A dynamic simulation method of heating pipeline based on differential transformation

Abstract

The present invention relates to the field of integrated energy system calculation and analysis, and specifically to a method for dynamic simulation of a heating pipeline based on differential transformation, comprising the following steps: S1, using a semi-discrete differential format with a total variation non-increasing property to differentiate the pipeline partial differential equation, thereby converting it into a set of ordinary differential equations; S2, using differential transformation to convert the set of ordinary differential equations into a linear recursive relationship; S3, recursively solving the differential transformation coefficients of the state quantities of the set of ordinary differential equations according to the initial value conditions and boundary value conditions of the equations, obtaining the approximate analytical expression of each state quantity, and then obtaining the change trajectory of each state quantity over time. This method can effectively eliminate the dispersion and dissipation errors in the simulation process of the heating pipeline. At the same time, due to the linear recursive nature of the differential transformation, this method can easily obtain a solution with high time accuracy, thereby ensuring the accuracy and reliability of the pipeline dynamic simulation.

Description

Technical Field

[0001] The invention relates to the field of calculation and analysis of integrated energy systems, and specifically to a dynamic simulation method for a heating pipeline based on differential transformation. Background Art

[0002] Heating pipes are an important part of the district heating system. Accurate characterization of the thermal dynamic process of the heating pipes helps to accurately analyze the dynamic characteristics of the district heating system, thereby helping to improve the operating economy and safety of the district heating system. Therefore, it has received widespread attention.

[0003] The thermodynamic process of the heating pipeline is manifested in the delay of temperature propagation, heat loss and heat storage effect of the heat medium, which is usually described by a partial differential equation with time-varying coefficients. However, the existing finite difference-based methods have large dissipation and dispersion errors, and the approximation accuracy of the time derivative is low. The difference format with the total variation non-increasing property can effectively suppress the dispersion error and avoid the dissipation error. The semi-discrete difference format can transform the partial differential equation into an ordinary differential equation by discretizing the derivatives of the variables in the partial differential equation with respect to space. Differential transformation is a method for solving ordinary differential equations. By converting the original differential equation into a linear recursive relationship, a solution with arbitrary time accuracy can be obtained. Summary of the invention

[0004] In order to solve the deficiencies mentioned in the above background technology, the purpose of the present invention is to provide a dynamic simulation method of a heating pipeline based on differential transformation, reduce the dissipation and dispersion errors in the process of solving partial differential equations, and improve the time accuracy of solving partial differential equations, thereby realizing accurate dynamic simulation of the heating pipeline.

[0005] The purpose of the present invention can be achieved through the following technical solutions:

[0006] A dynamic simulation method for a heating pipeline based on differential transformation, the simulation method comprising the following steps:

[0007] S1. Using a semi-discrete difference format with the total variation non-increasing property to differentiate the pipeline partial differential equation, thus transforming it into a system of ordinary differential equations;

[0008] S2. Use differential transformation to transform the ordinary differential equations into linear recursive relations;

[0009] S3. Recursively solve the differential transformation coefficients of the state quantities of the ordinary differential equation group according to the initial value conditions and boundary value conditions of the equation, obtain the approximate analytical expression of each state quantity, and then obtain the change trajectory of each state quantity over time.

[0010] Furthermore, the simulation method is directed to the following one-dimensional hyperbolic convection partial differential equation:

[0011]

[0012] Where τ is the distribution of temperature in time and space (℃); is the mass flow rate of the heating pipeline (kg / s); γ is the cross-sectional area of ​​the heating pipeline (m 2 );ρ is the density of the heat medium (kg m -3 ); λ is the total thermal conductivity (Wk -1 m -1 );C p is the specific heat capacity of the heat medium (J kg -1 K -1 );

[0013] In addition, the initial value condition of the equation is:

[0014]

[0015] The boundary conditions are:

[0016] τ(0,t)=ψ(t),t≥0. (3)

[0017] Furthermore, the specific steps of S1 are: differentiating the spatial derivative of Formula 1, thereby converting the partial differential equation into an ordinary differential equation as shown in Formula 4

[0018]

[0019] Where, Δh is the spatial differential step length of the heating pipeline; M is the total number of differential pipe sections; (τ j ) x Represents τ j Derivatives with respect to space;

[0020] For the case of 2<j<M, the minmod slope limiter is used to suppress the growth of total variation, thereby reducing the dispersion and dissipation errors in the partial differential equation solution process;

[0021] The minmod slope limiter has the form

[0022]

[0023] Among them, θ∈[1,2],

[0024]

[0025] When θ=1, the dispersion error of this format is the smallest; when θ=2, the dissipation error of this format is the smallest;

