Page 306 - Mathematical Models and Algorithms for Power System Optimization

298 Chapter 8

8.3.3.2 Composition and characteristics of observation decoupled state space
As stated in the previous section, the difference of δ i minus δ ei can be defined as a new state
variable:

δ i ¼ δ i δ ei i 2 1, NŠÞ (8.9)
ð
½
where δ i is a local decoupled variable from the observation perspective.
Obviously, when δ i ¼ 0, the power of the local system is balanced, that is, the point 0 of state
variable δ i is the local power equilibrium point of the system.
In a mathematical sense, the meaning of δ i is equivalent to a translation transformation of δ i .
Each δ i translates the original point by δ ei from the uniform coordinate center and is set up in
their respective local position. Due to the dynamic time-varying characteristics of δ ei , the
coordinate system where each δ i located is dynamic as well, but it does not matter. What is
important is that the respective origins of δ i will be the constant power equilibrium points for
each part of the system after such a translation transformation.

Likewise, a new state variable could be redefined:
_ _ _
½
δ i ¼ δ i δ ei ð i 2 1, NŠÞ (8.10)
_
n o
A new state space δ, δ could also be constructed, which is similar to the state space {δ, ω}
in the study of the dynamic behavior of the system. In line with the nature of the state variable
in the state space, the new state space could be defined as observation decoupled state space,
and its origin is the steady equilibrium point of system, which is a fairly important
characteristic of observation decoupled state space. Because the steady-state position in the
original state space is not at the coordinate origin of the system but in {δ 0i , ω 0 ji 2 [1, N]}, and
δ 0i is related to the operation mode of the system, it could be any value within the given scope;
thus, it is not easy to determine the stability goal of the system. However, by observation
decoupled state space, if the system fails, the state of the deviation from the origin can be
restored to the origin {0,0} of the decoupled state space. Regardless of the way the system
fails, the system will operate in a steady-state. In the original state space, the state variable ω i
is a local variable. Therefore, the form of the second state variable can be flexible when
observation decoupled state space is constructed by the original state space structure. For
example, when the system capacity is large relative to each local part, the variable can also be
defined:
_
½
δ i 50ðÞ ¼ ω i ω 0 ð i 2 1, NŠÞ (8.11)
as the second state variable, where ω 0 ¼2πf 0 ¼2π 50¼100π (rad/s), and the rated frequency

of system is 50Hz, that is, δ, δ 50ð Þ could be used to constitute an observation decoupled state
space. In addition, if ω i of each bus in the system can be derived: