Wind-light-pumped energy storage combined power generation dispatching method based on improved bat algorithm

Abstract

The invention discloses a wind-light-pumped energy storage combined power generation dispatching method based on an improved bat algorithm, and belongs to the technical field of renewable energy power generation dispatching. Internet surfing income, operation cost, pumping cost of pumping energy storage and pollutant emission punishment cost of a thermal power generating unit are considered, and the method aims at optimizing combined dispatching of wind power, photovoltaic, thermal power and pumping energy storage units and improving overall income and stability of a system. The improved bat algorithm is provided, chaotic mapping initialization is adopted, and adaptive step length updating and Cauchy variation factors are introduced, so that the convergence speed and the global search capability of the algorithm are remarkably improved, and local optimal traps can be more effectively avoided. According to the scheduling scheme formulated based on the improved bat algorithm, the peak regulation capacity of pumped energy storage is fully released, the receiving capacity of renewable energy sources is enhanced, and impact on a conventional thermal power generating unit is relieved.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of renewable energy power generation dispatching, and particularly relates to a wind power, photovoltaic, thermal power and pumped storage combined power generation dispatching method based on an improved bat algorithm. BACKGROUND

[0002] With the intensification of global energy shortage and air pollution problems, new energy power generation (such as wind power and photovoltaic power generation) has gradually become the main component of the power system. However, due to the randomness and volatility of wind power and photovoltaic power generation, it brings challenges to the safe and stable operation of the power system. In order to balance the volatility of wind power and photovoltaic power generation, pumped storage units as an important energy storage means can effectively regulate the load of the power grid, reduce the phenomenon of abandoned wind and light, and stabilize the power supply.

[0003] The traditional dispatching method has problems such as slow convergence speed and easy to fall into local optimum when dealing with the optimization of wind-light-storage combined system, and cannot meet the needs of modern large-scale renewable energy system dispatching. Therefore, a new dispatching method is needed to improve the economy and stability of the system. SUMMARY

[0004] The wind-light-pumped storage combined power generation dispatching method based on the improved bat algorithm includes the following steps, and the following steps are performed in order: Step 1: Construct a combined power generation system containing wind power, photovoltaic, thermal power and pumped storage units, the pumped storage units including water pumps and water turbines; Step 2: Analyze and model the loss mechanism of the pumped storage unit; Step 3: Perform uncertainty modeling, scenario generation and stochastic optimization of the combined power generation system to ensure the robustness of the dispatching and distribution process of the combined power generation system under different uncertainty scenarios; Step 4: The combined power generation system also includes a transient analysis module, which models the dynamic behavior of the power system through physical models and mathematical formulas, for real-time monitoring and analysis of the dynamic response of the combined power generation system, and provides a basis for dispatching; Step 5: Establish a dispatching model with the goal of maximizing the total revenue of the combined power generation system, which includes the power generation revenue, operating cost of various types of power generation units, pumped storage cost of pumped storage, and power generation cost and pollutant emission penalty cost of thermal power units, and imposes conditions on the combined power generation system to ensure the rationality and feasibility of the dispatching process; Step 6: Use the improved bat algorithm to solve the dispatching model to optimize the output dispatching of each power generation unit.

[0005] The improved bat algorithm includes the following steps: a. Initialize the population using chaotic mapping to increase the coverage of the initial solution space; The Bat Algorithm initially uses a random method for population initialization, which cannot cover the entire solution space. Therefore, a chaotic mapping method is used for population initialization to improve the coverage of the initial solution space. The calculation formula is as follows: ; In the formula, It is a chaotic sequence. ; For the rounds of the sequence; To initialize the population dimension; To initialize the population size, facing Perform the inversion operation to obtain the solution space initialization population pairs, calculated as follows: ; In the formula; and These are the minimum and maximum values ​​of the variable's range of values, respectively. Indicates the first Only bats Position of the round; b. Introduce an adaptive step size update method to accelerate the convergence process of the algorithm; The adaptive step size update method uses a large initial step size, which improves the convergence speed. In the later stages of the algorithm, the step size becomes smaller, and the search becomes more refined. The formula for the bat velocity update method based on adaptive step size is as follows: ; ; In the formula, , The first Only bats and Flight speed per round; For adaptive step size; For individuals in the optimal position; For the minimum step size, The maximum number of iterations, As a regulating factor; It is a natural constant; c. Add a Cauchy mutation factor during the search process to enhance the algorithm's ability to escape local optima; right t The bat's flight speed variable at time t is modified by adding a Cauchy mutation factor to change the bat's flight speed and enhance its ability to escape local optima. The mutation formula is shown below: ; ; In the formula, is the first i Only bats in The position of the round; Is a uniformly distributed random number in the interval [0, 1]; Is a coefficient vector, Is a vector that decreases linearly from 2 to 0; Is a uniformly distributed random vector in the interval [0, 1].

[0006] The specific method for analyzing and modeling the loss mechanism of the pumped storage unit in step two is: The loss mechanism includes hydraulic loss and electrical loss, wherein the hydraulic loss is represented by the efficiency coefficient of the water pump and the water turbine, and the electrical loss is represented by the efficiency coefficient of the generator and the motor. Finally, the overall energy conversion efficiency of the pumped storage system is calculated through the comprehensive efficiency coefficient formula, which is as follows: ; Wherein, Is the comprehensive efficiency coefficient, Is the efficiency coefficient of the water turbine, Is the efficiency coefficient of the generator, Is the efficiency coefficient of the motor; ; ; ; In the formula, Is the hydraulic power after the water pump converts electrical energy into potential energy of water, Is the input power of the water pump, Is the electric power of the generator, Is the mechanical power of the generator, Is the electric power of the motor, Is the mechanical power of the motor.

[0007] The uncertainty modeling in step three includes using Weibull distribution and Beta distribution to model the uncertainty of wind power and photovoltaic power, and using historical data statistical analysis to obtain the probability density function of wind speed and illumination intensity; The probability density function of wind speed is: ; Wherein, ; ; In the formula, Is the wind speed; For shape parameters; For scale parameters; It is a gamma function; Average wind speed; The standard deviation of wind speed reflects the degree of fluctuation of wind speed around the average value; The probability density function of solar radiation intensity is: ; In the formula, Normalized light intensity; and For shape parameters; These are Beta parameters.