[0026] For the case of j = 2 or j = M, assuming (τ j ) r=(τ j-1 ) r = 0, we get the following ordinary differential equation

[0027]

[0028] Furthermore, the specific step of S2 is: using differential transformation to transform the ordinary differential equation group of formula 4 into a linear recursive relationship, and the differential transformation is used to solve the coefficient of the Taylor series of the state quantity of the ordinary differential equation with respect to time, which is also called the differential transformation coefficient, and is defined as follows:

[0029]

[0030] After obtaining the 0th to Kth order differential transformation coefficient X(0:K), the explicit expression of the state quantity with respect to time can be obtained, which is as follows:

[0031]

[0032] Use the following transformation rules to convert the original ordinary differential equation into a linear recurrence relation;

[0033] 1)x(0)→X(0);

[0034] 2)cx(t)→cx(k);

[0035] 3)x(t)±y(t)→X(k)±Y(k);

[0036] 4)

[0037] 5)c→cδ(k)

[0038] in,

[0039]

[0040] 6)dx(t) / dt→(k+1)X(k+1);

[0041] Applying the above transformation rule to formula 4, we get

[0042]

[0043]

[0044] in,

[0045]

[0046] Applying the above transformation rule to the initial value conditions of Formula 2 and the boundary value conditions of Formula 3, we get

[0047]

[0048] T 1 (k)=Ψ(k) (k≥1) (14)

[0049] Among them, x j represents the position of the control volume j, and Ψ(k) is the differential transformation of ψ(t).

[0050] Furthermore, in S3, the right-hand side of Formula 10 and Formula 11 contains differential transformation coefficients of no more than k order. Right now Then, T can be directly obtained by using formula 10 and formula 11. j (k+1)(2≤j≤M);

[0051] According to T j (0)(2≤j≤M) and Find T j (1)(2≤j≤M);

[0052] According to T j (0:1)(2≤j≤M) and Find T j (2)(2≤j≤M);……;

[0053] According to Tj(0:K-1)(2≤j≤M) and Find T j (K)(2≤j≤M);

[0054] This gives an explicit expression of the pipe section temperature with respect to time:

[0055]

[0056] Beneficial effects of the present invention:

[0057] The simulation method of the present invention uses a semi-discrete difference format with the total variation non-increasing property, which effectively reduces the dissipation and dispersion errors of dynamic simulation while converting partial differential equations into ordinary differential equations; at the same time, the present invention uses differential transformation to solve ordinary differential equations, which converts ordinary differential equations into linear recursive relations, so that solutions with high-order time accuracy can be easily obtained, thereby achieving accurate characterization of the thermal process of the heating pipeline. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, for those skilled in the art, other drawings can be obtained based on these drawings without creative work.

[0059] Figure 1 It is a schematic diagram of the overall structure flow of the present invention;

[0060] Figure 2 is a structural schematic diagram of a heating system in an embodiment of the present invention;

[0061] Figure 3 It is a schematic diagram of the dynamic simulation results of the outlet temperature after the inlet temperature of the heating pipeline in the embodiment of the present invention undergoes a step change;

[0062] Figure 4 It is a schematic diagram of the dynamic simulation results of the outlet temperature after the inlet temperature and the pipeline mass flow rate of the heating pipeline change simultaneously in the embodiment of the present invention. DETAILED DESCRIPTION

[0063] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0064] A dynamic simulation method for a heating pipeline based on differential transformation comprises the following steps:

[0065] S1. Using a semi-discrete difference format with the total variation non-increasing property to differentiate the pipeline partial differential equation, thus transforming it into a system of ordinary differential equations;

[0066] S2. Use differential transformation to transform the ordinary differential equations into linear recursive relations;

[0067] S3. Recursively solve the differential transformation coefficients of the state quantities of the ordinary differential equation group according to the initial value conditions and boundary value conditions of the equation, obtain the approximate analytical expression of each state quantity, and then obtain the change trajectory of each state quantity over time.

[0068] like Figure 1 As shown, the simulation method is aimed at the following one-dimensional hyperbolic convection partial differential equation

[0069]

[0070] Where τ is the distribution of temperature in time and space (℃); is the mass flow rate of the heating pipeline (kg / s); γ is the cross-sectional area of ​​the heating pipeline (m 2 );ρ is the density of the heat medium (kg m -3 ); λ is the total thermal conductivity (Wk -1 m -1 );Cp is the specific heat capacity of the heat medium (J kg -1 K -1 );

[0071] In addition, the initial value condition of the equation is:

[0072]

[0073] The boundary conditions are:

[0074] τ(0,t)=ψ(t),t≥0 (3)

[0075] Specifically, Figure 1 As shown, the dynamic simulation method of the heating pipeline based on differential transformation includes the following steps:

[0076] S1. Differentiate the spatial derivative of formula 1 to transform the partial differential equation into the following set of ordinary differential equations:

[0077]

[0078] Where, Δh is the spatial differential step length of the heating pipeline; M is the total number of differential pipe sections; (τ j ) x Represents τ j Derivatives with respect to space.

