi)/dPi = = bi+2*ci*Pi

min and max are calculated by the following eqns

22


23/53

= min at Pi =Pmin

= max at Pi =Pmax

and Pi = [-bi ] / [2*ci ] ---------------------------------------- (2)

Then PPD = Pi i=1,2,K ---------------------- (3)

And the slope is given by

Slope = [i+1-i] / [PPDi+1 PPDi ] ------------------------ (4)

Scan PPDi and identify the interval i and ( i+1)

For given power demand, Pd

P= Pd - PPD --------------------------------- (5)

Then find

new = slope*(P) + i

5.2.2 ALGORITHM FOR SOLVING ELD USING NON-ITERATIVE

METHOD:

Step1:Start

Step2:Calculate prepaid data using

i min = dFi / dPi at Pi=Pi min

i max = dFi / dPi at Pi=Pimax

23


24/53

Step3: Arrange obtained values in ascending order

Step4:Now calculate total power demand at each use following fundamental observation

of ED condition.

(i) min(i) then P(i) =Pmin(i)

(i) max(i) then P(i) = Pmax(i)

min(i) (i) < max(i)

Step5:Calculate slope between any two intervals of power demand and vector.

Step6: Calculate P1,P2.Pk for specified pd using equation 1 where

= new

new = slope (P)+ j

Step7:Repeat the same process for different power demands.

Step8:Stop.

5.2.3 FLOWCHART FOR SOLVING ELD PROBLEM USING NON-

ITERATIVE LAMBDA BASED METHOD

24

Start

Read Fi(P

gi), Pd

i=1,2ng

I=

and Pd, i=1.ng

and Pd, i=1.ng


25/53

25

Differentiate Fi(Pgi)

and identify ai, bi

Calculate

min=dFi /dPgi at Pi=Pmin(i)

max= dFi /dPg(i) at Pi=Pmax(i)

Arrange lambda vector in

ascending order

If

j

min(i

)

j>

max(

i)

Pi=Pmin(i)Pi =Pmax(i)

Pi =[-bi ]/[2*ai


26/53

FIG 5.1 : FLOWCHART FOR ELD USING NON-ITERATIVE METHOD

26

Find the sum of power

demands at each j as

Calculate slope from

Scan PPD(j) and identify

interval

P=Pd- PPDj,

new

If j

min(i)

j

>max(i)

Pi = Pmin(i)PI = Pmax(i)

and Pd=PPDi

stop


27/53

CHAPTER 6

PARTICLE SWARM OPTIMIZATION

INTRODUCTION

ALGORITHM

FLOWCHART

27


28/53

PARTICLE SWARM OPTIMIZATION

6.1 INTRODUCTION : In this chapter the new approach of solving economic load

dispatch problem has been discussed. The basic parameters and how the swarm searches for the

food are clearly explained.

Particle swarm optimization (PSO)[4] is a population based stochastic optimization technique

developed by Dr.Ebehart and Dr.Kennedy in 1995,inspired by social behavior of bird flocking or

fish schooling. Pso shares many similarities with evolutionary computation techniques such as

Genetic Algorithms(GA).The system is initialized with a population of random solutions and

searches for optima by updating generations. However, unlike GA,PSO has no evolution

operators such as crossover and mutation In PSO, the potential solutions, called particles ,fly

through the problem space by following the current optimum particles. The detailed informationwill be given in following sections. Compared to GA the advantages of PSO are that PSO is easy

to implement and there are few parameters to adjust.PSO has been successfully applied in many

areas: function optimization, artificial neural network training, fuzzy system control, and other

areas where GA can be applied.

PSO simulates the behaviors of bird flocking. Suppose the following scenario: a group of

birds are randomly searching food in an area. There is only one piece of food in the area being

searched. All the birds do not know where the food is. But they know how far the food is in each

iteration. So whats the best strategy to find the food? The effective one is to follow the bird,

which is nearest to the food.PSO learned from the scenario and used it to solve the optimization

problems. In PSO, each single solution is a bird in the search space. We call it particle. All

of particles have fitness values, which are evaluated by the fitness function to be optimized, and

have velocities, which direct the flying of the particles. The particles fly through the problem

space by following the current optimum particles.

PSO is initialized with a group of random particles (solutions) and then searches for optima

by updating generations[3]. In every iteration, each particle is updated by following two best

values.

