CN118499148B - Power piston displacement prediction method of free piston Stirling generator - Google Patents

Abstract

A method for predicting the displacement of a power piston of a free piston Stirling generator comprises the following steps: S1: establishing the dynamic equations of the valve piston and the power piston; S2: calculating the pressure wave of the working chamber ; S3: Establish an equivalent circuit when the free piston Stirling generator load changes; S4: Establish a circuit differential equation of the free piston Stirling generator; S5: Calculate the electromagnetic torque of the linear motor; S6: Calculate the internal potential of the free piston Stirling generator; Step S7: Calculate the relationship between the output voltage of the free piston Stirling generator and the displacement of the power piston; S8: Calculate the maximum possible instantaneous value of the internal potential of the free piston Stirling generator; S9: Calculate the maximum possible instantaneous value of the power piston displacement. The present invention can quickly predict the maximum possible instantaneous value of the power piston displacement under a sudden load change, and then predict whether the amplitude of the power piston displacement may exceed the rated value.

Description

A method for predicting the displacement of the power piston of a free-piston Stirling generator

Technical Field

The invention relates to a piston displacement prediction of a piston type Stirling generator, and in particular to a method for predicting the power piston displacement of a free piston type Stirling generator.

Background Art

With the massive consumption of fossil fuels, the reliance on sustainable energy such as solar energy and wind energy is gradually increasing. Solar thermal power generation is an important way to utilize solar energy efficiently and environmentally friendly. Free Piston Stirling Generator (FPSG) is a thermoelectric conversion device based on the Stirling cycle, which has great application potential in solar thermal power generation. Free Piston Stirling Generator has many advantages, such as compatibility with a variety of heat sources, high reliability and significantly low noise. At the same time, the free piston Stirling generator also has the advantage of maintenance-free, because the various components of the engine are assembled in a high-pressure closed container, and there is no mechanical friction between the components. In addition, the free piston Stirling generator does not have the detonation phenomenon of the internal combustion engine during operation, and the noise level is extremely low.

The free piston Stirling generator mainly consists of two sets of elastic systems, each consisting of a piston and several leaf springs. The two elastic systems work together to shuttle high-pressure gas between the heater and the cooler, generating pressure fluctuations. The pressure fluctuations exhibit sinusoidal characteristics and form stable self-excited oscillations with the two elastic systems, driving the linear motor to generate electrical energy. Due to the mechanical movement of the piston and the thermal energy storage at the hot end, the free piston Stirling generator has the potential to stabilize the system voltage fluctuations and frequency fluctuations. However, there is no mechanical connection between the two elastic systems. Therefore, the free piston Stirling generator is difficult to control directly, and its output voltage characteristics are directly related to the load conditions. Compared with traditional photovoltaic power stations with converter interfaces, the dynamic characteristics of CSP power stations vary greatly, which brings great challenges to the analysis and control of the output transient characteristics of the free piston Stirling generator.

When the load of the free piston Stirling generator is suddenly reduced, or a serious open circuit fault occurs, the electromagnetic damping of the linear motor will decrease instantly, and the stroke of the power piston connected to the mover will suddenly increase. In severe cases, the power piston will collide with the cylinder and the valve piston, causing output voltage and current distortion, and damage to the components of the free piston Stirling generator. Therefore, it is very important to analyze the output voltage characteristics of the free piston Stirling generator when the load changes suddenly or the open circuit fault occurs, and then derive the dynamic change characteristics of the power piston, which can further guide the optimization design and scenario application of the free piston Stirling generator.

Since the free piston Stirling generator couples electromagnetic effects and thermodynamic effects, load mutations bring great difficulties to the transient characteristics analysis. Therefore, few studies can accurately describe the transient output voltage characteristics of the free piston Stirling generator. Traditional methods can only evaluate the parameter selection range of the engine during stable operation, and cannot describe the dynamic changes of the output voltage characteristics of the free piston Stirling generator during load switching or open circuit faults. On the other hand, although the transient characteristics of the inverter have received more attention, the method of inverter transient characteristics analysis is difficult to directly apply because the free piston Stirling generator is very different from the fully controlled inverter in terms of working principle.

