Rushdi Research Paper

7/29/2019 Rushdi Research Paper

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CONSTRANED OPTIMUM ECONOMIC DISPATCH OF ELECTRIC POWER SYSTEM

USING LINEAR PROGRAMMING TECHNIQUE

AND

IT'S APPLICATON IN WAPDA POWER SYSTEM OPERATIONAL PLANNING

ENGR. NAVED RUSHDI TAHIR NADEEM MALIK DR.AFTAB AHMAD Phd

M.Sc. ENGINEERING ASSOCIATE PROFESSOR DEPUTY DIRECTOR

ABSTRACT

Power system operational planning, its scope and

procedures have been reviewed in general. Present

practice for the operational planning i.e. economic

dispatch using classical techniques based on

unconstrained optimization and its application in

integrated operation of electric network have beendiscussed. The reasons that these classical

approaches may "fall through the crack" is obviousi.e. financially fragmented ownership structure.

Which definitely guides us to propose an alternate

strategy for operational planning procedures keeping

in view the new directions in power system

competitive operation and recent trends in

operational planning of the electric power industry.

To pursue the universally acceptable objective of

cost minimization, this research work is a break

through with regard to application of operations

research technique known as linear programming (inshort "L.P") for the practical solution to the optimum

economic dispatch problem under constrained

environment. This research work dates back to 1940

form Krons theory to Kuhn-Tucker multipliers for

the solution of constrained non-linear economic

dispatch problem under the umbrella of classical

approach, and the associated complications/

difficulties in applications and low level of itsvalidity in newly emerging competitive environment

leads us through the greatest achievement of Dentzig

in establishing linear programming technique.

INTRODUCTION

The Electric power system engineer is faced with the

challenging task of planning and successfullyoperating one of the most complex systems oftoday's civilizations. The efficient planning and

optimum economic operation of power system has

always occupied an important position in Electric

power industry. The operational planning of the

power system involves the best utilization of

available energy resources subject to various

practical constraints to transfer electrical energy

from far furlong generating stations to the load

centers and scattered consumers with maximum

safety of personal/equipment without interruptionand satisfactory quality of supply (i.e. close to

nominal frequency and declared voltage profile) at

minimum cost.

In modern complex and highly integrated power

systems, the operational planning involves many

steps such as load forecasting, unit commitment,scheduling of active/ reactive generation,

maintaining system frequency and voltage profileclose to declared level, maintaining spinning reserve

and cold reserve for stability issues, interchange

scheduling to neighboring utilities, etc.

The real world problem of optimum economy and

profit maximization under constrained environment

can best be addressed and practically implement

able as guided by the most popular technique of

operations research, namely mathematical linear

programming technique. Generally it was thought

that highly structured and systematic approach oflinear programming couldnt be applied to optimum

economic dispatch problem and especially in

constrained environment where the subject touches

the most complex boundaries. Little mention of

applications and validity for economic dispatch

optimization using linear programming in electric

power industry is found in the literature.

This research work is aimed with this challenge. Theextensive research has led to the development

whereby operations research technique can be

applied most effectively to optimum economic

dispatch problem. The objective function of

optimization problem can be extended easily to n-

dimensions and can handle thousands of applied

constraints both equality and inequality typeimplicitly. Initially a two-dimensional proto typemathematical model has been formed and algorithm

is formulated in Two-Phase method, coded in

FORTRAN-77, implemented in DOS environment

(P.C version). Results are exactly in confirmation

with by-hand calculations by the two different

approaches to the solution of linear programming

i.e. by corner solution method and the Big-M

method. Extended and tested for IEEE standard

formats for 5-Bus, 14-Bus, 30-Bus and finally for


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54-Bus set-up. Significant results are obtained.

WAPDA power system is represented by 54-Bus

system (in 54 dimensions) with both equality andinequality constraints and optimum economic

dispatch solution determined provides additional

features include optimum Hydro-Thermalcoordination, capability to handle transmission

system constraints implicitly. Significant results

represent a saving worth billion of dollars and easy

access to the source code.

Completion of this research work employs

indigenous resources and contributes to alleviate

future problems in economic operation of the powersystem. This research work definitely provides very

much insight to many of the researchers struggling

to enhance this area for the well being of the

humanity and bridges the inherent gap between

financial aspects and engineering jargons, which are

apparently two divergent fields and are usually nottalked about jointly.

Linear programming is the well established, mosteffective and highly efficient mathematical

programming technique from the field of operations

research seeking global optima pertains to linear zed

objective functions under the constraint environment

for the solution to real world problems. Its

application ranges from defense industry to Govt

sector, public sector, business, refinery, chemical,

transportation (including land, sea, air, and even

space) etc.

LINEAR PROGRAMMING AND OPTIMUM

ECONOMIC DISPATCH

PROBLEM

At times, the objectives of an organization are stated

in an unqualified way such as maximization of

profits or minimization of costs. In may more cases,

however, organizations attempt to pursue objectives

within a set of limitations properly termed asconstraints. For example, an organization may

attempt to maximize profits within a given available

pool of resources (Labour or machinery). Similarly,

organizations may attempt to minimize costs while

meeting various internally or externally imposed

quality specifications. Thus, organizational

objectives often are pursued within a constrainedenvironment. This reflects the reality of resourcescarcity and societal demands.

