A method for calculating transient power angle trajectory and stability boundary of a ship PPL power supply system

Abstract

The application provides a kind of ship PPL power supply system transient power angle trajectory and stability boundary calculation method, belong to ship power system transient stability analysis technical field, including the construction containing PPL ship power system period variable step length variable order Dormand-Prince analytical model, transient power angle phase space trajectory is described;Constitute Lyapunov periodic index as the criterion of the convergence of system period time-varying state variable, extract state trajectory characteristics and identify the convergence stability, sub-stable and divergent unstable region boundary of system parameter;Based on the criterion, a parameter sensitivity calculation method is proposed, and the multi-time scale parameter characteristics of the system transient phenomenon are quantitatively analyzed.The application adopts the above-mentioned ship PPL power supply system transient power angle trajectory and stability boundary calculation method, which overcomes the limitation of the existing method in dealing with the complex pulse load dynamic characteristics and nonlinear behavior of ship power system, and provides a technical solution for the safe and stable operation of ship power system under pulse load impact.

Description

Technical Field

[0001] This invention relates to the field of transient stability analysis technology for ship power systems, and in particular to a method for calculating the transient power angle trajectory and stability boundary of a ship PPL power supply system. Background Technology

[0002] Shipboard integrated electrical systems (SIPS) consist of power plants, power grids, and various types of loads. They suffer from a lack of high-power rotating equipment and limited energy storage capacity, resulting in low system inertia and weak power support, exhibiting the characteristics of an independent power system characterized by "high load - small grid - weak source." Driven by multi-mission capability requirements, ships need to carry high-power pulse loads (PPLs) such as radar. The power cycle, duty cycle, and peak power of these PPLs vary over a wide range of operating conditions, exhibiting strong nonlinear time-varying impact characteristics. This reduces the system's transient stability margin and shrinks the safe operating boundary. In particular, low-frequency, wide-duty-cycle PPLs are prone to causing impact oscillations in low-inertia SIPS, potentially leading to generator disconnection and shutdown. Therefore, to ensure the stable and safe operation of SIPS and the reliability of power supply, it is urgent to conduct research on the system's transient stability under nonlinear periodic disturbances to accurately assess the PPL's ​​accessibility.

[0003] The periodic power disturbances (PPL) and the dynamic interaction between power-frequency control in the ship's integrated electric power system (ESP) result in a nonlinear periodic time-varying (NLTP) characteristic. The PPL disturbance power surges from a low value to a high value within the period, then drops sharply back to a low value, causing significant jumps in the system state and complex dynamic characteristics. Linearized analytical methods struggle to capture the characteristic information at these state jump moments, leading to distorted characterization of the state trajectory. Furthermore, under periodic disturbances, the NLTP system's state may exhibit not only asymptotic convergence at the equilibrium point but also bounded periodic oscillations. Its stability boundary no longer possesses a regular shape and does not satisfy the traditional assumptions based on linearized or conserved equilibrium structures. In summary, transient stability assessment of NLTP systems faces modeling complexity due to frequent nonlinear state jumps, and the non-static and irregular nature of the stability boundary influenced by periodic disturbances.

[0004] The problem of modeling the dynamic response of a ship's integrated electric power system (UPS) under power phase displacement (PPL) disturbances is addressed. The PPL exhibits strong nonlinearity and periodic time-varying coupling characteristics, making fixed-structure analytical models inadequate for adaptability and unable to accurately characterize the system's dynamic response. Existing techniques employ Fourier decomposition to represent the PPL, but reconstructing the harmonic basis functions is necessary when load parameters change dynamically. Furthermore, approximating the step signal with mid-to-high frequency truncation can easily lead to Gibbs free energy, spectral leakage, and oscillation distortion. Alternatively, finite integral techniques and vector fitting algorithms can be used to improve the accuracy of the PPL model, but practical applications are limited by dynamic changes and nonlinear characteristics. Existing models often employ static structures and fixed parameters, making it difficult to respond to frequent state transitions under PPL disturbances. This results in the loss of periodic dynamic characteristics of the operating trajectory, reducing the accuracy of characterizing NLTP features.

[0005] Numerical methods can accurately simulate the dynamic response of a system after disturbances, evaluate short-circuit fault recovery capability, and identify critical fault conditions. These methods mainly include the Euler method, the Runge-Kutta (RK) method, and the Dormand-Prince (DP) method. The implicit RK method evaluates the transient stability of nonlinear systems, but its ability to resolve short-time dynamic responses to high-frequency disturbances is limited, and it is prone to phase shifts and analytical errors. Compared to the fixed-step RK method, the DP method uses an adaptive step size, making it more suitable for complex nonlinear systems. The DP method is applied to voltage sag fault analysis, but in-depth transient stability research is still lacking in PPL (Power Line-Lowering) scenarios. Given the high accuracy and efficiency requirements for model solving in integrated shipboard power systems with PPLs, it is necessary not only to use variable-step-size analytical methods to accurately describe the time-series response of the system's nonlinear state variables, but also to construct stability criteria that match its periodic evolution characteristics.