Summary of the invention

The technical problem to be solved by the present invention is to overcome the shortcomings of the above-mentioned background technology and provide a method for predicting the power piston displacement of a free-piston Stirling generator, which can quickly predict the maximum possible instantaneous value of the power piston displacement under a sudden load change, and then predict whether the power piston displacement amplitude may exceed the rated value.

The technical solution adopted by the present invention to solve the technical problem is a method for predicting the displacement of a power piston of a free piston Stirling generator, comprising the following steps:

Step S1: establishing the dynamic equations of the valve piston and the power piston;

Step S2: Calculate the working chamber pressure wave p ;

Step S3: establishing an equivalent circuit of the free piston Stirling generator when the load changes;

Step S4: establishing a circuit differential equation of a free piston Stirling generator;

Step S5: Calculate the electromagnetic torque of the linear motor;

Step S6: calculating the internal potential of the free piston Stirling generator;

Step S7: Calculating the relationship between the output voltage of the free piston Stirling generator and the displacement of the power piston;

Step S8: Calculate the maximum possible instantaneous value of the potential inside the free piston Stirling generator;

Step S9: Calculate the maximum possible instantaneous value of the power piston displacement.

Further, in step S1, the kinetic equation is expressed as:

(1)

(2)

In the formula, is the gas piston acceleration, is the power piston acceleration, is the valve piston speed, is the power piston speed, is the displacement of the valve piston, is the power piston displacement,

is the total dynamic mass of the valve piston, is the total dynamic mass of the power piston, is the spring stiffness of the gas piston leaf. is the power piston leaf spring stiffness, is the cross-sectional area of the power piston, is the cross-sectional area of the gas piston rod, is the working chamber pressure wave, To buffer the cavity pressure, To distribute the piston damping, is the power piston damping, is the electromagnetic torque of the linear motor.

Further, in step S2, the working chamber pressure wave It is expressed as:

(3)

in:

(4)

(5)

In the formula, is the mass of the working fluid in the working chamber, is the gas constant of the working fluid, , , , , are the volumes of the heater, expansion chamber, regenerator, compression chamber, and cooler, respectively. and Respectively represent the volumes of the expansion chamber and the compression chamber in the initial state, , , , are the temperatures of the heater, expansion chamber, compression chamber, and cooler, respectively. is the cross-sectional area of the valve piston.

Further, in step S3, the internal potential of the free piston Stirling generator It is expressed as:

(6)

In the formula, is the power generation coefficient of the linear motor;

Assuming the initial phase of the working chamber pressure wave is 0, the displacement of the valve piston, the power piston, the working chamber pressure wave and the internal potential can be expressed by a sinusoidal expression:

(7)

(8)

(9)

(10)

is the displacement amplitude of the valve piston, is the displacement amplitude of the power piston, is the amplitude of the pressure wave, is the magnitude of the internal potential, for The initial phase of for The initial phase of for The initial phase of is the operating angular frequency of the free piston Stirling generator;

Combining formula (6), we can get .

Further, in step S4, the circuit differential equation of the free piston Stirling generator is expressed as:

(11)

In the formula, represents the internal resistance of the linear motor, represents the inductance inside the linear motor, represents the equivalent load resistance, represents the output current of the free-piston Stirling generator;

When the free piston Stirling generator is in a steady state, the output current is expressed as:

(12)

In the formula, is the displacement amplitude of the power piston when the free piston Stirling generator is in a stable operating state;

When the load suddenly changes, the load resistance suddenly changes from Reduce to After that, the output current is expressed as:

(13)

In the formula, is the output current of the free piston Stirling generator after a sudden load change, for The periodic component of for The non-periodic component of

Periodic Component It is expressed as:

(14)

Non-periodic component It is expressed as:

(15)

in,

(16)

(17)

In the formula, Represents the non-periodic component The time constant, Indicates time, Indicates the equivalent load resistance after a sudden load change.

