Stability Prediction Techniques for Electric Power Systems based on Identification Models and Gramians 1

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IFAC PapersOnLine 52-13 (2019) 481–485 Stability Prediction Techniques for Electric Power Systems Stability Prediction Techniques for Electric Power Systems Stability Prediction Techniques for Electric Power Systems 11 Stability Prediction Techniques for Electric Power Systems based on Identification Models and Gramians 111 Stability Prediction Techniques for Electric Power Systems based on Identification Models and Gramians 11 based on Identification Models and Gramians Stability Prediction Techniques for Electric Power Systems 1 based Models and Gramians * ** based on on Identification Identification Models and Gramians 1 ** N. ,,, I. ** ** N. Bakhtadze Bakhtadze*****Models I. Yadykin Yadykin ** based on Identification and Gramians N. Bakhtadze I. Yadykin N.  *,, I. ** N. Bakhtadze Bakhtadze I. Yadykin Yadykin**

 **,, I. N. Bakhtadze Yadykin** ** N. Bakhtadze  , I. N.of Bakhtadze I. Yadykin Yadykin * **  V.A. Trapeznikov Institute Control Sciences of the  V.A. Trapeznikov Institute of Control Sciences of the Russian Academy of Science, N. Bakhtadze , I. Yadykin V.A. Trapeznikov Trapeznikov Institute Institute of of Control Control Sciences Sciences of of the the Russian Russian Academy Academy of of Science, Science, V.A. Russian Academy of Science, V.A. Trapeznikov Institute of Control Sciences of the Russian Academy of Science, Moscow, Russia , 65 Profsoyuznaya, Moscow 117997, Russia  Moscow, Russia , 65 Profsoyuznaya, Moscow 117997, Russia V.A. Trapeznikov Institute of Control Sciences of the Russian Academy of Science, Moscow, Russia , 65 Profsoyuznaya, Moscow 117997, Russia V.A. Trapeznikov Trapeznikov Institute of Control Sciences of the Russian Academy of Science, * ** Moscow, Russia , 65 Profsoyuznaya, Moscow 117997, Russia V.A. Institute of Control Sciences of the Russian Academy of Science, * (e-mail: ** (e-mail Moscow, Russia , 65 Profsoyuznaya, Moscow 117997, Russia [email protected]), [email protected]) * ** * (e-mail: ** (e-mail (e-mail: [email protected]), (e-mail [email protected]) Moscow, Russia , 65 Profsoyuznaya, Moscow 117997, Russia * ** [email protected]), [email protected]) , V.A. Trapeznikov Institute of Control Sciences of the Russian Academy of Science, Moscow, Russia , 65 Profsoyuznaya, Moscow 117997, Russia * (e-mail: ** (e-mail [email protected]), [email protected]) Moscow, Russia , 65 Profsoyuznaya, Moscow 117997, Russia * ** [email protected]), [email protected]) * (e-mail: ** (e-mail [email protected]), [email protected]) * (e-mail: ** (e-mail Moscow, Russia , 65 Profsoyuznaya, Moscow 117997, Russia (e-mail: [email protected]), (e-mail [email protected]) (e-mail: [email protected]), ** (e-mail [email protected]) * Abstract: The methods for developing predictive models control systems and decision-making (e-mail: [email protected]), (e-mailin [email protected]) Abstract: The methods for developing predictive models in control systems and decision-making Abstract: The methods for developing predictive models in control systems and decision-making Abstract: The methods for developing predictive models in control systems and decision-making Abstract: The methods for developing predictive models in control systems and decision-making support for nonlinear non-stationary objects are proposed. The methods are based on the application of support for nonlinear non-stationary objects are proposed. The methods are based on the application of Abstract: The methods for developing predictive models in control systems and decision-making support for nonlinear non-stationary objects are proposed. The methods are based on the application of Abstract: The methods for developing predictive models in control systems and decision-making support for nonlinear non-stationary objects are proposed. The methods are based on the application of Abstract: The methods for developing predictive models in control systems and decision-making support for nonlinear non-stationary objects are proposed. The methods are based on the application of associative search procedure to virtual model identification as well as Gramian techniques. The associative search procedure to virtual model identification as well as Gramian techniques. The support for nonlinear non-stationary objects are proposed. The methods are based on the application of associative search procedure to virtual model identification as well as Gramian techniques. The Abstract: The methods for developing predictive models in control systems and decision-making support for nonlinear non-stationary objects are proposed. The methods are based on the application of associative search procedure to virtual model identification as well as Gramian techniques. The support for nonlinear non-stationary objects are proposed. The methods are based on the application of associative search procedure to virtual model identification as well as Gramian techniques. The search methods use intelligent process knowledge analysis. The knowledge base is created associative search methods use intelligent process knowledge analysis. The knowledge base is created search methods procedure tointelligent virtual model identification as well wellThe as based Gramian techniques. The associative search methods use intelligent process knowledge analysis. The knowledge base is created support for search nonlinear non-stationary objects are proposed. The methods are on the application of search procedure to virtual model identification as as Gramian techniques. The associative use process knowledge analysis. knowledge base is created search procedure to operation. virtual model identification asarewell as knowledge Gramian techniques. The associative search methods use process knowledge analysis. The base is and extended in real-time process Intelligent algorithms offered for predicting power plant and extended in process operation. Intelligent algorithms offered for plant associative search methods use intelligent intelligent process knowledge analysis. The knowledge basepower is created created and extended in real-time real-time process operation. Intelligent algorithms arewell offered for predicting predicting power plant search procedure virtual model identification asare as knowledge Gramian techniques. The associative search methods use process knowledge analysis. The base is created and extended in real-time process operation. Intelligent algorithms are offered for predicting plant associative search methods usetointelligent intelligent process knowledge analysis. The knowledge basepower isused created and extended in real-time process operation. Intelligent algorithms are offered for predicting power plant dynamics in optimization tasks. Gramian technique of stability analysis for discrete system is for dynamics in optimization tasks. Gramian technique of stability analysis for discrete system is used for and extended extended in real-time real-time process operation. Intelligent algorithms are offered offered for predicting predicting power plant dynamics in optimization tasks. Gramian technique of stability analysis for discrete system is used for associative search methods use intelligent process knowledge analysis. The knowledge base is created and in process operation. Intelligent algorithms are for power plant dynamics in optimization tasks. Gramian technique of stability analysis for discrete system is used for and extended in real-time process operation. Intelligent algorithms are offered for predicting power plant dynamics in optimization tasks. Gramian technique of stability analysis for discrete system is used for investigating linear virtual model stability. It is shown that the bilinear Lyapunov equation solutions can investigating linear virtual model stability. It is shown that the bilinear Lyapunov equation solutions can dynamics in optimization tasks. Gramian technique of stability analysis for discrete system is used for investigating linear virtual model stability. It is shown that the bilinear Lyapunov equation solutions can and extended in real-time process operation. Intelligent algorithms are offered for predicting power plant dynamics in optimization tasks. Gramian technique of stability analysis for discrete system is used for investigating linear virtual model stability. It is shown that the bilinear Lyapunov equation