[0006] Existing stability criteria for nonlinear periodic time-varying systems suffer from conservatism. For example, eigenvalue analysis, the Nyquist criterion, and Floquet theory all rely on linearization modeling assumptions. On one hand, approximating the NLTP system as a linear periodic time-varying system and extracting dominant modes using Floquet theory leads to an exponential increase in the decomposition order due to periodic decoupling of the state transition matrix, and the cumulative error caused by high-frequency mode truncation severely impacts the reliability of the criteria. On the other hand, while the dynamic averaging method simplifies the NLTP system to a linear time-invariant model, this model cannot analyze the periodic behavior of the trajectory, and its averaging process obscures the dynamic characteristics of the time-varying parameters, resulting in fundamental deviations in the critical stability margin prediction. Furthermore, to compensate for the shortcomings of linear methods, energy functions such as the mixed potential function and Hamiltonian function are used for stability assessment based on energy dissipation theory. While Hamiltonian surface formation and power flow calculations analyze the dynamic behavior of the system, improving the accuracy of the criteria, the energy functions they rely on are essentially still based on static simplified models, making it difficult to characterize the nonlinear coupling and high-dimensional path evolution characteristics of the system state under PPL perturbations.

[0007] To meet the stability assessment needs of systems under nonlinear periodic perturbations, existing research has used state trajectory convergence characteristics for stability determination. Among these, the Lyapunov exponent, which assesses stability by calculating the long-term average convergence exponent of the system's state trajectory, has advantages such as not relying on linearization modeling and adapting to time-varying characteristics, and is widely used in the dynamic stability analysis of nonlinear systems. However, constructing a stability criterion based on the maximum Lyapunov exponent for the system's perturbation response is essentially a time-averaged quantity, which is difficult to accurately reflect the system's response characteristics under periodic perturbations, limiting its accuracy and practicality in high-dimensional, rapidly changing systems. In summary, existing stability criteria for NLTP systems under PPL perturbations suffer from problems such as model simplification, neglect of dynamic information, and lack of apparent nonlinear time-varying characteristics, making it difficult to support accurate analysis of complex state trajectories and effective identification of stability boundaries.

[0008] To accurately characterize the nonlinear periodic time-varying features and dynamic behavior of a system and clarify the transient mechanism of periodic disturbances, this patent proposes a method for calculating the transient power angle trajectory and stability boundary. This method constructs a Dormand-Prince analytical model with variable step size and order to solve the modeling problem of nonlinear periodic time-varying state variables, and derives the system's transient power angle trajectory curve. It constructs a Lyapunov periodic exponent as a criterion for measuring the convergence of the system's periodic time-varying state variables, extracts state trajectory features, and identifies the boundaries of convergent, stable, metastable, and divergent unstable regions of system parameters. Based on this criterion, a parameter sensitivity calculation method is proposed to quantify the influence of multiple time-scale parameters on stability. Summary of the Invention

[0009] The purpose of this invention is to provide a method for calculating the transient power angle trajectory and stability boundary of a ship's PPL power supply system. It constructs a variable-step Dormand-Prince analytical model for a pulse load DC power supply system, solving the problem of insufficient accuracy in existing methods when dealing with nonlinear time-varying problems caused by high-power periodic pulse loads. Furthermore, it constructs a Lyapunov periodic exponent criterion to accurately analyze complex state trajectories and effectively identify stability boundaries. Through parameter sensitivity analysis, it quantifies the impact of key parameters on system stability to optimize control strategies. This overcomes the limitations of existing methods in representing the complex dynamic characteristics and nonlinear behavior of ship power systems under pulse loads, providing a technical solution for the safe and stable operation of ship power systems under pulse load impacts.

[0010] To achieve the above objectives, this invention provides a method for calculating the transient power angle trajectory and stability boundary of a ship's PPL power supply system, comprising the following steps:

[0011] S1. Construct a DC power system model for a ship with pulsed loads. Use the periodic variable step size and variable order Dormand-Prince method to analyze the DC power system model. Specifically, a fourth-order small step size Dormand-Prince model is used during the load power change phase, and a fifth-order large step size Dormand-Prince model is used after the load power remains stable at the peak or trough, to capture the nonlinear dynamic response of the pulsed load in different dynamic phases.

[0012] S2. Determine the transient stability mechanism of the ship's DC power system, divide the ship's DC power system into three states: convergent stability, metastability and divergent instability, construct the Lyapunov periodic index as a criterion for measuring the convergence of the system's periodic time-varying state variables, extract the system state trajectory features, and avoid the loss of dynamic information caused by time averaging by measuring the difference between two state points that are half a cycle apart in the system state trajectory.

[0013] S3. Calculate the transient power angle trajectory and stability boundary of the ship's DC power system. Based on the ship's DC power system model after analysis by the Dormand-Prince method, draw the phase space operation trajectory of the ship's DC power system state quantities in the full time domain. Using the Lyapunov periodic exponent as the stability criterion, gradually adjust the key parameters, extract the stable and metastable boundary points of the ship's DC power system, and draw the parameter stability domain.

[0014] S4. Introduce a normalized parameter sensitivity analysis method for the ship's DC power system to quantify the impact of key parameters on the stability of the ship's DC power system at different time scales.

[0015] Preferably, the periodically variable step-size and variable-order Dormand-Prince method in S1 includes a pulse load access phase where the local error of the fourth-order small step-size is... Precisely captures the dynamic response during sudden changes in pulsed load power; during the stable phase when the load power remains at its peak or trough, the fifth-order large-step local error is... Improve computational accuracy and efficiency.

