Treatment Of Inequality Constraints

The constrained minimization problem is solved by transforming it not an unconstrained minimization problem. There are many procedures to transforming:

1-Kuhn-Tucker Method

Here , the system model can be written as:

Where:

C is the cost function to be minimized.

      F is equality constraints

are dual variables

There are computation problem due to the existence of inequality constraints an x, it's usually difficulty to choose successive value of u that produces smaller and smaller values of c without violating the inequality constraint on x and hence it's difficult to foresee what effect a given change in u will have on x.

The necessary conditions that the optimal value of u most satisfy are the following eqs

 (6-2): 

Using eqs. (6-2) for optimal is hardly straight forward since it's not usually known which inequality constraints  are included in the necessary conditions many possibilities exist because for optimal u, some control and state variables may be forced to their upper limit and other to lower limit depending on which constraints are active.

Since u is needed to determine which unequal is active, the problem become hardly to handily computationally.

To overcome these problem using  the following steps:

1) Choose feasible control vector that satisfies all constraints but not necessarily optimal and let

2) Make load flow to find the corresponding value of x simply (temporarily) eq.(6-1) by assume

and let the system unconstraint w.r.t. x, then eq.(6-1) becomes: 

and S  is determined as w.r.t 

3) Calculate  using eq(6-4) and using this value to define the up date  u     as follows

4) Using the value

calculate the new value of x  as:       

where:

 calculated from the eqn(5-7).

5) Update

 and calculate the value of x and repeated the procedure up to no possible change of  u     without violating the state inequality constraint, then: 

and the process stop and calculate the x corrected too.

         

2-Penalty Factor Method

This is a method to solve the constrained problem. In this method, the inequality constraints are considered as:

And are incorporated in the original cost function to give the penalized cost function as:

Where the factor:

 and the penalty function is defined as:

Then the problem is solved as unconstrained problem for

It can be shown that the penalty factors approach will converge to the true solution of the original optimization problem. The convergence, however, may be slow if the simple gradient method is used. Convergence speed is greatly enhanced if the matrix of second partial approach is used. In general, the inclusion of the penalty factors will not comprise the sparsely of the Hussein matrix. However, some numerical problems may arise as the factor

 approach large values.

An extension of the above-mentioned part, A variable penalty-factor approach is employed to incorporate the functional inequality constraints. Further, a simple and effective method to find the step-size length is used in transforming it into an unconstrained problem using Lagrange.

Multipliers for the equality constraints solves the constrained problem. The treatment of the inequality constraints is achieved by a penalty factor method. The Lagrange function is then defined by:

Where W is the penalty function for the violated inequality constraints. The

necessary conditions for the optimization of L in eq. (6-11) are:

The system of eq's (6-11),(6-12) are solved with the same interactive

 scheme has used to solve 

   Algorithm Of Penalty Factor Method In Treatment Of Inequality Constraints:

The violation of the functional inequality constraints related to the voltage magnitudes at load nodes and reactive power at generating nodes is treated using the penalty function method, in which the inequality constraints are included to the objective 

           function to get a penalized cost function  

The penalty function w is defined as:

Where:

     are the penalty factors related to the reactive and the voltage constraints. The differences in eq (6-13) are treated as follows:

The idea of the exact transformed decoupling is applied to the solution of eqs.

(6-12) which were derived by Dommel  and Tinny for the optimal power flow. 

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Ahmad Atiia

Hi. I’m Designer of Engineering Topics 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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