solutions can dynamics in optimization tasks. Gramian technique ofthat stability analysis forthe discrete system is Faddeev used can for investigating linear virtual model stability. It is shown the bilinear Lyapunov equation solutions be calculated as an infinite sum of the matrix quadratic forms made up by products of the be calculated as an infinite sum of the matrix quadratic forms made up by the products of the Faddeev investigating linear virtual model stability. It is shown that the bilinear Lyapunov equation solutions can be calculated as an infinite sum of the matrix quadratic forms made up by the products of the Faddeev dynamics in optimization tasks. Gramian technique ofthat stability analysis forthe discrete system is Faddeev used can for investigating linear virtual model stability. It is isquadratic shown that the bilinear bilinear Lyapunov equation solutions can be calculated as anby infinite sum of the matrix forms made up by products of the investigating linear virtual model stability. It shown the Lyapunov equation solutions be as infinite sum the matrix quadratic forms made up by the of the matrices obtained decomposing of linear subsystem dynamic matrix resolvents. Copyright © 2019 matrices obtained by decomposing of linear subsystem dynamic matrix resolvents. Copyright © 2019 be calculated calculated as an anby infinite sum of ofstability. the matrix quadratic forms made upLyapunov by the products products ofsolutions the Faddeev Faddeev matrices obtained decomposing of linear dynamic matrix resolvents. Copyright © 2019 investigating linear virtual model It issubsystem shown that the bilinear equation can be calculated as an infinite sum of the matrix quadratic forms made up by the products of the Faddeev matrices obtained by decomposing of linear subsystem dynamic matrix resolvents. Copyright © 2019 be calculated as an infinite sum of the matrix quadratic forms made up by the products of the Faddeev matrices obtained by decomposing of subsystem dynamic matrix resolvents. Copyright © IFAC IFAC matrices obtained by decomposing of linear linear subsystem dynamic matrix resolvents. Copyright © 2019 2019 IFAC be calculated as anby infinite sum of the matrixsubsystem quadratic forms made up by the products of the Faddeev matrices obtained decomposing of dynamic matrix resolvents. Copyright © IFAC matrices obtained by decomposing of linear linear subsystem dynamic matrix resolvents. Copyright © 2019 2019 IFAC IFAC © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: process identification, knowledgebase, associative search models, wavelet analysis, Gramian matrices obtained by decomposing of linear subsystem dynamic matrix resolvents. Copyright © 2019 IFAC Keywords: process identification, knowledgebase, associative search models, wavelet analysis, Gramian Keywords: process knowledgebase, associative search models, wavelet analysis, Gramian IFAC Keywords: process identification, identification, knowledgebase, associative search models, wavelet analysis, Gramian Keywords: identification, knowledgebase, associative search models, wavelet analysis, Gramian technique. IFAC  process technique. Keywords: process identification, knowledgebase, associative search models, wavelet analysis, Gramian technique. Keywords: process identification, knowledgebase, associative search models, wavelet analysis, technique. process identification, knowledgebase, associative search models, wavelet analysis, Gramian Keywords: Gramian technique.  technique. Keywords: process identification, knowledgebase, associative search models, wavelet analysis, Gramian technique.  technique. technique.1. 1. INTRODUCTION INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION 1. For For nonlinear nonlinear objects, objects, aaa linear linear virtual virtual model model is is developed developed 1. 1. INTRODUCTION 1. For nonlinear objects, linear virtual model is developed 1. INTRODUCTION Industrial enterprises implementing modernization and 1. For nonlinear objects, a this linear virtual model is developed Industrial enterprises implementing 1. For nonlinear objects, a linear virtual model is developed Industrial enterprises enterprisesINTRODUCTION implementing modernization modernization and and 1. at each time step. For purpose, input vectors most Industrial implementing modernization and at time For purpose, input most 1. For nonlinear objects, linear virtual model is developed at each each time step. step. Foraaa this this purpose, input vectors vectors most Industrial enterprises implementing modernization and 1. INTRODUCTION 1. For nonlinear objects, linear virtual model is developed equipment replacement programs aimed at higher capacity at each time step. For this purpose, input vectors most equipment replacement programs aimed at higher capacity 1. at For nonlinear objects, linear virtual model isthe developed Industrial enterprises implementing modernization and each time step. For this purpose, input vectors most equipment replacement programs aimed at higher capacity close (subject to the specified criterion) to current Industrial enterprises implementing modernization and equipment replacement programs aimed at higher capacity close (subject to specified criterion) to current at each time step. For purpose, input most Industrial enterprises implementing modernization and 1. For close (subject to the the specified criterion) tovectors the current equipment replacement programs aimed at higher nonlinear objects, a this linear virtual model isthe developed at each time step. For this purpose, input vectors most and product quality need high-reliability power close (subject to the specified criterion) to the current and product quality consistency need high-reliability power at each timevector step. Forspecified this purpose, input vectors most equipment programs aimed at higher capacity capacity close (subject to the criterion) to the current and productreplacement quality consistency consistency need high-reliability power Industrial enterprises implementing modernization and equipment replacement programs aimed at capacity input data are being selected from the process input data vector are being selected from the process and product quality consistency need high-reliability power close (subject to the specified criterion) to the current equipment replacement programs aimed at higher higher capacity input data vector are being selected from the process and product quality consistency need high-reliability power at each time step. For this purpose, input vectors most close (subject to the specified criterion) to the current supply. The problem is becoming increasingly important for input data vector are being selected from the process supply. The problem is becoming increasingly important for close (subject to the specified criterion) to the current and product quality consistency need high-reliability power input data vector are being selected from the process supply. The problem is becoming increasingly important for equipment replacement programsneed aimed at higher capacity historian. The prediction is being being furtherfrom donethe by process solving and product product quality consistency consistency need high-reliability power historian. The prediction is further done by solving supply. The problem is becoming increasingly important for input data vector are being selected and quality high-reliability power historian. The prediction is being further done by solving supply. The problem is becoming increasingly important for close (subject to the specified criterion) to the current input data vector are being selected from the process the sites where AC motors are key consumers. historian. The prediction is being further done by solving the sites sites where AC motors motors are key keyneed consumers. input dataThe vector are being selected from the process supply. The problem is becoming increasingly important for historian. prediction is being further done by solving the where AC are consumers. and product quality consistency high-reliability power the appropriate appropriate systems of linear equations by means of supply. The problem is increasingly important for the systems of linear equations by means of the sites where AC motors are key consumers. historian. prediction is being further done by solving supply. The problem is becoming becoming increasingly important for the appropriate systems of linear equations by means of the