28


29/53

The first one is the best solution (fitness) it has achieved so far.(The fitness value is also stored.)

This value is called pbest. Another best value that is tracked by the particle swarm optimizeris the best value, obtained so far by any particle in the population. This best value is a global best

and called g-best. When a particle takes part of the population as its topological neighbors, the

Best value is a local best and is called p-best. After finding the two best values, the particle

updates its velocity and positions with following equation.

In the above equation,

The term rand()*(pbest i -Pi(u)) is called particle memory influence.

The term rand()*( gbesti -Pi(u)) is called swarm influence.

Vi(u) which is the velocity of ith particle at iterations u must lie in the range

Vmin Vi(u) Vmax

The parameter Vmax determines the resolution, or fitness, with which regions The are to

be searched between the present position and the target position.

If Vmax is too high, particles may fly past good solutions. If Vmin is too small, particles

may not explore sufficiently.

In many experiences with PSO, Vmax was often set at 10-20% of the dynamic range on

each dimension.

The constants C1 and C2 pull each particle towards pbest and gbest positions.

Low values allow particles to roam far from the target regions before being tugged back.

On the other hand, high values result in abrupt movement towards, or past, target regions.

The acceleration constant C1 and C2 are often set to be 2.0 according to past experiences.

29


30/53

Suitable selection of inertia weight provides a balance between global and local

explorations, thus requiring less iteration on average to find a sufficiently optimal

solution.

In general,the inertia weight w is set according to the following equation,

Where

w -is the inertia weighting factor

Wmax - maximum value of weighting factor

Wmin - minimum value of weighting factor

ITERmax - maximum number of iteration

ITER - current number of iteration

6.2 SOLVING ELD BY PSO METHOD

6.2.1 PROBLEM FORMULATION :

The Economic Load Dispatch (ELD) is generating adequate electricity to meet the continuously

varying consumer load demand at the least possible cost under a number of constraints.

Practically, while the scheduled combination of units at each specific period of operation arelisted, the ELD planning must perform the optimal generation dispatch among the operating units

to satisfy the load demand, spinning reserve capacity, and practical operation constraints of

generators.

The objective of the ELD problem is to minimize the total fuel cost.

Where

Ft = ai + biPi + ciPi2 $/hr ----------------(1)

The ELD problem is subjected to the following constraints,

30


31/53

The power balance equation

gi = Pd+ Pl ------------------- (2)

The total Transmission loss,

Pl= Pm Bmn Pn --------------------- (3)

In addition, power output of each generator has to fall with in the operation limits of the

Generator

Pgimin Pgi Pgimax for i=1,2n

Constraints Satisfaction Technique:

To satisfy the equality constraint of equation (3), a loading of any one of the units is selected

as the dependent loading Pd and its present value is replaced by the value calculated according to

the following equation,

Pd=Pd+Pl- i

Where Pd can be calculated directly from the equation (5.a) with the known power demand P D

and the known values of remaining loading of the generators. Therefore the dispatch solution

will always satisfy the power balance constraint provided that Pd also satisfies the operation limit

constraint as given in equation (5). An infeasible solution is omitted and above procedure is

repeated until Pd satisfies its operation limit. Because Pl also depends on Pd, we can substitute an

expression for Pl in terms of P1,P2Pd..Pn and Bmn coefficients. After substituting it in the

equation (5.a), separate the independent and dependent generator terms to obtain a quadratic

equation for Pd. Solving the quadratic equation for Pd, the power balance equality condition is

exactly satisfied.

31


32/53

6.2.2 ALGORITHM FOR SOLVING ELD USING PSO METHOD :

STEP 1: Initialize randomly the individuals of the population according to the limit of each

unit including individual dimensions, searching points, and velocities. These intial individuals

must be feasible candidate solutions that satisfy the practical operation constraints.

STEP 2: To each chromosome of the population the dependent unit output p d will be

calculated from the power balance equation and Bmn Coefficient matrix.

STEP 3: Calculate the evaluation on value of each individual P g1 , in the population using the

elevation function f given by

FT=i =1 Fi(Pi)

STEP 4: Compare each individuals evaluation value with its pbest. The best evaluation value

among the pbest is denoted as gbest.