Further, in step S5, the electromagnetic torque of the linear motor is expressed as:

(18)

In the formula, is the electromagnetic torque of the linear motor, is the electromagnetic torque coefficient of the linear motor.

Further, in step S6, the internal potential of the free piston Stirling generator It is expressed as:

(19)

In the formula, for The periodic component of for The non-periodic component of

Among them, the periodic component for:

(20)

Non-periodic component for:

(twenty one)

in:

(twenty two)

(twenty three)

(twenty four)

(25)

In the formula, Represents the non-periodic component The time constant.

Further, in step S7, the relationship between the output voltage of the free piston Stirling generator and the displacement of the power piston is expressed as:

(26)

In the formula, Represents the amplitude of the output voltage of the free-piston Stirling generator.

Further, in step S8, the maximum possible instantaneous value of the internal potential of the free piston Stirling generator is expressed as:

(27)

In the formula, Represents the maximum possible instantaneous value of the potential within a free-piston Stirling generator.

Further, in step S9, the maximum possible instantaneous value of the power piston displacement is:

(28)

In the formula, Indicates the maximum possible instantaneous value of the power piston displacement.

Compared with the prior art, the advantages of the present invention are as follows:

The present invention can quickly predict the maximum possible instantaneous value of the power piston displacement under a sudden load change, and further predict whether the power piston displacement amplitude is likely to exceed the rated value. If overtravel is predicted, emergency control of the free piston Stirling generator is immediately started, which helps to implement emergency control more quickly and prevent overtravel.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG1 is a schematic diagram of the structure of a free piston Stirling generator.

FIG. 2 is an equivalent circuit topology diagram of a free piston Stirling generator when the load changes.

FIG. 3 is a diagram of the displacement of the gas distribution piston of the present invention. , power piston displacement , working chamber pressure wave , internal potential , output current , output voltage The phasor diagram of .

In the figure, 1 is a heater, 2 is a heat regenerator, 3 is a cooler, 4 is an expansion chamber, 5 is a compression chamber, 6 is a buffer chamber, 7 is a gas distribution piston, 8 is a power piston, 9 is a gas distribution piston leaf spring, 10 is a power piston leaf spring, and 11 is a linear motor.

DETAILED DESCRIPTION

The present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

The present invention predicts the displacement of the power piston of a free piston Stirling generator. The free piston Stirling generator is a free piston Stirling generator with an existing structure. Referring to FIG1 , the existing free piston Stirling generator is mainly composed of a heater 1, a regenerator 2, a cooler 3, an expansion chamber 4, a compression chamber 5, a buffer chamber 6, a gas distribution piston 7, a power piston 8, a gas distribution piston leaf spring 9, a power piston leaf spring 10 and a linear motor 11. The expansion chamber 4 and the compression chamber 5 constitute a working chamber.

The method for predicting the displacement of a power piston of a free piston Stirling generator of the present invention comprises the following steps:

Step S1: establishing the dynamic equations of the valve piston and the power piston;

Perform force analysis on the valve piston and power piston. Assuming that upward is the positive direction, the dynamic equation can be expressed as:

(1)

(2)

In the formula, is the gas piston acceleration, is the power piston acceleration, is the valve piston speed, is the power piston speed, is the displacement of the valve piston, is the power piston displacement,

is the total dynamic mass of the valve piston, is the total dynamic mass of the power piston, is the spring stiffness of the gas piston leaf. is the power piston leaf spring stiffness, is the cross-sectional area of the power piston, is the cross-sectional area of the gas piston rod, is the working chamber pressure wave, To buffer the cavity pressure, To distribute the piston damping, is the power piston damping, is the electromagnetic torque of the linear motor.