[0008] The objective function of the scheduling model considers maximizing the expected return for different scenario probability distributions, and the formula is as follows: ; In the formula, To maximize the expected revenue of the combined power generation system; For the first One scenario; For the scene The probability of; For the scene The following benefits; Total number of scenes; Revenue generated by a combined power generation system that includes wind power, photovoltaic power, thermal power units and pumped storage units; For the operating costs of wind power and photovoltaic units; The cost of pumping water for pumped storage units; The cost of generating electricity from thermal power units; The punitive costs of pollutant emissions; Among them, the power generation revenue of the combined power generation system including wind power, photovoltaic, thermal power units and pumped storage units It is the sum of the following parts: The feed-in tariff for solar power is multiplied by the output power of the solar power unit. The on-grid electricity price of wind power multiplied by the output power of the wind turbine; The grid-connected electricity price of pumped hydro storage is multiplied by the output power of the pumped hydro storage unit. The on-grid electricity price of thermal power units multiplied by the output power of thermal power units; The scenario involves using scenario generation technology to create multiple possible wind power and photovoltaic unit power generation output scenarios. Each scenario is randomly generated based on the probability distribution of wind speed and solar irradiance, reflecting power generation fluctuations under different weather conditions and time periods, thereby improving the robustness and accuracy of the scheduling model. A scenario As shown below: ; wherein, and are the output power of the wind turbine and photovoltaic generator at time t in the first

[0009] The transient analysis module establishes dynamic simulation models for the following parts: Based on the mechanical and electromagnetic characteristics of the generator, the transient model of the generator is expressed as: ; wherein, is the moment of inertia of the generator; is the mechanical angular velocity; and are the mechanical input torque and electromagnetic output torque, respectively; is the damping coefficient; ; wherein, is the stator voltage; is the stator resistance; is the stator current; is the stator inductance; is the transient electromotive force of the generator; is the electrical angular frequency on the stator side; In order to ensure the stability of the system during the sudden change of power supply and demand, the voltage response and frequency response modeling are used to deal with the fault burst event, and in order to describe the voltage recovery process after the fault, the voltage response is modeled as: ; wherein, is the voltage before the fault; is the decay coefficient; is the damped oscillation frequency; is the phase angle; is the time variable after the fault; The frequency response is used to describe the change of the system frequency after the disturbance, which is modeled by the following equation: ; wherein, is the maximum value of the frequency deviation; is the natural oscillation frequency.

[0010] ​The transient analysis module further comprises a time domain simulation and frequency domain analysis module, which simulates the changes of voltage, frequency and phase angle of the system through a system of differential equations and a numerical integration method, and performs frequency domain analysis on the established dynamic simulation model using fast Fourier transform (FFT), to evaluate the stability and oscillation mode of the system.

[0011] Through the above design scheme, the present application can bring the following beneficial effects: The present application constructs a scheduling model aiming at maximizing the expected revenue of the combined power generation system, and considers the power generation revenue, operation cost of various types of generating units, pumping cost of pumped storage energy, and penalty cost of pollutant emission of thermal power units, aiming to optimize the combined scheduling of wind power, photovoltaic, thermal power and pumped storage units, and improve the overall revenue and stability of the system. In order to solve the problems of poor convergence and easy falling into local optimal solution of traditional scheduling algorithms in dealing with complex optimization problems, the present application proposes an improved bat algorithm, which uses chaotic mapping initialization, introduces adaptive step updating and Cauchy mutation factor, significantly improving the convergence speed and global search ability of the algorithm. Case analysis shows that the improved algorithm not only improves the convergence speed, but also effectively avoids the local optimal trap. The scheduling scheme based on the improved bat algorithm fully releases the peak shaving capacity of pumped storage energy, significantly alleviating the impact of renewable energy output uncertainty on the power grid during large-scale distributed energy grid connection. This scheme not only enhances the accommodation capacity of renewable energy, but also reduces the impact on conventional thermal power units.

[0012] Through the innovative algorithm improvement, the present application significantly improves the scheduling efficiency and system revenue. This method optimizes the output scheduling of wind power, photovoltaic and pumped storage units, reduces the phenomenon of wind and light abandonment, and improves the accommodation capacity of renewable energy. At the same time, the system can effectively regulate the power grid load, alleviate the impact of new energy fluctuation on the power grid, reduce the scheduling pressure of thermal power units and reduce pollutant emission. In addition, the improved bat algorithm has strong robustness and can adapt to complex power grid environment and uncertain power generation output, ensuring stable operation of the system under different conditions. Through multi-objective optimization, the present application realizes the balance between economic benefit and environmental benefit, providing an effective solution for the optimal scheduling of green power system.

[0013] Furthermore, this invention deeply analyzes the loss mechanism of pumped storage systems, including hydraulic and electrical losses, and accurately calculates the overall energy conversion efficiency of the system using a comprehensive efficiency formula. Addressing the uncertainties of wind and solar power generation, scenario generation technology is employed to create multiple possible power output scenarios to maximize expected returns and improve the robustness and accuracy of the scheduling model. To further optimize the scheduling scheme, this invention introduces a highly integrated transient analysis module, employing time-domain simulation and frequency-domain analysis methods to evaluate the dynamic response of the power system in real time and provide a basis for scheduling strategies. Simulation results show that the improved bat algorithm, compared to traditional algorithms, reaches the optimal solution in fewer iterations, improves scheduling returns, optimizes the absorption capacity of new energy sources, and effectively reduces the scheduling burden on thermal power units. This invention provides an effective solution for the optimized scheduling of wind-solar-pumped storage combined power generation systems, possessing significant application value and promising prospects for widespread adoption. Attached Figure Description

[0014] The present invention will be further described below with reference to the accompanying drawings and specific embodiments: Figure 1 This is a structural block diagram of a distributed combined power generation system containing pumped hydro storage in the wind-solar-pumped hydro storage combined power generation scheduling method based on the improved bat algorithm of the present invention. Figure 2 This is a flowchart of the scheduling model solution based on the improved bat algorithm in the wind-solar-pumped hydro storage joint power generation scheduling method based on the improved bat algorithm of the present invention. Figure 3 This is a wind power and photovoltaic output curve diagram in an embodiment of the wind-solar-pumped hydro storage joint power generation scheduling method based on the improved bat algorithm of the present invention. Figure 4 This is a simulation iterative convergence curve in an embodiment of the wind-solar-pumped hydro storage joint power generation scheduling method based on the improved bat algorithm of the present invention; Figure 5 This is a revenue curve of the joint power generation system at each time step in an embodiment of the wind-solar-pumped hydro storage joint power generation scheduling method based on the improved bat algorithm of the present invention, after using the bat algorithm, particle swarm algorithm and improved bat algorithm. Figure 6 This is an example of the wind-solar-pumped hydro storage joint power generation and dispatching method based on the improved bat algorithm of the present invention. A schematic diagram of the Pareto optimal solution for the constraint method. Detailed Implementation