STEP 5: Modify the member velocity v of each individual pg, according to

Where 1=1,2,n d=1,2,..m

STEP 6: Check the velocity components constraint occurring in the limits from the following

conditions,

If V id(u+1) > Vdmax then Vid(u+1)=Vdmax

If V id(u+1) > Vdmin then Vid(u+1)=Vdmin

Where

Vd min = -0.5 Pgmin

32


33/53

Vdmax= +0.5 Pgmax

STEP 7: Modify the member position of each individual pg according to

STEP 8: If the evaluation value of each individual is better than previous pbest, the current value

is set to be pbest. If the best pbest is better than gbest, the value is set to be gbest.

STEP 9: If the number of iterations reaches the maximum, then go to step10. Otherwise, go to

step2 .STEP 10: The individual that generates the latest gbest is the optimal generation power of each

unit with the minimum total generation cost.

33


34/53

6.2.3 FLOWCHART FOR SOLVING ELD PROBLEM USING PARTICLE

SWARM OPTIMIZATION TECHNIQUE

34

Start

Initialize the particles with random position

and velocity vectors

For each particle position (p)evaluate

If fitness (p) is better than fitness of

pbest then

Set best of pbest as gbest

Update particle velocity and

osition

If Gbest

is the

optimal

solution

solution

Stop


35/53

FIG 6.1: FLOWCHART FOR ELD USING PSO

CHAPTER 7

RESULTS AND DISCUSSIONS

ECONOMIC LOAD DISPATCH OF THREE UNITSYSTEM

ECONOMIC LOAD DISPATCH OF SIX UNIT SYSTEM

35


36/53

RESULTS AND DISCUSSIONS

Programs are coded for the developed algorithms in MATLAB language. Thedeveloped programs are used to solve the problem of ELD for a system with three generators and

a system with six generating units and the data is given in Appendix-I. Results obtained by using

Particle Swarm Optimization (PSO) are compared with conventional lambda iteration method

including the losses. Also, the programs developed for lambda iterative method and non-

iterative lambda method are tested on the systems mentioned above while neglecting losses. The

results obtained are tabulated below.

7.1 ECONOMIC LOAD DISPATCH OF THREE UNIT SYSTEM

7.1.1 LAMBDA ITERATIVE METHOD (WITH LOSSES)

In this method initial value of lambda is guessed in the feasible region that can be

calculated from derivative of cost function. For the convergence of the problem the delta lambda

should be selected small. Here delta lambda is selected 0.0001 and the value of lambda must be

chosen near the optimum point.

S.NO

Power

demand(MW)

P1(MW) P2(MW) P3(MW) Loss(MW) Cost(Rs/Hr)

Time(sec)

1 450 184.80 198.36 68.16 1.36 4665.1 10.52

2

585 241.29 255.30 90.78 2.33 5844.7 6.67

3 700 289.20 304.02 110.13 3.36 6872.2 7.06

4 800 330.70 364.59 127.03 4.41 7783.3 7.71

5 900 372.22 389.39 144.01 5.06 8711.8 5.07

36


37/53


38/53

7.1.3 COMPARISION OF RESULTS OBTAINED IN LAGRANGE

METHOD AND PSO METHOD ( WITH LOSSES)

It has been observed that when transmission line losses are included the minimum cost was

found in PSO method and the execution time is minimum for Lagrange method.

Method

Power

demand

(MW)

P1(MW) P2(MW) P3(MW) Loss(MW) Cost

(Rs/Hr)

Time

(sec)

PSO 450 204.71 188.59 58.06 1.37 4664.1 12.58

LAGRANGE 450 184.80 198.36 68.16 1.36 4665.1 10.52

TABLE 7.3 : RESULTS OBTAINED IN LAGRANGE METHOD AND PSO METHOD( WITH LOSSES) FOR Pd= 450

TABLE 7.4 : RESULTS OBTAINED IN LAGRANGE METHOD AND PSO METHOD( WITH LOSSES) FOR Pd= 900

38

Method

Power

demand

(MW)

P1(MW)

P2(MW)

P3(MW)

Loss

(MW)

Cost

(Rs/Hr)

Time

(sec)

PSO

900 416.70 365.90 122.90 5.64 8705.8 6.30

LAGRANGE900 372.22 389.39 144.01 5.06 8711.8 5.07


39/53

7.2 ECONOMIC LOAD DISPATCH OF SIX UNIT SYSTEM

7.2.1 LAMBDA ITETARIVE METHOD (WITH LOSSES)

The initial value of lambda is guessed in the feasible region that can be calculated

from the derivative of cost function. For the convergence of the problem must be selected

small. The convergence is largely affected by the selection of lambda value. The time taken for

convergence increases than the three unit system. It is also observed that the time taken for

convergence of six unit with losses case is more than without losses case.

