Nonlocal symmetry, optimal systems, and explicit solutions of the mKdV equation

The nonlocal symmetry of the mKdV equation is obtained from the known Lax pair; it is successfully localized to Lie point symmetries in the enlarged space by introducing suitable auxiliary dependent variables. For the closed prolongation of the nonlocal symmetry, the details of the construction for a one-dimensional optimal system are presented. Furthermore, using the associated vector fields of the obtained symmetry, we give the reductions by the one-dimensional sub-algebras and the explicit analytic interaction solutions between cnoidal waves and kink solitary waves, which provide a way to study the interactions among these types of ocean waves. For some of the interesting solutions, the figures are given to show their properties.

Lie Symmetries,1-Dimensional Optimal System and Optimal Reductions of(1+2)-Dimensional Nonlinear Schrodinger Equation

For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, classical symmetry algebra is found and 1-dimensional optimal system, up to conjugacy, is constructed. Its symmetry reductions are performed for each class, and someexamples of exact invainvariant solutions are given.

Lie Symmetries,One-Dimensional Optimal System and Optimal Reduction of(2+1)-Coupled nonlinear Schrodinger Equations

For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, the infinite dimensional Lie algebra of the classical symmetry group is found and the one-dimensional optimal system of an 8-dimensional subalgebra of the infinite Lie algebra is constructed. The reduced equations of the equations with respect to the optimal system are derived. Furthermore, the one-dimensional optimal systems of the Lie algebra admitted by the reduced equations are also constructed. Consequently, the classification of the twice optimal symmetry reductions of the equations with respect to the optimal systems is presented. The reductions show that the (1 + 2)-dimensional nonlinear Schrodinger equations can be reduced to a group of ordinary differential equations which is useful for solving the related problems of the equations.

Optimal System and Invariant Solutions on ((U_yy(t,s,y)-U_t(t,s,y))y-2sU_sy(t,s,y))y+(s²+1)U_ss(t,s,y)+2sU_s=0

The purpose of this paper is to find the invariant solutions of the reduction of the Navier-Stokes equations where s=z/y ((Uyy(t,s,y)-Ut(t,s,y))y-2sUsy(t,s,y))y+(s2+1)Uss(t,s,y)+2sUs=0 This equation is constructed from the Navier-Stokes equations rising to a partially invariant solutions of the Navier-Stokes equations. Group classification of the admitted Lie algebras of this equation is obtained. Two-dimensional optimal system is constructed from classification of their subalgebras. All invariant solutions corresponding to these subalgebras are presented.

Optimal System of Subalgebras for the Reduction of the Navier-Stokes Equations

The purpose of this paper is to find the admitted Lie group of the reduction of the Navier-Stokes equationswhere using the basic Lie symmetry method. This equation is constructed from the Navier-Stokes equations rising to a partially invariant solutions of the Navier-Stokes equations. Two-dimensional optimal system is determined for symmetry algebras obtained through classification of their subalgebras. Some invariant solutions are also found.

Potential Symmetries, One-Dimensional Optimal System and Invariant Solutions of the Coupled Burgers’ Equations

In this paper, we discuss one-dimensional optimal system and the invariant solutions of Coupled Burgers’ equations. By using Wu-differential characteristic set algorithm with the aid of Mathematica software, the classical symmetries of the Coupled Burgers’ equations are calculated, and the one-dimensional optimal system of Lie algebra is constructed. And we obtain the invariant solution of the Coupled Burgers’ equations corresponding to one element in one dimensional optimal system by using the invariant method. The results generalize the exact solutions of the Coupled Burgers’ equations.

Lie Symmetry Analysis, Optimal Systems and Explicit Solutions of the Dispersive Long Wave Equations

In this paper, the dispersive long wave equation is studied by Lie symmetry group theory. Firstly, the Lie symmetries of this system are calculated. Secondly, one dimensional optimal systems of Lie algebra and all the symmetry reductions are obtained. Finally, based on the power series method and the extended Tanh function method, some new explicit solutions of this system are constructed.