[0015] This invention relates to a wind-solar-pumped hydro storage joint power generation scheduling method based on an improved bat algorithm, which is implemented through the following technical solutions: 1. Design of a distributed combined generation system including pumped hydro storage Pumped storage is to use the excess power of the power grid to drive the water pump, and the water pump will pump water from the lower reservoir to the upper reservoir to convert the potential energy of the upper reservoir water into storage. When the power demand is insufficient, the water's potential energy is converted into electrical energy by the water turbine generator. The low suction and high generation function of pumped storage realizes the effective storage of electrical energy, effectively regulates the production, supply and use of the power system, and maintains the dynamic balance among the three.

[0016] The access of pumped storage units can control the charging and discharging process to some extent according to the demand of the power grid, that is, to constitute a wind-solar-storage integrated system. The platform control mode in the wind-solar-storage integrated system is more convenient and flexible, and the platform control mode has been intelligentized at present. The platform control mode can adjust the action ability of the energy storage system according to the power situation. The typical platform mainly includes a wind farm, a photovoltaic device, an energy storage system, a control center, a power transmission line and a signal feedback line. The structure of the distributed combined power generation system containing pumped storage is shown in Figure 1

[0017] The main part of the pumped storage unit is the water pump and the water turbine. During the low power consumption period, the water pump will pump the water from the lower reservoir to the upper reservoir, and the excess power in the power grid will be converted into potential energy for storage. During the peak power consumption period, the pumped storage unit will guide the water from the upper reservoir to the lower reservoir to drive the water turbine in the main plant to generate electricity, and the potential energy will be converted into electrical energy. The mathematical model of the energy storage and release process can be expressed as: (1); (2); In the formula, Pp is the power of the water pump; is the density of water; is the acceleration of gravity; is the pressure head of the water pump; is the water volume flow rate pumped by the water pump to the reservoir; is the comprehensive efficiency coefficient; Pt is the power generated by the water turbine; is the water volume flow rate from the reservoir to the water turbine; is the efficiency coefficient of the water turbine.

[0018] A detailed analysis of the loss mechanism of the pumped storage unit is the key to accurately evaluate the system performance. The present invention further deepens the discussion of various loss mechanisms in the pumped storage unit, and details how these losses are modeled in the efficiency calculation and their impact on the overall performance of the system. The process is as follows: ​The efficiency of pumped hydro storage is affected by various loss mechanisms. The thermal losses in a pumped hydro storage unit mainly originate from the energy conversion processes in the hydraulic system. When water flows through the pump and the turbine, due to friction, turbulence, and flow resistance, part of the mechanical energy is converted into heat energy, causing the liquid temperature to rise. This portion of heat energy cannot be converted back into mechanical energy, thus directly reducing the system's efficiency. In the model of the invention, thermal losses are modeled through the efficiency coefficients of the hydraulic system, specifically the efficiency reduction factors of the pump and the turbine. In practical operation, the efficiency of the pump is usually between 80% and 90%, while the efficiency of the turbine can be slightly higher. In short, thermal losses mainly occur in the hydraulic system when water flows through the pump and the turbine, due to friction and turbulence, part of the energy is lost in the form of heat energy. These losses can be expressed through the efficiency coefficients of the turbine : (3); where is the hydraulic power after the pump converts electrical energy into potential energy of water, is the input power of the pump.

[0019] The electrical losses of a pumped hydro storage unit refer to the electrical energy loss due to factors such as electrical resistance, hysteresis, and eddy current during the energy conversion process of the generator and motor. Specifically, when the motor converts electrical energy into mechanical energy to drive the pump, and when the generator converts hydro-mechanical energy into electrical energy, electrical losses occur. These losses not only affect the power generation efficiency of the system, but also have a negative impact on the energy conversion during the charging and discharging processes. In the model of the invention, electrical losses are represented by the efficiency coefficients of the generator and motor. The efficiency coefficient of the generator ηg and the efficiency coefficient of the motor ηm can be expressed as: (4); (5); where is the electrical power of the generator, is the mechanical power of the generator, is the electrical power of the motor, is the mechanical power of the motor.

[0020] After detailed modeling of the above-mentioned losses, the invention uses a comprehensive efficiency formula to calculate the overall efficiency of the pumped hydro storage unit. The comprehensive efficiency coefficient η is the product of the efficiency coefficients of each part, representing the overall energy conversion efficiency of the entire pumped hydro storage unit, i.e.: (6); The impact of various losses on the overall system performance manifests as a decrease in energy conversion efficiency and adverse economic effects. The existence of loss mechanisms means that the actual energy recovery rate is lower than the theoretical value, which directly affects the system's profitability and economic feasibility. Furthermore, higher losses can lead to decreased system stability under high load conditions, especially in scenarios with significant fluctuations in wind and solar power generation. Therefore, this invention places particular emphasis on accurate modeling of losses when optimizing scheduling strategies to ensure that the system can maintain high efficiency and stability under various operating conditions.

[0021] Uncertainty surrounding renewable energy is one of the major challenges facing modern power systems. To effectively address this uncertainty, the model in this invention employs multiple methods, including uncertainty modeling, scenario generation, and stochastic optimization, to ensure the robustness of scheduling and allocation processes under different uncertainty scenarios. This invention uses historical data to statistically analyze wind speed and solar irradiance, establishing corresponding probability distribution models. For wind speed, it is typically assumed to follow a Weibull distribution; for solar irradiance, it is assumed to follow a Beta distribution. The probability density function of wind speed is calculated as follows: (7); in, (8); (9); In the formula, Wind speed; For shape parameters; For scale parameters; It is a gamma function; Average wind speed; This represents the standard deviation of wind speed, reflecting the degree of fluctuation in wind speed around its average value.

[0022] The probability density function of solar radiation intensity is calculated as follows: (10); In the formula, Normalized light intensity; and For shape parameters; These are Beta parameters.