Power

demand

(MW)

P1

(MW)

P2

(MW)

P3

(MW)

P4

(MW)

P5

(MW)

P6

(MW)

Loss

(MW)

Cost

(Rs/Hr)

Time

(sec)

600 23.86 10 95.62 100.69 202.8 181.17 14.23 32132.1 6.84

700 28.30 10 118.96 118.68 230.76 212.75 19.43 36914.1 6.74

800 32.11 14.22 141.60 136.09 257.72

243.09

25.33 41927.1

9.90

850 34.74 17.45 152.78 144.67 270.97 257.96 28.56 44452.1 6.74

950 39.03 23.97 175.30 161.95 297.57 287.77 35.64 49683.1 9.9

TABLE 7.5: RESULTS OBTAINED IN LAGRANGE MEHTOD ( WITH LOSSES)

39


40/53


41/53

For Power Demand of 600MW

Method P1

(MW)

P2

(MW)

P3

(MW)

P4

(MW)

P5

(MW)

P6

(MW)

Loss

(MW)

Cost

(Rs/Hr)

Time

(sec)

PSO 23.80 10 95.70 100.00 202.6 181.2 14.24 32091.6

8

7.30

LAGRANGE 23.86 10 95.62 100.69 202.8 181.17 14.23 32132.1 6.84

TABLE 7.7: RESULTS OBTAINED IN LAGRANGE METHOD AND PSO METHOD( WITH LOSSES) FOR Pd= 600

For Power Demand of 950MW

Method P1

(MW)

P2

(MW)

P3

(MW)

P4

(MW)

P5

(MW)

P6

(MW)

Loss

(MW)

Cost

(Rs/Hr)

Time

(sec)

PSO 39.05

24.40 191.80

172.56

294.50

262.40

34.90

49681.38

9.40

LAGRANGE 39.03 23.97 175.30 161.95 297.57 287.77 35.64 49683.10 9.9

TABLE 7.8: RESULTS OBTAINED IN LAGRANGE METHOD AND PSO METHOD

( WITH LOSSES) FOR Pd= 950

41


42/53

7.3 ECONOMIC LOAD DISPATCH OF THREE UNIT SYSTEM

7.3.1 LAMBDA ITERATIVE METHOD (NEGLECTING LOSSES)

The initial value of lambda is guessed in the feasible region that can be calculated

from the derivative of cost function. For the convergence of the problem must be selected

small. The convergence is largely affected by the selection of lambda value.

S.NO

Power

demand

(MW)

P1(MW) P2(MW) P3(MW)

Cost

(Rs/Hr)

Time

(sec)

1 600 418.16 138.54 50.00 5692.0 0.1880

2

800 516.83 210.77 72.38 7341.5 0.1920

3 950 590.85 264.95 94.00 8657.4 0.2030

TABLE 7.9: RESULTS OBTAINED IN LAGRANGE MEHTOD ( NEGLECTINGLOSSES)

7.3.2 NON-ITERATIVE LAMBDA BASED METHOD

This is a direct or non iterative method it does not demand any initial guess value of for ED of

units for a given P demand . many of the existing conventional methods fail to impose P-limits at

voilating units. Improper selection of initial value of may cause slow convergence or at atimes

leads to divergence for conventional algorithms

42


43/53

S.NO

Power

demand

(MW)

P1(MW) P2(MW) P3(MW) Cost

(Rs/Hr)

Time

(sec)

1 600 424.25 132.70 50.00 5701.0 0.1720

2

800 522.94 204.94 72.11 7351.3 0.1870

3 950 596.96 259.11 93.91 8669.8 0.1560

TABLE 7.10 : RESULTS OBTAINED IN NON ITERATIVE MEHTOD

( NEGLECTING LOSSES)


METHOD AND NON ITERATIVE METHOD ( NEGLECTINGLOSSES)

It has been observed that when transmission line losses are not included the minimum cost

was found in lambda based method when compared to non-iterative lambda based method but

the time of execution is minimum for non iterative method.