Step S2: Calculate the working chamber pressure wave ;

According to the ideal gas state equation, the working chamber pressure wave in (1) and (2) is It is expressed as:

(3)

in:

(4)

(5)

In the formula, is the mass of the working fluid in the working chamber, is the gas constant of the working fluid, , , , , are the volumes of the heater, expansion chamber, regenerator, compression chamber, and cooler, respectively. and Respectively represent the volumes of the expansion chamber and the compression chamber in the initial state, , , , are the temperatures of the heater, expansion chamber, compression chamber, and cooler, respectively. is the cross-sectional area of the valve piston.

The free piston Stirling generator is a thermodynamic coupling system. The displacement of the valve piston and the power piston depends on the combined effect of the thermodynamic cycle and the mechanical motion performance. After the load changes suddenly, the piston movement and the internal potential cannot change suddenly. Therefore, the current changes first, causing the electromagnetic torque of the linear motor to change. Then, the motion state of the power piston also changes accordingly, which will eventually be reflected in the change of the output voltage. When the load resistance is small or an open circuit fault occurs, the power piston will hit the cylinder due to the small electromagnetic damping. Therefore, the output voltage characteristics of the linear motor can reflect the changes in the circuit and the changes in the internal thermodynamic cycle state of the free piston Stirling generator.

Step S3: establishing an equivalent circuit of the free piston Stirling generator when the load changes;

When the load changes, the equivalent circuit topology of the free piston Stirling generator is shown in Figure 2.

In Figure 2, represents the internal resistance of the linear motor, represents the inductance inside the linear motor, Represents the internal impedance of the free piston Stirling generator linear motor. is the equivalent load resistance. is the equivalent load resistance after the load changes suddenly. Under open circuit fault, can be considered to be infinite. is the internal potential of the free piston Stirling generator, which is proportional to the speed of the power piston and can be expressed as:

(6)

In the formula, is the power generation coefficient of the linear motor.

Therefore, assuming that the initial phase of the working chamber pressure wave is 0, the displacement of the valve piston and the power piston and the working chamber pressure wave can be written in the form of a sinusoidal expression:

(7)

(8)

(9)

(10)

is the displacement amplitude of the valve piston, is the displacement amplitude of the power piston, is the amplitude of the pressure wave, is the magnitude of the internal potential, for The initial phase of for The initial phase of for The initial phase of is the operating angular frequency of the free piston Stirling generator.

Combining formula (6), we can get .

Displacement of valve piston , power piston displacement , working chamber pressure wave , internal potential , output current , output voltage The phasor diagram is shown in Figure 3.

Step S4: establishing a circuit differential equation of a free piston Stirling generator;

The output current of the free piston Stirling generator can be expressed by the circuit differential equation:

(11)

When the free piston Stirling generator is in a stable state, the output current can be expressed as:

(12)

In the formula, is the displacement amplitude of the power piston when the free-piston Stirling generator is in stable operation.

It can be seen from equation (12) that the output current of the free piston Stirling generator is It is mainly related to the displacement amplitude of the power piston when the free-piston Stirling generator is in a stable operating state, the operating angular frequency of the free-piston Stirling generator, and the equivalent load resistance.

When the load suddenly changes, the load resistance suddenly changes from Reduce to Since the power piston has mechanical inertia, the amplitude, speed and internal potential of the active piston will not change instantaneously. Therefore, the output current of the free piston Stirling generator after a sudden load change can be expressed as:

(13)

In the formula, is the output current of the free piston Stirling generator after a sudden load change, for The periodic component of for The non-periodic component of .

Periodic Component It is expressed as:

(14)

Non-periodic component It is expressed as:

(15)

In the formula, Represents the non-periodic component The time constant, Indicates time;

(16)

(17)

Step S5: Calculate the electromagnetic torque of the linear motor;

The change of current will cause the electromagnetic torque of the linear motor to change, which can be expressed as:

(18)

In the formula, is the electromagnetic torque coefficient of the linear motor.

The electromagnetic torque of the linear motor changes the motion of the power piston, which leads to changes in the thermodynamic cycle and thus changes the pressure wave. The pressure wave can further feed back into the power piston motion. Ultimately, the internal potential changes with the coupled thermodynamic-kinetic interaction.

