208 results about "System of differential equations" patented technology

Methods and systems for finding optimal or near optimal solutions for generic optimization problems by an approach to minimizing functions over high-dimensional domains that mathematically model the optimization problems. Embodiments of the disclosed invention receive a mathematical description of a system, in symbolic form, that includes decision variables of various types, including real-number-valued, integer-valued, and Boolean-valued decision variables, and that may also include a variety of constraints on the values of the decision variables, including inequality and equality constraints. The objective function and constraints are incorporated into a global objective function. The global objective function is transformed into a system of differential equations in terms of continuous variables and parameters, so that polynomial-time methods for solving differential equations can be applied to calculate near-optimal solutions for the global objective function. Embodiments of the present invention also provide for distribution and decomposition of global-gradient-descent- field-based optimization methods, by following multiple trajectories, and local-gradient- descent-field-based optimization methods, by using multiple agents, in order to allow for parallel computation and increased computational efficiency. Various embodiments of the present invention further include approaches for relatively continuous adjustment of solutions to optimization problems in time, to respond to various events, changes in priorities, and changes in forecasts, without needing to continuously recalculate optimization solutions de novo. While many embodiments of the present invention are specifically directed to various classes of optimization problems, other embodiments of the present invention provide a more general approach for constructing complex hierarchical computational processes and for optimally or near optimally controlling general computational processes.

A pulse circuit to solve a system of differential equations in the pulse domain based on analog or inputs or time-encoded pulse inputs. Intrinsically linear 1-bit digital to analog converters are used as feedback elements within circuits implementing solutions to a differential equation or to a system of differential equations. The circuits may be used to implement filters. While the input to the circuit may be an analog signal, the internal signals of the circuit are time-encoded current or voltage pulses. Output of the circuit is a time-encoded pulse or a series of time-encoded pulses from which an analog output may be recovered by time decoding.

The present invention provides a fluorescence-based modeling method that is capable of capturing the dynamic changes of different membrane foulant fractions that occur in fluid filtration operations. Principal component analysis is utilized to de-convolute spectral information captured within fluorescence EEMs into principal component scores that are related to different known foulant groups. The principal component scores are then used as states within a system of differential equations representing approximate mass balances of the main foulant groups to obtain a dynamic forecasting of membrane fouling. Based on the fouling dynamics forecasted by this modeling method, an optimization strategy can be developed for estimating the optimal membrane back-washing scenario for minimizing energy consumption while maximizing clean fluid production.

The invention belongs to the technical field of aerospace modelling and discloses a method for building the six degrees of freedom movement model of a spinning aircraft. During spinning, the special geometric configuration of an aircraft and air mutually act to provide elevating force for the aircraft, and the aircraft skilfully combines a pulsed detonation engine to provide thrust. Based on classical aerodynamics theorem of momentum and theorem of moment of momentum, a nonlinear mathematical simultaneous differential equation which can descript the center of mass motion and the center of mass surrounding motion is deduced. According to known engine thrust, the track of spatial movement of the aircraft is solved with the Runge-Kutta method; according to the track of spatial movement, the corresponding thrust of the engine is solved with a difference method; the two different methods can obtain the same result, i.e. the correctness of the movement model algorithm built by the invention is proved. The invention has the effect and benefits that the invention can realize the control of flexible space movement of the spinning aircraft, and has important application value in various application fields.

The present invention relates to an attack and defense differential game-based network security defense decision determination method and a device. The method comprises the steps of constructing a state evolution model NIRM for network nodes according to an SIR model during the network attack and defense process; according to a network node state migration path, acquiring a node state change differential equation set based on the state evolution model NIRM; according to the node state change differential equation set, constructing an attack and defense differential game model ADDG; according to an attack and defense return strategy and a corresponding execution cost, executing a revenue function of the attack party and the defense party during the differential game process; according to the revenue function and the attack and defense differential game model ADDG, solving out a saddle point strategy for the attack party and the defense party through the dynamic planning method, determining an optimal defense strategy and outputting the optimal defense strategy. According to the technical scheme of the invention, the problem that actual requirements cannot be met through the traditional dynamic game analysis method can be solved. Compared with the existing network security defense strategy, the attack and defense process in the continuous and real-time engagement condition is analyzed. The timeliness, the pertinence and the guiding significance of network defense decision results are better.