Symmetries,optimal system,exact and soliton solutions of(3+1)-dimensional Gardner-KP equation

In this research article,the(3+1)-dimensional nonlinear Gardner-Kadomtsov-Petviashvili(Gardner-KP)equation which depicts the nonlinear modulation of periodic waves,is analyzed through the Lie group-theoretic technique.Considering the Lie invariance condition,we find the symmetry generators.The pro-posed model yields eight-dimensional Lie algebra.Moreover,an optimal system of sub-algebras is com-puted,and similarity reductions are made.The considered nonlinear partial differential equation is re-duced into ordinary differential equations(ODEs)by utilizing the similarity transformation method(STM),which has the benefit of yielding a large number of accurate traveling wave solutions.These ODEs are further solved to get closed-form solutions of the Gardner-KP equation in some cases,while in other cases,we use the new auxiliary equation method to get its soliton solutions.The evolution profiles of the obtained solutions are examined graphically under the appropriate selection of arbitrary parameters.

Applications of cnoidal and snoidal wave solutions via optimal system of subalgebras for a generalized extended (2+1)-D quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering

The nonlinear evolution equations have a wide range of applications,more precisely in physics,biology,chemistry and engineering fields.This domain serves as a point of interest to a large extent in the world’s mathematical community.Thus,this paper purveys an analytical study of a generalized extended(2+1)-dimensional quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering.The Lie group theory of differential equations is utilized to compute an optimal system of one dimension for the Lie algebra of the model.We further reduce the equation using the subalgebras obtained.Besides,more general solutions of the underlying equation are secured for some special cases of n with the use of extended Jacobi function expansion technique.Consequently,we secure new bounded and unbounded solutions of interest for the equation in various solitonic structures including bright,dark,periodic(cnoidal and snoidal),compact-type as well as singular solitons.The applications of cnoidal and snoidal waves of the model in oceanography and ocean engineering for the first time,are outlined with suitable diagrams.This can be of interest to oceanographers and ocean engineers for future analysis.Furthermore,to visualize the dynamics of the results found,we present the graphic display of each of the solutions.Conclusively,we construct conservation laws of the understudy equation via the application of Noether’s theorem.

Distributed Minimum-Energy Containment Control of Continuous-Time Multi-Agent Systems by Inverse Optimal Control

Dear Editor,This letter focuses on the distributed optimal containment control of continuous-time multi-agent systems(CTMASs)with respect to the minimum-energy performance index over fixed topology.To achieve this,we firstly investigate the optimal containment control problem using the inverse optimal control method,where all states of followers asymptotically converge to the convex hull spanned by the leaders while some quadratic performance indexes get minimized.A sufficient condition for existence of the distributed optimal containment control protocol is derived.By introducing the parametric algebraic Riccati equation(PARE),it is strictly proved that the global performance index can be used to approximate the standard minimumenergy performance index as the parameters tends to infinity.In consequence,the standard minimum-energy cooperative containment control can be solved by local steady state feedback protocols.

A Special Issue“Planning and Optimal Operation of New-Type Power System”of Global Energy Interconnection

The power system,as an energy hub,plays a crucial role in the transformation of energy production and consumption.On July 19,2023,the International Energy Agency(IEA)released a Global Electricity Market Report for 2023-2024.This report indicates that the development of the world’s energy production is rapidly moving towards the critical point where the proportion of electricity generated from renewable sources surpasses that from non-renewable sources.

Adaptive Optimal Output Regulation of Interconnected Singularly Perturbed Systems With Application to Power Systems

This article studies the adaptive optimal output regulation problem for a class of interconnected singularly perturbed systems(SPSs) with unknown dynamics based on reinforcement learning(RL).Taking into account the slow and fast characteristics among system states,the interconnected SPS is decomposed into the slow time-scale dynamics and the fast timescale dynamics through singular perturbation theory.For the fast time-scale dynamics with interconnections,we devise a decentralized optimal control strategy by selecting appropriate weight matrices in the cost function.For the slow time-scale dynamics with unknown system parameters,an off-policy RL algorithm with convergence guarantee is given to learn the optimal control strategy in terms of measurement data.By combining the slow and fast controllers,we establish the composite decentralized adaptive optimal output regulator,and rigorously analyze the stability and optimality of the closed-loop system.The proposed decomposition design not only bypasses the numerical stiffness but also alleviates the high-dimensionality.The efficacy of the proposed methodology is validated by a load-frequency control application of a two-area power system.