43

Method

Power

demand

(MW)

P1(MW) P2(MW) P3(MW) Cost

(Rs/Hr)

Time

(sec)

LAGRANGE 600 418.16 138.54 50.00 5692.0 0.1880

NONITERATIVE

600 424.25 132.70 50.00 5701.0 0.1720


44/53


45/53

TABLE 7.13 : RESULTS OBTAINED IN LAGRANGE MEHTOD( NEGLECTING LOSSES)

7.4.2 NON-ITERATIVE LAMBDA BASED METHOD(NEGLECTING LOSSES)

This is a direct or non iterative method it does not demand any initial guess value of for ED of

units for a given P demand . many of the existing conventional methods fail to impose P-limits at

voilating units. Improper selection of initial value of may cause slow convergence or at atimes

leads to divergence for conventional algorithms

TABLE 7.14 : RESULTS OBTAINED IN NON ITERATIVE MEHTOD( NEGLECTING LOSSES)


METHOD AND NON ITERATIVE METHOD ( NEGLECTINGLOSSES)

It has been observed that when transmission line losses are not included the minimum cost was

found in lambda based method when compared to non-iterative lambda based method but the

time of execution is minimum for non iterative method.

45

Power

demand

(MW)

P1(MW) P2(MW) P3(MW) P4(MW) P5(MW) P6(MW) Cost

(Rs/Hr)

Time

(sec)

550 103.46 173.49 74.57 62.97 130 125 8171.2 0.1720

600 114.18 191.46 82.30 69.49 130 125 8878.6 0.1870

700 135.62 227.41 97.75 82.53 130 125 10503.2 0.1410


46/53

Method Powerdemand

(MW)

P1

(MW)

P2

(MW)

P3

(MW)

P4

(MW)

P5

(MW)

P6

(MW)

Cost

(Rs/Hr)

Time

(sec)

LAGRANGE

550 109.56 150 78.96 66.67 130 125 8056.7 0.1870

NONITERATIVE

550 103.46 173.49 74.57 62.97 130 125 8171.2 0.1720

TABLE 7.15 : RESULTS OBTAINED IN LAGRANGE METHOD AND NON ITERATIVEMETHOD (NEGLECTING LOSSES) FOR Pd= 550

Method Powerdemand

(MW)

P1

(MW)

P2

(MW)

P3

(MW)

P4

(MW)

P5

(MW)

P6

(MW)

Cost

(Rs/Hr)

Time

(sec)

LAGRANGE 700 125.00 150 100.49 84.85 130 125 9079.5 0.2030

NON

ITERATIVE700 135.62 227.41 97.75 82.53 130 125 10503.2 0.1410

TABLE 7.16 : RESULTS OBTAINED IN LAGRANGE METHOD AND NON ITERATIVEMETHOD (NEGLECTING LOSSES) FOR Pd=700

46


47/53

CHAPTER 8

47


48/53

CONCLUSION

SCOPE OF FUTURE WORK

CONCLUSION

In both study cases of three generating and six generating systems considering losses, PSO

method gives the better cost than Lambda iterative method. In Lambda iterative method the

number of iterations to converge increases. But in PSO method the no of iterations are not

affected. In PSO method the selection of parameters c1,c2 and w are very important in the

convergence of the system. And for the case of neglecting losses though Lambda iterativemethod gives the best result but the non-iterative method provides the result in less execution

time.

8.1 SCOPE OF FUTURE WORK

Here the loss coefficients are prescribed in the problem the work may be extended for the

problem when transmission loss coefficients are not given. In that case the loss coefficients can

be calculated by solving the load flow problem.In PSO method selection of parameters are important. So, the parameters may be optimized by

using ANN method. Any other method can be applied with PSO to improve the performance.

48


49/53

REFERENCES/ BIBILOGRAPHY

[1] D.P.Kothari , J.S.Dhillon Power System Optimization Published by Prentice

Hall of India Pvt.Ltd , NewDelhi.