The invention relates to a simulation analysis method of the vibratory response of a variable working condition single-stage heavy gear case containing a crackle fault, which comprises the following steps: establishing a gear pair 3D solid model containing a crackle fault on the basis of 3D design software SolidWorks; leading the model into finite element analysis software ansys, programming a program capable of calculating the gear meshing rigidity of any meshing positions by combining menu operations of the ansys and APDL languages and outputting a result data file of the meshing rigidity; establishing a damping gear case system kinetic model of 8 degrees-of-freedom and kinetic simultaneous differential equations considering factors of the rigidity and the damping of a shaft and the case, an input torque and a load torque of a motor, and the like and utilizing MATLAB to solve the equations so as to simulate the system vibratory response of the single-stage heavy gear case containing a crackle fault. The invention integrates the advantages of the SolidWorks, the ansys and the MATLAB and can quickly and accurately simulate the vibratory responses of variable working condition heavy single-stage heavy gear cases containing various different forms of crackle faults.

The invention relates to a city-wide infectious disease simulation method and a city-wide infectious disease simulation device. The method includes the steps of: 1) partitioning the city and classifying the people; 2) building a 4p ordinary simultaneous differential equation set which describes the dynamic process of the transmission of the disease in p (number) areas; 3) discretizing the equation set to obtain a difference equation set; 4) converting the discretized difference equation set into an open equation set by taking the connections of other cities into consideration; and 5) resolving the parameters of the equation set and visually expressing the infected people in the manner of dot density and piled bar graph. According to the epidemiological characteristics of the infectious disease and based on the data of interregional population flow across the city, the city-wide infectious disease simulation method and the city-wide infectious disease simulation device can predict and simulate the dynamic contagion process of the disease spatially in a period of time in the future, and the spatial propagation process of a typical infectious disease can be analyzed on a map in a visual way. The city-wide infectious disease simulation method and the city-wide infectious disease simulation device provide a decision support to infectious disease propagation prevention and control.

The invention discloses an evaluation method for unconventional reservoir volume transformation multi-pore media productivity contribution. The evaluation method comprises the following steps that firstly, based on the principle of effective stress and flow characteristics of a transformation zone, and a fluid-solid coupling mathematical model of interaction between a geotechnical deformation field (stress field) and a fluid flow field (seepage field) of the unconventional reservoirs is established; then a perfectly coupled method solution of the seepage field and the stress field is realizedby solving a fluid-solid perfectly coupled system of partial differential equations in a general formula mode; and finally, fluid-solid coupling interactions under different combinations of pore mediaare simulated, and the contribution ratio of three systems of base materials, natural cracks and grid cracks to the cumulative production of a volume transformation well is obtained. A set of evaluation method for unconventional reservoir volume transformation multi-pore media productivity contribution is established eventually. The evaluation method for unconventional reservoir volume transforming multi-pore media productivity contribution has the advantages that the model factor consideration is comprehensive, the evaluation method is simple and results are quantifiable, an oil and gas field can be guided timely to be subjected to development measure adjustment and comprehensive management, and a theoretical basis and technical support are provided to the unconventional reservoir to realize economic recovery.

The invention relates to a vibration response simulation analyzing method for a rolling bearing with single-point failures. On the basis of a Hertz contact theory, related knowledge about kinematics and kinetics are utilized to establish a five-degree-of-freedom nonlinear vibration mode of a rolling bearing and kinetic differential equations of the rolling bearing while influencing factors like slipping of rolling balls, oil slick rigidity and the like are taken into consideration, according to that the rolling balls change due to contact deformation is caused when the rolling balls roll through partial failures, a partial failure model of the rolling bearing is introduced into the kinetic differential equations of the rolling bearing, and the single-point failures of the rolling bearing are classified according to length-width ratios of the failures and size ratios between the failures and the rolling balls so as to establish different failure shape functions; the differential equations are solved by an ode solver in MATLAB (matrix laboratory) software, and vibration responses of the rolling bearing are simulated when an inner ring, an outer ring and the rolling balls of the rolling bearing are suffered from the single-point failures. By the aid of the vibration response simulation analyzing method, vibration responses of the bearings with failures in different sizes can be simulated. Compared with traditional methods for acquiring vibration responses of failed bearings, the vibration response simulation analyzing method has the advantages that an experimental period is short and experimental cost is saved. The computation for the vibration responses for the failed rolling bearings can lay the foundation of late failure diagnosis for the rolling bearings.

