2. Knowledge-Based Virtual Identification Models Research Methodology
The use of adjustable identification models has been a long-term trend in the identification theory and applications. The convergence of the empirical distribution functional to the theoretical one with the sample size increase has been analyzed [
19].
Identification methods based on data mining [
20] not only helped to overcome the challenges posed by nonlinear and non-stationary behavior but also increased significantly the accuracy of analysis and forecasting. Associative search algorithms develop point linear models of nonlinear processes based on process history analysis. Thus, the dynamic adjustment of identification models is carried out on the basis of e-learning, knowledge extraction and replenishment.
These intelligent identification algorithms have shown high accuracy results in various applications, such as chemistry, oil refining, smart grids, and transport systems [
21]. To analyze and predict the stability of dynamical systems, an approach based on the study of multiscale wavelet expansion can be used [
22].
2.1. Predictive Intelligent Model Design Problem Statement
In knowledge-based modeling methods, knowledge is used for the most accurate reproduction of the object’s image from its fragment [
23]. In the associative search method, knowledge is understood as an associative relationship between images. Set patterns of input and output variables are considered as images. In [
22], a model of associative thinking was presented. The memorization process is interpreted as a sequential formation of associations of image pairs.
The model can be considered as an intermediate stage between logical models and neural networks. In our case (associative search), we use pairs of patterns of input and output variables of a dynamic system. At each time step, a new virtual model is created. In order to build a model for a specific time step, a temporary “ad hoc” database of historic and current process data is generated. After calculating the output forecast based on plant’s current state, this database is deleted without saving. The linear dynamic prediction model looks as follows:
where
is the input vector value;
is the predicted output value for the next time step;
r and
s are the output and input memory depth, accordingly;
P is the input vector length.
The identification algorithm forms an approximating hypersurface of the input vector space and one-dimensional outputs of the dynamic object (
Figure 1). The values of the coefficients of the linear model and the predicted output value are determined on the basis of the least squares method.
In fact, the associative search method simulates the decision-making process of a human operator. The sum
can be considered as a metric in the
P-dimensional input space, where
j is typically less than
t;
are the components of the input vector at the time step
t. Source: authors’ illustration.
2.2. Development of a Virtual Model on the Basis of Associative Search Technique
Assume that for the current input vector
:
In order to build an approximating hypersurface for
, we select such vectors
,
j = 1, …,
s from the input data archive that for a given
the following conditions will hold:
The 2-D case is presented in
Figure 2. Source: authors’ illustration.
If the domain selected on the basis of knowledge does not allow constructing a simultaneous system of linear algebraic equations, the inputs selection criterion is weakened by increasing the threshold
The purpose of associative search is to restore all features of the object based on the accumulated set of its images, the “dynamic twins”. Let be the image that initiates the search; R is the resulting image of the associative search. The pair (, R) may be called association. The set of all associations on a set of images makes the content of the intelligent system’s knowledgebase.
To apply the associative search technique, a preliminary training stage is required, for which an archive of images is created. The algorithm that implements the image reconstruction procedure based on R0a can generally be described by the predicate (R0,ia, Ria, Ta), where R0ia ⊆ R0, Ria ⊆ R. In particular, this predicate can be a function that asserts the truth or falsity of the input vector membership in a certain area of the input space.
2.3. Associative Search Technique in Short-Term Prediction
The associative search technique consists of two sequential stages: (i) the hypersurface is selected in such way that it contains input vector
,
j = 1, …,
s at the current time step
t, and (ii) the hypersurface is selected from the historical data archive corresponding to
,
j = 1, …,
s, contains the input vector at the previous time step
t − 1. The predicate describing the choice will have the form:
Within the framework of this approach, there is a possibility to improve the accuracy of the procedure by increasing the memory, for example, to
m steps (
m <
t):
2.4. Solving the System of Linear Equations for the LS Method
The development of identification models with the help associative search algorithms in the closed loop is, however, not always possible: The control faces the same challenges as the traditional methods do. Dependent values are used in the closed-loop control; optimal controllers generate linear state feedbacks that results in the degenerate problem [
24]. In this case, one can apply the Moore–Penrose method and the Singular Value Decomposition (SVD) [
25,
26] that allows to obtain pseudo-solutions of a system of linear equations in order to apply the associative search method [
22].
2.5. Associative Search Based on Clustering
The search of data making the best fit to the current values of the input variables may be exhaustive. In order to improve the algorithm performance, one of the clustering methods can be applied. Such methods allow to determine for the current time step the membership of the current input vector in a certain area of the multidimensional space. Further on, when using the associative search technique, the input vectors close to the current one are selected within a specific cluster.