Step S6: calculating the internal potential of the free piston Stirling generator;

In order to simplify the analysis, the effect of the non-periodic component of the current on the electromagnetic torque of the linear motor is ignored, and only the periodic component of the current is considered. By combining (2), (6), (8), (9), (13) and (18), the internal potential It can be calculated as:

(19)

In the formula, for The periodic component of for The non-periodic component of .

Among them, the periodic component for:

(20)

Non-periodic component for:

(twenty one)

In the formula, Represents the non-periodic component The time constant of

in:

(twenty two)

(twenty three)

(twenty four)

(25)

From the above analysis, it can be seen that the internal potential of the free piston Stirling generator The non-periodic component is not affected by the pressure fluctuation amplitude of the working chamber The influence of is only related to the initial motion state and system parameters. Therefore, the non-periodic component of the free piston Stirling generator can be calculated when the load changes suddenly. However, for the periodic component, when the load changes suddenly, due to the thermodynamic-dynamic coupling effect, the amplitude of the pressure wave in the working chamber will change slowly, so the periodic component of the voltage will change gradually.

Step S7: Calculating the relationship between the output voltage of the free piston Stirling generator and the displacement of the power piston;

Based on the calculation of the output voltage characteristics of the free piston Stirling generator, we can evaluate the motion of the power piston during load switching or open circuit failure. As a high-voltage closed external combustion engine, the free piston Stirling generator usually has only one electrical interface output. Therefore, we can only sample the output voltage and current. In the steady state, according to the circuit principle and formulas (6) and (12), the relationship between the output voltage and the power piston displacement is expressed as:

(26)

In the formula, Represents the amplitude of the output voltage of the free-piston Stirling generator.

Under transient conditions, the current change in the RL circuit lags behind the voltage change. There is a large delay in judging the power piston displacement change by sampling the output voltage and current. Therefore, predicting the maximum possible instantaneous value of the power piston displacement in advance, rather than directly measuring the terminal voltage, helps to implement emergency control faster and prevent overtravel.

Step S8: Calculate the maximum possible instantaneous value of the potential inside the free piston Stirling generator;

When the displacement of the power piston reaches its maximum possible instantaneous value, the corresponding internal potential is also the maximum possible instantaneous value. In the transient process, the internal potential is equivalent to the periodic component superimposed on the non-periodic component. Therefore, when the periodic component takes the amplitude at the moment of sudden load change, the maximum possible instantaneous value will appear when the non-periodic component is superimposed. The maximum possible value occurs about half a cycle after the load mutation. Therefore, according to formula (19), the maximum possible instantaneous value of the internal potential of the free piston Stirling generator is for:

(27)

Step S9: Calculate the maximum possible instantaneous value of the power piston displacement;

At this time, combining equations (6) and (27), the maximum possible instantaneous value of the power piston displacement is:

(28)

Under load mutation, the power piston displacement should not exceed the rated value. Using equation (28), it is possible to quickly predict in advance whether the power piston displacement amplitude is likely to exceed the rated value. If overtravel is predicted, the emergency control of the free piston Stirling generator is immediately started.

Those skilled in the art may make various modifications and variations to the present invention. If these modifications and variations are within the scope of the claims of the present invention and their equivalents, then these modifications and variations are also within the protection scope of the present invention.

The contents not described in detail in the specification are prior art known to those skilled in the art.

Claims (3)

