Transmission - Linearized DC-OPF

Contents

1. Definitions
2. Motivation
3. Formulation

The following is a derivation and motivation for the linearized DC-OPF. This discussion is adapted from (Broad. 19996).

1. Definitions

r+jx	Impedance represented by the resistance respectively.
G+jB	Real and imaginary terms of admittance (inverse of impedance). Note that the approximations of the imaginary term, 1/x is called susceptance.
bus	Constant voltage construction to which transmission lines are connected.
|V|q	Indicates voltage magnitude and phase angle at a system bus.
p.u.	"per unit" indicates that the quantity has been scaled down for power calculations, where the scaling is specified by both a kilovolt and megawatt rating for any given section of the power system.

2. Motivation

The linearized DC load flow model is motivated by the observations that in large high voltage power systems:

1. Line reactance is significantly larger than line resistance x/r 1.0;
2. Bus voltage magnitudes are very similar |V| 1.0 p.u.;
3. The phase angle difference over transmission lines is mall e.g. one would expect that any given line would have an angular difference less than 30 degrees.

Hence the following approximations can be made:

• G = ^r ⁄ _{(r² + x²)} ≅ 0;


• B = ^-x ⁄ _{(r² + x²)} ≅ ¹ ⁄ _x; often the term g=1/x (susceptance) is used in power equations;


• cos(qk - q1 ) ≅ 1; this is the first term of the power series equivalent of cos(): and


• sin(qk - q1 ) ≅ qk - q1 this is the first term of the power series equivalent of sin().

(Broad, 1996) shows that these assumptions lead to the equation for power flow on a line:

f_kl = −B_kl(q_k − q_l) = g_kl(q_k − q_l)

where g_kl is the susceptance such that g_kl = -B_kl = 1/ x

Thus the power flow equations are entirely linear and can be represented directly in a linear programming (LP) framework.

3. Formulation

To formulate the OPF in PLEXOS, reactance is converted to susceptance using the formula:

g_kl = 1/x

The susceptance determines the power flow on a line using the equation:

f_kl = g_kl (q_k - q_l)

In this equation q_k - q_l is the phase angle difference between the buses at each end of the line, measured in radians.