General formulation

Benders decomposition is a decomposition technique developed by J.F Benders [57] for mixed integer optimization problems exhibiting a special structure that involves complicating variables. The principle of this technique is to take advantage of the problem structure and split the large original problem into a master and a subproblem. The master problem is solved

79 while approximating the subproblem’s optimal value. The master’s optimal solution of the complicating variable constitutes a trial value that is passed to the subproblem. The subproblem is then solved with respect to the proposed trial value and the dual variable of the coupling constraint3 is used to construct a linear constraint (also known as Benders cut) that is appended to the master problem. The set of the appended Benders cuts constitutes a linear piecewise representation of the subproblem. This process is repeated in an iterative manner where additional Benders cuts are added until the master’s subproblem approximation accurately represents the subproblem. Generally, problems suitable for Benders decomposition have the following structure:



c x d y



zmin T  T (3.55) s.t. b Ax (3.56) h Ey Fx  (3.57) 0 , 0   y x (3.58)

The objective function is a linear combination of the two decision vectors x and y, which are coupled through equation (3.57), while the complicating variable is

x

. We proceed by showing how this general problem can be solved using Benders decomposition.

Nomenclature

v Iteration index.



Variable approximating the subproblem value.

) (v

x Master problem trial decision for iteration v.

)

(x

(v)



Subproblem optimal value at iteration v for the trial decisionxv.

) (v



Subproblem dual variable associated with trial decisionxv.

) (v

upper

z Upper bound of optimal solution of the original problem.

) (v

lower

z Lower bound of optimal solution of the original problem.



Convergence tolerance.

3 _{The coupling constraint is the constraint in the subproblem that includes the complicating variable, thus} coupling the trial value to the subproblem objective function.

80

Benders Decomposition algorithm

The original problem is split into a master (3.59) and a subproblem (3.60). The solution strategy is as follows:

Step 1. Iteration index v=1.

Step 2. The master problem is solved subject to all appended Benders cuts (none for v=1)

and the optimal complicating variablex(v)is identified.

Step 3. The subproblem is solved subject to the trial complicating variablex(v).

Step 4. An optimality check (3.64) takes place by evaluating the difference between the

lower and upper bounds of the problem, z_lower(v) andz_upper(v) . If the bounds have converged, the master’s approximation of the subproblem value is accurate and the algorithm goes to Step 7.

Step 5. If convergence has not been achieved, the Benders cut (3.61) related to the current

iteration is constructed and appended to the master problem.

Step 6. Iteration index v=v+1. The algorithm returns to Step 2. Step 7. End

Master problem

The master problem is independent of non-complicating variables and approximates the subproblem value using the scalar variable



. Solving the master problem in the absence of any Benders cuts gives the lower bound of the original problem. As Benders cuts are being added to the master problem, the subproblem approximation is built up from below [54] until

) (v

z is equal to the value of the original problem.







c

x

z

(v)

min

T (3.59) s.t. b Ax 0  x Subproblem

The subproblem contains all non-complicating variables y _{and utilizes the master’s trial}

81 variable (v) can be interpreted as the value change in the subproblem’s objective function

)

(x

(v)



following a unit change in the trial solution v

x .

 d

y

x

v

)

min

T

(

( )





(3.60) s.t. ) ( ) ( : v v Fx h Ey   0  y Benders Cut

The Benders cut to be appended to the master problem at iteration v is given by equation (3.61). It is a linear constraint that utilizes the dual variable (v) and the trial solution x(v) to approximate the subproblem value.

) ( ) ( ) (

)

(

)

(x

v

x

x

v T

F

T



v









(3.61) Lower Bound

At iteration v, the master problem is a relaxed version of the original problem since it includes a limited number of all the potential Benders cuts that accurately describe the subproblem. As a result, the optimal value of the master problem is a lower bound to the optimal value of the original problem.



  ( ) ) (v T v lower c x z (3.62) Upper Bound

At iteration v, the subproblem constitutes a more constrained version of the original problem because the complicating variable is fixed at the non-optimal trial valuex(v). As a result, the value of the original problem at iteration v is an upper bound to the true optimal value of the original problem. ) ( ( ) ) ( ) (v T v v upper c x x z   (3.63) Convergence Criterion

The Benders decomposition algorithm is terminated subject to the convergence criterion (3.64), where



is a small positive number. When this holds true, the master’s approximation

82 to the subproblem value



is very close to the subproblem optimal value

(x

(v)

)

and the trial solution x(v) _{is the optimal solution to the original problem.}

   ( ) ) ( v lower v upper z z (3.64)

It is important to note that the general form of Benders decomposition includes an additional step ensuring that decisions x(v) result in a feasible subproblem solution. In the case that the subproblem is infeasible, instead of an optimality cut, a suitable feasibility cut is generated and appended to the master problem. However, through the use of slack variables in the sub problem, problem feasibility can be guaranteed and feasibility cuts are not necessary as long as an infeasibility measure is passed to the master problem through the dual variables.

The transmission planning problem structure lends itself to Benders decomposition due to the distinct separation between capital and operation costs, with the complicating variables being the transmission investment decisions. In the context of the above formulation,

x

is the investment decision vector, yis the operation decision vector (including dispatch, power flow decision variables etc.) and the original objective function is a linear combination of capital and operation costs as in (3.55). In addition, investment and operation decisions are coupled through the power flow constraints as in (3.57). Following this analogy, the above Benders decomposition scheme can be used to solve the transmission expansion problem. The original problem is split into a master problem that models only investment variables and a subproblem that models only operation variables having implemented the trial expansion plan suggested by the master. Through the iterative algorithm, the master investment problem informs the subproblem about capacity addition decisions, while the operation subproblem informs the master about the need for additional capacity through the Benders cuts.

3.8.1.1 Benefits of Benders decomposition

The benefits of employing Benders decomposition to the TEP can be summarized as follows: i. Problem de-scaling. The computational complexity of linear problems grows non-

linearly with the addition of variables and constraints. By decomposing the original problem in a large number of small-scale programs, complexity is reduced significantly. Thus, studies involving large systems that would otherwise have prohibitive memory and CPU time requirements become manageable.

83 ii. Parallel computing. In the absence of inter-temporal constraints related to the system

operation, it is possible to solve several operational subproblems in parallel, gaining a very substantial computational advantage.

In document A stochastic optimization framework for anticipatory transmission investment (Page 78-83)