[0016] Preferably, constructing the Lyapunov periodic index in step S2 includes the following steps:

[0017] S21. Select the transient power angle δ as the main state variable of the dynamic evolution characteristics of the ship's DC power system under periodic disturbance;

[0018] S22. Using 2m pulse waves as the analysis interval, select the m-th period of the middle segment as the reference point, extract the difference between the state of the middle segment and the state of the first half of the period, and introduce the numerical integration results of the Dormand-Prince model to define the Lyapunov periodic exponent expression.

[0019] Preferably, the parameter sensitivity analysis method in S4 includes: introducing normalization processing to improve the comparability and evaluation reliability of sensitivity indicators, wherein the expression for the normalized parameter sensitivity is:

[0020]

[0021] Where x is the state variable, z is the parameter, Δz is the relative change of the parameter, [||f(x,z+Δz)||-||f(x,z)||] / ||f(x,z)|| represents the relative change of the system state response when the parameter undergoes a relative change of Δz; |Δz| / |z| is the relative disturbance of the parameter itself.

[0022] Preferably, the ship DC power system model with pulse load in S1 includes a diesel generator, a supercapacitor, a DC voltage regulator, and a Boost converter. The supercapacitor is connected to the DC bus, the Boost converter is located between the pulse load and the DC bus, and the diesel generator is connected to the DC bus through a phase-shifting transformer and a 24-pulse rectifier.

[0023] Preferably, in S2, the metastable state is that the ship's DC power system operates in a periodic dynamic equilibrium, neither fully converging to a steady state nor infinitely diverging to instability. The ship's DC power system maintains periodic oscillations within a finite amplitude range. The dynamic coupling between the high-frequency pulse power disturbance of the pulse load and the low inertia characteristics of the ship's DC power system causes the energy of the ship's DC power system to continuously exchange between the electromechanical time scale and the electromagnetic time scale, forming a periodic dynamic equilibrium.

[0024] Preferably, the expression for constructing the Lyapunov periodic exponent is:

[0025]

[0026] Where λ represents the Lyapunov periodic exponent, n represents the period number, and ||·||2 represents the 2-norm;

[0027] In S22, the Lyapunov periodic exponent expression is defined as follows:

[0028]

[0029] Where, δ m Let δ be the transient power angle state of the ship's DC power system at time mT. m-0.5 Let b be the transient power angle state of the ship's DC power system at time mT-0.5T. i k i This represents the analytical increment at time mT corresponding to the Dormand-Prince method. The increment for parsing the Dormand-Prince method at time mT-0.5T is given.

[0030] Preferably, the Lyapunov periodic index classifies the evolution characteristics of the ship's DC power system into three types: when λ = 0, the ship's DC power system converges and stabilizes; when λ is a constant, the ship's DC power system exhibits periodic behavior but is not unstable, which is defined as metastable; when λ ≠ constant and grows exponentially, the ship's DC power system tends to become chaotic with an exponential growth trend throughout the entire pulse load cycle, and the system becomes unstable.

[0031]

[0032] Therefore, the present invention employs the above-mentioned method for calculating the transient power angle trajectory and stability boundary of a ship PPL power supply system, and the technical effects are as follows:

[0033] 1. This invention proposes the Dormand-Prince method with variable step size and variable order, which adaptively adjusts the solution accuracy and efficiency according to the dynamic stage of the system, and realizes the continuous and accurate description of the system state variables throughout the pulse disturbance process, effectively overcoming the problem of poor adaptability of traditional fixed step size or local linearization methods to nonlinear coupling characteristics.

[0034] 2. Based on the periodic evolution characteristics of the system state trajectory in phase space, this invention constructs the Lyapunov periodic index as a stability indicator, which can accurately classify the three operating states of convergent stability, metastability and divergent instability. It overcomes the shortcomings of the averaging method ignoring dynamic details and the energy function method lacking evolutionary expression, and significantly improves the dynamic response sensitivity and boundary identification accuracy of stability determination.

[0035] 3. This invention analyzes the influence weight of key parameters under three typical time scales—current control, voltage control, and power control—by using normalized parameter sensitivity indices. It identifies the dominant role of voltage control parameters in transient stability and clarifies their dynamic coupling mechanism with pulse period signals, providing theoretical support for control strategy design and system parameter optimization. Attached Figure Description

[0036] Figure 1 This is a schematic diagram of a ship DC power system model containing pulse load according to the present invention;

[0037] Figure 2 This is a diagram of the pulse load power supply structure based on the Boost converter of the present invention;

[0038] Figure 3 This is a schematic diagram of the DC generator set control structure of the present invention;

[0039] Figure 4This is a schematic diagram illustrating the physical meaning of the transient stability interval of the system after the pulse load is connected according to the present invention.

[0040] Figure 5 This is a schematic diagram illustrating the relationship between power transfer and inertia support in the pulsed load connection of the present invention.