[0023] To capture the volatility of power generation, this invention uses scenario generation technology to create multiple possible wind and photovoltaic power output scenarios. Each scenario is randomly generated according to a different probability distribution, reflecting different weather conditions and time periods.

[0024] (11); in, For the first a scenario; and are the output power of wind turbines and photovoltaic units at time t in the first is the total number of scenarios.

[0025] The present invention maximizes the expected revenue by considering the probability distribution of different scenarios. The goal of stochastic optimization is to maximize the expected revenue, which can be formulated as follows: (12); where, is the expected revenue; is the probability of scenario is the revenue under scenario The probability of each scenario is estimated from historical data, reflecting the likelihood of power generation under different weather conditions. By considering these probabilities, the present invention can prioritize high-probability scenarios in the dispatch scheme, thereby improving the actual operational performance of the system.

[0026] The rapid fluctuations of wind and photovoltaic power generation are one of the main challenges faced by renewable energy power systems, especially in cases where load demand and power generation fluctuate dramatically. To address the rapid fluctuations of wind and photovoltaic power generation in load demand and power generation changes, the system considered in the present invention includes a highly integrated transient analysis module, which models the dynamic behavior of the power system in detail through various physical models and mathematical formulas. This module not only monitors and analyzes the dynamic response of the system in real time, but also provides reliable basis for the optimization of dispatch and control strategies.

[0027] The rapid fluctuations of wind and photovoltaic power generation are one of the main challenges faced by renewable energy power systems, especially in cases where load demand and power generation fluctuate dramatically. To address the rapid fluctuations of wind and photovoltaic power generation in load demand and power generation changes, the system considered in the present invention includes a highly integrated transient analysis module, which models the dynamic behavior of the power system in detail through various physical models and mathematical formulas. This module not only monitors and analyzes the dynamic response of the system in real time, but also provides reliable basis for the optimization of dispatch and control strategies.

[0028] ​​Time-domain simulation and frequency-domain analysis predict and evaluate the dynamic response of power systems under contingencies, including voltage and frequency changes, power oscillations, and system stability. The module architecture includes a data acquisition module, a dynamic simulation module, a frequency-domain analysis module, and a decision support module. The data acquisition module acquires system operation data in real time, including power generation, load demand, voltage, current, frequency, etc. These data are collected by Phasor Measurement Units (PMU) and Supervisory Control and Data Acquisition (SCADA) at high frequency, ensuring the timeliness and accuracy of the data. The dynamic simulation module simulates the response behavior of the power system under disturbance based on the electrical parameters and topology of the system. This module uses differential equations and numerical integration methods to describe the dynamic changes of voltage, frequency, and phase angle in detail. The frequency-domain analysis module: through Fast Fourier Transform (FFT), the power output and voltage signal of the system are analyzed in frequency domain, the possible resonance frequency and low frequency oscillation are identified, and the oscillation mode of the system is evaluated. Based on the decision support module, according to the transient analysis results, the invention provides automatic scheduling and control suggestions, including starting standby generator sets, adjusting reactive power compensation devices, and implementing load shedding measures.

[0029] The dynamic simulation model is the core part of the transient analysis module, which mainly uses the following physical model for system modeling. The transient model of the generator can be expressed as: (13); In the formula, is the moment of inertia of the generator; is the mechanical angular velocity; and are the mechanical input torque and electromagnetic output torque, respectively; is the damping coefficient.

[0030] (14); In the formula, is the stator voltage; is the stator resistance; is the stator current; is the stator inductance; is the transient electromotive force of the generator; is the electrical angular frequency on the stator side.

[0031] To ensure system stability during power supply and demand mutations, the system of the present application has the ability to respond to emergencies (such as short-circuit faults, sudden load changes, etc.) in a short time through voltage response and frequency response modeling. To describe the voltage recovery process after the fault, the voltage response can be modeled as: (15); where, is the voltage before the fault; is the damping coefficient; is the damped oscillation frequency; is the phase angle, is the time variable after the fault occurs, which is a continuous period of time.

[0032] The frequency response is used to describe the change of system frequency after the disturbance , which is modeled by the following equation: (16); where, is the maximum value of frequency deviation; is the natural oscillation frequency.

[0033] To improve the robustness of the system in extreme conditions, the present application is based on a decision support module that provides automated scheduling and control recommendations based on transient analysis results, including starting backup generator sets, adjusting reactive power compensation devices, and implementing load shedding measures. Through voltage response and frequency response modeling, the ability to respond to emergencies in a short time, when extreme weather (such as storms, strong winds, or intense sunlight) is predicted, the standby generator set is started in advance or the standby capacity of the energy storage system is increased. These strategies enhance the system's regulation margin, ensuring that the system remains balanced in the event of an emergency. Through the above detailed modeling and analysis, the transient analysis module of the present application can effectively handle the rapid changes in wind and photovoltaic power generation in the process of load demand and power generation, ensuring the dynamic stability of the power system. These technical details are fully described in the revised version.

[0034] 2. Joint power generation system optimization scheduling model construction Pumped storage units can serve as a special type of electrical load and store excess electricity in the power system, significantly reducing wind and solar power curtailment, ensuring economic benefits, and playing a crucial role in stabilizing the power grid. This invention considers the power generation revenue and pollutant emission penalty costs of wind power, photovoltaic power, thermal power units, and pumped storage units, establishing a scheduling model with the goal of maximizing the total revenue of the combined power generation system. The model's objective function includes the power generation revenue and operating costs of various power sources, the pumping cost of pumped storage, and the power generation cost and pollutant emission penalty costs of thermal power units. Combining equation (12), the objective revenue function that maximizes the expected revenue can be expressed as: (17); In the formula, Revenue generated by a combined power generation system that includes wind power, photovoltaic power, thermal power units and pumped storage units; For the operating costs of wind power and photovoltaic units; The cost of pumping water for pumped storage units; The cost of generating electricity from thermal power units; The punitive costs of pollutant emissions.

[0035] Revenue from combined power generation systems including wind power, solar power, thermal power units, and pumped hydro storage It can be represented as: (18); In the formula, For photovoltaic feed-in tariffs; For wind power grid connection price; Grid connection price for pumped hydro storage power generation: The on-grid electricity price for thermal power units; Let t be the power transmitted from the photovoltaic unit to the grid. Let t be the power delivered to the grid by the wind turbine. for t The power that is constantly pumped and stored in water and transmitted to the power grid; for t The power transmitted from thermal power units to the power grid at all times.

