Optimal Power Flow Report

8/12/2019 Optimal Power Flow Report

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Sasmita Tripathy

M.Tech 1stYear

12040036(PSE)


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OUTLINE

Introduction

OPF Formulation

Different Methodologies of OPF solution

Applications

Conclusion

References


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INTRODUCTION

The concept of OPF was first introduced by Carpentier in 1962 as

an extension of the conventional ELD problem.

An OPF problem schedules the power system controls to optimize

the specific objective function while satisfying the physical and

operational constraints of the electric network.

Basically an OPF is a static, constrained, non-linear optimization

problem.

Optimal power flow analysis is highly important as it helps provide

accurate calculations in the future expansion of power systems. Italso helps in determining the best type of operation of existing

systems.


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OPF FORMULATION

o Objective Function:o The most common OPF objective is the minimization of

generation costs, with or without consideration of systemlosses.

o Variables:o Control Variables

o Dependent Variables

o Constraints:o Equality Constraints

o Inequality Constraints


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OPF FORMULATION(Contd)

OPF is formulated mathematically as a general constrained optimizationproblem.

Minimize f(u,x)

Subject to g(u,x)=0 (1)

h(u,x) 0 (2)

The objective function f(u,x) represents the systems goal which is a scalar

function. Vector functions g(u,x) and h(u,x) are the equality and inequality

constraints respectively.

Where u=the set of controllable quantities in the system

x=the set of dependent variables


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DIFFERENT METHODOLOGIES

The optimization techniques are mainly of two types:Conventional Methods

1)Nonlinear Programming Method

2)Quadratic Programming Method

3)Newtons Method

4)Linear Programming Method

5)Interior Point Method

Artificial Intelligence Methods

1)Genetic Algorithm Method

2)Particle Swarm Optimization Method


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NLP Method

This is the earliest formulation category. In 1962, Carpentier first introduced

a generalized nonlinear programming formulation of the economic

dispatch problem, including voltage and other constraints.

The optimality conditions using Kuhn-Tucker method can be derived by

formulating the Lagrange function (L):

The conditions for an optimum for the points

1)

2)

3)

4)

0),,(000

x

x

L

0)( 0

xg

0)( 0

xh

0

0)(

0

00

xh


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Merits and Demerits

Merits:

o Capture system behavior accurately.

o Formulation is simple.

Demerits:

o The sufficiency conditions are difficult to verify.

o Slow in Convergence.o Difficult to solve in presence of inequality

constraints.


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QP Method

Quadratic programming method is a special form of nonlinear

programming whose objective function is quadratic with linear constraints.In 1973, Reid and Hasdorf presented a quadratic programming method to

solve QP problems.

Step1: Initial solution Hessian approximation and penalty parameter r >

0 are set up. and k = 0.

Step2: Quadratic programming problem is defined by objective function andconstrained condition.

Minimize

Subjects to:

Where

x0

B0

dxddxf kkkT

kk

T

Lk

),(2

)(

0)(

)( dxcxc kT

ki

ki

0)( )( dxcxc kT

ki ki

xxd kkk

1


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QP Method (Contd)

Step6: When convergence condition is fulfilled, this calculation is finished.

Otherwise k=k+1 and returns to step 2.

Merits:o In this QP method, convergence is very fast.

o The method can solve both the load flow and economic dispatch problems.

o The accuracy of QP method is much higher compared to other established

methods.

Demerits:o Difficulties in obtaining solution of quadratic programming in large

dimension of approximating QP problems.o Complexity and reliability of quadratic programming algorithms.


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NEWTONs Method

To speed up the convergence, in 1974, M.H. Rashed presented a method using

Lagrange multiplier and Newtons method, the method also introduced an

acceleration factor to compute the update controls.

Step1: Make initial guesses of vector and which inequality

constraints to enforce.

Step2: Create the Lagrangian given active inequality constraints.

Step3: Calculate the Hessian and gradient of the Lagrangian.

Step4: Solve the equation,

Step5: Calculate new z ,


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NEWTONs Method(Contd)

Step6: Check tolerance , , if not then go to step 3.

Step7: If correct inequalities are enforced, then problem completed, else go

to step2 by determining new set of inequalities to enforce using Lagrange

multipliers.

Merits:

o The method has the ability to converge fast.

o It can handle inequality constraints very well.

Demerits:

o It is not possible to develop practical OPF programs without employing

sparsity techniques.

o Sensitive to the initial conditions and they may even fail to converge due

to inappropriate initial conditions.


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LP Method

Linear Programming method treats problems having constraints and objective

functions formulated in linear form with non negative variables. In 1968,

Wells developed a linear programming approach to determine an economical

schedule. The cost objective and its constraints were linearized and solved

using the simplex method.

Step1: Initial power conditions.

Step2: Solve power flow equations(DC or AC power flow).

Step3: Create linear objective function.

Step4: Obtain linearized constraint sensitivity coefficients.

Step5: Set up and solve LP for new control variable settings

Step6: Check for convergence test, if not then goto step1 by adjusting one or more

control variables else stop the problem.


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LP Method(Contd)

Merits:o The LP method easily handles Non linearity constraints.

o The LP solution is completely reliable.

o It has the ability to detect infeasible solution.

o The LP solution can be very fast.

o Complete computational reliability and very high speed enables it, suitablefor real time or steady mode purposes.

Demerits:o It suffers lack of accuracy.


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IP MethodKarmarkar proposed a new method in 1984 for solving large-scale linear

programming problems very efficiently. It is known as an interior methodsince it finds improved search directions strictly in the interior of thefeasible space.

The barrier function is in the form;

After introducing the barrier function, we can write the modified OPFformulation:

The Lagrange function for this problem is:


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IP Method(Contd)

The KKT conditions are

These nonlinear equations are then solved iteratively by

Newtons method, and the value is adjusted toward zero.