[0079] For the case of 2<j<M, a minmod slope limiter is used to suppress the growth of total variation, thereby reducing the dispersion and dissipation errors in the partial differential equation solution process. The minmod slope limiter has the following form

[0080]

[0081] Among them, θ∈[1,2],

[0082]

[0083] When θ=1, the dispersion error of the format is minimal; when θ=2, the dissipation error of the format is minimal.

[0084] For the case of j = 2 or j = M, assuming (τ j ) x =(τ j-1 ) x = 0, we can get the following ordinary differential equation

[0085]

[0086] S2. Use differential transformation to transform the ordinary differential equation system of the form of Formula 4 into a linear recursive relationship. Among them, differential transformation is used to solve the coefficient of the Taylor series of the state quantity of the ordinary differential equation with respect to time. This coefficient is also called the differential transformation coefficient, which is defined as follows

[0087]

[0088] After obtaining the 0th to Kth order differential transformation coefficients X(0:K), an explicit expression of the state quantity with respect to time can be obtained, and its form is shown in Formula 9.

[0089]

[0090] The following transformation rules are usually used to convert the original ordinary differential equation into a linear recurrence relationship.

[0091] 1)x(0)→X(0);

[0092] 2)cx(t)→cx(k);

[0093] 3)x(t)±y(t)→X(k)±Y(k);

[0094] 4)

[0095] 5)c→cδ(k)

[0096] in,

[0097]

[0098] 6)dx(t) / dt→(k+1)X(k+1.

[0099] Applying the above transformation rule to formula 4, we can get

[0100]

[0101]

[0102] in,

[0103]

[0104] Applying the above transformation rule to the initial value conditions of Formula 2 and the boundary value conditions of Formula 3, we can obtain

[0105]

[0106] T 1 (k)=Ψ(k) (k≥1) (14)

[0107] Among them, x jrepresents the position of the control volume j, and Ψ(k) is the differential transformation of ψ(t).

[0108] S3. Since the right-hand sides of Formula 10 and Formula 11 contain differential transformation coefficients of no more than order k, given Right now (express to After the linear combination of ), T can be directly obtained by 10 and formula 11 j (k+1)(2≤j≤M).

[0109] Therefore, according to T j (0)(2≤j≤M) and We can find T j (1)(2≤j≤M); According to T j (0:1)(2≤j≤M) and We can find T j (2)(2≤j≤M); ...; According to T j (0:K-1)(2≤j≤M) and We can find T j (K)(2≤j≤M), thus obtaining an explicit expression of the pipe section temperature with respect to time:

[0110]

[0111] This enables dynamic simulation of the heating pipeline.

[0112] The simulation method of the present invention uses a semi-discrete difference format with the property of total variation non-increasing, which effectively reduces the dissipation and dispersion errors of dynamic simulation while converting partial differential equations into ordinary differential equations; the simulation method of the present invention uses differential transformation to solve ordinary differential equations, which converts ordinary differential equations into linear recursive relations, so that solutions with high-order time accuracy can be easily obtained. Thus, accurate characterization of the thermal process of the heating pipeline is achieved.

[0113] Example:

[0114] Figure 2 The structural diagram of the district heating system used in this embodiment is shown in Figure 1. The actual system includes a heat source and a heat exchange station, which are connected by a water supply pipe and a return pipe with a length of 9250m. The proposed dynamic simulation method is verified in the matrix laboratory (MATLAB for short).

[0115] Simulations were performed for two situations: a step change in inlet temperature while the mass flow rate remained unchanged; a change in inlet temperature and mass flow rate simultaneously to verify the simulation accuracy of the method of the present invention. The time step was set to 180s and the space step was set to 370m. Based on differential transformation, a 20th-order time polynomial approximation of the outlet temperature was constructed and compared with the simulation results of the first-order upwind format and the second-order Yao format.

[0116] In the first case, the heat source inlet temperature changes from 90.1725℃ to 92℃ in 10 hours. The dynamic simulation results of the outlet temperature of the heating pipeline are as follows: Figure 3 As shown. Compared with the analytical solution of the partial differential equation of the heating pipeline under the condition of fixed mass flow, the upwind format has a larger dissipation error, and the high-frequency component of its solution is significantly attenuated. The Yao format has a larger dispersion error, and its solution has obvious false numerical oscillation phenomenon. The simulation results based on the present invention have small dissipation and dispersion errors at the same time, ensuring the simulation accuracy.

