STATE ESTIMATION

STATE ESTIMATION

State Estimation is a study of the Electrical Power System which enables us to estimate the state of the Electrical Power System at any given time using state variables which may or may not be measurable, observable or accessible. The state of a system is described by a set of state variables, which at time t(0) contains all information about the system which has any influence on the future (t>t(0)) system behaviour. For a given physical system a number of different state variables can be chosen, for example temperature, positions, speeds, voltages, currents, etc. Aconvinient choice is the selection of a minimum set of variables, thus obtaining a minimum, but sufficient, set of state variables. It is important to emphasize that the state variables are not necessarily directly accessible, measurable or observable.

The system model that is used for the Electrical Power System is a node based model, ie. only the conditions in the nodes are described by the model. In such a model, Power - flows, nodal voltages, line  - impedances etc. are the interesting variables. The choice of state variables is rather obvious: If the line -impedances are known, all other values are uniquely defined by the set of complex nodal voltages. Thus the state variables are the absolute voltage magnitudes of the nodes, and the angles between the nodes.

The state variables are the variables that are directly calculated by the state estimation function. All other variables, such as Power - Flows, can then be calculated from the set of State Variables.

SCADA   :   Supervisory Control And Data Acquisition.

There are various steps to be performed before any State Estimation can be initiated. The measurements can also be more or less pre-checked. The allocation of measurements can leave several lines or zones empty of measurements. There may be no measurements available from a certain part of the Power System due to, eg. , a failure in a communication link to an RTU. (Remote Terminal Unit). A necessary condition for the existence of a solution to the state estimation problem is that the system is observable, ie a possibility exists of calculating all state variables from the available measurements. If some part of the Power System is unobservable, the state estimation program must be able to treat this accordingly

:- include all observable parts of the system in the estimation algorithm,
:- exclude the unobservable parts or treat each one separately
- Exclude non-observable part of the system. Use some method to identify what part(s) of the Electrical Power network which must be excluded and perform State Estimation on the rest of the network.
- Include pseudo-measurements to achieve observability. Sometimes it (is) possible to replace real measurements by intelligent "guesses" and go on with the State Estimation.

The first method should be preferred, since it is not advisable to mix real measurements with "guessed" measurements. If, however, many measurements are lost from an important part of the network, the second approach is reasonable.

Another basic necessity for state estimation is that there must be a redundancy in measurements. If we denote the numbeer of measurements by 'm' and the state variables (two times the number of nodes minus 1{sometimes minus 2, where the reference bus or node has two variables eg the voltage magnitude and angle} ie. - the reference angle)by 'n' the redundancy must be greater than one.     

In Practice, it is often 1.5 - 2.0

Redundancy = (m/n ) > 1

Mathematical Formulation of State Estimation

The mathematical formulation of State estimation in Electrical Power Systems is based on the assumption that the Electrical Power System is more or less static.

Assume that we have a system which is characterised by "n" state variables which completely describe the "System State". The state variables are denoted by 'X'.  'X' is merely a vector which holds all state variables X= x(i) where i = 1,2,3,4........,n.

The 'm' measurements are denoted by 'z'. However, each measurement is corrupted by noise. If the noise is denoted by 'v' the expression relating the measurement to the state can be formulated. The relationship is denoted by h.

Z(1) = h(1){x(1),x(2),.......,x(n)} + v(1)

ie.

Z(1) = h(1){x} + v(1)
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.
.
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Z(m) = h(m){x} + v(m)


More compactly we can write

Z = h{x} + v

If h(x) is linear,

Z = HX + V

where: H is independent of the state variables x(i). H is called the Measurement Matrix.