[0036] Operating costs of wind power and solar power It can be represented as: (19); In the formula, and These are the power generation cost coefficients for wind turbines and photovoltaics, respectively; For a moment t Inner The actual power generation of the typhoon generators; For a moment t Inner The actual power generation of the photovoltaic system in Taiwan; The number of wind turbine units; This refers to the number of photovoltaic (PV) units.

[0037] Pumping cost of pumped storage unit It can be represented as: (20); In the formula, The pumping electricity price for pumped storage; Pumping power for pumped storage.

[0038] Power generation cost of thermal power units It can be represented as: (twenty one); In the formula: For the first t In the first time period The power output of the thermal power unit; The total time of the scheduling cycle; This refers to the number of thermal power units. , , For the first The power generation cost coefficient of a thermal power unit; and For the first The startup cost and startup status of a thermal power unit.

[0039] During system scheduling, the planned output of thermal power units with high pollutant emission characteristic coefficients should be reduced as much as possible, while the output of thermal power units with low pollutant emission characteristic coefficients should be increased. Therefore, the penalty cost of the pollutant emission coefficient of thermal power units should be considered. It can be represented as: (twenty two); In the formula, , and For the first Pollutant emission coefficients of thermal power units in Taiwan.

[0040] Constraints: Taking into account the output limitations of various units in the combined power generation system, power balance constraints, thermal power unit ramping rate, and pumped storage unit output constraints, the rationality and feasibility of the dispatching process are ensured.

[0041] (1) Power balance constraint The mathematical relationship between the output of thermal power units, wind power clusters, photovoltaic power, and energy storage charging and discharging power in a combined power generation system can be expressed as follows: (23); wherein, t the actual power generation of the i-th wind power unit in the time interval; the actual power generation of the i-th wind power unit in the time interval; t the actual power generation of the i-th wind power unit in the time interval; t the actual power generation of the i-th wind power unit in the time interval; the actual power generation of the i-th wind power unit in the time interval; t the power delivered to the grid by the pumped storage power plant in the time interval; the load power, the number of wind power units, the number of photovoltaic units.

[0042] (2) Upper and lower limits of wind power output (24); wherein, , are respectively the upper and lower limits of wind power output.

[0043] (3) Upper and lower limits of photovoltaic output (25); wherein, , are respectively the upper and lower limits of photovoltaic output.

[0044] (4) Upper and lower limits of thermal power output (26); wherein, , are respectively the upper and lower limits of thermal power output.

[0045] (5) Climbing constraint of thermal power The size of the power that can be increased or decreased by different thermal power units in a unit time depends on the operating adjustment parameters of the generator, so the speed of output scheduling of the thermal power unit needs to consider the output climbing rate. Different thermal power units have different climbing rates, and it is not possible to excessively increase or decrease the output in a unit time, so the climbing constraint of each thermal power unit must be considered in the scheduling plan, that is: (27); (28); (29); (30);​​​​​​ wherein: is the t-1 is the is the power output of the is the t is the start-up state of the is the is the shut-down state of the t is the is the operation state of the is the t is the operation state of the is the is the operation state of the t-1 is the is the operation state of the and are the upward and downward ramp rates of the t is the is the

[0046] (6) Pumped storage unit power output constraint (31); wherein, and are the maximum and minimum power outputs of the pumped storage unit.

[0047] (7) Grid acceptance capability constraint: (32); wherein , are the upper and lower limits of the grid-acceptable power; is the power delivered to the grid by the photovoltaic unit; is the power delivered to the grid by the wind power unit; is the power delivered to the grid by the pumped storage unit, is the power delivered to the grid by the thermal power unit.

[0048] As an effective energy storage method, pumped hydro storage can effectively regulate the dynamic balance between production, supply and consumption of the power system. The integration of pumped hydro storage units can control the charging and discharging process to some extent according to the needs of the power grid, thereby forming a wind-solar-storage integrated system. In the wind-solar-storage integrated system, the platform control mode is more convenient and flexible, and tends to be intelligent with the development of technology. The platform integrates a model predictive control (MPC) algorithm, which predicts the future behavior of the system using the mathematical model of the system and generates an optimal control strategy based on the prediction results. In the present invention, MPC focuses on optimizing the scheduling and energy management of the energy storage system to maximize economic benefits and ensure grid stability. In each control cycle, the MPC optimization objective function can be expressed as: (33); wherein, represents the system state; is the control input; is the reference trajectory, which is the system reference state vector at the discrete control time; and are weight matrices; is the prediction time domain length; is the time step number.

[0049] The system state space model is usually expressed as the following linear or nonlinear discrete-time system: (34); (35); wherein, , , and are system matrix parameters; is an external disturbance (such as load change or weather change), representing the disturbance vector acting on the system at the discrete control time; is the system output, representing the system output vector at the discrete control time; is the system state vector at the discrete control time, representing the control input vector applied at the discrete control time; is a disturbance input matrix.

[0050] By accessing real-time data from the power grid and weather forecast data, MPC can dynamically adjust the control strategy to respond to changes in external conditions. The MPC algorithm obtains key parameters of the power grid in real time, such as the current load level, power generation output, voltage, and frequency. These data are obtained through the SCADA system or the phasor measurement unit (PMU) and are used to adjust the state variables and control variables in the MPC. Weather forecast data, such as wind speed and solar irradiance, have a direct impact on renewable energy generation, so in the MPC, the application estimates future power generation based on predicted weather data and adjusts the operation strategy of the energy storage system accordingly. The impact of weather data on the energy storage system can be represented as: (36); is the predicted power generation; is divided into the predicted value of wind speed and the predicted value of solar irradiance (also known as light intensity) at time t; is a nonlinear function that maps the weather forecast , to the predicted value of power generation.

[0051] Based on these predicted data, the MPC can pre-schedule the energy storage system to respond to possible power fluctuations. Adaptive control is one of the key features of MPC, especially when responding to sudden changes in the system. The MPC algorithm of the system platform is designed to immediately adjust the control strategy when a sudden event in the system is detected (such as a sudden increase in load or a sudden drop in power generation), ensuring stable operation of the system. In each control period, the MPC algorithm monitors the system state and detects whether there is a disturbance (such as frequency deviation or voltage fluctuation) that exceeds the normal range. Once a disturbance is detected, the MPC recalculates the optimal control strategy based on the new state. The disturbance model and adaptive control strategy can be represented as: (37); (38); where, is the state deviation at the th discrete control time; is the control adjustment at the th discrete control time; is the disturbance at the th discrete control time; is the adaptive control input at the th discrete control time; is the adaptive gain matrix, ensuring that the system can quickly recover to the normal operating state after a disturbance occurs; is the​​​ discrete control time.