If the input vector is known to belong to a certain cluster, “close” (similar) vectors are selected within the cluster
, and the associative search procedure for
is applied, where
P is the dimension of the input vector
r = 1, …;
R is the number of the cluster in the sub-space X
X is the set of inputs values:
This approach does not require the knowledge of the nonlinear object structure.
2.6. Clusterizaton and Associative Search
Clustering acts as a convenient learning and pre-learning tool that allows to increase the computational performance of associative search.
Crisp and fuzzy approaches may be considered. In the first case, each input vector belongs to only one of the disjoint sets (clusters) of the input space. In case of fuzzy clustering, an object can belong to several clusters simultaneously with various degrees of confidence. The degree of confidence is determined by the selected membership function.
The associative search technique is as follows. The current input vector refers to a certain cluster subject to the criteria of the minimum distance from the center:
where
is the current input vector of the control plant;
is the center of the cluster
k.
For associative search, the vectors close to the current input vector are selected within this cluster. If they are not enough, the cluster can be expanded using single-channel methods that combine two clusters with a minimum distance between members.
2.7. Wavelet Analysis of Time-Varying Processes
The recent years have been seeing the growing popularity of time-variant dynamic process analysis based on the wavelet transform. The wavelet transform is a generalization of spectral analysis with respect to Fourier transform. The first works on wavelet analysis examined time series [
27]; the method was then considered as an alternative to Fourier transform with frequency localization.
Today, wavelet analysis is extensively used in many areas [
28]. The most popular applications include processing and synthesis of non-stationary signals, information compression and coding, image processing, and pattern recognition, particularly in medicine. The method is effective for studying geophysical fields and meteorological time series, as well as for earthquake prediction.
Wavelet analysis is based on a linear transform (called a wavelet transform) made by means of soliton-like functions (wavelets) that form an orthonormal basis in L2. These basis functions are localized in a limited area. Therefore, the wavelet transform allows, as against Fourier transform, to obtain information on local properties of the signal. Wavelets also provide a powerful approximation tool. They may be used with a minimal number of basis functions for synthesizing the functions that are poorly approximated by other methods. Wavelet analysis allows you to investigate the properties of a signal in the time and frequency domains.
The wavelet transform may be used in systems with identifier [
29]. The expediency of using it for the identification of nonlinear systems with unknown time-varying coefficients, which can be represented as a linear combination of basic wavelet functions, was shown in [
30].
Moreover, to solve identification problems, various wavelet types are used (biorthogonal wavelets, wavelet frames, wavelet networks, spline wavelets) [
31].
2.8. Criteria of Linear System Stability in the Sence of the Spectrum of Multi-Scale Wavelet Expansion Analysis
A multiscale wavelet expansion of an associative predictive model of a nonlinear time-varying object (1) for the selected detailing level
looks as follows [
32]:
where
is the depth of the multi-scale expansion (
, where
and
is the power of the set of states of the system in the system dynamics knowledgebase);
are scaling functions;
are the wavelet functions obtained from mother wavelets by means of tension/compression and shift:
Haar wavelets are chosen as mother wavelets;
is the level of data detailing;
are the scaling coefficients,
are the detailing coefficients. The coefficients are calculated by use of the Mallat algorithm. The object equation is as follows:
By considering the detailing and approximating parts of (7) separately, we have
The sufficient conditions of the object (1) stability for
for the detailing and approximating coefficients respectively are as follows [
33]:
- -
if
:
- -
if
, then:
- -
if
, then the condition of the stability for the detailing coefficients:
for the approximating coefficients:
- -
if
, then the condition of the stability for the detailing coefficients:
for the approximating coefficients:
3. Determining Static Stability Degree by Gramian Method
The Gramian method [
16] provides an effective tool for analyzing the stability degree of power systems. It enables the investigation of system dynamics on the basis of a new mathematical technique for solving Lyapunov and Sylvester equations [
16]. The method is based on the decomposition of the Gramian matrix, which is the solution of Lyapunov or Sylvester equations, into the spectrum of the matrices of these equations. To study the stability of differential-algebraic equations describing a power system, the system Gramian is calculated in real time using the asymptotic Frobenius norms [
15].
From the methods used for solving the discrete Lyapunov equation [
34], we have chosen the one offered in [
35]. It applies the Fourier transform and
z-transform to the discrete Lyapunov equation. The solution of the Lyapunov equation is an integral in the complex area of the product of resolvents of two matrices: the dynamics matrix and its transposed and adjoint one.