The invention discloses a method for improving simulation capacity of HVDC transmission system in the technical field of electric power system operation and control technique. The technical program includes: singlet dynamic phasor is expanded to polymorphic dynamic phasor; switch function is used for indicating the characteristic of a converter valve, polymorphic dynamic phasor model of HVDC system is built considering the non-ideal state that alternating current fundamental voltage is asymmetric and direct current is mixed with ripples; curves indicating that each variable changes with time are obtained according to algebraic differentiation equations of the model; complete curves indicating that direct voltage at rectification side of the HVDC transmission system, direct voltage at contravariance side, direct line current and current at alternating current side change with time are calculated respectively by inverse Fourier transform; and finally values at the alternating current side and the direct current side of the HVDC transmission system are calculated, thus realizing the simulation of the HVDC transmission system. The invention has the advantages that the adaption to unsymmetrical fault is good, computational efficiency is high and computation is accurate.

Embodiments provide methods and systems for constructing surrogate models for use in interactive experiences. One such embodiment begins by defining a model that includes a parametric state vector and a design variable vector and represents a real world system. Next, a first and second experiment are performed to determine a response over time of the parametric state vector and to produce a dataset of the parametric state vector and the design variable vector as a function of time. The dataset is then modified with one or more derivatives of the parametric state vector and a set of surrogate differential equations is constructed that approximates a higher derivative of the parametric state vector relative to that in the dataset and the set of surrogate differential equations is stored as a surrogate model. The surrogate model is in turn provided from memory in a manner accelerating simulated behavior in response to user-interaction with the model.

The invention provides a method for solving solution of elastic foundation beams and for back analysis of value m, belonging to the technical field of geotechnical engineering. The method comprises the following steps of: 1, firstly, converting a fourth order partial differential equation of the elastic foundation beam into a low order partial differential equation set, and setting iteration times and admissible errors; 1, taking a minimum error function of a displacement value sol.y obtained in the step one and an actually-measured displacement value as an objective function, wherein the minimum error function is the sum of squares of difference values; 3, optimizing the objective function in the second step by the application of MATLAB nonlinear optimization algorithm fmincon functions, when reaching the iteration times or the admissible errors set in the first step, stopping iteration, and in this case, the objective function reaching the minimum value; and 4, the parameters corresponding to the minimum value of the objective function being the optimum solution of horizontal resistance factor value m of the to-be-determined soil layer foundation soil. The method provided by the invention is simple in analysis process and easy for programming.

A method use line-frequency disturb the zero-sequence impedance parameter of resource charge measure mutual inductance circuit, cuts the power of a certain circuit in the mutual inductance circuit in order, puts one three-phase of the power cut circuit to earth, measures the induction voltage of the other one after it is in short-circuit; at the same time measures the zero-sequence voltage pressure loss and zero-sequence current in the other run circuit; uses the GPS technology sample the zero-sequence voltage, zero-sequence current in the mutual inductance circuit in-phase, gets the zero-sequence voltage, zero-sequence current data in the mutual inductance circuit; collects the data of each measurement point to the center computer; gets the zero-sequence impedance parameter of the mutual inductance circuit by using the least square law solute the algebraic equations, differential equations or integral equations; this equipment is composed of GPS antenna, signal input connection interface, signal transducer, embedded DPS synchromesh data collection card, launch amount card, relay group, embed PC card, the electrical source card.

The invention discloses a power distribution network CPS protective method based on bounded rationality. Equipment data of a power distribution network, running data of the power distribution network,the residence time of a historical attacker, the average reaction time, the average restoration time and other types of power grid multi-source data are fully utilized, and comprehensive power distribution network CPS risk assessment indexes are established on the basis of two aspects including a physical system and an information system, so that all line risk assessment results effectively helpa defense building party to select defense building strategies through a game learning method based on bounded rationality. A defense building efficiency value corresponding to each attack and defensebuilding strategy is obtained from two aspects of economic benefits and safety, a logarithm synthetic method is adopted to obtain final expected efficiency and average efficiency when the defense building party employs different strategies, dynamic differential equations are adopted to represent dynamic change rates when the attacker and the defense building party select certain strategies, and then a dynamic differential equation group of a game learning model is established to obtain power distribution network CPS protective satisfactory strategies.

