Page 13 - Mathematical Models and Algorithms for Power System Optimization

Introduction 3

1.2 Ideas about the Setting of the Variable and Function

(1) Ideas about the variable in conventional power system analysis:
Two kinds of the basic components, single-ended and double-ended (which could also be
briefly represented as node and branch), are included in power system analysis, in which
the former can represent loads, generators, capacitors, reactors, and other grounded
components, whereas the latter can represent lines, switches, transformers, and other
branch components. The conventional calculation model of AC power flow for each node i
is as follows:
X

U j G ij cosθ ij + B ij sinθ ij ¼ 0
P Gi P Li U i
j2i
X

U j G ij sinθ ij B ij cosθ ij ¼ 0
Q Gi Q Li U i
j2i
where θ ij =θ i θ j , which is the angle difference between node i and j. The assigned values
include the load (P L , Q L ), some generations (P G , Q G ), and some voltages and phase angles
(U and θ). That is, there are only two variables and two equations for each node.
The basic variables in the analysis and calculation for the steady-state of a power
system can be classified into four types: active power, reactive power, voltage, and phase
angle (namely: P, Q, U, and θ). Among them, active power and reactive power can
be divided into active power generation and reactive power generation (P G , Q G ), and
active load and reactive power load (P L , Q L ), respectively. Sometimes, the “P” and “Q”on
the node can be considered as the corresponding impedance rather than the variable. In the
transient calculation of a power system, besides the basic variables previously described,
the power angle δ and the angular frequency or rotational speed of the generator ω¼2πf
are also included (where f is the system frequency).
When all of P, Q, U, and θ are treated as variables, with their upper and lower limits added,
the conventional optimization method can be applied to search for an optimal solution. In
addition, all these variables can be subscripted to indicate changes over time (such as
seconds, minutes, hours, months, or years). For example, the power generation output of
unit (i) in a different time period (t) can be expressed as P Gi (t).
(2) The parameter and variable can be transformed from one to another:
If the parameters of components are taken as variables with upper and lower limits, then
they can be adjusted in the optimization calculation. For example, if the expression of the
transformers and capacitors, are expanded as variables with limits for the capacitor bank
number C and the transformer tap ratio T, then they can be optimized by way of an
optimization method.
The idea to optimize the parameters of the components is to taken component parameters
as variables is to optimize the parameters of the components. However, in optimization