The particular objective of an organization can be

expressed in terms of a mathematical model in

equation form. Constraints on this objective also

may be stated mathematically. Often the most

realistic description of constraints is through the use

of in-equality statements. Examples of constraints

stated as in-equalities are that organization must use

no more than specified labor-hours per week for

production of a particular good or that the

government has stated that a particular drug must

contain at least five milligrams of folic acid pertablet.

Mathematical programming includes techniques for

evaluating problems of optimization underconstraints. When the objective and constraints are

expressed in linear form, a type of mathematical

programming called linear-programming is applied

to the problem. So first, we should discuss linear

programming and then various methods of its

solution and finally its application to situation of

constrained optimization (optimum active powerdispatch problem in this case).

LINEAR PROGRAMMING.

A linear programming [11], [25] problem is a

mathematical problem in which the objective

function and the constraints including equality andinequality constraints are linear function of the

unknowns (that is function of x raised to the powers1 and 0. Mathematically, it can be stated as:

Minimize f = Cj xj : j = 1,2,3,.......n variables.

(Objective)

Subject to: aij xj ( , = , or )bi ,: i = 1,2,3... m

constraints

(Secondary Constraints)

and xj 0 : j = 1,2,3,.......n

(Non-Negativity or primary constraints)

where Cj, aij , bi are known constants (called

coefficients) for all i and j.

And xj are non-negative variables called non-

negativity constraints, or primary constraints, whichrestricts the problem in first quadrant only.

The constraints of the problem can be converted into

equations by adding a (non-negative) slack variable

xn+1 if the ith in-equality is of the type (, and

subtracting a (non-negative) surplus variable xn+k if

the kth in-equality is of the type . Assume that

Augmentation of the slack and surplus variables willresult in a total of p variables, the problem can be

written in matrix form as:

Minimize: f = [C] [x]

Subject to [A][x] = [b]

and [x] 0

Eq-1

Where

[C] is a "p" dimensional row vector[A] is an m * p matrix represents coefficients

[b] is an m-dimensional column vector


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and

[x] is a "P" dimensional column vector represents

variables

Note that the elements of [C] corresponding to the

slack and surplus variables are all zero.

According to fundamental theorem of linear

programming, the optimal solution of a linear

programming problem, if finite (i.e. possible as

opposed to infinite), is always obtained at one of the

basic feasible solution (i.e. corner point). A linear

programming algorithm then improves the objectivefunction in successive iterations from one basic

feasible solution point to an adjoining one, until the

optimum solution is reached.

At each iteration , one previously non-basic variable

becomes basic in exchange with one of the basic

variables, which then becomes non-basic.This is called variable exchange, usually "Pivot

operations'" and is the main linear programmingmechanism.

The way in which the exchanges are handled in

various problem formulations are the important

characterizing features of the different solution

methods of linear

programming , based on Gauss-Jordan elimination

technique employing upper-triangularization and

back substitution method to reach the most

appropriate /feasible optimum solution . the advent

of the algorithmic iterative technique for computerbased solution further seeks the optima within the

lapse of few seconds with reasonable accuracy and

can handle the problem comprising hundreds of

dimensions of objective function under thousands of

constraints. which includes ;

Simplex method

Revised Simplex method

Big-M methodTwo phase method

Primal-dual method

Minimization by way of -Z maximization.

To establish the basis, the idea came about fromtransformation in primal-dual technique of L.P,

where by primal problem of maximization can be

transformed to duel problem of minimization andvice versa , in order to opt for easy solution

methodology for maximization problem.

The application to electric power engineering

industry involves transformation of L.P from real

life domain to augmented domain similar to Fourier

or Laplace transforms, where by in Laplace

transform the problem of simultaneous solution ofdifferential equations from "t" (time) domain to

"s"(secondary) domain, where the problem

simplifies to solution of simultaneous ordinary

algebric equations and after having the solution in

"s" domain the solution is re-transformed back to "t"

domain for realization . The transformation toaugmented domain is very simple using only

arithmetic subtraction of lower bound in case ofupper bound problem and vice versa. The

augmented optimization problem can be treated as

auxiliary problem to the original problem and

solution of the auxiliary problem is a transformed

version of the original problem. The transformed

optimum results can easily be re-transformed back

to real life domain merely by a reverse process of

arithmetic addition of requisite bounds .The

augmentation process of transformation carries

along with it additional benefits like machine upperand lower bounds once incorporated at problem

formulation stage are dealt automatically i.e.

implicitly and need not be bothered again at

algorithmic or coding stage. On contrary these limits

are difficult to handle in non-linear optimization and

needs additional "u" multipliers (Kuhn-Tucker

multipliers) and algorithm should be designed to

look after these inequality constraints.


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REFERENCES.

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