Therefore, we investigate the stability of the linear model described above. Let the linear stationary discrete time-invariant system be as follows:
where
Suppose the matrices , are the real ones, where m, n integer positive numbers, . Suppose that the system (19) is stable, fully controllable and observable; all matrix A eigenvalues are distinct ones.
The system characteristics in the frequency domain are defined by the transfer function
The methods of selective modal analysis (SMA) and normal forms (NF) are the closest ones to our approach. The most generic approach is based on the modal analysis (eigenvalue decomposition, EVD) and the use of a linearized power system model in the system operating point (SEP).
(1) SMA and NF employ a linear model of an autonomous system, while for the Gramian method, the model of a system with input actions is used.
(2) The main difficulty of calculations in the NF method is a formation of the initial conditions [
15]. In the Gramian method, initial conditions are formed as the values of electrical and/or mechanical moments; some known input functions may be specified instead.
(3) In the Gramian method, the calculating of stability loss risk is reduced to calculating the sums of the energy functionals of dominant modes, while in the NF method, nonlinear interaction indices, which depend on the initial conditions, play the similar role.
(4) Dominant modes in the Gramian method are determined by the participation factor of the energy functional of the mode in the total value of the square of H
2 norm of the power system’s discrete transfer function [
4,
9]. The NF method uses nonlinear modal persistence indices for estimating the extent of dominance of the mode combinations for the third-order continuous approximating model.
(5) Modern electric power systems feature high dimension of the tasks being solved. The main method for constructing an approximating model for such systems is the interpolation based on the use of controllability and observability Gramians for linear and bilinear systems [
8,
9,
10,
34].
Suppose that transfer function (20) is strictly proper. Consider the following algebraic discrete Lyapunov (Stein) equation of the form [
34]
It is known that the matrix
A resolvent decomposition has the form:
where
is characteristic polynomial of
A. The matrices
called
Fadeev matrices [
36], can be defined by means of Fadeev–Leverie algorithms [
37]. Suppose that
.
Then the algorithm has the form
The solutions of Equation (19) in the time domain can be defined in the following way [
38]:
The solution in frequency domain looks as follows:
The following variable change
is made in the integrals:
where
γ is a unit circle, moving counter-clockwise. Therefore, all eigenvalues are distinct ones, and we have the following resolvent decomposition:
We introduce the following designations:
After substituting this formula in Equations (27) and (28) we obtain:
Let us introduce the following designations for each sub-Gramian:
The integrands in Equations (34) and (35) are the analytic functions over the whole complex plane with the exclusion of particular points
. By means of the Caushy residue theorem, we have the Gramians spectral semi-decomposition in following way:
We transform the matrix
by using matrix resolvent decomposition to vulgar fractions [
39]:
By substituting the Equations (36) and (37) to the above formulae, we obtain the full decomposition of the Gramians in the form:
Theorem 1. Consider LTI MIMO—the real discrete system with (A, B, C) presentation in the form where the matrices A, B, C are the real ones. Suppose the system is stable, fully controllable and observable; its transfer function is strictly proper, and the matrix A eigenvalues are distinct ones.
Then the following matrix equalities hold:
where
is the residue of the matrix
A resolvent in the point equal to the matrix eigenvalue. The Equations (43)–(46) define the semi-decomposition of the infinite controllability and observability Gramians on the eigenvalues set to belong of unit circle:
and the Equations (39)–(42) define full decomposition of the infinite Gramians on the eigenvalues set to belong to the interior of the unite circle on complex plane. Expressions for Gramians spectral decomposition one can simplify
Corollary [
38]. If all eigenvalues of the matrix
A are distinct, then the matrix can be transformed to diagonal form by means of the similarity transform:
or
where matrix
consists of the right eigenvectors
, and matrix
T consists of the left eigenvectors
corresponding to the eigenvalue
. The last equality is a condition for eigenvectors normalization.
The Gramians of diagonalized system are the solution of Lyapunov equations:
which are defined from the formulae:
The controllability Gramian
is linked with the Gramian
by equation
The observability Gramian
is linked with Gramian
by similar equation
We introduce the new designation
for the matrix with all zeros except for the element “
ij”, which is equal to one (1):
For diagonalized matrix
A, the following expressions are valid:
Consider the spectral decomposition of the controllability and observability Gramians by pairwise combinational spectrum of the dynamics matrix. In this case, the Equations (55) and (56) have the form:
Note that the premultiplication of the matrix
by the matrix
and the post-multiplication by the matrix
allows us to cut from the matrix its element located in the intersection of the column “
k” and the row “
p”. For the diagonalized system, we have the following formulae:
These simple and compact expressions allow to compute sub-Gramians by computing their n2 elements. They are simpler than the common Equations (47)–(50) for Gramian spectral expansion. We have got spectral separable Gramians decomposition in the form of a direct sum of n2 sub-Gramians corresponding to the decomposition of controllability and observability Gramians by pairwise combinational eigenvalues of the dynamics matrix’s spectrum.