sites where AC motors are key consumers. input dataThe vector are being selected from the process historian. The prediction is further done by solving the appropriate systems of linear equations by means of historian. The prediction is being being further done by solving the sites where AC motors are key consumers. the appropriate systems of linear equations by means of supply. The problem is becoming increasingly important for the sites where AC motors are key consumers. least squares technique. Modern production processes need reliable power supply least squares Modern production processes need reliable supply the systems of linear equations by means of the sites where AC motors are key consumers. leastappropriate squares technique. historian. Thetechnique. prediction is being done solving Modern production processes need reliable power supply the appropriate systems linear equations by means of Modern production processes need reliablethepower power supply least squares technique. the appropriate systems of of linear further equations byby means of least squares technique. Modern production processes need reliable power supply the sites where AC motors are key consumers. with consistent power quality. In particular, prediction of with consistent power quality. In particular, the prediction of least squares technique. Modern production processes need reliable power supply the appropriate systems of linear equations by means of with consistent power quality. In particular, the prediction of least squares technique. 2. The stability of the linear dynamic object state under Modern production processes need reliable power supply 2. The stability of the linear dynamic object state under with consistent power quality. In particular, the prediction of least squares technique. Modern production processes need reliable power supply 2. The stability of the linear dynamic object state under with consistent power quality. In particular, the prediction of electric power system’s state approach to stability threshold 2. The stability of the linear dynamic object state under electric power system’s state approach approach to stability stability threshold with power quality. In particular, the prediction of The stability of the linear dynamic object state under least squaresis technique. electric power system’s state to threshold Modern production reliable supply prediction is further investigated with the help of with consistent consistent powerprocesses quality. Inneed particular, thepower prediction of 2. prediction further investigated with the help of electric power system’s state approach to stability threshold 2. The stability of the linear dynamic object state under with consistent power quality. In particular, the prediction of prediction is of further investigated with the help of electric power system’s state approach to stability threshold 2. The stability the linear dynamic object state under is prediction is further investigated with the help of is important. The stability offurther the (Antoulas, linear dynamic object state under electric power system’s state approach to stability threshold is investigated with the help of is important. important. with consistent power quality. In particular, the prediction of 2. prediction electric power system’s state approach to stability threshold Gramian techniques 2005). Unlimited growth Gramian techniques (Antoulas, 2005). Unlimited growth is important. prediction is further investigated with the help of electric power system’s state approach to stability threshold Gramian techniques (Antoulas, 2005).with Unlimited growth is important. 2. The stability offurther the (Antoulas, linear dynamic object state under prediction is investigated the help of Gramian techniques 2005). Unlimited growth prediction isnorm further with the help of is important. Gramian techniques 2005). Unlimited growth electric power system’s state approach to stability threshold of Frobenius Frobenius in(Antoulas, theinvestigated controllability Gramian is used used is important. The following of norm in the controllability Gramian is The following scheme is used for predicting process’s state Gramian techniques (Antoulas, 2005). Unlimited growth is important. of Frobenius norm in the controllability Gramian is used prediction is further investigated with the help of The following scheme scheme is is used used for for predicting predicting process’s process’s state state Gramian techniques (Antoulas, 2005). Unlimited growth The following scheme is used for predicting process’s state of Frobenius norm in the controllability Gramian is used Gramian techniques (Antoulas, 2005). Unlimited growth of Frobenius norm in the controllability Gramian is used The following scheme is used for predicting process’s state is important. as an indicator of the approaching dangerous limit. approach to critical limits. First, we offer a predictive model as an indicator of the approaching dangerous limit. approach to critical limits. First, we offer aa predictive model of Frobenius norm in the controllability Gramian is used The following scheme is used for predicting process’s state as an indicator of the approaching dangerous limit. Gramian techniques (Antoulas, 2005). Unlimited growth approach to critical limits. First, we offer predictive model of Frobenius norm in the controllability Gramian is used The following scheme is used for predicting process’s state as an indicator of the approaching dangerous limit. approach to critical limits. First, we offer a predictive model of Frobenius norm in the controllability Gramian is used The following scheme is used for predicting process’s state as an indicator of the approaching dangerous limit. approach to critical limits. First, we offer a predictive model development procedure based on associative search technique development procedure based on associative search technique as an indicator of the dangerous limit. approach to limits. First, we offer model of Frobenius norm inapproaching the Gramian is used development procedure based onfor associative search technique as an indicator of the approaching dangerous limit. The following scheme is used predicting process’s state approach to critical critical limits. First, we offer aaa predictive predictive model Gramian method based oncontrollability new mathematical mathematical technique Gramian method on aaaa new technique development procedure based on associative search technique as an indicator of based the approaching dangerous limit. approach to critical limits. First, we offer predictive model Gramian method based on new mathematical technique development procedure based on associative search technique (Bakhtadze et al., 2012). The predictive model structure at Gramian method based on new mathematical technique (Bakhtadze et al., 2012). The predictive model structure at development procedure based on associative search technique Gramian method based on a new mathematical technique as an indicator of the approaching dangerous limit. (Bakhtadze et al., 2012). The predictive model structure at approach to critical limits. First, we offer a predictive model of solving Lyapunov and Silvester differential and development procedure based on associative search technique of Lyapunov and Silvester differential and (Bakhtadze et al., 2012). The model structure at Gramian method based on aa new mathematical technique development procedure based onpredictive associative search technique of solving solving Lyapunov and Silvester differential and (Bakhtadze et al., 2012). The predictive model structure at Gramian method based on mathematical technique each time step is based on system’s knowledge. of solving Lyapunov and Silvester differential and each time step is based on system’s knowledge. Gramian method based on a new new mathematical technique (Bakhtadze et al., 2012). The predictive model structure at of solving Lyapunov and Silvester differential and each time step is based on system’s knowledge. development procedure based on associative search technique algebraic equations that was developed over the past ten (Bakhtadze et al., 2012). The predictive model structure at algebraic equations that was developed over the past ten each time step on system’s knowledge. of solving Lyapunov and Silvester differential (Bakhtadze et is al.,based 2012). The predictive model structure at algebraic equations thaton was developed over thetechnique past and ten