[0052] To cope with sudden changes, MPC algorithms employ real-time optimization and iterative updating strategies. By solving optimization problems online, MPC can generate new control strategies at each time instant to cope with the latest system states and external conditions. The iterative optimization formula is: (39); where, is the control input at the discrete control time; is the system state vector at the discrete control time; is the system reference state vector at the discrete control time.

[0053] At each control period, MPC calculates new control inputs through iterative optimization to ensure optimal operation of the system under dynamic changing conditions.

[0054] 3. Multi-objective problem transformation To solve the trade-off problem between maximizing economic benefits and minimizing environmental emissions, the invention transforms the multi-objective problem between pollutant emission cost and economic benefits into a single-objective problem based on the constraint method. The maximization of economic benefits is optimized as the total objective function, and the pollutant emission amount is taken as the constraint condition, which is mathematically expressed as: (40); In the interval [0, 1], a fuzzy decision maker is used to assign a fuzzy membership degree to each point, and its membership function is: (41); where, is the membership value of the o-th objective function, and is the minimum value and the maximum value that the o-th objective function can reach in the feasible solution space, is the actual value of the o-th objective function under the current solution, is the satisfactory upper limit (i.e., the expected value) of the o-th objective function, is the tolerance upper limit (i.e., the maximum value allowed) of the o-th objective function.

[0055] The best compromise solution is obtained by the min-max method, and the maximum of the minimum is selected as the best compromise solution, represents the membership or satisfaction of the first objective function under the current solution, represents the membership or satisfaction of the second objective function under the current solution.

[0056] 4. A scheduling model solving method based on an improved bat algorithm The improved bat algorithm (BA) is used to solve the scheduling model converted into a single objective problem. The traditional bat algorithm has slow convergence speed and is prone to falling into a local optimal solution when dealing with large-scale optimization problems, therefore, the bat algorithm is improved in the application: (1) Chaos mapping initialization: the population is initialized through chaos mapping method to improve the coverage rate of the initial solution space.

[0057] (2) Adaptive step size update: an adaptive step size update strategy is introduced to speed up the convergence speed.

[0058] (3) Cauchy mutation factor: a Cauchy mutation factor is added in the search process to enhance the ability of the algorithm to jump out of the local optimum.

[0059] According to the bat bionics principle, the bat algorithm includes three elements of search pulse frequency, pulse sound intensity and frequency of emitting pulse. Among them, the speed of flight is determined by the search pulse frequency, and the probability of accepting the position update is determined by the pulse sound intensity and frequency. The steps of the bat algorithm are as follows: Step 1: set the size of the bat population , the maximum pulse sound intensity , the maximum pulse frequency , the upper limit of pulse frequency , the lower limit of pulse frequency , the sound intensity attenuation coefficient , the frequency increase coefficient , set the dimension of the position vector and the maximum number of iterations.

[0060] Step 2: initialize the unit bat position At the same time, according to the objective function, the fitness value of the bat position is calculated, and the individual in the best position is found . The bat position initialization is shown in formula (42).

[0061] (42) In the formula: , represent the upper and lower limit values of the position area of the bat; represents a one-dimensional vector matrix containing a number of values ranging from (0, 1) randomly generated.

[0062] Step 3: Initialize the pulse rate of unit bat, calculate the flight speed of unit bat, update the position of bat, and update the formula as shown in equations (43)~(45).

[0063] (43); (44); (45); wherein, is a random factor, uniformly distributed in the interval (0, 1); , is the flight speed of the i-th bat in the t-th iteration; and is the position of the i-th bat in the t-th iteration. , respectively represents the position of the i-th bat in the t-th iteration.

[0064] Step 4: In each iteration, generate a random number for unit bat, if , select the current best solution and perform local disturbance. Wherein, is the pulse rate of the i-th bat in the t-th iteration. The local disturbance is shown in equation (46).

[0065] (46); Step 5: Calculate the new fitness of the bat after disturbance, if the new fitness is better than the individual optimal fitness or , replace the old position with the new position after disturbance and save it, and update the pulse rate and sound intensity as shown in equations (47), (48).

[0066] (47); (48); wherein, is the pulse rate of the i-th bat in the t-th iteration; is the pulse sound intensity of the i-th bat in the t-th iteration; is the pulse sound intensity of the i-th bat in the t-th iteration.

[0067] ​​​​​​​​​​​Step 6: Stop the search when the maximum number of iterations is reached, and output the position and fitness value of the bat corresponding to the global optimal solution; otherwise, jump back to step 3 to continue the search.

[0068] As the number of iterations increases in the basic Bat Algorithm, the individual differences within the population become increasingly pronounced in later iterations, leading to a decrease in population diversity, which eventually approaches zero. Therefore, this invention employs chaotic mapping for population initialization to improve the coverage of the initial solution space. The Bat Algorithm is improved through adaptive step-size updates and the introduction of the Cauchy mutation inverse cumulative distribution function. The improvement strategy is as follows: Population initialization operation using chaotic mapping The Bat Algorithm initially uses a random method for population initialization, which cannot cover the entire solution space. Therefore, this invention employs a chaotic mapping method for population initialization to improve the coverage of the initial solution space. The calculation formula is as follows: (49); In the formula, , It is a chaotic sequence; To initialize the population dimension; To initialize the population size. Facing Perform the inversion operation to obtain the solution space initialization population pairs, calculated as follows: (50); In the formula; Indicates the first Only bats The position of the round, and These represent the minimum and maximum values ​​of the variable's range of values, respectively.

[0069] Adaptive step size update Based on equations (47) and (48) to update bat velocity, a bat velocity update method based on adaptive step size is proposed. The algorithm has a relatively large initial step size, which improves the convergence speed. In the later stages of the algorithm, the step size becomes smaller, and the search becomes more refined.

[0070] (51); (52); In the formula, For the minimum step size, The maximum number of iterations, It is a regulating factor.

