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COMMON PART

Project Number22-29-01619

Project titleAdvanced Methods for Nonlinear Crisp and Fuzzy Dynamical Models: Theory and Applications

Project LeadSidorov Denis

AffiliationFederal State Budget Educational Institution of Higher Education "Irkutsk National Research Technical University",

Implementation period | 2022 - 2023 |

Research area 09 - ENGINEERING SCIENCES, 09-402 - Hydropower engineering, new and renewable power sources

Keywordsnonlinear analysis, storage control, electrical load leveling, integral equations, fuzzy logic, semi-analytical methods, homotopy perturbation method, Adomian decomposition method, branching solution, solution blow-up, CESTAC method

PROJECT CONTENT

Annotation

The aim of this project is to study the mathematical models of power systems with energy storage and electrical load coverage using modern methods of nonlinear analysis. The investigated models are based on Volterra integral equations with discontinuous kernels. The project is aimed at solving these classes in linear, nonlinear, and multidimensional cases, as well as system of integral equations. In recent years, fuzzy mathematics has found wide applications in various fields of science and technology. It is known that fuzzy dynamical systems allow one to describe a wider range of problems than classical crisp systems. Thus, in this project, using the capabilities of this research team, the fuzzy dynamical systems will be investigated to solve the energy problems of covering the load. Some of numerical and semi-analytical methods will be developed to solve these problems in both crisp and fuzzy cases, such as the homotopy analysis method, homotopy perturbation method, expansion method, Adomian decomposition method, variational iteration method and collocation method. In addition, the CESTAC method and CADNA library will be used to dynamically estimate the round-off error. These methods will be used to solve linear and nonlinear problems of large dimensions. The effectiveness of the theory and numerical methods proposed in the project will be demonstrated on the management of hybrid networks, including those based on fuzzy controllers.

Expected results

Methods for the analysis of crisp and fuzzy linear and nonlinear Volterra integral models with parameters: basic provisions and qualitative theory. Numerical and semi-analytical methods for solving fuzzy and crisp Volterra integral equations (and systems) of the first kind with discontinuous kernels in linear and nonlinear cases and in fractional cases. The scientific and applied significance of the study of this project is due to the lack of theory, analytical and numerical methods of fuzzy linear and nonlinear Volterra integral equations of the first kind with discontinuous kernels arising in the modeling of developing systems, and is also due to the need to develop effective methods of control and optimization
of energy storage and coating processes and electrical load. As applied results, it is planned to develop new methods for optimal control of the normal mode of traditional and hybrid EPS, namely: 1. Adaptive assessment of the state at the rate of the process, which does not require interactive human intervention. 2. Prediction of the EPS operation mode, taking into account RES and energy storage units. 3. Dynamic optimization of the mode on a given forecast time horizon by choosing the composition and time of the control action. In addition, as applied results, the development of new methods for optimal control of the dynamic mode of traditional and hybrid EES are planned: 1. Implementation of local control devices for normal and transient modes using the developed methods of fuzzy mathematics. 2. Layered emergency control using fuzzy mathematics.

REPORTS

Annotation of the results obtained in 2023

The theory is developed and effective numerical methods for solving crisp and fuzzy Volterra integral equations with piecewise continuous kernels are constructed. The existence and uniqueness theorems of solutions of equations of such new classes are proved. A theoretical and empirical study of convergence of numerical methods has been carried out. Namely, the sync-collocation method for the numerical solution of fuzzy Volterra integral equations with piecewise continuous kernels is theoretically justified and implemented. To evaluate and improve the accuracy of the numerical solution of fuzzy Volterra integral equations with piecewise continuous kernels, a stochastic arithmetic methodology based on the Controle et Estimation Stochastique des Arrondis de Calculs (CESTAC) method and the Control of Accuracy and Debugging for Numerical Applications (CADNA) library is employed. An estimate of the accuracy of the approximate solution is obtained, with a stopping rule using the Hausdorff criterion. Thus, the method takes into account both the uncertainty of the initial data setting and rounding errors. The issues of construction and justification of stable numerical methods for solving crisp linear and nonlinear Volterra integral equations of the second kind with a piecewise continuous kernel are investigated. A new iterative method is developed to solve this problem. The convergence theorem is proved and error estimates of the method are obtained. A stable numerical method for solving linear and nonlinear Volterra integral equations of the second kind with a piecewise continuous kernel is developed, which allows us to achieve the optimal results. Namely, to obtain the minimum error achieved at the optimal iteration of the method and the optimal approximation. The main theorem of the CESTAC method is proved: the equality of the number of common significant digits (NCSD) of two consecutive iterations with exact and approximate solutions is established. Thus, to show the accuracy and efficiency of the method, a new stopping rule was involved. An integral dynamic fuzzy linear Volterra model of the first kind was developed and validated on real data to determine the mode parameters of energy storage devices (charge level and alternating power function of storage devices). In the field of development of theory and methods of fuzzy control of nonlinear dynamic systems with delays, Takagi-Sugeno fuzzy models capable of approximating nonlinear systems with a given accuracy by means of accessory functions have been investigated. This allows the application of classical methods of control techniques for linear systems and Lyapunov-Krasovskii functional theory for fuzzy dynamic systems.