[0041] Figure 6 A flowchart is drawn for the parameter stability region of this invention;

[0042] Figure 7 This is a schematic diagram of the phase space trajectory of the power angle with different parameters according to the present invention; Figure 7 (a) is a schematic diagram of the power angle phase space trajectory under different pulse load duty cycles; Figure 7 (b) is a schematic diagram of the power angle phase space trajectory under different pulse load cycles; Figure 7 (c) is a schematic diagram of the power angle phase space trajectory under different pulse load peak power;

[0043] Figure 8 This is a schematic diagram of the calculation results of the stability boundary of the pulse load parameters in this invention; Figure 8 (a) is a schematic diagram of the convergent stable and metastable boundary ε; Figure 8 (b) represents the metastable and divergent instability boundary γ;

[0044] Figure 9 This is a schematic diagram illustrating the time scale division of transient events in this invention;

[0045] Figure 10 This is a schematic diagram showing the parameter sensitivity results of different key stabilizing factors in this invention;

[0046] Figure 11 This is a schematic diagram of the simulation and LDP evaluation results of the present invention;

[0047] Figure 12 This is a schematic diagram showing the verification results of the stability boundary ε and γ of the present invention; Figure 12 (a) is a schematic diagram of the boundary ε verification under operating condition A1; Figure 12 (b) is a schematic diagram of the boundary ε verification under operating condition B1; Figure 12 (c) is a schematic diagram of the boundary ε verification under operating condition C1; Figure 12 (d) is a schematic diagram of the verification boundary γ under operating condition A2; Figure 12 (e) is a schematic diagram of the verification boundary γ under operating condition B2; Figure 12 (f) is a schematic diagram of the verification boundary γ under operating condition C2. Detailed Implementation

[0048] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0049] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.

[0050] Example 1

[0051] This invention provides a method for calculating the transient power angle trajectory and stability boundary of a ship's PPL power supply system, comprising the following steps:

[0052] S1. Construct a DC power system model for a ship with pulsed loads. Use the periodic variable step size and variable order Dormand-Prince method to analyze the DC power system model. Specifically, a fourth-order small step size Dormand-Prince model is used during the load power change phase, and a fifth-order large step size Dormand-Prince model is used after the load power remains stable at the peak or trough, to capture the nonlinear dynamic response of the pulsed load in different dynamic phases.

[0053] S2. Determine the transient stability mechanism of the ship's DC power system, divide the ship's DC power system into three states: convergent stability, metastability and divergent instability, construct the Lyapunov periodic index as a criterion for measuring the convergence of the system's periodic time-varying state variables, extract the system state trajectory features, and avoid the loss of dynamic information caused by time averaging by measuring the difference between two state points that are half a cycle apart in the system state trajectory.

[0054] S3. Calculate the transient power angle trajectory and stability boundary of the ship's DC power system. Based on the ship's DC power system model after analysis by the Dormand-Prince method, draw the phase space operation trajectory of the ship's DC power system state quantities in the full time domain. Using the Lyapunov periodic exponent as the stability criterion, gradually adjust the key parameters, extract the stable and metastable boundary points of the ship's DC power system, and draw the parameter stability domain.

[0055] S4. Introduce a normalized parameter sensitivity analysis method for the ship's DC power system to quantify the impact of key parameters on the stability of the ship's DC power system at different time scales.

[0056] like Figure 1 As shown, the ship's DC power system for pulsed load (PPL) consists of a diesel generator (DG), a supercapacitor, a DC regulated capacitor, and a boost converter. The supercapacitor is connected to the DC bus via a bidirectional DC / DC converter, which is equivalent to adding a capacitor C to the DC bus. bus The value of .

[0057] like Figure 2As shown, the PPL power supply relies on the preceding Boost converter, which is isolated from the DC bus to prevent the PPL surge from directly affecting the DC bus voltage U. bus , Figure 2 In this context, i1 represents the input current of the Boost converter, and U... dc1 U represents the output voltage. ref G represents a given voltage. v (s), G i (s) represent the voltage loop and current loop transfer functions, respectively, D1 represents the duty cycle, C1 is the capacitor, and L1 is the inductor.

[0058] To investigate the impact of pulse load (PPL) connection on the stability of the ship's DC power system, the PPL must first be analyzed. Taking radar pulse load as an example, its high-power PPL is used in the transmission signal mode. paek In the receive signal mode, it is low power P pul It exhibits strong periodic fluctuation characteristics and can be approximated as a rectangular wave. Its time-domain expression is:

[0059]

[0060] Among them, D p T is the duty cycle of PPL. p Let T be the period, T be the real-time point, and t be the ratio of T to T. p Take the remainder, when the remainder t falls within [0, D] p T p When within the specified range, it is in radar pulse divergence signal mode.

[0061] Boost converter U dc1 The dynamic equations for i1 are:

[0062]

[0063] Introduce variable m v With m i Determine the duty cycle D1, m v With m i Characterize the integral quantities of voltage and current errors respectively:

[0064]

[0065] Among them, i ref For a given current, k iu k ii k pu k pi All are proportionality coefficients, k iu Used to adjust voltage-related control quantities; their specific values ​​depend on the system design and parameter settings; k ii Used to adjust the response characteristics of current control; kpu Used to convert voltage difference into current reference value; k pi Used to adjust the effect of current error on duty cycle.

[0066] The diesel generator is connected to the DC bus via a phase-shifting transformer and a 24-pulse rectifier, such as... Figure 3 As shown, the 24-pulse rectifier uses uncontrolled diode rectification, which effectively reduces ripple. Since there is no control unit, it is equivalent to a gain model, i.e.:

[0067]

[0068] Among them, C bus For the bus capacitance, U bus P is the DC bus voltage. L U L K represents the power and peak voltage on the AC side, respectively. m This represents the equivalent gain of the rectifier.

[0069] The equivalent transfer functions of the exciter and governor of the DG can be represented by first-order functions:

[0070]

[0071] Among them, K e For the equivalent gain of the exciter, τ e K is the exciter time constant. g For the equivalent gain of the speed controller, τ g This is the time constant of the speed controller.