[0071] Introducing the Cauchy inverse cumulative distribution function Cauchy mutation originates from the Cauchy distribution of a continuous probability distribution. Appropriate random variables are used to generate the Cauchy mutation factor, which is then applied in step 3 of the Bat Algorithm. The speed variable in each round is modified by adding a Cauchy mutation factor to change the bat's flight speed and enhance its ability to escape local optima. The mutation formulas are shown in equations (53)-(54).

[0072] (53); (54); In the formula, It is a uniformly distributed random number within the interval [0, 1]. For the coefficient vector, It is a vector that decreases linearly from 2 to 0; Let be a uniformly distributed random vector within the interval [0, 1]. The solution process of the scheduling model based on the improved bat algorithm is as follows: Figure 2 As shown.

[0073] 5. Examples The combined power generation system consists of three thermal power units, one pumped storage unit, one wind farm, and one photovoltaic unit. The installed capacity of each power source is shown in Table 1. The feed-in tariff is shown in Table 2. The parameters of the thermal power units are shown in Table 3. The wind and photovoltaic power output curves are shown below. Figure 2 As shown.

[0074] Table 1 Model Parameters

[0075] Table 2 Grid Connection Price

[0076] Table 3 Parameters of Thermal Power Units

[0077] To verify the robustness of the Improved Bat algorithm, the Bat algorithm, PSO algorithm, and the improved Bat algorithm were used in three typical test functions. The results of each algorithm were tested and compared in MATLAB software. These six test functions are: Schaffer function: (55); Ackley function: (56); Salomon function: (57); For vectors The Each component, when considering the two-dimensional case. , denote the variables in the 1st, 2nd dimension respectively; is a constant pi , used to control the periodicity of cosine term; is the dimension of Salomon function, i.e. the number of decision variables.

[0078] The maximum iteration number and population size of three different test functions are set to different values, as shown in Table 4.

[0079]

[0080] When performing optimal value and average optimal value analysis, the specific parameter settings are as follows: the iteration number is 200 times, and the population size is 40. In order to avoid the error caused by the randomness of the algorithm, the three functions are independently run 30 times in different dimensions, and the worst value, optimal value and average value are taken. The results are shown in Table 5.

[0081] As can be seen from Table 5, compared with the BA algorithm and the PSO algorithm, the improved bat algorithm has better comprehensive optimization performance in solving low-dimensional test functions. The optimal value obtained is close to the theoretical value, and the robustness of the algorithm is good, which further illustrates that the improved bat algorithm has good adaptability and robustness.

[0082] Now set the bat algorithm, particle swarm optimization algorithm and improved bat algorithm for simulation comparison. The maximum iteration number of the algorithm is set to 200, and the population size is 100. The simulation iteration convergence curve is shown in Figure 4 .

[0083] As can be seen from Figure 4 , the improved bat algorithm adopted in the application can tend to the optimal solution in fewer iteration times in the optimization and solution process. The optimization and solution efficiency is obviously better than that of the traditional bat algorithm and particle swarm optimization algorithm. The necessity and effectiveness of the improved bat algorithm for optimization and scheduling solution are verified. After adopting the bat algorithm, particle swarm optimization algorithm and improved bat algorithm, the benefit curve of the joint power generation system at each time is shown in Figure 5 .

[0084] As can be seen from Figure 5It can be seen that the improved bat algorithm has higher comprehensive income than the original bat algorithm and the improved bat algorithm before 15:00. After 15:00, the partial comprehensive income of the improved bat algorithm is lower than that of the other two algorithms, but the overall comprehensive income is slightly higher than that of the other two algorithms. After optimization, the bat algorithm improves the overall comprehensive income by 2.7% compared with the particle swarm algorithm, and the improved bat algorithm improves the overall comprehensive income by 21.9% compared with the bat algorithm. The scheduling results of the thermal power unit based on the method of the application are shown in Table 6.

[0085] As can be seen from Table 6, the output change of unit 3 is more sensitive to the introduction of the pollution cost, and the reason is that the pollution emission coefficient of this unit is higher. After introducing the penalty cost, unit 3 with the highest pollution emission coefficient will be reduced. In this process, the thermal power unit does not need to frequently adjust the output plan, and the system equivalent load it bears is basically unchanged. This not only greatly weakens the additional impact of traditional large-scale new energy grid connection on the system, but also fully utilizes the pumped storage unit to participate in the peak shaving and valley filling task of the system in a more reasonable way.

[0086] By The Pareto optimal solution of the constraint method is shown in Table 5. Figure 6 It can be seen that the Pareto optimal solution is obtained after 20 iterations, and the maximum membership function is 0.750, corresponding to the 13th solution. Therefore, considering the minimum-maximum fuzzy satisfaction method, the 13th solution is selected as the best compromise solution.

[0087] According to the simulation results, the improved bat algorithm proposed in the application can approach the optimal solution with fewer iterations in the optimization solution process, and can significantly improve the scheduling income compared with the traditional bat algorithm and the particle swarm algorithm (PSO). The simulation shows that the optimized scheduling scheme not only improves the new energy consumption capacity, but also effectively reduces the scheduling burden of the thermal power unit.

Claims

1. A wind-solar-pumped hydro storage joint power generation scheduling method based on an improved bat algorithm, characterized in that: The steps include the following steps, and the following steps are performed in sequence: Step 1: Construct a combined power generation system that includes wind power, photovoltaic power, thermal power and pumped storage units, wherein the pumped storage units include water pumps and water turbines; Step 2: Analyze and model the loss mechanism of pumped storage units; Step 3: Perform uncertainty modeling, scenario generation, and stochastic optimization of the joint generation system to ensure the robustness of the scheduling and allocation process of the joint generation system under different uncertainty scenarios; Step 4: The combined power generation system also includes a transient analysis module. The transient analysis module models the dynamic behavior of the power system using physical models and mathematical formulas, and is used to monitor and analyze the dynamic response of the combined power generation system in real time, and to provide a basis for dispatching. Step 5: Establish a scheduling model with the goal of maximizing the total revenue of the combined power generation system. The scheduling model includes the power generation revenue and operating costs of various generator units, the pumping cost of pumped storage, and the power generation cost and pollutant emission penalty cost of thermal power units. Conditional constraints are imposed on the combined power generation system to ensure the rationality and feasibility of the scheduling process. Step 6: Solve the scheduling model using the improved bat algorithm to optimize the output scheduling of each generator unit.