each is on knowledge. Gramian method based a new mathematical of solving Lyapunov and Silvester differential and algebraic equations that was developed over the past ten of solving Lyapunov and Silvester differential and each time time step step isal.,based based on system’s system’s knowledge. algebraic equations that was developed over the past ten (Bakhtadze et 2012). The predictive model structure at each time step is based on system’s knowledge. years in the Institute of Control Sciences for assessment Automatic control systems based on knowledge processing years in Institute of Sciences for assessment Automatic control systems based knowledge algebraic equations that was over past ten each time step is based on system’s knowledge. years in the the Institute of Control Control Sciences for the assessment of solving Lyapunov and developed Silvester differential Automatic control systems based on on knowledge processing processing algebraic equations that was developed over the past and ten years in the Institute of Control Sciences for assessment Automatic control systems based on knowledge processing algebraic equations that was developed the past ten years in the Institute of Control Sciences for assessment Automatic control systems based on knowledge processing each time step is based on system’s knowledge. of linear linear dynamic systems stability. The over method operates have been progressing quickly. Their application in control of dynamic systems stability. The method operates have been progressing quickly. Their application in control years in the Institute of Control Sciences for assessment Automatic control systems based on knowledge processing of linear dynamic systems stability. The method operates algebraic equations that was developed over the past ten have been progressing quickly. Their application in control years in the Institute of Control Sciences for assessment Automatic control systems systems basedTheir on knowledge knowledge processing of linear dynamic systems stability. The method operates have been progressing quickly. application in control years in the Institute of Control Sciences for assessment Automatic control based on processing of dynamic systems stability. operates have been progressing quickly. Their application in control by linear decomposing the Gramian matrixThe thatmethod is aaa solution solution of tasks looks justified under insufficient aaa priori information by decomposing the Gramian matrix that is of tasks looks justified under insufficient priori information of linear dynamic systems stability. The method operates have been progressing quickly. Their application in control by decomposing the Gramian matrix that is solution of years in the Institute of Control Sciences for assessment tasks looks justified under insufficient priori information of linear dynamic systems stability. The method operates Automatic control systems based on knowledge processing have been progressing quickly. Their application in control by decomposing the Gramian matrix thatmethod is aa asolution solution of tasks looks justified under insufficient aa weakly priori information of linear dynamicthe systems stability. The operates have been progressing quickly. Their of application in control by decomposing Gramian matrix that is of tasks looks justified under insufficient priori information the Lyapunov-Silvester-Krein equation into spectrum about the plant as well as in cases formalized the Lyapunov-Silvester-Krein equation into spectrum about the as as in formalized by decomposing Gramian matrix that is aa aasolution of tasks looks justified under aa weakly priori information thelinear Lyapunov-Silvester-Krein equation into spectrum of dynamicthe systems stability. The operates about been the plant plant as well well as insufficient in cases cases of weakly formalized by decomposing the Gramian matrix that is of have progressing quickly. Their of application in control tasks looks justified under insufficient priori information the Lyapunov-Silvester-Krein equation into spectrum about the plant as well as in cases of formalized by decomposing the Gramian matrix thatmethod is a aasolution solution tasks looks justified under insufficient a weakly priorimodels. information the Lyapunov-Silvester-Krein equation into spectrum about the plant as well as in cases of weakly formalized of dynamics dynamics matrices that make make up these these equations. It of is signals, semi-structured systems, and nonlinear of matrices that up equations. It is signals, semi-structured systems, and nonlinear models. the Lyapunov-Silvester-Krein equation into aasolution spectrum about the plant as well as in cases of weakly formalized of dynamics matrices that make up these equations. It is by decomposing the Gramian matrix that is a signals, semi-structured systems, and nonlinear models. the Lyapunov-Silvester-Krein equation into spectrum tasks looks justified under insufficient a priori information about the plant as well as in cases of weakly formalized of dynamics matrices that make make up these these equations. It of is signals, semi-structured and nonlinear models. the Lyapunov-Silvester-Krein equation into a spectrum about the plant as wellsystems, as in cases of weakly formalized of dynamics matrices that up equations. It is signals, semi-structured systems, and nonlinear models. more effective comparing to modal analysis in some more effective comparing to modal analysis in some of dynamics matrices that make up these equations. It is signals, semi-structured systems, and nonlinear models. more effective comparing to equation modal analysis in some some the Lyapunov-Silvester-Krein into a spectrum of dynamics matrices that make up these equations. It about the plant as great wellsystems, as in cases ofmade weakly formalized signals, semi-structured and nonlinear models. In the recent years, advances were made in the fields of of more effective comparing to modal analysis in of dynamics that make up the these equations. It is is In years, advances were in the fields signals, semi-structured and nonlinear models. more effective comparing to modal analysis in In the the recent recent years, great greatsystems, advances were made in the fields of domains as it itmatrices allows to prescribe prescribe criterion of some EPS domains as allows to the criterion of EPS more effective comparing to modal analysis in some In the recent years, great advances were made in the fields of domains as it allows to prescribe the criterion of EPS of dynamics matrices that make up these equations. It is more effective comparing to modal analysis in some In the recent years, great advances were made in the fields of signals, semi-structured systems, and nonlinear models. intelligent systems (of various types), including decisiondomains as it allows to prescribe the criterion of EPS more effective comparing to evaluate modal analysis inof some intelligent systems (of various types), including decisionIn the recent years, great advances were made in the fields of domains as it allows to prescribe the criterion EPS intelligent systems (of various types), including decisionstability loss risk, and also to the interaction of In the recent years, great advances were made in the fields of stability loss and also to the interaction of domains as itrisk, allows prescribe the criterion EPS intelligent systems (of various types), including decisionIn the recent years, great advances were made in the fields of stability loss risk, and to also to evaluate the interaction of more effective comparing to evaluate modal inof domains as allows to prescribe the criterion of EPS intelligent systems (of types), including decisionmaking and decision-making support, information processing stability loss risk, and also to evaluate the interaction of domains as it itrisk, allows to prescribe the analysis criterion of some EPS making and decision-making support, information processing intelligent systems (of various various types), including decisionstability loss and also to evaluate the interaction of making andsystems decision-making support, information processing their ill-stable ill-stable system modes (Grobovoi et al., 2013). In the recent years, great advances were made in the fields of intelligent (of various types), including decisiontheir system modes (Grobovoi et al., 2013). stability loss risk, and also to evaluate the interaction of making and decision-making support, information processing intelligent systems (of various types), including decisiontheir ill-stable system modes (Grobovoi et al., 2013). domains as it allows to prescribe the criterion of EPS stability loss risk, and also to evaluate the interaction of making and decision-making support, information processing and pattern recognition, diagnosis and automatic monitoring. their ill-stable system modes (Grobovoi et al., 2013). stability loss risk, and also to evaluate the interaction of and recognition, and monitoring. making andsystems decision-making support, information processing their ill-stable system modes (Grobovoi et al., 2013). and pattern pattern recognition, diagnosis and automatic automatic monitoring. intelligent (of diagnosis various types), including decisionmaking and decision-making support, information processing their system modes (Grobovoi et al., 2013). and pattern diagnosis and automatic monitoring. making andrecognition, decision-making support, information processing loss risk, and also evaluate the interaction of their ill-stable system modes (Grobovoi et 2013). and pattern recognition, diagnosis and monitoring. 