Optimal Power Flow Modeling


The optimal power flow, in general, is expressed as a nonlinear static optimization problem, the cost function (Criterion) has various forms as follows:


1-Economic cost, where the cost function is C





Where bus 1 is the slack bus and buses 2, 3, ….., ng are ng generation buses. X = vector of state variables (normally all bus angles except the slack bus angle and voltage magnitudes of load buses). U = vector of control variables, (active generation power except the slack generation and voltage of all generators). Here the slack power generation is used as output variable.

2-Load Shedding Criterion, (the cost function is defied as "c")



where:  



is the given load at bus i before load shedding.


 is the control value of the load at bus i.


NL is the number of load buses and Wi are assigned weights to different load buses, the criterion is used if the loads can not be met.


3-Pollution Criterion

                                                                 
Where Pi(PGi) shows the level of pollution of generation i as a function of generation level.
The above functions eq's (5-3)are minimized under set of equality and inequality constraints such as:

a) The Equality Constraints

The equality constraints of the problem can be expressed as the load flow equations plus slack generation as:


The number of equations is equal to the number of system state variables.

    b) The Inequality Constraints

 1.Real generation constraints for all generators 

 2.Voltage magnitude constraints for all generators plus buses controlled by other control devices.


  0.9 < Vi < 1.1                       ( Normal mode )        
        0.95 < Vi < 1.05                   ( High quality mode )
         0.98 < Vi < 1.08                  ( Very High quality mode )
3. Reactive generation constraints for all generation buses and other controlled buses.


4. Security constraints on lines flows for all or specified lines.


The optimal power flow is solved as constrained (all constraints are considered) or unconstrained (where the inequality constraints are ignored) problem. The mathematical form of unconstrained problem is:


The necessary optimality conditions are given the Lagrange


 The suggested iterative scheme to solve the eq's (5-13)is:

  
A convergence is an curtained, the optimum solution and optimal conditions are attained. Every iteration needs a load flow solution, this may lead to a very computation time consuming. To over come this problem and reduce the computation time, a matrix of second partial derivation, Hussein matrix algorithm, is used.
Here, we define the vector Z as:

  
As a result, the Lagrange is a function of Z, i.e.,


The necessary optimality conditions become simply


Defining the Hussein matrix H(Z) of partial derivations as:


Then the Newton-Raphson iterative procedure becomes


By choosing a good initial vector Z, this approach should converge very quickly. For a large system, the matrix H is sparse. Hence, sparse matrix methods are applicable. Since Z is >> 2.5 times larger than the vector X of state variables, solution times for this OPF approach should be two to three times greater than the corresponding load flow problem.

OPF Algorithm's Flow Chart :  





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Ahmad Mohammad

Hi. I’m Designer of Mawsoo3a Blog. I’m Electrical Engineer And Blogger Specializing In Electrical Engineering Topics. I’m Creative.I’m Working Now As Maintenance Head Section In An Industrial Company.

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