Sequential Inverse Optimal Control of Discrete-Time Systems

This paper presents a novel sequential inverse optimal control(SIOC)method for discrete-time systems,which calculates the unknown weight vectors of the cost function in real time using the input and output of an optimally controlled discrete-time system.The proposed method overcomes the limitations of previous approaches by eliminating the need for the invertible Jacobian assumption.It calculates the possible-solution spaces and their intersections sequentially until the dimension of the intersection space decreases to one.The remaining one-dimensional vector of the possible-solution space’s intersection represents the SIOC solution.The paper presents clear conditions for convergence and addresses the issue of noisy data by clarifying the conditions for the singular values of the matrices that relate to the possible-solution space.The effectiveness of the proposed method is demonstrated through simulation results.

Optimal scheduling of a township integrated-energy system using the adjustable heat-electricity ratio model

With the expansion and implementation of rural revitalization strategies,there is a constant need for new energy sources for the construction of new townships.Consequently,integrated energy systems with the interconnection and interaction of multiple energy sources are developing rapidly.Biomass energy,a renewable green energy source with low pollution and wide distribution,has significant application potential in integrated energy systems.Considering the application of biomass energy in townships,this study established an integrated biomass energy system and proposed a model to optimize its operation.Lowest economic cost and highest clean energy utilization rate were considered as the objective functions.In addition,a plan was suggested to adjust the heat-electricity ratio based on the characteristics of the combined heat and power of the biomass.Finally,a simulation analysis conducted for a town in China was discussed,demonstrating that the construction of a township integrated-energy system and the use of biomass can significantly reduce operating costs and improve the energy utilization rate.Moreover,by adjusting the heat-electricity ratio,the economic cost was further reduced by 6.70%,whereas the clean energy utilization rate was increased by 5.14%.

Multi-Time Scale Optimal Scheduling of a Photovoltaic Energy Storage Building System Based on Model Predictive Control

Building emission reduction is an important way to achieve China’s carbon peaking and carbon neutrality goals.Aiming at the problem of low carbon economic operation of a photovoltaic energy storage building system,a multi-time scale optimal scheduling strategy based on model predictive control(MPC)is proposed under the consideration of load optimization.First,load optimization is achieved by controlling the charging time of electric vehicles as well as adjusting the air conditioning operation temperature,and the photovoltaic energy storage building system model is constructed to propose a day-ahead scheduling strategy with the lowest daily operation cost.Second,considering inter-day to intra-day source-load prediction error,an intraday rolling optimal scheduling strategy based on MPC is proposed that dynamically corrects the day-ahead dispatch results to stabilize system power fluctuations and promote photovoltaic consumption.Finally,taking an office building on a summer work day as an example,the effectiveness of the proposed scheduling strategy is verified.The results of the example show that the strategy reduces the total operating cost of the photovoltaic energy storage building system by 17.11%,improves the carbon emission reduction by 7.99%,and the photovoltaic consumption rate reaches 98.57%,improving the system’s low-carbon and economic performance.

Robust optimal dispatch strategy of integrated energy system considering CHP-P2G-CCS

Integrated energy systems(IESs)can improve energy efficiency and reduce carbon emissions,essential for achieving peak carbon emissions and carbon neutrality.This study investigated the characteristics of the CHP model considering P2G and carbon capture systems,and a two-stage robust optimization model of the electricity-heat-gascold integrated energy system was developed.First,a CHP model considering the P2G and carbon capture system was established,and the electric-thermal coupling characteristics and P2G capacity constraints of the model were derived,which proved that the model could weaken the electric-thermal coupling characteristics,increase the electric power regulation range,and reduce carbon emissions.Subsequently,a two-stage robust optimal scheduling model of an IES was constructed,in which the objective function in the day-ahead scheduling stage was to minimize the start-up and shutdown costs.The objective function in the real-time scheduling stage was to minimize the equipment operating costs,carbon emission costs,wind curtailment,and solar curtailment costs,considering multiple uncertainties.Finally,after the objective function is linearized with a ψ-piecewise method,the model is solved based on the C&CG algorithm.Simulation results show that the proposed model can effectively absorb renewable energy and reduce the total cost of the system.