3. stability The ill-stable authors also offer new technique for discrete 3. The authors also offer aaaa to new technique for discrete their ill-stable system modes (Grobovoi et al., al., 2013). and pattern recognition, diagnosis and automatic automatic monitoring. 3. The authors also offer new technique for discrete making and decision-making support, information processing and pattern recognition, diagnosis and automatic monitoring. In our method, the knowledgebase is a set of patterns 3. The authors also offer new technique for discrete In our method, the knowledgebase is aa set of patterns and pattern recognition, diagnosis and automatic monitoring. 3. The authors also offer a new technique for discrete their ill-stable system modes (Grobovoi et al., 2013). In our method, the knowledgebase is set of patterns analysis of electric power system’s stability. The analysis of electric power system’s stability. The 3. authors offer aa new technique for discrete In our method, the knowledgebase is aa set of patterns analysis of also electric power system’s stability. The 3. The The authors also offer technique for In our method, the knowledgebase isprocess of patterns and pattern recognition, diagnosis and of automatic monitoring. resulting from the intelligent analysis process history. This analysis of electric power system’s stability. The The authors also offer a new newsystem’s technique for discrete discrete resulting from intelligent analysis history. This In our method, knowledgebase aa set set of patterns of electric power stability. The resulting from the the the intelligent analysis of ofis process history. This 3. analysis technique is based on considering the contribution to the In our method, the knowledgebase is set of patterns technique is based on considering the contribution to the analysis of electric power system’s stability. The resulting from the intelligent analysis of process history. This In our method, the knowledgebase is a set of patterns technique is based on considering the contribution to the 3. The authors also offer a new technique for discrete analysis of electric power system’s stability. The resulting from the intelligent analysis of process history. This allows to use all available a priori information about the technique is based on considering the contribution to the analysis of electric power system’s stability. toThe allows to use all available aa priori information about the resulting from the intelligent analysis of process history. This technique is based on considering the contribution the allows to use all available priori information about the system’s total energy of its part, which is accumulated in In our method, the knowledgebase is a set of patterns resulting from the intelligent analysis of process history. This system’s energy of its which is in technique is based on considering the contribution the allows to use all available aa priori information about the resulting from the intelligent analysis of process history. This system’s total total energy of its part, part,system’s which is accumulated accumulated in analysis of electric power stability. to technique is on considering the to the allows to use all available information about the plant. The method can be briefly stated as follows: system’s total energy of its part, which is accumulated in technique is based based onstable considering the contribution contribution toThe the plant. The method can be briefly stated as follows: allows to use all available aa priori priori information about the system’s total energy of its part, which is accumulated in plant. The method can be briefly stated as follows: the dominant weakly modes. resulting from the intelligent analysis of process history. This allows to use all available priori information about the the dominant weakly stable modes. system’s total energy of its part, which is accumulated in plant. The method can be briefly stated as follows: allows to use all available a priori information about the the dominant weakly stable modes. technique is based on considering the contribution to the system’s total energy of its part, which is accumulated in plant. method can be stated as the dominant dominant weakly stable modes. system’s totalweakly energy stable of its part, which is accumulated in plant. The The method can be briefly briefly statedinformation as follows: follows: about the the modes. allows to method use all can available a priori plant. The be stated the dominant weakly stable modes. plant. The method can be briefly briefly stated as as follows: follows: system’s total energy of modes its part, which is accumulated in 11 the dominant weakly stable modes. The energy of weakly stable modes is calculated by means means of of The energy of weakly stable is by the dominant weakly stable modes. This work was partially supported by the research projects 1 The energy energy of of weakly weakly stable stable modes modes is is calculated calculated by means of plant. can be briefly by stated as follows: work was supported the projects The calculated by means of 111 This This The workmethod was partially partially supported by the research research projects The energy of weakly stable modes is calculated by means of the dominant weakly stable modes. spectral decomposition of Gramians or spectral This work was partially supported by the research projects 17-08-01107-a and No. No. 17-07-00235 work was partially supported by the research projects 1No. spectral decomposition of spectral The energy of weakly stable modes is calculatedor means of 1 This spectral decomposition of Gramians Gramians orby spectral No. 17-08-01107-a and 17-07-00235 The energy of is by of work partially supported by the projects No. 17-08-01107-a and No. 17-07-00235 1 This spectral decomposition of Gramians or spectral This work was wasFoundation partially supported by the research research projects The energydecomposition of weakly weakly stable stable modes modes is calculated calculatedor by means means of spectral of Gramians spectral No. 17-08-01107-a and No. 17-07-00235 work was partially supported the research projects ofThis the Russian for Basic by Research (RFBR). No. 17-08-01107-a and No. 17-07-00235 spectral decomposition of Gramians or spectral of the Russian Foundation for Basic Research (RFBR). 1 The energydecomposition of weakly stable modes is calculatedor of spectral decomposition of Gramians Gramians orby means spectral No. 17-08-01107-a and No. 17-07-00235 ofThis the Russian Foundation for Basic by Research (RFBR). No. 17-08-01107-a and No. 17-07-00235 of the Russian Foundation for Basic Research (RFBR). work was partially supported the research projects spectral of spectral No. 17-08-01107-a and No. 17-07-00235 of the Russian Foundation for Basic Research (RFBR). Copyright © 2019, 2019 IFAC 486 of the Russian Foundation for Basic Research (RFBR). spectral decomposition of Gramians or spectral Copyright 2019 IFAC 486 Hosting of the Russian Foundation for Basic Research (RFBR). 2405-8963 © IFAC (International Federation of Automatic Control) by Elsevier Ltd. All rights reserved. No. 17-08-01107-a and No. 17-07-00235 Copyright 2019 IFAC 486 of the Russian Foundation for Basic Research (RFBR). Copyright © 2019 IFAC 486 ** **,, **, , **,, *, , *