Publications

**1.** *Noeiaghdam S., Sidorov D.N.* **Application of the CESTAC method and CADNA library to control of accuracy of fuzzy Volterra integral equations with discontinuous kernels** Динамические системы и компьютерные науки: теория и приложения (DYSC 2023) : материалы 5-й Международной конференции., Иркутск, 18–23 сентября 2023 г. / ФГБОУ ВО «ИГУ» ; [отв. ред. В. Г. Антоник]. – Иркутск : Издательство ИГУ, 2023. (year - 2023)

**2.** *Noeiaghdam S., Sidorov, D., Dreglea, A.* **A novel numerical optimality technique to find the optimal results of Volterra integral equation of the second kind with discontinuous kernel** Applied Numerical Mathematics, том 186, апрель 2023, стр 202-212 (year - 2023) https://doi.org/10.1016/j.apnum.2023.01.011

**3.** *Yang T., Zou R., Liu F., Liu C., Sidorov D.* **Improved stabilization condition of delayed T-S fuzzy systems via an extended quadratic function negative-determination lemma** Chaos, Solitons & Fractals, том 175, 2, 2023, 114055 (year - 2023) https://doi.org/10.1016/j.chaos.2023.114055

**4.** *Samad Noeiaghdam, Aliona Dreglea, Denis Sidorov* **Fuzzy Volterra integral equations with piecewise continuous kernels: theory and numerical solution** arXiv, arXiv ID submit/5275685 (year - 2023)

Annotation of the results obtained in 2022

1) A theory and efficient numerical methods for solving Volterra fuzzy integral equations with piecewise continuous kernels have been developed. Existence and uniqueness theorems for solutions of such a class of fuzzy integral equations are proved. The convergence of the method of successive approximations is established and an estimate of the error of the method is obtained. An efficient stable numerical method for solving such fuzzy linear Volterra integral equations based on sinc-collocation with double exponential and single exponential decay has been developed.
2) To solve crisp linear Volterra integral equations of the first kind with piecewise continuous kernels, the polynomial spline collocation method is proposed. At the same time, to approximate the integrals during discretization, the proposed projection method uses the Gaussian quadrature formula. An estimate for the accuracy of the approximate solution is obtained. Stochastic arithmetic is used based on the CESTAC method (Contrôle et Estimation STochastique des Arrondis de Calculs) and the CADNA library (Control of Accuracy and Debugging for Numerical Applications). Using this approach, one can find the optimal parameters of the proposed projection method.
3) A new inverse nonlinear problem is posed and solved for the crisp Volterra integral equations with piecewise continuous kernels. For such Volterra integral equations of the first kind, it is assumed that the desired ones are the kernel's discontinuity curves, but the rest of the information is known. The resulting integral equation is nonlinear with respect to discontinuity curves corresponding to the integration limits. For its numerical solution, a direct discretization method with a posteriori verification of calculations has been developed and tested.
4) Questions of stability and stabilization of a class of fuzzy systems of the Takagi-Sugeno type with a time-varying delay are studied. The corresponding Lyapunov-Krasovskii functional containing special integral terms is constructed. Based on the methodology of generalized free-matrix-based integral inequality, a criterion for ensuring the stability of such fuzzy systems is formulated. The validity and feasibility of the stability criterion, as well as the method of designing fuzzy controllers, are demonstrated using the model examples.
5) An efficient strategy for charging and discharging storages based on data is proposed, taking into account wind energy forecasting intervals. Two concrete scientific results can be here distinguished. First, a power interval prediction model implemented on a long-term memory network with lower and upper bound estimation (LUBE) to quantify wind energy uncertainty. Secondly, it is an energy storage management method implemented as a Markov decision process and solved using a deep reinforcement learning method. The state space, action space, and agent-environment interaction reward function are used, and the value function is approximated via a deep Q-network. Then, according to the real-time state (such as wind power, power prediction intervals, local load, dynamic electricity price, and state of charge), the proposed approach enables an automatic charge/discharge scheduling.

Publications

**1.** *Tynda A.N., Noeiaghdam S., Sidorov D.N.* **Метод полиномиальной сплайн-коллокации для решения слабо регулярных интегральных уравнений Вольтерра I рода** «ИЗВЕСТИЯ ИРКУТСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА». СЕРИЯ «МАТЕМАТИКА», том 39, стр. 62—79 (year - 2022) https://doi.org/10.26516/1997-7670.2022.39.62

**2.** *Tynda A.N., Sidorov D.N.* **Inverse Problem for the Integral Dynamic Models with Discontinuous Kernels** Mathematics, том 10, 3945 (year - 2022) https://doi.org/10.3390/math10213945