Jordan Form modeling and Entropybased modeling

1
Power systems modeling
User- oriented models
Models for computer analysis
Simulation Technology
Hierarchical (Jordanian) Models, Hardware
models, Structural Synthesis Parametric
Identification, Models Reduction
Modular analysis (analysis by parts)
Entropy Models (Fluctuation, Thermodynamic, Hamilt
onian models)
2
STRUCTURAL SYNTHESIS BASED ON JORDAN CANONICAL
FORM
3
The structural synthesis based on Jordan models
simple eigenvalues
eigenvalues of algebraic multiplicity
eigenvalues of geometrical multiplicity
l
t
e
b
1
1
l
t
e
b
2
2
l
l
l
t
t
t
b
e
e
e
3
3
3
3
S
b
4
b
5
l
t
e
b
4
6
l
t
e
b
4
7
v(t)
4
The basics of a Jordanian approach and its
applications
Transformation of operator with given
input-output map for power systems and its
components to Jordan form is equivalent to
construction of system with such structure that
processes having the different speed should be
located in different parts of that system. In
this case we have system with time and
space-separated movements. The processes in
different parts of the such system are
independent one each another.
State equations
Object-oriented realization with support of
parallel computations
Analysis
Plots or tables, entered by designer
Synthesis

Time or frequency response
Identification
Reduction
Transfer function Transfer matrix
5
The main properties and advantages of the Jordan
approach
Mathematical model of any electrical network in
Jordan form can be created in form of
parallel-sequential structure. In the general
case this structure can be composed from
six base primitive circuits which are determining
by type and multiplicity of correspondent
eigenvalues of mathematical model. Thus, the
problem of identification and synthesis can be
reduced to determination of number, type and
parameters of primitive circuits.
Unification of synthesis and identification
processes and simple implementation in the
object-oriented software
Creation of hierarchical families of system
models
Easy model reduction in time as well as in
frequency domains.
Applicable for analysis methods, based on time
decomposition (movements separation) as well as
algorithms, based on space decomposition of the
system (relaxation methods, iterative
algorithms of type Gauss-Jacobi and Gauss-Seidel).
6
Main results
Opened an uniform way of circuits and systems
synthesis. Data for synthesis may be entered in
form of frequency or time characteristics, state
equation or transfer function (matrix).
Developed algorithms and software for
identification of multi-input - multi-output
circuits and systems in time as well as in
frequency domains
Proposed new approach of reduction in time and
frequency domains simultaneously
Created object-oriented methodology of
miscellaneous models of energetic systems blocks
generation
Methods of parallel computations based on a
movements separation (direct integration
methods, adaptive algorithms) as well as
algorithms, based on a space decomposition was
adopted to a Jordan approach
Developed methodology and software for quick
access to a storage of energetic systems
components models based on different queries,
including graphical
7
The further evolution of a Jordan approach
ANALYSIS Modeling of distributed systems, based
on RDO (Remote Data Objects)
technologyEstimation of performance properties
of an energetic objects
Development ofActiveX libraries of energetic
systems components Investigations of the
inheritance methods for energetic system
models derivation from the Jordan root
class Specifications elaborationfor the uniform
storage of electrical and energetic systems
models
SYNTHESIS IDENTIFICATION Expansion of a Jordan
models to the problems of essentially non-linear
and parametric systems synthesis and
identification
Investigation of Lyapunov functions approach to
reduction of non-linear and parametric models of
energetic systems
8
ENTROPY (FLUCTUATION) MODELING
9

Entropy (Fluctuation) Model
The basics of a model
1. Electrical circuit is represented as an
multidimensional object by analogy with such
thermodynamic objects as gas, liquid, solid
body, etc. 2. Circuit analysis is based on
the Power Balance Theorem. 3. Statistical
description of circuits properties is based on
introduction of states probability density
function, finding an entropy, measure of power
fluctuations and external macro-parameters.
10
Using of entropy model for projecting of PS
Estimation
Reliability parameters Express-analysis
Electric values during operational period
Reliability (mean time between failures) The
stream of failures Stability, operational
reliability, etc. Responses on environment
influences
Mean values of currents, voltages and
powers Responses on electrical influences
Currents, voltages, powers fluctuations 1/f noise
Estimation of parameters during
exploitation Analogs of thermodynamic equalities
11

Estimation of electrical values and their
fluctuations
Schematic in RLCEJ basis
Deriving of State Equations Matrixes of
parameters
Distribution function Statistical weight
Time constant of the circuit Circuits
quality factor
Measure of fluctuations Temperature
analogFunction of power, overheat and quality
Mean values Current, voltage, power
Fluctuations Root mean squares Current,
voltage, power Mutual fluctuations (correlation)
12
Estimation of reliability parameters
Analysis (parameters prediction)
Initial data
Energy fluctuations WLC (U,I) having 1/f spectra
for every block
Fluctuation (entropy) models of a blocks
Minimal frequency fmin of a blockFrequency of
destructionEnergy WLC (U,I) gt then critical
energy
Critical values of current, voltage and
overheat of every block
Failure time tfailure 1/fmin of a block 1/
tfailure fmin - failures intensity
Constant time of a circuit Circuits quality
factor of every block
Failure time of whole PSFailures intensity of PS
is equal to sum of failures intensity of single
blocks
Measure of fluctuations for every block and
whole PS
Another exploitation parametersAnalogs of
thermodynamic equalities
13
Estimation of failure time of blocks and whole PS
Circuit breaker
Load
Torque source
Synch gen.
Circuit breaker
(t60.2 year)
t44 years
Cable
t225 years
Circuit breaker
Load
t34 years
t10.4 year
t54 years
(t70.2 year)
With loads - 1 month
Without loads - 4 months
14

Further development
Fundamental problems of non-equilibrium.
processes and systems
Application to Power system Modeling
The problem of interactionbetween blocks.Steady
State processes
Non-equilibrium processes in objects of different
nature.Analysis, based on power balance
Transient processes
1/f noise - cause and effects.Correlation with
system behavior.Degradation and destruction
Control of PS operation parameters. The maximum
principle, basedon power balance
Non-equilibrium systems.Analysis, based on
circuits theory.Application to biological
objects
Software Development and debugging