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IP Method(Contd)

Merits:

o The Interior Point Method can solve a large scale linear

programming problem by moving through the interior, rather

than the boundary as in the simplex method, of the feasible

region to find an optimal solution.

o The Interior Point Method is preferably adapted to OPF due

to its reliability, speed and accuracy.

o IP provides user interaction in the selection of constraints.

Demerits:o Limitation due to starting and terminating conditions.

o Infeasible solution if step size is chosen improperly.


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GA Method

GAs search within a population of points, not a single point. Therefore

GAs can provide a globally optimal solution. GAs use only objective

function information, not derivatives or other auxiliary knowledge.

Step-by-step algorithm for GA Method:

Step1: Read the database.Step2:Assume suitably population size(pop_size) , maximum no. of

generations or population(gen_max).

Step 3: Set valid number of population counter. Pop_vn=0.

Step 4: Randomly generate the chromosomes.

Step 5: Run the power flow using the Newton-Raphson method.

Step 6: Check all inequality constraints,if any of the limits is violated, go to

step 4.


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GA Method(Contd)

Step 7: If all the constraints are satisfied, increment pop_vn by 1. If pop_vn

less than than or equal to pop_size, go to step 4, otherwise go to next step.

Step8: Calculate and store the total cost of generation corresponding to each

valid generation pattern of chromosome.

Step 9: Find and store minimum cost among all valid indivisual parents and

corresponding generation pattern.

Step 10: Check if random no. (crossover rate) for i=1 to pop_size. Select the

ith chromosome. Apply the crossover operator to that indivisual.

Step 11: Run power flow using Newton-Raphson method and Check system

constrains as in step-6.


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GA Method(Contd)

Step 12: If all the constraints are satisfied, the individual of the newpopulation becomes valid otherwise it becomes invalid.

Step 13: Apply the mutation operator to the calculated generation patterns.

Step 14: Run power flow using Newton-Raphson and check all the constraints.

Step 15: If all the constraints are satisfied go to next step otherwise go to step

4.

Step 17: Calculate the total cost of all valid patterns.

Step 18: Find the optimum solution of among all.


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GA Method(Contd)

Merits:

o GAs can handle the Integer or discrete variables.

o GAs has the potential to find solutions in many different areas of the

search space simultaneously there by multiple objectives can be achieved

in single run.

o GAs are adaptable to change, ability to generate large number of solutions

and rapid convergence.

o GAs can be easily coded to work on parallel computers.

Demerits:

o The execution time and the quality of the solution, deteriorate with the

increase of the chromosome length, i.e., the OPF problem size.o If the size of the power system is increasing, the GA approach can

produce more in feasible springs which may lead to wastage of

computational efforts.


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PSO MethodParticle swarm optimization (PSO) is a population based stochastic

optimization technique inspired by social behavior of bird flocking or fish

schooling. In PSO, the search for an optimal solution is conducted using a

population of particles, each of which represents a candidate solution to

the optimization problem.

Step 1: Each member is called a particle, and each particle (i-th particle) is

represented by d-dimensional vector and described as

Step 2: The set of n particles in the swarm are called population and described

as pop=

Step 3: The best previous position for each particle (the positions giving the

best fitness value) is called particle best and described as


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PSO Method(Contd)

Step 4: The best position among all of the particle best position

achieved so far is called global best and described as

Step 5: The rate of position change for each particle is called theparticle velocity and it is described as

Step 6: At iteration k the velocity for d-dimension of ith particle isupdated by:

Step 7: Check the optimum case, if not then update individuals and goto step 2.


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PSO Method(Contd)Merits:

o These methods are simple concept, easy implementation, relativerobustness to control parameters and computational efficiency.

o The prominent merit of PSO is its fast convergence speed.

o PSO algorithm can be realized simply for less parameter adjusting.

o PSO has the flexibility to control the balance between the global and local

exploration of the search space.

Demerits:o Real coding of these variables represents limitation of PSO methods as

simple round-off calculations may lead to significant errors.

o Slow convergence in refined search stage (weak local search ability).


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APPLICATIONS

Security Constrained Economic Dispatch

Preventive and Corrective Rescheduling

Reactive Power Planning and Voltage Control

Power Wheeling and Wheeling Loss calculation

Pricing of Real and Reactive Power


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CONCLUSION

In general, key developments in the OPF formulation

have been accompanied by developments in solution

techniques. Moreover, the application of new solution

techniques has paralleled increases in computing ability

such that recent OPF algorithms are much more

computationally intensive than the original gradient

methods.


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REFERENCES

1. J. A. Momoh, M. E. El-Hawary and R. Adapa, "A review of selected optimal

power flow literature to 1993. Part-I: Non-linear and quadratic programming

approaches", IEEE Transactions on Power Systems, Feb.'99, pp. 96-104.

2. J. A. Momoh, M. E. El-Hawary and R. Adapa, "A review of selected optimal

power low literature to 1993. Part-II: Newton, linear programming and interior

point methods",IEEE Transactions on Power Systems, Feb.'99, pp. 105-111.

3. M. Huneault and F. D. Galiana, "A survey of the optimal power flow literature", IIEEE Transactions on Power Systems, May '91, pp. 762-770.

4. J. Carpentier, "Contribution e I'etude do Dispatching Economique", Bulletin

Society Francaise Electriciens, August 1962.

5. A. J. Wood and B. F. Wollenberg,Power Generation. Operation and Control, 2nd

Edition, John Wiley and Sons, Inc. 1996.

6. H. W. Dommel and W. F. Tinney, "Optimal power flow solutions",

IEEE Transactions on Power Apparatus and Systems, October 1968, pp.1866-

1876.


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