2. The wind-solar-pumped hydro storage joint power generation scheduling method based on the improved bat algorithm according to claim 1, characterized in that: The improved bat algorithm includes the following steps: a. Initialize the population using chaotic mapping to increase the coverage of the initial solution space; The Bat Algorithm initially uses a random method for population initialization, which cannot cover the entire solution space. Therefore, a chaotic mapping method is used for population initialization to improve the coverage of the initial solution space. The calculation formula is as follows: ； In the formula, It is a chaotic sequence. ; For the rounds of the sequence; To initialize the population dimension; To initialize the population size, facing Perform the inversion operation to obtain the solution space initialization population pairs, calculated as follows: ； In the formula; and These are the minimum and maximum values ​​of the variable's range of values, respectively. Indicates the first Only bats Position of the round; b. Introduce an adaptive step size update method to accelerate the convergence process of the algorithm; The adaptive step size update method uses a large initial step size, which improves the convergence speed. In the later stages of the algorithm, the step size becomes smaller, and the search becomes more refined. The formula for the bat velocity update method based on adaptive step size is as follows: ； ； In the formula, , The first Only bats and Flight speed per round; For adaptive step size; For individuals in the optimal position; For the minimum step size, The maximum number of iterations, As a regulating factor; It is a natural constant; c. Add a Cauchy mutation factor during the search process to enhance the algorithm's ability to escape local optima; right t The bat's flight speed variable at time t is modified by adding a Cauchy mutation factor to change the bat's flight speed and enhance its ability to escape local optima. The mutation formula is shown below: ； ； In the formula, For the first i Only bats Position of the round; It is a uniformly distributed random number within the interval [0, 1]. For the coefficient vector, It is a vector that decreases linearly from 2 to 0; Let be a uniformly distributed random vector within the interval [0, 1].

3. The wind-solar-pumped hydro storage joint power generation scheduling method based on the improved bat algorithm according to claim 1, characterized in that: The specific method for analyzing and modeling the loss mechanism of pumped storage units in step two is as follows: The loss mechanism includes hydraulic losses and electrical losses. Hydraulic losses are represented by the efficiency coefficients of the water pumps and turbines, while electrical losses are represented by the efficiency coefficients of the generators and motors. Finally, the overall energy conversion efficiency of the pumped storage system is calculated using a comprehensive efficiency coefficient formula, which is as follows: ； in, The overall efficiency coefficient, This represents the efficiency coefficient of the water turbine. It is the efficiency coefficient of the generator. It is the efficiency coefficient of the electric motor; ； ； ； In the formula, It is the hydraulic power of a water pump after it converts electrical energy into the potential energy of water. It is the input power of the water pump. It is the generator's electrical power. It is the mechanical power of the generator. It is the electric power of the motor. It refers to the mechanical power of the electric motor.

4. The wind-solar-pumped hydro storage joint power generation scheduling method based on the improved bat algorithm according to claim 1, characterized in that: The uncertainty modeling in step three includes using Weibull and Beta distributions to model the uncertainty of wind power and photovoltaic power generation, and using historical data statistical analysis to obtain the probability density functions of wind speed and light intensity. The probability density function of wind speed is: ； in, ； ； In the formula, Wind speed; For shape parameters; For scale parameters; It is a gamma function; Average wind speed; The standard deviation of wind speed reflects the degree of fluctuation of wind speed around the average value; The probability density function of solar radiation intensity is: ； In the formula, Normalized light intensity; and For shape parameters; These are Beta parameters.

5. The wind-solar-pumped hydro storage joint power generation scheduling method based on the improved bat algorithm according to claim 1, characterized in that: The objective function of the scheduling model considers maximizing the expected return for different scenario probability distributions, and the formula is as follows: ； In the formula, To maximize the expected revenue of the combined power generation system; For the first One scenario; For the scene The probability of; For the scene The following benefits; Total number of scenes; Revenue generated by a combined power generation system that includes wind power, photovoltaic power, thermal power units and pumped storage units; For the operating costs of wind power and photovoltaic units; The cost of pumping water for pumped storage units; The cost of generating electricity from thermal power units; The punitive costs of pollutant emissions; Among them, the power generation revenue of the combined power generation system including wind power, photovoltaic, thermal power units and pumped storage units It is the sum of the following parts: The feed-in tariff for solar power is multiplied by the output power of the solar power unit. The on-grid electricity price of wind power multiplied by the output power of the wind turbine; The grid-connected electricity price of pumped hydro storage is multiplied by the output power of the pumped hydro storage unit. The on-grid electricity price of thermal power units multiplied by the output power of thermal power units; The scenario involves using scenario generation technology to create multiple possible wind power and photovoltaic unit power generation output scenarios. Each scenario is randomly generated based on the probability distribution of wind speed and solar irradiance, reflecting power generation fluctuations under different weather conditions and time periods, thereby improving the robustness and accuracy of the scheduling model. A scenario As shown below: ； in, and The first The output power of wind turbines and photovoltaic units at time t in each scenario.

6. The wind-solar-pumped hydro storage joint power generation scheduling method based on the improved bat algorithm according to claim 1, characterized in that: The transient analysis module establishes dynamic simulation models for the following components: Based on the mechanical and electromagnetic characteristics of the generator, the transient model of the generator is represented as follows: ； In the formula, The moment of inertia of the generator; It is the mechanical angular velocity; and These are the mechanical input torque and the electromagnetic output torque, respectively. The damping coefficient; ； In the formula: It is the stator voltage; It is the stator resistance; It is the stator current; It is a stator inductor; It is the transient electromotive force of the generator; The stator-side electrical angular frequency; To ensure system stability during sudden changes in power supply and demand, voltage response and frequency response modeling are used to address sudden fault events. Voltage response is used to describe the voltage recovery process after a fault. The model is as follows: ； In the formula, It is the voltage before the fault; It is the attenuation coefficient; It is the damped oscillation frequency; It is the phase angle; The time variable is the period after the fault occurs; Frequency response The change in system frequency after a disturbance is described by the following equation: ； in, It is the maximum value of the frequency offset; It is the natural oscillation frequency.

7. The wind-solar-pumped hydro storage joint power generation scheduling method based on the improved bat algorithm according to claim 6, characterized in that: The transient analysis module also includes time-domain simulation and frequency-domain analysis modules. It simulates the changes in voltage, frequency and phase angle of the system through differential equations and numerical integration methods, and uses Fast Fourier Transform (FFT) to perform frequency-domain analysis on the established dynamic simulation model to evaluate the stability and oscillation modes of the system.

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