The invention provides an under-actuated release control method based on a nonlinear space tether system. The method comprises the following steps: 1, a dumbbell model is adopted, system dynamics differential equations are built according to a second kind Lagrange equation, and a dimensionless form is introduced in the differential equations; 2, based on a normal form expression for the above dynamics differential equations, expected values for a system in-plane pitching angle and an out-plane rolling angle are set, and the balance position of the system in the release process is obtained; 3, starting from the balance position of the system, a tether length change control law for realizing tether release is deduced and obtained; and 4, the range for the expected in-plane pitching angle is determined to ensure the tether release process to be asymptotically stable under effects of the control law. Numerical simulations show that the space tether can be released asymptotically stably under effects of the release control method provided by the invention, and an attractor capable of covering the balance point and an attraction domain with enough thickness can be found out.

The invention relates to a locating diagnosis method for malfunctions in an outer ring of a rolling bearing. The method includes the steps that firstly, a non-linear kinetic model of a rolling bearingsystem is created, and according to a second-type Lagrange equation, kinetic differential equations of the bearing system are deducted; secondly, on the basis of the kinetic differential equations ofthe system, vibration acceleration signal is simulated in the horizontal and vertical directions of the bearing system when the bearing system is positioned at different malfunctioned corners; thirdly, according to simulation data, a least square method is adopted for fitting a linear relation expression of the position of the malfunctioned corner at the outer ring and a synchronous root-mean-square value in the horizontal and vertical directions; fourthly, locating diagnosis base data of a bearing system to be diagnosed is measured and obtained, a synchronous root-mean-square value in the horizontal and vertical directions of the locating diagnosis base data is calculated, and therefore the relation expression of the position theta of the malfunction corner at the outer ring of the bearing system to be diagnosed and the synchronous root-mean-square value in the horizontal and vertical directions can be determined; finally, by adopting a position locating law for malfunctioned cornersof outer rings, the position theta d of the malfunctioned corner of the outer ring to be diagnosed is diagnosed. Precise locating diagnosis of the malfunctions in the outer ring of the bearing is achieved, and the method has significant theoretical meaning and application value.

The invention discloses an electric power system transient stability calculation method based on a semi-implicit Runge-Kutta method. The method includes: establishing a dynamic element model for transient stability simulation calculation of the power system; forming a whole system differential algebraic equation set; inputting raw data and information, carrying out tide calculation, obtaining an operation parameter initial value before system disturbance; calculating an initial value of the state variable; generating a Jacobian matrix and a network admittance matrix of each dynamic element; performing system disturbance judgment; generating a Jacobian matrix of the differential equation set of the whole system; solving and obtaining the value of each state variable at the next moment; carrying out machine network alternate iterative computation, obtaining an operation parameter value at the next moment until a convergence condition is met, and taking the operation parameter value as aninitial value to calculate the next step length. The calculation is simple and stable, the larger step length can be used, the problem that the numerical stability and the calculation efficiency of the algorithm are difficult to consider at the same time in the transient stability simulation calculation is solved, and the method is more efficient, accurate and stable.

Recursive digital pre-distortion (DPD) techniques are provided. Digital pre-distortion is performed by applying a signal to a recursive system to generate a state vector; providing the state vector as a feedback value to the recursive non-linear system; and applying the state vector to a second function to generate an output signal, wherein at least one of the recursive system and the second function comprise a non-linear function. The recursive non-linear system can be initialized to a known initial value. The recursive system is defined by a system of non-linear differential equations.

A method and system for image segmentation by evolving radial basis functions (RBFs) is disclosed. A set of RBFs define a contour on an image. The contour is the zero level set of an implicit function defined by the RBFs. The RBFs are weighted and parameterized by multiple parameters, such as center point, standard deviation, and orientation. The contour is iteratively deformed by updating the weight and each of the parameters of each of the RBFs based on each pixel of the contour, until the contour converges. The parameters and the weight of each RBF is updated using coupled set of differential equations derived based from a region-based or boundary-based image segmentation energy formulation. The final contour at a convergence defines boundaries of a target object or region in the image.

