Extended OPF Formulation

6.3 Extended OPF Formulation

Matpower employs an extensible OPF structure to allow the user to modify or augment the problem formulation without rewriting the portions that are shared with the standard OPF formulation described above. The standard formulation is modiﬁed by introducing additional variables, user-deﬁned costs, and/or user-deﬁned constraints. The full extended formulation can be written as follows:

min f(x) + fu(ˆx) ˆx

(6.34)

subject to

ˆg(ˆx) = 0 (6.35) ˆ h(ˆx) ≤ 0 (6.36) ˆxmin ≤ ˆx ≤ ˆxmax (6.37) l ≤ A ˆx ≤ u (6.38)

The ﬁrst diﬀerence to note is that the optimization variable $x$ from the standard OPF formulation has been augmented with additional variables $z$ to form a new optimization variable $ˆx$ , and likewise with the lower and upper bounds.

[ ] [ ] [ ] x xmin xmax xˆ= z ˆxmin = zmin ˆxmax = zmax

(6.39)

Second, there is an additional user-deﬁned cost term $fu(ˆx)$ in the objective function. This cost consists of three pieces that will be described in more detail below.

fu(ˆx) = fq(ˆx) + fnln(ˆx) + flegacy(ˆx)

(6.40)

Third, the nonlinear constraints $g$ and $h$ are augmented with user deﬁned additions $gu$ and $hu$ to give $gˆ$ and $ˆh$ .

[ ] [ ] ˆg(ˆx ) = g(x) , ˆh (ˆx) = h (x) (6.41) gu(ˆx) hu(ˆx)

And ﬁnally, a new set of linear constraints are included in (6.38).

Up through version 6.0 of Matpower, the OPF extensions were handled via optional input parameters that deﬁne any additional variables,³⁵ linear constraints,³⁶ and costs of the pre-speciﬁed form deﬁned by $f legacy$ .³⁷ This preserved the ability to use solvers that employ pre-compiled MEX ﬁles to compute all of the costs and constraints. This is referred to as Matpower’s legacy extended OPF formulation [27].

For the AC OPF, subsequent versions also include the general nonlinear constraints $gu$ and $hu$ , and the quadratic and general nonlinear costs $fq$ and $fnln$ . The new quadratic cost terms can be handled by all of Matpower’s AC OPF solvers, but the general nonlinear costs and constraints require a solver that uses Matlab code to implement the function, gradient and Hessian evaluations.³⁸

Section 7 describes the mechanisms available to the user for taking advantage of the extensible formulation described here.

6.3.1 User-deﬁned Variables

The creation of additional user-deﬁned $z$ variables can be done explicitly or implicitly based on the diﬀerence between the number of columns in $A$ and the dimension of the standard OPF optimization variable $x$ . The optional vectors $zmin$ and $zmax$ are available to impose lower and upper bounds on $z$ , respectively.

6.3.2 User-deﬁned Constraints

Linear Constraints – The user-deﬁned linear constraints (6.38) are general linear restrictions involving all of the optimization variables and are speciﬁed via matrix $A$ and lower and upper bound vectors $l$ and $u$ . These parameters can be used to create equality constraints ( $l = u i i$ ) or inequality constraints that are bounded below ( $ui = ∞$ ), bounded above ( $li = ∞$ ) or bounded on both sides.
Nonlinear Constraints – The user-deﬁned general nonlinear constraints take the form
$j gu(x) = 0 ∀j ∈ 𝒢u (6.42 ) hju(x) ≤ 0 ∀j ∈ ℋu, (6.43 )$
where $𝒢u$ and $ℋu$ are sets of indices for user-deﬁned equality and inequality constraint sets, respectively.
Each of these constraint sets is deﬁned by two M-ﬁle functions, similar to those required by MIPS, one that computes the constraint values and their gradients (Jacobian), and the other that computes Hessian values.

6.3.3 User-deﬁned Costs

The user-deﬁned cost function $fu$ consists of three terms for three diﬀerent types of costs: quadratic, general nonlinear, and legacy. Each term is a simple summation over all of the cost sets of that type.

∑ j fq(ˆx ) = j fq(ˆx) (6.44) fnln(xˆ) = ∑ fj (xˆ) (6.45) ∑ j nln flegacy(xˆ) = j fjlegacy(ˆx) (6.46)

Quadratic Costs for a cost set $j$ are speciﬁed by parameters $Qj$ , $cj$ and $kj$ that deﬁne a quadratic function of the optimization variable $ˆx$ .
$j T T fq(ˆx ) = xˆ Qj ˆx + cj ˆx + kj$ (6.47)
General Nonlinear Costs for a cost set $j$ consist of a cost function $fjnln(ˆx)$ provided in the form of a function handle to an M-ﬁle function that evaluates the cost and its gradients and Hessian for a given value of $ˆx$ .
Legacy Costs $j flegacy(ˆx)$ are speciﬁed in terms of parameters $Hj$ , $Cj$ , $Nj$ , $ˆrj$ , $kj$ , $dj$ and $mj$ . For simplicity of presentation, we will drop the $j$ subscript for the rest of this discussion. All of the parameters are $nw × 1$ vectors except the symmetric $nw × nw$ matrix $H$ and the $nw × nˆx$ matrix $N$ . The legacy user cost function takes the form
$1 T T flegacy(ˆx ) = -w Hw + C w 2$ (6.48)

where $w$ is deﬁned in several steps as follows. First, a new vector $u$ is created by applying a linear transformation $N$ and shift $ˆr$ to the full set of optimization variables

$u = N ˆx − ˆr,$ (6.49)

then a scaled function with a “dead zone” is applied to each element of $u$ to produce the corresponding element of $w$ .

$({ m f (u + k ), u < − k i di i i i i wi = ( 0, − ki ≤ ui ≤ ki mifdi(ui − ki), ui > ki$ (6.50)

Here $ki$ speciﬁes the size of the “dead zone”, $mi$ is a simple scale factor and $fdi$ is a pre-deﬁned scalar function selected by the value of $di$ . Currently, Matpower implements only linear and quadratic options:

${ α, ifd = 1 fdi(α ) = 2 i α , ifdi = 2$ (6.51)

as illustrated in Figure 6-1 and Figure 6-2, respectively.

Figure 6-1:Relationship of $wi$ to $ri$ for $di = 1$ (linear option)

Figure 6-2:Relationship of $wi$ to $ri$ for $di = 2$ (quadratic option)

This form for $flegacy$ provides the ﬂexibility to handle a wide range of costs, from simple linear functions of the optimization variables to scaled quadratic penalties on quantities, such as voltages, lying outside a desired range, to functions of linear combinations of variables, inspired by the requirements of price coordination terms found in the decomposition of large loosely coupled problems encountered in our own research.
Some limitations are imposed on the parameters in the case of the DC OPF since Matpower uses a generic quadratic programming (QP) solver for the optimization. In particular, $ki = 0$ and $di = 1$ for all $i$ , so the “dead zone” is not considered and only the linear option is available for $fdi$ . As a result, for the DC case (6.50) simpliﬁes to $wi = miui$ .

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