1. A method for predicting the displacement of a power piston of a free piston Stirling generator, characterized in that it comprises the following steps: Step S1: establishing the dynamic equations of the valve piston and the power piston; Step S2: Calculate the working chamber pressure wave p ; Step S3: establishing an equivalent circuit of the free piston Stirling generator when the load changes; Step S4: establishing a circuit differential equation of a free piston Stirling generator; Step S5: Calculate the electromagnetic torque of the linear motor; Step S6: calculating the internal potential of the free piston Stirling generator; Step S7: Calculating the relationship between the output voltage of the free piston Stirling generator and the displacement of the power piston; Step S8: Calculate the maximum possible instantaneous value of the potential inside the free piston Stirling generator; Step S9: Calculate the maximum possible instantaneous value of the power piston displacement; In step S1, the kinetic equation is expressed as: (1) (2) In the formula, is the gas piston acceleration, is the power piston acceleration, is the valve piston speed, is the power piston speed, is the displacement of the valve piston, is the power piston displacement, is the total dynamic mass of the valve piston, is the total dynamic mass of the power piston, is the spring stiffness of the gas piston leaf. is the power piston leaf spring stiffness, is the cross-sectional area of the power piston, is the cross-sectional area of the gas piston rod, is the working chamber pressure wave, To buffer the cavity pressure, To distribute the piston damping, is the power piston damping, is the electromagnetic torque of the linear motor; In step S3, the internal potential of the free piston Stirling generator is It is expressed as: (6) In the formula, is the power generation coefficient of the linear motor; Assuming the initial phase of the working chamber pressure wave is 0, the displacement of the valve piston, the power piston, the working chamber pressure wave and the internal potential can be expressed by a sinusoidal expression: (7) (8) (9) (10) is the displacement amplitude of the valve piston, is the displacement amplitude of the power piston, is the amplitude of the pressure wave, is the magnitude of the internal potential, for The initial phase of for The initial phase of for The initial phase of is the operating angular frequency of the free piston Stirling generator; Combining formula (6), we can get ; In step S4, the circuit differential equation of the free piston Stirling generator is expressed as: (11) In the formula, represents the internal resistance of the linear motor, represents the inductance inside the linear motor, represents the equivalent load resistance, represents the output current of the free-piston Stirling generator; When the free piston Stirling generator is in a steady state, the output current is expressed as: (12) In the formula, is the displacement amplitude of the power piston when the free piston Stirling generator is in a stable operating state; When the load suddenly changes, the load resistance suddenly changes from Reduce to After that, the output current is expressed as: (13) In the formula, is the output current of the free piston Stirling generator after a sudden load change, for The periodic component of for The non-periodic component of Periodic Component It is expressed as: (14) Non-periodic component It is expressed as: (15) in, (16) (17) In the formula, Represents the non-periodic component The time constant, Indicates time, Indicates the equivalent load resistance after a sudden load change; In step S6, the internal potential of the free piston Stirling generator It is expressed as: (19) In the formula, for The periodic component of for The non-periodic component of Among them, the periodic component for: (20) Non-periodic component for: (twenty one) in (twenty two) (twenty three) (twenty four) (25) In the formula, Represents the non-periodic component The time constant of In step S7, the relationship between the output voltage of the free piston Stirling generator and the displacement of the power piston is expressed as: (26) In the formula, It represents the amplitude of the output voltage of the free piston Stirling generator; In step S8, the maximum possible instantaneous value of the potential inside the free piston Stirling generator is expressed as: (27) In the formula, It represents the maximum possible instantaneous value of the potential in a free-piston Stirling generator; In step S9, the maximum possible instantaneous value of the power piston displacement is: (28) In the formula, Indicates the maximum possible instantaneous value of the power piston displacement. 2. The method for predicting the displacement of a power piston of a free piston Stirling generator according to claim 1, characterized in that: in step S2, the pressure wave of the working chamber It is expressed as: (3) in (4) (5) In the formula, is the mass of the working fluid in the working chamber, is the gas constant of the working fluid, , , , , are the volumes of the heater, expansion chamber, regenerator, compression chamber, and cooler, respectively. and Respectively represent the volumes of the expansion chamber and the compression chamber in the initial state, , , , are the temperatures of the heater, expansion chamber, compression chamber, and cooler, respectively. is the cross-sectional area of the valve piston. 3. The method for predicting the displacement of a power piston of a free piston Stirling generator according to claim 1, characterized in that: in step S5, the electromagnetic torque of the linear motor is expressed as: (18) In the formula, is the electromagnetic torque of the linear motor, is the electromagnetic torque coefficient of the linear motor.