[0072] The time-domain expression of the DG model considering the exciter and speed governor is as follows:

[0073]

[0074] Among them, P n The governor outputs active power, Δω is the change in angular velocity, ω0 is the initial angular velocity, and P e Q e Q represents the output active and reactive power. n Where J is the rated reactive power, D is the equivalent inertia, and K is the equivalent damping. w K is the frequency modulation coefficient. q V is the reactive power coefficient. ref Here, U0 is the rated voltage at the DG terminal, and δ represents the power angle of the DG. It's worth noting that the power angle δ represents the difference between the motor terminal voltage U0 and the load terminal voltage U. L The phase difference between them.

[0075] Neglecting line impedance, the equivalent impedance of the phase-shifting transformer can be expressed as X. T Electromagnetic power time-domain expression:

[0076]

[0077] Therefore, the time-domain dynamic model of the ship's DC power system is obtained as follows:

[0078]

[0079] Based on the time-domain dynamic model of the ship's DC power system established above, the ship's DC power system can be expressed as a system of nonlinear differential equations:

[0080]

[0081] Among them, the state quantity y n =[i1,U dc1 ,m i ,m v U bus U o ,P n ,Δω,δ] T P represents the PPL power. Due to the short duration of PPL connection and significant transient disturbances, a high-precision analytical method is needed to characterize its dynamic behavior. The Dormand-Prince (DP) method, with its variable step size and order characteristics, can balance high accuracy in the transient phase with computational efficiency in the steady phase. It should be noted that step size and order are key parameters in numerical solution methods, directly affecting the analytical accuracy and computational efficiency of the system state trajectory. Generally, a higher order and a smaller step size result in higher system accuracy. However, a small step size affects solution efficiency; this can be mitigated by reducing the solution order to maintain both accuracy and efficiency.

[0082] When the system is at a moment of power change, i.e., mod(t) n ,T p ) = 0 or mod(t) n ,T p ) = D p T p PPL power is supplied by P pul Suddenly rises to P peak Or by P peak Sudden drop to P pul This leads to the system state variable y n The power change is significant due to sudden power fluctuations. To accurately capture the dynamic response of the PPL at the moment of a step change, this section uses the fourth-order small-step DP method to describe the moment of the power fluctuation, with the small step size h1 set to 10. -4 The local error is O(h1) 5 The fourth-order increment corresponding to the nonlinear differential equation system is:

[0083]

[0084] tn+1 The system state at time t can be represented as:

[0085]

[0086] When the system is in the pulse power duration range, i.e., mod(t) n ,T p )∈(0,D p T p ) or mod(t n ,T p )∈(D p T p ,T p PPL power remains at P pul or P peak At this point, the step term is zero and no longer has any additional effect on the system. Therefore, using a large step size can improve computational efficiency, but to maintain analytical accuracy, a fifth-order large-step DP method is required, with h2 set to 10. -3 Its incremental expression is as follows:

[0087]

[0088] t n+2 The system state at time t can be represented as:

[0089]

[0090] The fifth-order DP model adds k5 and k6 increments to the fourth-order model. In particular, k6 is a weighted average of the information between k1 and k5, providing additional supplementary information, making the local error O(h2). 6 In addition, the maximum step size h2 is 10. -3 .

[0091] This invention achieves accurate modeling of systems containing transient pulse loops (PPLs) through a periodic variable step size and variable order dynamic modeling (DP) method. A fourth-order small step size DP model is used during the PPL initiation phase to accurately capture transient changes; once the system enters a stable phase, a fifth-order large step size DP model is employed to improve computational efficiency and accuracy. This method effectively balances accuracy and computational efficiency, ensuring accurate modeling of the system at different dynamic stages.

[0092] When conventional loads are connected to a ship's integrated electrical system, the system's operating state can generally be classified as either stable or unstable. However, the periodic impact characteristics of PPL loads can induce a third operating state, namely a metastable state, such as... Figure 4As shown. This state can be defined as the system operating in a periodic dynamic equilibrium, neither fully converging to a steady state nor diverging infinitely to instability. It is characterized by the system maintaining periodic oscillations within a finite amplitude range. The cause is the dynamic coupling between the PPL high-frequency pulse power disturbance and the system's low inertia characteristics, which leads to the continuous exchange of system energy between the electromechanical time scale and the electromagnetic time scale, forming a periodic dynamic equilibrium.

[0093] like Figure 4 As shown, a metastable state can be represented as a circle with center O and radius γ. Then there exists another circle with center O and radius ε. Motion on circle ε starts from any point and ends within circle γ; this is a metastable state, characterizing a transitional state where the system can withstand short-term periodic disturbances without exceeding the critical energy threshold. It can be described as:

[0094]

[0095] A convergent stable state can be represented as follows: If a state is not only metastable but also has a neighborhood from which points gradually converge to the equilibrium point O, it is called convergent stability. The corresponding system state retains convergence after the PPL perturbation and can be described as follows:

[0096]

[0097] Divergent stability can be expressed as follows: if a certain motion state does not meet the above conditions, in other words, there exists a circle γ, and no matter how the circle ε is chosen, its trajectory will eventually exceed the circle γ. This can be described as the PPL disturbance exceeding the system's comprehensive regulation capability, causing the system state to deviate from the stable region and continue to diverge, making it difficult to maintain safe operation.