As it is known [
34], the necessary and sufficient condition for the energy stability of the system in terms of the square of the
H2 norm of the transfer function has the form:
We define the stability loss risk functional as follows:
As the system approaches the stability threshold caused by the approaching of the characteristic equation roots to the imaginary axis, the risk functional approaches infinity. Let us define the acceptable risk of stability loss in the form
We will consider any system as
conditionally unstable if all its roots are in the left half-plane, but the functional of the stability loss risk exceeds the established acceptable risk value. Accordingly, we will consider the system
conditionally stable if
The square of the
H2 norm of the system transfer function can be calculated by solving the Lyapunov matrix algebraic equation by means of substituting the known matrices
A,
B,
C into it, while the spectrum of the matrix
A is not required to be calculated. On the other hand, the spectral expansions of the square of the
H2 norm of the system transfer function characterize the separability property of the stability loss risk functional: It is equal to the sum of terms, each one corresponding either to a separate eigenvalue of the dynamics matrix, or to their pairwise combination. The energy functional, which allows for only the weakly stable components of the quadratic forms
Jδ, makes it possible to determine the overall risk of stability loss as well as to estimate the energy stability margin in decibels:
The mathematical model of the system is linear; however, the spectral decomposition of the square of H2—norm of the power system discrete transfer function takes into account the nonlinear interaction of modes. The group interaction of modes is limited by taking into account only pairwise combinations of eigenvalues of the dynamics matrix.
The Gramians method can be used simultaneously for state monitoring and control of large-scale power systems, in particular, for static stability analysis; for developing stability estimator; for detecting dangerous free and forced oscillations; and for assessing the resonant interaction of dangerous oscillations [
38,
39].
5. Conclusions
In this paper, we investigated the possibility of combining associative search identification technique with the Gramian method for predicting of the approaching of process dynamics to the stability threshold. A new technique for solving discrete Lyapunov equations based on matrix equations and semi-expansions of controllability Gramians was developed.
Features and novelty of the Gramian method are as follows:
(1) New methods of elementwise computation of the algebraic Lyapunov equation solution, based on separable spectral expansions of the controllability and observability Gramians;
(2) New energy criteria for assessing the risk of the loss of electric power systems stability, based on the identification and analysis of nonlinear effects caused by modes interaction;
(3) The method for identifying any potential swing centers and the forecast of the evolution of swing processes caused by the interaction of the modes in the linear model.
In the analysis of static and transient stability in power systems, the Gramian method occupies an intermediate position between the methods of selective modal analysis and normal forms. From the first one, it differs by the fact that it is actually a nonlinear modal analysis technique because the spectral expansions of Gramians in pairwise mode combinations include the products of second order infinitesimals. It also differs from the second one owing to the capability of (i) obtaining a direct assessment of stability loss risk based on the use of energy functionals and (ii) predicting the evolution of the stability loss process.
A linear discrete model is developed for predicting the approach of the state of a nonlinear system to the stability threshold. To build such a model, identification methods and algorithms based on knowledge formation and analysis are proposed and named associative search algorithms.
It is shown how the proposed method uses data mining for performing dynamic predictive remote diagnostics for the general primary frequency control in a power system. The application cases discussed in the paper demonstrate the high accuracy of the estimates obtained with help of associative search algorithm. The results of the theoretical investigation of identification models, algorithms, and methods developed by the authors were applied in the Scientific and Technical Center of the Russian Federal Grid Company.
Before applying the Gramian method, the approach of the system to the stability threshold can be predicted on the basis of multiresolution wavelet decompositions. The features and novelties of the proposed associative search algorithms are as follows:
(1) They allow to obtain linear models of nonlinear objects (at any given time step a new model is developed based on data mining);
(2) The version of algorithms for non-stationary objects is developed on the basis of wavelet analysis;
(3) New identification models feature high accuracy because they use the maximum of available information about object operation.
The difficulties of predictive model development and the analysis of their stability for electric power systems requires a systematic approach. Therefore, combining the methods proposed by the authors for constructing identification models with Gramians methods for stability studying at the stages of planning, monitoring, management, and optimization provides a synergistic effect and opens up a wider application outlook.