[0117] In the second case, the heat source inlet temperature and the pipe mass flow rate change simultaneously. The dynamic simulation results of the outlet temperature of the heating pipe are as follows: Figure 4 Compared with the measured temperature data at the pipeline outlet, the result of the Yao format has obvious false numerical oscillation, while the simulation result of the present invention does not have false numerical oscillation.

[0118] The root mean square error based on the measured outlet temperature data is further calculated. The error of the Yao format is 0.1773, while the error of the simulation result of the present invention is 0.1253.

[0119] It can be seen from this embodiment that after adopting the simulation method of the present invention, the dissipation and dispersion errors of the simulation results are significantly reduced. At the same time, due to the high time accuracy, the present invention can more accurately characterize the thermal dynamic process of the heating pipeline.

[0120] The above shows and describes the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments, and the above embodiments and descriptions are only for explaining the principles of the present invention. Without departing from the spirit and scope of the present invention, the present invention may have various changes and improvements, and these changes and improvements all fall within the scope of the present invention to be protected.

Claims

1. A dynamic simulation method for heating pipeline based on differential transformation, characterized in that: The simulation method comprises the following steps: S1. Using a semi-discrete difference format with the total variation non-increasing property to differentiate the pipeline partial differential equation, thus transforming it into a system of ordinary differential equations; S2. Use differential transformation to transform the ordinary differential equations into linear recursive relations; S3. Recursively solve the differential transformation coefficients of the state quantities of the ordinary differential equation group according to the initial value conditions and boundary value conditions of the equation, obtain the approximate analytical expression of each state quantity, and then obtain the change trajectory of each state quantity over time; The specific steps of S1 are: differentiating the spatial derivative of formula 1, thereby converting the partial differential equation into an ordinary differential equation as shown in formula 4 Where, Δh is the spatial differential step length of the heating pipeline; M is the total number of differential pipe sections; (τ j ) x Represents τ j Derivatives with respect to space; For the case of 2<j<M, the minmod slope limiter is used to suppress the growth of total variation, thereby reducing the dispersion and dissipation errors in the partial differential equation solution process; The minmod slope limiter has the form Among them, θ∈[1,2], When θ=1, the dispersion error of this format is the smallest; when θ=2, the dissipation error of this format is the smallest; For the case of j = 2 or j = M, assuming (τ j ) x =(τ j-1 ) x = 0, we get the following ordinary differential equation The specific steps of S2 are: using differential transformation to transform the ordinary differential equation group of formula 4 into a linear recursive relationship. The differential transformation is used to solve the coefficient of the Taylor series of the state quantity of the ordinary differential equation with respect to time. The coefficient is also called the differential transformation coefficient, which is defined as follows: After obtaining the 0th to Kth order differential transformation coefficient X(0:K), the explicit expression of the state quantity with respect to time can be obtained, which is as follows: Use the following transformation rules to convert the original ordinary differential equation into a linear recurrence relation; 1)x(0)→X(0); 2)cx(t)→cx(k); 3)x(t)±y(t)→X(k)±Y(k); 4) 5)c→cδ(k) in, 6)dx(t) / dt→(k+1)X(k+1); Applying the above transformation rule to formula 4, we get in, Applying the above transformation rule to the initial value conditions of Formula 2 and the boundary value conditions of Formula 3, we get T 1 (k)=Ψ(k)(k≥1)(14) Among them, x j represents the position of the control volume j, Ψ(k) is the differential transformation of ψ(t); In S3, the right-hand side of Formula 10 and Formula 11 contains differential transformation coefficients of no more than k order. Right now Then, T can be directly obtained by using formula 10 and formula 11. j (k+1)(2≤j≤M); According to T j (0)(2≤j≤M) and Find T j (1)(2≤j≤M); According to T j (0:1)(2≤j≤M) and Find T j (2)(2≤j≤M);……; According to T j (0:K-1)(2≤j≤M) and Find T j (K)(2≤j≤M); This gives an explicit expression of the pipe section temperature with respect to time:

2. A heating pipeline dynamic simulation method based on differential transformation according to claim 1, characterized in that: The simulation method is for the following one-dimensional hyperbolic convection partial differential equation: Among them, τ is the distribution of temperature in time and space, in °C; is the mass flow rate of the heating pipeline, in kg / s; γ is the cross-sectional area of ​​the heating pipeline, in m 2 ; ρ is the heat medium density, unit is kgm -3 ; λ is the total thermal conductivity, in Wk -1 m -1 ; C p is the specific heat capacity of the heat medium, in Jkg -1 K -1 ; In addition, the initial value condition of the equation is: In addition, the initial value condition of the equation is: The boundary conditions are: τ(0,t)=ψ(t), t≥0(3).

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