[0098] like Figure 5 As shown, the power change ΔP generated when PPL is connected load It acts directly on capacitor C1 and produces a change ΔU dc1 Since C1 provides short-term inertia support, the change in residual power is ΔP. in1 The Boost converter will then generate a voltage change ΔU on the common bus. bus With power change ΔP L The relationship between DC-side power transmission and inertia support can be expressed as:

[0099]

[0100] Since the electromotive force and phase inside the prime mover of a diesel generator cannot change abruptly, it needs to rely on the kinetic energy stored in the rotor to provide short-term load buffering, which can be described as:

[0101]

[0102] The transient power angle δ is selected as the main state variable of the dynamic evolution characteristics of the ship DC power system under periodic disturbance, and the boundary ε and γ are determined.

[0103] Under convergent and stable conditions, the ΔP caused by PPL connection load When the capacitance is small, the DC-side capacitor can completely absorb the periodic load power fluctuations, and the convergence stability boundary ε is:

[0104]

[0105] In a metastable state, the DC-side capacitor cannot absorb periodic load power fluctuations, requiring short-term power supply from the rotor's kinetic energy. Limited by the inherent characteristic of rotor inertia, there exists an upper limit, which is the metastable boundary γ.

[0106]

[0107] Based on the dynamic evolution characteristics of the aforementioned transient stable region and the power-inertia coupling mechanism of PPL disturbances, a quantitative index needs to be constructed to accurately characterize the periodic convergence characteristics of the system state trajectory. The Lyapunov exponent, as an important indicator for evaluating the convergence of system state variables, judges stability by calculating the long-term average exponential convergence characteristics of the system state trajectory; however, its time-averaged characteristics are difficult to accurately characterize the dynamic response under periodic disturbances. Therefore, a Lyapunov periodic exponent is constructed as a criterion for measuring the convergence of the system's periodic time-varying state variables. The characteristics of the system state trajectory are extracted, and the trend of system state change is measured by comparing the adjacent periodic values ​​of the state variables on the trajectory.

[0108] The boundary between the convergent stable boundary ε and the metastable boundary γ depends on the PPL perturbation power ΔP. load With system inertia support capability J s ·dδ 2 / d 2 t The dynamic equilibrium is reached. To quantify this equilibrium state, the transient work angle δ is chosen as the state variable, and a Lyapunov periodic exponent is constructed:

[0109]

[0110] Where λ represents the Lyapunov periodic exponent, n represents the period number, and ||·||2 represents the 2-norm, which geometrically represents the straight-line distance from the origin to the vector. For describing changes in system state or the distance between adjacent orbits, the 2-norm provides a more intuitive measure.

[0111] To improve the construction accuracy of the Lyapunov periodic exponent λ and reduce the traversal calculation of the complete periodic state sequence, the state y of the NLTP system is analyzed based on the periodic variable step size and variable order DP analytical model. nThe analytical results, combined with the selection of the reference period and the local state difference, are used to construct a periodic disturbance response criterion. This patent selects the transient power angle δ as the main state variable for the dynamic evolution characteristics of the system under periodic disturbances. Using 2m pulse waves as the analysis interval, the m-th period of the middle segment is selected as the reference point. The difference between this point and the state of the first half of the period is extracted, and the numerical integration results of the DP model are introduced. The Lyapunov periodic exponent expression is defined as follows:

[0112]

[0113] Where, δ m Let δ be the transient power angle state of the ship's DC power system at time mT. m-0.5 Let b be the transient power angle state of the ship's DC power system at time mT-0.5T. i k i This represents the analytical increment at time mT corresponding to the Dormand-Prince method. The increment for parsing the Dormand-Prince method at time mT-0.5T is given.

[0114] The above classification of transient stable regions into three categories corresponds to the classification of system evolution characteristics into three types based on the Lyapunov periodic exponent: when λ = 0, the system is convergent and stable; when λ is a constant, the system exhibits periodic behavior but is not unstable, defined as metastable; when λ ≠ constant and grows exponentially, the system exhibits exponential growth and tends towards chaos throughout the entire PPL period, and the system becomes unstable. It should be noted that λ is rounded to two decimal places.

[0115]

[0116] The system state evolution process under PPL perturbation exhibits nonlinear coupling and path dependence, causing the transient stability boundary to lose its static, regular shape. To address this, this invention precisely analyzes the system's dynamic response under PPL perturbation, extracts state trajectory features, and calculates the convergent, metastable, and divergent instability region boundaries of the system parameters. The flowchart is shown below. Figure 6 As shown. The steps are as follows:

[0117] Based on the periodic variable step size and variable order analytical model of the ship DC power system with PPL, the phase space operation trajectory of the system state variables in the whole time domain is plotted. Using the Lyapunov periodic exponent as the criterion for stability, key parameters are gradually adjusted to extract the stable and metastable boundary points of the system, and then the parameter stability domain is plotted to characterize the distribution characteristics of the stability region in the parameter space.

[0118] To further quantify the influence of key system parameters on transient stability characteristics, this invention proposes a normalized parameter sensitivity analysis method. This method reflects the influence weight of each parameter during the dynamic evolution of the system state by analyzing the response of state variables to parameter disturbances. Considering the coexistence of variables of different scales in a multidimensional state space, differences in parameter dimensions may distort the sensitivity results. Therefore, this patent introduces normalization processing to improve the comparability and reliability of the sensitivity indicators. Taking state variable x as an example, with parameter z, the normalized parameter sensitivity expression is:

[0119]

[0120] Where x is the state variable, z is the parameter, Δz is the relative change of the parameter, [||f(x,z+Δz)||-||f(x,z)||] / ||f(x,z)|| represents the relative change of the system state response when the parameter undergoes a relative change of Δz; |Δz| / |z| is the relative disturbance of the parameter itself.