OptimizationA computer-implemented method and system are disclosed for solving an optimization problem in which nodes of a population have a probability of undergoing a state transition in response to an input. Transition probabilities are modelled in a matrix T, where T is an N×N matrix, N being the number states, and Tab is the transition probability from state a to state b. The matrix T is multiplied by a vector of coupled differential equations to determine a system of differential equations. From an initial state of nodes of the population, the system of differential equations is solved for each of a plurality of time increments.

The invention discloses a backup structure-oriented fault tree analysis method, wherein a dynamic fault tree model is built based on working principles and failure processes of a backup structure, and the failure process of the backup structure is transformed to a fuzzy Markov model; Kolmogorov state transfer differential equations corresponding to the fuzzy Markov model are built and solved, so as to obtain fuzzy state probabilities of the backup structure at any time point. The invention efficiently resolves the problems of "small sample" and "inaccuracy" in fault tree analysis processes, and dynamic characteristic and sequential characteristic of the backup structure and the problem of imperfect coverage of backup switching are also considered, thereby a relatively accurate reliability quantitative analysis result is obtained, which helps to find the weak link affecting the reliability of the system and provides a guidance in optimizing design of the system.

The invention disclose a method for analyzing dynamic transferred water quality based on a fugacity theory, relating to a water quality analysis method, and solving the problem of larger deviation between a simulated result and an actual condition caused by actually unobtainable required parameters and randomly non-modifiable inner parameters in a traditional water quality analysis method. The method comprises the following steps of: firstly, confirming environment facies of a water area to be detected and acquiring geographic information parameters of the water area to be detected, environmental parameters of various environment and materialization parameters of a target pollutant; then computing fugacity capacity of each environmental facies, speed constant of reaction dynamics and transfer parameters among each environmental face; then establishing a system of differential equation by taking the fugacity of each environmental face as an unknown and solving the system of differential equation through a Runge-Kutta algorithm to obtain the fugacity of each environmental face; and calculating out pollutant concentration by the fugacity of each environmental facies and the fugacity capacity. The invention overcomes defects of the prior art, and can be applied to the field of analyzing transferring transformation of organic pollutants in the water area and distribution law of multi-medium environment.

The invention discloses a real time parameter identification method of an air-conditioning belonged building second order equivalent thermal parameter model, and a real time built air-conditioning belonged building second order equivalent thermal parameter model, which are used to build and real time control an air-conditioning load equivalent polymerization model; a second order differential equation group of the air-conditioning load belonged building is built and the input & output quantity and the parameter needs to be identified are determined; the real time data acquisition of the input & output quantity is conducted through a high-grade metering terminal of an intelligent power grid; data transmission to a load aggregator is conducted; and the real time parameter identification is conducted through the cooperation particle swarm algorithm by the load aggregator. According to the invention, research insufficient in domestic air-conditioning load real time modeling is made up; an accurate air-conditioning load model is constructed; and technical support for demand response is provided.

The invention discloses a decoupling design method for helicopter control, relating to the technical field of helicopters. The decoupling design method for a helicopter comprises the following steps: 1, establishing an aerodynamic force and moment model for a helicopter component, and substituting the obtained force and moment into a movement equation of six freedom degree to obtain a nonlinear differential equation set; 2, solving trim control amounts of control mechanisms in helicopter flight according to the nonlinear differential equation set; 3, determining a coupling relationship between the control mechanisms, and substituting the trim control amounts in the step 2 into the coupling relationship; 4, verifying that whether the design improves the driving quality or not via a flight simulation test; 5, carrying out flight testing. The decoupling design method disclosed by the invention has the advantage that in the flight process, the tail rotor tension and the horizontal trim can be automatically adjusted just by the total pulling moment. The control correction precision is improved, the workload of a pilot is reduced, and the requirements on driving skills of the helicopter are reduced.

The invention discloses a social network double-rumor model-based rumor propagation range estimation method, which comprises the following steps: according to an SI model, considering propagation of two rumor and interaction between the two rumor, constructing a double-rumor SI model, and obtaining a differential equation set of time-varying user number proportions in different states under the double-rumor SI model; on the basis of the SI model, considering the condition of an immunization person, constructing a double rumor model based on the SIR model, and respectively considering a degreeuniformity network and a degree nonuniformity network to obtain a differential equation set of time-varying user number proportions under the double rumor SIR model; and the equation is simplified, propagation threshold expressions in the two networks are derived, and the relationship between the propagation threshold and the propagation range in the final state steady state is obtained, so that the propagation range of the rumor is estimated. The method can be applied to finding appropriate time to control diffusion of the rumor under different network conditions, the influence of the rumor is reduced, the rumor propagation range can be estimated, and rumor propagation is effectively inhibited.