[2] D.P.Kothari , I.J.Nagrath Modern Power System Analysis Published by Mc.Graw-Hill

[3] Dr.Maheswarapu Sydulu A Very Fast And Effective Non-Iterative -Logic

Based Algorithm For Economic Dispatch Of Thermal Units 1999 IEEE TENCON

[4] M.Sudhakaran , P.Ajay-D-Vimal Raj and T.G.Palanivelu Application Of Particle Swarm

Optimization For Economic Load Dispatch Problems

[5] S.S.Rao Power System Optimization

[6] D.C.Waltersw and G.B.Scheble , Genetic Algorithm Solution of Economic Dispatch with

valve point loading, IEEE Trans. Power system

[7] J.Tippayachai, W.Ongsakul and I.Ngamroo, Parallel micro genetic algorithm for

constrained economic dispatch IEEE Trans Power system.

[8] N.Sinha, R.Chakrabarthi and P.K.Chattopadhyay, Evolutionary programming techniquesfor economic load dispatch IEEE Evol. Comput.,7(February(1))( 2003).

[9] H.T.Yang, P.C.Yang and C.L.Huang, evolutionary programming based economic dispatch

For units with nonsmooth fuel cost functions,IEEE trans.power system.

[10] A.J.Wood and B.F.Wollenberg, Power Generation, Operation, and Control (2nd ed.),

Wiley, New York(1996).

[11] W.M.Lin,F.S.Cheng and M.T.Tsay, An improved Tabu search for economic dispatch with

multiple minima, IEEE Trans , Power systems[12] A.J. Wood and B.F. Wollenberg, Power Generation, Operation, and

Control, John Wiley

and Sons., New York (1984).

[13] P. Aravindhababu and K.R. Nayar, Economic dispatch based on optimal

lambda using

49


50/53

radial basis function network, Elect. Power Energy Syst,. 24 (2002), pp.

551556.

[14] IEEE Committee Report, Present practices in the economic operation

of power systems,

IEEE Trans. Power Appa. Syst., PAS-90 (1971) 17681775.

APPENDIX- I

CASE STUDY-1: THREE UNIT SYSTEM

The three generation units considered are having different characteristics. Their cost function

characteristics are given by the following equations

F1=0.00156P12+7.92P1+561 Rs/Hr

F2=0.00194P22+7.85P2+310 Rs/Hr

F3=0.00482P32+7.97P3+78 Rs/Hr

According to the constraints considered in this work among inequality constraints only active

power constraints are considered. Their operating limits of maximum and minimum power are

also different. The unit operating ranges are

100MWP1600MW

100MWP2400MW

50MWP3200MW

The transmission line losses can be calculated by knowing the loss coefficient. The B mn loss

coefficient matrix is given by

Bmn = 0.000075 0.000005 0.0000075

0.001940 0.000015 0.0000100

50


51/53

0.004820 0.000100 0.0000450

CASE STUDY-2: SIX UNIT SYSTEM

The cost function of six units are given as follows

F1= 0.15240P12+38.53P1+756.79886 Rs/Hr F2= 0.10587P22+46.15916P2+451.32513 Rs/Hr

F3= 0.02803P32+40.39655P3+1049.9977 Rs/HrF4= 0.03546P42+38.30553P4+1243.5311 Rs/HrF5= 0.02111P52+36.3278P5+1658.5596 Rs/HrF6= 0.01799P62+38.27041P6+1356.6592 Rs/Hr

The unit operating ranges are

10 MW P1 125 MW10 MW P2 150 MW35 MW P3 225 MW35 MW P4 210 MW130 MW P5 325 MW125 MW P6 315 MW

Bmn = 0.000022 0.000020 0.000019 0.000025 0.000032 0.0000850.000026 0.000015 0.000024 0.000030 0.000069 0.0000320.000019 0.000016 0.000017 0.000071 0.000030 0.0000250.000015 0.000013 0.000065 0.000017 0.000024 0.0000190.000017 0.000060 0.000013 0.000016 0.000015 0.0000200.000140 0.000017 0.000015 0.000019 0.000026 0.000022

51


52/53

CASE STUDY-3: THREE UNIT SYSTEM

( NON- ITERATIVE LAMBDA BASED METHOD)

The cost function of three units are

F1=0.00142P12+7.20P1+510 Rs/Hr

F2=0.00194P22+7.85P2+310 Rs/Hr

F3=0.00482P32+7.97P3+78 Rs/Hr

According to the constraints considered in this work among inequality constraints only

active power constraints are considered. Their operating limits of maximum and minimum power

are also different. The unit operating ranges are

150MWP1600MW

100MWP2400MW

50MWP3200MW

52


53/53

economic load dispatch using pso methode

Documents