[0121] Parameter sensitivity reflects the degree of relative change in state variables caused by a unit relative parameter disturbance, and can be used to identify key parameters that significantly affect the convergence and transient stability of a system. The results can be positive or negative, which can be understood as follows: a positive sensitivity indicates a positive correlation, meaning that increasing the parameter increases its equivalent negative impedance, leading to a greater risk of system instability; a negative sensitivity indicates the opposite.

[0122] To verify the effectiveness of the proposed stability assessment method and to determine the transient stability factors of the ship's DC power system containing PPL, and to characterize the parameter stability region, the simulation parameters of the ship's DC power system are shown in Table 1.

[0123] Table 1

[0124]

[0125]

[0126] This invention will apply the proposed transient stability assessment method to characterize the phase space trajectory of the system state variables under different operating conditions, and determine the boundary ε and γ based on the Lyapunov periodic exponent.

[0127] Different pulse load duty cycles D p Period T p and peak power P peak The trajectory diagrams of DG power angle change under the operating conditions are as follows: Figure 7 As shown. By Figure 7 (a) It can be seen that the degree of power angle fluctuation is not linearly related to the duty cycle, but rather increases first and then decreases as the duty cycle increases, with the most obvious fluctuation in the middle region; Figure 7 (b) It can be known that Tp Increasing the PPL cycle leads to a longer PPL duration, which exacerbates the adverse effects on the DG excitation system. The closer the PPL cycle is to the DG excitation speed regulation response time, the more severe the frequency fluctuations become. Figure 7 (c) It can be seen that the increase in peak power has a significant impact on the transient stability of the system. When the peak power exceeds the system's inertial support, the DG cannot respond quickly and absorb the influence of the high-frequency peak PPL, causing the DG power angle to tend to diverge chaotically.

[0128] Furthermore, the boundary values ​​of the PPL parameters ε and γ are explored using the Lyapunov periodic exponent, such as... Figure 8 As shown.

[0129] The multi-timescale dynamics of ship equipment also determine the multi-timescale dynamic characteristics of ship electrical systems. Based on the cascading relationships presented by the system structure control, the transient response times corresponding to different controls also roughly exhibit a stepped distribution, such as... Figure 9 As shown. This invention divides the time scale of transient phenomena into current control, voltage control, and power control, performs quantitative analysis on key stabilization links, and explores their coupling degree with PPL disturbances.

[0130] The main parameters under current control time scale are the Boost current loop bandwidth (BCL) and the parameters under voltage control time scale include the Boost voltage loop bandwidth (B). VL Capacitor C bus The parameters in the power control time scale include inductance L1 and the equivalent gain K of the excitation circuit. e The inertia J of DG s With damping D s Furthermore, since the equivalent delay caused by sampling and control is on the order of microseconds, it does not belong to the three major control time scales, but it is still closely related to system stability. Figure 10 The sensitivity analysis results for the above key parameters in the three stability intervals are listed, and it can be found that:

[0131] (1) The parameter sensitivity of the delay element varies significantly in different intervals. In the convergent stable interval, the delay has a weak impact on stability, but its impact is high in the metastable and divergent unstable intervals. The delay element can be equivalent to an integral term. After being subjected to frequent disturbances, its output will gradually accumulate errors, resulting in a slow or unstable system response.

[0132] (2)B VL With C busAs a voltage control time scale parameter, it has a significant impact on the transient stability of the system, and its selection value is directly related to the overall stability of the ship's electrical system. Since the voltage control time scale is within [10ms, 200ms], which is similar to the PPL periodic signal, the control response under the same time scale is prone to dynamic interaction. The stacking of the voltage control signal and the PPL signal can cause the system to fail to accurately identify the signal, thus leading to over-control or lag, or even resonance.

[0133] (3) Current control time scale parameter B CL With power control time scale parameters L1, K e The impact on system transient stability is negligible. Furthermore, the inertia J of DG... s With damping D s It provides inertial support and oscillation suppression for system disturbances, and has a high degree of influence on the overall stability of the system.

[0134] To facilitate a direct and intuitive analysis of the influence of each parameter, their absolute values ​​are taken. However, in reality, variables with positive sensitivity include time delay and B. CL B VL The variables with negative sensitivity are C bus L1, K e J s With D s In other words, increasing the positive sensitivity parameter is detrimental to system stability, while increasing the negative sensitivity parameter is beneficial to system stability.

[0135] This invention uses PPL operating condition P peak =0.5MW, D P =0.5, T P Taking a sudden change from 50ms to 80ms as an example, let's compare the results of simulation and the LDP transient stability evaluation method, such as... Figure 11 As shown, although there are errors between the simulation results and the LDP transient stability assessment results, their trends are consistent, and the errors are within 5%.

[0136] Different boundary points were selected to verify the stable and metastable boundary ε, and the metastable and unstable boundary γ, such as... Figure 12 As shown. P is tested at the verification boundary ε under operating conditions A1, B1, and C1 respectively. peak T p and D p The application of mutations, such as Figure 12 As shown in (a), (b), and (c), taking operating condition A1 as an example, when the power output abruptly changes from 0.3MW to 0.32MW (operating condition A'1), the power angle is in a convergent stable state in A1, and in a clearly oscillating metastable state in A'1. Similarly, Figure 12 (d)(e)(f) Verify the boundary γ, taking working condition B2 as an example, when T pThe system response changed from metastable oscillation to divergent instability mode when it suddenly changed from 50ms to 53ms (operating condition B'1).