Two-Stage Optimal Scheduling of Community Integrated Energy System

From the perspective of a community energy operator,a two-stage optimal scheduling model of a community integrated energy system is proposed by integrating information on controllable loads.The day-ahead scheduling analyzes whether various controllable loads participate in the optimization and investigates the impact of their responses on the operating economy of the community integrated energy system(IES)before and after;the intra-day scheduling proposes a two-stage rolling optimization model based on the day-ahead scheduling scheme,taking into account the fluctuation of wind turbine output and load within a short period of time and according to the different response rates of heat and cooling power,and solves the adjusted output of each controllable device.The simulation results show that the optimal scheduling of controllable loads effectively reduces the comprehensive operating costs of community IES;the two-stage optimal scheduling model can meet the energy demand of customers while effectively and timely suppressing the random fluctuations on both sides of the source and load during the intra-day stage,realizing the economic and smooth operation of IES.

A new optimal adaptive backstepping control approach for nonlinear systems under deception attacks via reinforcement learning

In this paper,a new optimal adaptive backstepping control approach for nonlinear systems under deception attacks via reinforcement learning is presented in this paper.The existence of nonlinear terms in the studied system makes it very difficult to design the optimal controller using traditional methods.To achieve optimal control,RL algorithm based on critic–actor architecture is considered for the nonlinear system.Due to the significant security risks of network transmission,the system is vulnerable to deception attacks,which can make all the system state unavailable.By using the attacked states to design coordinate transformation,the harm brought by unknown deception attacks has been overcome.The presented control strategy can ensure that all signals in the closed-loop system are semi-globally ultimately bounded.Finally,the simulation experiment is shown to prove the effectiveness of the strategy.

Optimal Operation Strategy of Electricity-Hydrogen Regional Energy System under Carbon-Electricity Market Trading

Given the“double carbon”objective and the drive toward low-carbon power,investigating the integration and interaction within the carbon-electricity market can enhance renewable energy utilization and facilitate energy conservation and emission reduction endeavors.However,further research is necessary to explore operational optimization methods for establishing a regional energy system using Power-to-Hydrogen(P2H)technology,focusing on participating in combined carbon-electricity market transactions.This study introduces an innovative Electro-Hydrogen Regional Energy System(EHRES)in this context.This system integrates renewable energy sources,a P2H system,cogeneration units,and energy storage devices.The core purpose of this integration is to optimize renewable energy utilization and minimize carbon emissions.This study aims to formulate an optimal operational strategy for EHRES,enabling its dynamic engagement in carbon-electricity market transactions.The initial phase entails establishing the technological framework of the electricity-hydrogen coupling system integrated with P2H.Subsequently,an analysis is conducted to examine the operational mode of EHRES as it participates in carbon-electricity market transactions.Additionally,the system scheduling model includes a stepped carbon trading price mechanism,considering the combined heat and power generation characteristics of the Hydrogen Fuel Cell(HFC).This facilitates the establishment of an optimal operational model for EHRES,aiming to minimize the overall operating cost.The simulation example illustrates that the coordinated operation of EHRES in carbon-electricity market transactions holds the potential to improve renewable energy utilization and reduce the overall system cost.This result carries significant implications for attaining advantages in both low-carbon and economic aspects.

Tilt Angle Optimality Criteria for Stand Alone PV Systems

The conventional approach to optimizing tilt angles for fixed solar panels aims to maximize energy generation over the entire year. However, in the context of a supply controlled electric grid, where solar energy availability varies, this criterion may not be optimal. This study explores two alternative optimization criteria focused on maximizing baseload supply potential and minimizing required storage capacity to address seasonality in energy generation. The optimal tilt angles determined for these criteria differed significantly from the standard approach. This research highlights additional factors crucial for designing solar power systems beyond gross energy generation, essential for the global transition towards a fully renewable energy-based electric grid in the future.