Constrained Cost Variable Formulation

As the formulation of standard OPF is unable to solve the non-smooth piecewise linear cost functions that results from discrete offers and bids, when such cost functions are convex they can be modeled by means of CCV method [19-21]. The piecewise linear cost function c(x) is substituted by a helper variable y and linear constraints that form a convex ‘‘basin’’ requiring the cost variable y to put into the function c(x). Figure 6.4 shows a convex n-segment piecewise linear cost function.

Ш1 (x — X1)+ C1 , x < X1

m2 (x — x2)+C2, x1 <x < x2

Подпись: c(x)Подпись: (6.14)

Подпись: Pd Pp2 + Qd
Подпись: cos u Подпись: constant Подпись: (6.13)


m„ (x — x„)+c„, x„—1<x

Constrained Cost Variable Formulation Подпись: n Подпись: (6.15)

A convex n-segment piecewise linear cost function is defined by a sequence of points (xj, cj), j = 0 ■ ■■ n where mj denotes the slope of the jth segment,

and x0 < xi < ■■■<xn and mi < m2 < ■■■<mn.

The ‘‘basin’’ corresponding to the cost function is formed by the following n constraints on the helper variable y.

Fig. 6.4 Constrained cost c


y > mj(x — xj) + cj, j = 1 ■ ■■ n (6.16)

The cost term added to the objective function instead of c(x) is the variable y. The CCV method can convert any piecewise linear costs of active or reactive power into the appropriate helper variable and the related constraints.

By applying CCV method, every piecewise function in the objective is substituted with a helper variable. A number of inequality constraints, one for each piece of the piecewise function, are placed on that variable. CCV is a way that formulates a piecewise linear cost function on a new variable that is linearly constrained.

In this chapter, the offers of WTs and bids of DLs are taken and treated as marginal cost and marginal benefit functions, respectively, then they are converted into the equivalent total cost and total benefit functions by integrating the marginal cost and benefit functions and plugged into a matrix as piecewise linear costs. CCV approach conquers the difficulty of disruptive Lagrange derivatives by expanding the optimization into a higher dimensional space and counting on good constrained optimization methods to resolve the transformed smooth optimization problem.

Updated: October 23, 2015 — 12:41 pm