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Copyright © 2019 IFAC 486 Copyright ©under 2019 responsibility IFAC 486Control. Peer review of International Federation of Automatic of the Russian Foundation for Basic Research (RFBR). Copyright © 2019 IFAC 486 Copyright © 486 Copyright © 2019 2019 IFAC IFAC 486 10.1016/j.ifacol.2019.11.108 Copyright © 2019 IFAC 486

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The goal of the paper is to obtain a representation of another solution in the form of spectral decomposition based on integral (2) and to study some applications of the resulting decompositions.

decomposition of quadratic Frobenius norm of the system’s transfer function. In this case, the matrices of the linearized power system’s model should be known along with the weakly stable eigenvalues of its dynamics matrix (Yadykin et al., 2016b).

3. THE MAIN RESULTS

By comparing this energy with the system’s total momentum energy one can estimate the system’s proximity to stability threshold. This allows, in particular, to detect rather quickly the power grid’s trajectory towards cascading failure and identify the potential center of its unstable oscillations. It was proved (Sukhanov et al., 2012) that the investigation of the Frobenius norm of the dynamic objects’s transfer function is sufficient for this purpose.

Consider decomposition of matrices �, �∗ in the form ���

��� � ���� � � � � �� � ���� ���, ���

∗ ��

���� � � �

In the paper (Bakhtadze et al., 2012) this technique was extended for discrete-time bilinear systems. The suggested approach has a transparent physical interpretation, i.e., stability loss risk corresponds to the energy of ill-stable modes group.

��� � 1� � ����� � ��������� � �����, ���� � �, ���� � � � ����,

where � � �� , � � �� , � � �� .

The bilinear system controllability Gramian is the limit solution obtained as a result of the following iterative procedure fulfillment: � � ��� �� , ���

2. PROBLEM STATEMENT

where

Consider the discrete algebraic Lyapunov equation

�∗ �1 � � �11 � ���∗ ,

(1)

�∗ �21 � � �21 � �Τ �11 � � �,

In (1), А, H and Y – are complex � � � matrices. It is a special case of the Sylvester-Lyapunov-Krein (Daleckii and Krein, 1974) equations:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

� ��� ��∗ �� ��� � �,

It is convenient to present the iterative procedure consisting of two stages, the first of which corresponds to the solution of the second equation (3), and the second to the solution of the remaining equations, with the iterations starting from the second stage.

�,���

in which the coefficients ��� are assumed to be real numbers. The discrete Lyapunov equation is obtained by setting ��� � 1, ��� � ��� � �, ��� � �1, � � 1.

At the first stage the solution of the discrete Lyapunov equation of the linear subsystem (3) is sought, the solution found is the Gramian of controllability of the linear system �� :

The solution of the Lyapunov equation can be represented as infinite series in products of the original matrices powers and their adjoint (Benner and Damm, 2011) �

��∗ �� ��� .

���

The solution can be represented in the form of the matrix integral �

�

� � ��� �� � �� � � �� � ���� .

���

1 � ��� ��� � � �∗ ��� � ���� �� � � ���� ��. �� 2�

(3)

� � � �Τ ���1 � � �, . . . � � �, . . . �. �∗ �� � � �

�

���

���

Consider the bilinear discrete stationary system of the form

In this paper the decomposition method disseminates to the transfer functions and energy functionals of bilinear systems based on the multi-dimensional Laplace transform. Based on these results, the associative search method can be applied to the study of bilinear systems.

� � � �� � �.�

� � � � ��∗ � ���� ∗ ���,

where �� ���, ��∗ ��� denote the characteristic polynomials of the matrices �, �∗ , which are known as Faddeev matrices. They can be found by applying the Faddeev-Leverrier algorithms.

Bilinear models for studying the dynamics of electric power systems, in particular, resistance in small and transient stability, are used to determine the effect on the stability of inter-district oscillations (Al-Bayet et al., 1993; Arroyo et al., 2005).

∗

���

���

In the (Alessandro, et al., 1974) it had been found that if the spectrum of the matrix A belongs to the interior of the unit circle, the iterative procedure (3) gives, in the limit, the solutions of the matrix Lyapunov equation:

(2)

�

�∗ �� � � � � � �� � ��� � �.� 487

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4. BILINEAR SYSTEM TRANSFER FUNCTION DECOMPOSITION

Theorem. Suppose that the linear subsystem is stable, the spectral radius of the matrix N is less than unity, and the subsystem transfer function is strictly proper.