The invention discloses a simulation computing method and device of a landslide surge disaster. The method comprises the steps of building a three-dimension computing domain model according to the obtained terrain, the water level and the landslide mass shape and the position information; according to a Navier-Stokes partial differential equation set which conducts integral in the thickness and water depth direction of the landslide mass, obtaining the motion of water and the landslide mass and the mutual effect of the water and the landslide mass, and simulating the process that surges are generated, propagated and climbs along a bank according to the change of the water pressure after the landslide mass enters the water; visualizing the landslide motion and the surge propagation rule, conducting numerical forecasting on the surge motion trail and the impact influence area, and evaluating the surge disaster. By means of the simulation computing method and device of the landslide surgedisaster, by conducting numerical forecasting on feature parameters and the influence area of the surge motion, the surge disaster can be evaluated, the method and device have great theoretical significance and high practical value in evaluating the surge disaster and predicting the landslide on the reservoir bank and the secondary disaster of the surges, and also have reference significance in numerical simulation of geological disasters like debris flow and barrier lakes.

The invention discloses a distributed energy-saving method based on an unlicensed frequency band D2D system. The method comprises the following steps: S1) establishing a system model with a minimum total power as a target function, restricting a throughput demand of each pair of D2D users, restricting the power of the k-th D2D user on the k-th unlicensed frequency band channel, restricting the proportion of distributing the frequency spectrum of the k-th unlicensed frequency band to the k-th D2D user time-division factor, and restricting the total interference on each licensed frequency band sub-channel n by all D2D users; S2) establishing differential equations to perform the solving, and solving the differential equations constructed by the target function based on a gradient projectionalgorithm; and S3) performing simulation analysis by adopting a simlink module in the matlab. The method disclosed by the invention has the advantages that each D2D user reuses the licensed frequencyband of an LTE system and the unlicensed frequency band on a Wi-Fi system station (APs), and a distributed algorithm is proposed to accomplish the power and spectrum resource allocation of the D2D users.

The invention provides an airplane automatic airdropped material guiding method for aiming at a movable object, wherein the method relates to the field of material airdropping by an airplane. A to-be-corrected angle deviation and a to-be-corrected distance deviation of a carrying airplane are obtained through a center-of-mass motion differential equation set. After the carrying airplane continuously perform correction of the angle deviation and the distance deviation, the carrying airplane drops the material, thereby finishing an airdropping task. The airplane automatic airdropped material guiding method is advantageous in that a relatively simple model and easy equation set resolving are realized; the airplane automatic airdropped material guiding method can be used for performing a dropping guiding task in various conditions of material airdropping by transporters, helicopter maritime rescue, etc.; a resolving method of the invention is suitable for various conditions and conditions of fixed target, movable target, windy environment and no-wind environment can be resolved through the resolving method. High adaptability and simple resolving method are realized. Convenient calculation through a simple iterative method is realized. An on-airplane computer can quickly calculate a result. Furthermore the airplane automatic airdropped material guiding method has relatively high precision.

A small signal model based on a single-phase cascaded island ACHB inverter system and a stability analysis method based on a floquet theory are provided. The single-phase cascaded island ACHB invertersystem includes a digital controller with double closed loop control, an LC filter, a load and three single-phase H-bridge inverters operating at low voltage, medium voltage and high voltage respectively. The digital controller is connected to a filter capacitor. The small signal model based on the single-phase cascaded island ACHB inverter system, in consideration of a linear resistive load withsmall signal disturbance, separates a disturbance component, obtains a modulation signal expression by coordinate transformation, and analyzes the system stability by using the floquet theory. The beneficial effect is that a multi-level inverter reduces the harmonic components in the output voltage due to the increase of the output level of a main circuit, and the stability analysis of the smallsignal model solves the difficulty of modeling the nonlinear circuit of a converter, the dynamic characteristics of steady-state operating point of the converter are described by unified differentialequations, the stability domain of the system is accurately verified by the floquet theory, and a reliable criterion is provided for the stable operation of the whole single-phase cascaded island inverter system.