[0137] Therefore, the present invention adopts the above-mentioned method for calculating the transient power angle trajectory and stability boundary of a ship's PPL power supply system. From multiple aspects such as model analysis, convergence judgment and parameter sensitivity analysis, it comprehensively improves the transient stability assessment and control level of the ship's power system under pulse load disturbance, and provides strong support for the safe and stable operation of the ship's power system.

[0138] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for calculating transient power angle trajectory and stability boundary of a ship PPL power supply system, characterized in that, The method comprises the following steps: S1, constructing a ship DC power system model containing pulse load, using the Dormand-Prince method with periodical variable step length and variable order to analyze the ship DC power system model, specifically, using the fourth-order Dormand-Prince model with small step length in the stage of sudden change of load power, and using the fifth-order Dormand-Prince model with large step length in the stable stage after the load power keeps at the peak value or the trough value, so as to capture the nonlinear dynamic response of the pulse load in different dynamic stages; S2, determining the transient stability mechanism of the ship DC power system, dividing the ship DC power system into three states of convergent stability, metastable state and divergent instability, constructing a Lyapunov periodic index as a criterion for measuring the convergence of the periodic time-varying state quantity of the system, extracting the characteristic of the state trajectory of the system, and avoiding the loss of dynamic information caused by time averaging by measuring the difference between two state points separated by half a period in the state trajectory of the system; S3, calculating the transient power angle trajectory and the stability boundary of the ship DC power system, based on the ship DC power system model analyzed by the Dormand-Prince method, drawing the state space running trajectory of the ship DC power system in the full time domain, taking the Lyapunov periodic index as the stability criterion, gradually adjusting the key parameters, extracting the stable and metastable boundary points of the ship DC power system, and drawing the parameter stability domain; S4, introducing a normalized parameter sensitivity analysis method into the ship DC power system, and quantifying the influence of the key parameters on the stability of the ship DC power system in different time scales.

2. The method of claim 1, wherein, The Dormand-Prince method with periodical variable step length and variable order in S1 comprises that in the stage of connecting pulse load, the local error of four-order small step length is accurately capturing the dynamic response at the moment of power mutation of pulse load; in the stable stage of keeping the peak or trough of load power, the local error of five-order large step length is improving the calculation accuracy and efficiency.

3. The method of claim 1, wherein, The step S2 of constructing the Lyapunov periodic index comprises the following steps: S21, selecting the transient power angle δ as the main state quantity of the dynamic evolution characteristic of the ship DC power system under periodic disturbance; S22, selecting the middle section mth period as a reference point in the analysis interval of 2m pulse waves, extracting the difference between the state of the mth period and the state of the previous half period, and introducing the numerical integration result of the Dormand-Prince model to define the expression of the Lyapunov periodic index.

4. The method of claim 1, wherein, The parameter sensitivity analysis method in the step S4 comprises: introducing normalization processing to improve the comparability and evaluation reliability of the sensitivity index, and the expression of the normalized parameter sensitivity is wherein x is a state variable, z is a parameter, Δz is a relative change of the parameter, [||f(x,z+Δz)||-||f(x,z)||] / ||f(x,z)|| represents the relative change of the system state response when the parameter has a relative change Δz, and |Δz| / |z| is the relative disturbance of the parameter itself.

5. The method of claim 1, wherein, The ship DC power system model containing pulse load in the step S1 comprises a diesel generator, a super capacitor, a DC voltage stabilizing capacitor and a Boost converter, the super capacitor is connected to the DC bus, the Boost converter is arranged between the pulse load and the DC bus, and the diesel generator is connected to the DC bus through a phase-shifting transformer and a 24-pulse rectifier device.

6. The method of claim 1, wherein, The metastable state in S2 is that the ship DC power system runs in periodic dynamic balance, neither converges completely to a steady state nor diverges infinitely to instability, and the ship DC power system maintains periodic oscillation in a limited amplitude range. The dynamic coupling of high-frequency pulse power disturbance of the pulse load and the low-inertia characteristics of the ship DC power system leads to the continuous exchange of energy of the ship DC power system between the mechanical and electrical time scales, forming a dynamic balance with periodicity.

7. The method of claim 3, wherein the method further comprises: The constructed Lyapunov periodic index expression is: Wherein, λ represents the Lyapunov periodic index, n represents the periodic sequence number, and ||·||2 represents the 2-norm; In S22, the Lyapunov periodic index expression is defined as: wherein δ m is the transient power angle state of the mT moment of the ship DC power system, δ m-0.5 is the transient power angle state of the mT-0.5T moment of the ship DC power system, b i k i is the Dormand-Prince method analytical increment corresponding to the mT moment, is the Dormand-Prince method analytical increment corresponding to the mT-0.5T moment.

8. The method of claim 7, wherein, The Lyapunov periodic index divides the evolution characteristics of the ship DC power system into three types, that is, when λ = 0, the ship DC power system converges stably; when λ is a constant, the ship DC power system presents a periodic behavior but does not lose stability, which is defined as metastable; when λ ≠ constant and exponentially increases, the ship DC power system presents an exponential growth trend to chaos in the whole cycle of the pulse load, and the system loses stability.

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