Consider the transfer function of the bilinear system Volterra kernels of order ��� Τ

�1

Then the following statements are true.

�1

�� ��1 , �2 , . . . �� � � � ��� � � �� � ��� ���1 � � �� �.

�

i)

���

��1

�� ��1 , . . . � � � � � � �

��� ��1 , . . . � � � �

We have introduced the designations ���� – the characteristic polynomial of the linear subsystem, �� , �� , �� – poles of the transfer function of the linear subsystem.

����� ����� � ��1 ��0 � �Τ Н�� ��1 , . . . � � ��. �

�� �

��� �1 ��1 , . . . � �

� �1

�,

Formulas (4), (5) allow us to calculate and analyze the transfer functions of a bilinear system based on the theory of linear dynamic systems.

In the first, we shall consider the spectral decomposition of the controllability Gramian of the linear subsystem (Benner and Damm, 2011).

The main method of analysis is based on the iterative construction of the transfer function and the use of Faddeev matrices for describing the transfer function as a linear subsystem. Its refinement can obtained by using a finite number of Volterra matrix kernels.

��0

�

��1 ��1

� � � � ��1 � ��1 ��0 ��0

�

� �� ����1 � �

����� �����1 � �

��∗ ��Τ �� .

The bilinear system first kernel is a solution of the equation

Formulas (4), (5) make it possible to calculate the square of the H2-norm of the bilinear system transfer function and its decomposition at the spectrum of the linear subsystem dynamics matrix. So, in turn, it makes possible to use tools of energy functionals of linear systems for analyzing the stability of bilinear systems (Yadykin, 2016a; Zhang and Lam, 2002).

�∗ ��1 � � ��1 � ��Τ ��0 �.

A convenient tool for an iterative procedure is the method of mathematical induction. To calculate the Gramian of controllability kernel with the number �� we have

�

�

� �� ����� �� ��� �� ���� � � ∗ � � � �� � �� � � � ���� � � � λ��� … λ��� λ�� � � ��� � � � �� � � � ��� � � � � ����������������������� � �� ������ � ��� � � �� ������ � � � � � � � ��������������������� ��� ��� ��� ��� ��� ��� ������������������������� ��� �����

For the observability Gramian, the formula is preserved with the last matrix product replaced by the matrix product ��� ���� �Τ ��∗� �Τ ��� �Τ ��∗ . � � � ������������������� �

� � � ���� � � �

�

� � 1,2. . . �� .

� ��� ���

(6)

��� �����

��� �����

2�� times

(5)

Suppose that the Volterra series converges uniformly for values �1 � �0, ��, �2 � �0, ��, . . . �� � �0, ��.

�

the spectral decomposition (5).

� ��� ���

��� �∗� �Τ

The simplest way to construct the spectral decomposition of the solution of the Lyapunov equation of the bilinear subsystem is to use the iterative procedures proposed in (Shaker and Takavori, 2014).

��� ��1 , . . . � � � is the matrix kernel of the Volterra series in

� ��� ���

(4)

���� , . . . �� . . � � � �� ��� , . . . �� �, �

ii)

483

� ��� ���

�

Corollary. Let give the expressions for the spectral expansions of the square H2-norm of the transfer function of the bilinear system Volterra kernels

�

� �� ����� �� ��� �� ���� � �

� � � �∗� �� ��� � � �� ��� �� � ��� � �� � , � � ���� � � � � � ������������������������� � ��� ������� � ��� � � ��� ������ � ��������������������� ��� ��� ��� ��� ��� ��� ����������������� ��� ����� ��� �����

��� �����

488

(7)

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By analogy with the definition of the functional for the linear subsystem we call the functional (7) the energy functional of the bilinear system (Yadykin, 2016a).

system inputs and outputs for a certain time depth at each time step. Since the wavelet transformation allows displaying the properties of signals not only in the frequency domain, but also displays changes in frequency characteristics over time, the use of such a transformation allows to explore objects with varying internal characteristics. For each �-th input or output signal of the system, the dynamic trace of the system is taken as a set of � previous time inputs and outputs and a wavelet transform is performed over this time series. The obtained wavelet transformation coefficients are the elements of the knowledge base.

5. PREDICTIVE INTELLIGENT MODEL DESIGN The dynamic algorithm consists in the design of an approximating hyper surface of current input vector space and the related one-dimensional outputs at every time step. To build a virtual model for a specific time step, the points close in a manner to the current input vector are selected (Fig. 1). The output value at the next step and corresponding model coefficients are further calculated using Least Mean Squares (LMS). For identification modeling of nonlinear nonstationary processes, which include a significant number of industrial processes in industry and energy, associative search identification method (Bakhtadze et al., 2012) have shown a high efficiency.

In this case, the discrete Haar-wavelet transformation is used: ��,� ��,� � ������ ���� � �, ���,� ��,� ��,� � ������ ���� � �, ���,

where: ��,� , ��,� are respectively the vectors of the approximating and detailing coefficients of the wavelet transformation of the input at time �, ��,� , ��,� are the vectors of the approximating and detailing coefficients wavelet transformation of output at time �.

In this case, the predictive identification model is as follows: �

�

�

�� � �� � � �� ���� � � � ��� ����,� , ���

Thus, the set of inputs and outputs of the system is converted into elements of the knowledge base, where dynamic imprints of the system behavior are stored at each moment of time. For multidimensional systems, the knowledge base is filled using the same method for each component.

��� ���

where �� is the prediction of the model at the moment �, �� is the input vector, � is the memory depth by outputs, � is the memory depth by inputs, � is the length of the output vector.

Unlike the usual regression model, in this linear model of a non-linear process there are only inputs (and corresponding outputs) selected from the knowledge base in accordance with a certain criterion of “similarity” and “associativity”.

7. CRITERIA FOR THE SELECTION OF KNOWLEDGE BASE ELEMENTS The criterion for selection of the elements of the knowledge base for constructing a local model is determined depending on the properties of the researched system, and can be of both a logical and metric type. Next shows two metric selection criteria for the inputs and outputs of the system and for dynamic traces. For the first selected criterion for the selection of M inputs and the corresponding outputs of the system, the Euclidean metric is used between the knowledge input vector of the knowledge base and the input vector for which the output is being predicted. Among all the elements of the knowledge base, only those are selected for which the Euclidean metric is minimal. �

�

Knowledge  Base

���

�

������������� � � ���� �� , ���

where l is the dimension of the input vector � �� , ����� ��� are input vectors which are selected from data base. The matrix of selected vectors is checked for conditionality, and if necessary, the set of selected elements is supplemented.

Fig. 1. Approximation of nonlinear dynamics by a virtual (locally linear) model. 6. KNOWLEDGE BASE

The second selection criterion considered for � inputs is based on the choice of such system inputs for which dynamic footprints would be the closest in terms of object dynamics. Using dynamic fingerprints constructed using wavelet transformation of the input history of a certain depth allows building predictive models in the case of non-stationary

The knowledge base stores the most complete set of inputs and outputs of the system throughout the whole lifecycle of the object, as well as its “dynamic casts”. The dynamic cast is a multiple-scale wavelet-wavelet decomposition of the 489

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485

internal properties of the system, since selection does not consider the system inputs themselves, which may not reflect the current properties of the system, but their dynamic properties.

The developed approach opens up the possibility of analyzing transient stability in large electric power systems (shortcircuit modes) using linear models of static stability of power systems (stability in small).

The wavelet analysis is based on applying a special linear transform of processes to study real data interpreted by these processes, characterizing processes and physical properties of real plants, in particular, technological processes. Such a linear transform is implemented by use of special soliton-like functions (wavelets) forming an orthonormal basis in �� .

REFERENCES Al-Baiyat, S., Farag, A.S., and M. Bettayeb (1993). Transient approximation of a bilinear two-area interconnected power system, Electric Power Systems Research, vol. 26, no 1, pp. 11-19. Alessandro, P.D., Isidori, A., and A. Ruberti, (1974). Realization and structure theory of bilinear dynamic systems, SIAM Journal on Control, vol. 12, pp. 517-535. Antoulas, T. (2005). Approximation of Large-Scale Dynamical Systems, SIAM, Philadephia, 479 pp. Arroyo, J., Betancourt, R., Messina, A.R., and E. Barocio (2007). Development of bilinear power system representations for small signal stability analysis, Electric Power Systems Research, vol. 77, no. 10, pp. 1239-1248. Bakhtadze, N., Lototsky, V., Vlasov, S., and E. Sakrutina (2012). Associative Search and Wavelet Analysis Techniques in System Identification, IFACPapersOnLine, vol. 45, no. 16, pp. 1227-1232. Benner, P. and T. Damm (2011). Lyapunov equations, Energy Functionals and Model Order Reduction of Bilinear and Stochastic Systems, SIAM Journal on Control and Optimization, vol. 49, no. 2, pp. 686-711. Daleckii, Yu.L., and M.G. Krein (1974). Stability of Solutions of Differential Equations in Banach Space. Am. Math. Soc., Providence, R.I. Grobovoi, A., Shipilova, V, Arestova, A., Yadykin, I., Afanasyev, V., Iskakov, A., and D. Kataev (2013). Application of Gramians method for Smart Grid investigations on the example of the Russky Island Power Network, Proceedings of IREP 2013 Symposium ”Bulk Power Systems Dynamics and Control”, Rethymno, Greece. Paper ID 93, pp. 1-6. Shaker, H.R. and M. Takavori (2014). Generalized Hankel Interaction Index Array for Control Structure Selection for Discrete-Time MIMO Bilinear Processes and Plants, Proceedings of IEEE 53rd Annual Conference on Decision and Control (CDC), Los Angeles, CA. Pp. 3149-3154. Sukhanov, O.A., Novitsky, D.A., and I. Yadykin (2011). Method of Steady-state Stability Analysis in Large Electrical Power Systems, Proceedings of 17th Power Systems Computation Conference. Stockholm. Curran Associates, Inc., vol. 2, pp. 934-941. Yadykin, I.B. (2016a). On Spectral Decompositions of Solutions to Discrete Lyapunov Equations, Doklady Mathematics, vol. 93, no. 3, pp. 344-347. Yadykin, I.B., Bakhtadze, N.N., Lototsky, V.A., Maximov, E.M., and E.A. Sakrutina, (2016b). Stability Analysis Methods of Discrete Power Supply Systems in Industry, IFAC-PapersOnLine, vol. 49, no. 12, pp. 355-359. Zhang, L., and J. Lam (2002). On H2 model order reduction of bilinear systems, Automatica, vol. 38, vol. 2, pp. 205216.

In comparison to the Fourier transform apparatus, when a function is used generating the orthonormal basis of the space by use of the scale transform, the wavelet transform is formed by use of a basis function localized in a bounded domain and belonging to the space, i.e. to the all numerical axis. In comparison to the “window”. Fourier transform, to obtain the transformation on one frequency all time information is not already required. The Fourier transform does not provide information on local properties of a signal under fast enough changes in the time of its spectral make-up. Thus, the wavelet transform may provide one with frequency-time information on a function, which in many practical situations is more actual than information obtained by the standard Fourier analysis. The criterion for selecting the elements of the knowledge base in this case is: �

�

�

���

���

� � � ������ �� ������ � ���� � � � ������ � ���� � �, ���

����

�

����

�

where and are approximating and detailing coefficients of the multi-scale wavelet decomposition of the input vector (Bakhtadze et al., 2012). 8. CONCLUSIONS The paper discusses the identification techniques and algorithms based on knowledge processing forming and analysis. The prediction model was developed on the base of the associative search algorithm – the intelligent identification one. Along with an overview of the research results obtained by the authors in the framework of this approach, it was investigated the possibility of joint use of associative search and Gramian method for the prediction of the process dynamic’s approximation to the boundaries of the stability area. The new technique of discrete Lyapunov equations solution was suggested, which is based on matrix equations and semiexpansions of controllability Gramians. The new results were obtained on the development of the Gramians' method for “weakly nonlinear” – bilinear dynamic systems based on the application of the multidimensional Laplace transform and Volterra matrix series. The spectral decompositions of multidimensional transfer functions, controllability and observability Gramians, energy functionals for bilinear control systems with one input and one output are obtained. 490