ANALYTICAL METHOD FOR DYNAMIC ANALYSIS OF NATURAL GAS NETWORK

Disclosed is an analytical method for dynamic analysis of a natural gas network in the field of energy system modeling and operational analysis, which includes establishing adynamic model of natural gas transmission according to the conservation equations, and reconstructing the dynamic model into the equations in a heat conduction equation form. The present disclosure directly constructs an analytical method for dynamic analysis of a natural gas network, avoiding approximation errors, numerical dispersion, and dissipation compared with the traditional numerical methods. The discretization process is avoided during the solution, greatly improving the computational efficiency and solution accuracy of dynamic analysis of the natural gas network.

TECHNICAL FIELD

The present disclosure relates to the field of energy system modeling and operational analysis, and in particular to an analytical method for dynamic analysis of a natural gas network.

BACKGROUND

The growing energy consumption and environmental pressures are driving the transformation of low-carbon green energy network technologies. Cities, as the main subjects of energy consumption and transformation, are transitioning towards complex energy networks with multiple energy flows and dynamics. As an important component of the urban energy network, a natural gas system can significantly improve the overall utilization efficiency of energy system and enhance the renewable consumption through the integration with the power system and heating system, as well as through optimized energy management. The time scales for the operation and management of different energy networks vary significantly, making it necessary to obtain real-time, reliable, and consistent network information based on accurate simulation models and technologies. However, since multi-energy networks are managed and operated by different companies, the exchange of information is highly limited, requiring special attention to information protection during co-simulation and operational optimization.

The simulation of the natural gas system essentially involves defining a set of state variables to describe the key characteristics of the system, followed by mechanism analysis to acquire the evolution of all state variables under a given boundary. Since the natural gas system model is composed of a set of partial differential-algebraic equations, one of the mainstream methods is to discretize the pipeline model using space-time segmentation and recursively calculate the state distribution within the system based on boundary and initial conditions. However, to ensure calculation efficiency, a large number of segments is typically required for each pipeline, which results in a low computational efficiency for the recursive process. Additionally, it is difficult to directly and quantitatively characterize the response of the state variables in the natural gas system to external boundaries.

In light of the above shortcomings, it is necessary to develop an analytical model for the transmission dynamics of the natural gas system that balances both accuracy and complexity from a modeling perspective. This model would allow for the rapid and precise description of the system's dynamic processes, making it more conducive to multi-agent simulation and optimization operation, and better suited to the practical application conditions in engineering.

SUMMARY

In view of the deficiencies of the prior art, the present disclosure proposes an analytical method for dynamic analysis of a natural gas network. The disclosure utilizes a constant variation method to derive an equivalent form of a dynamic transmission model of a natural gas system. Furthermore, based on the superposition, time-invariant, and causal properties of natural gas transmission, practical analytical forms of the dynamic model under different initial conditions and boundary conditions, as well as the equivalent network model, are constructed, providing a modeling basis for the operational analysis of an integrated energy system.

An object of the present disclosure may be achieved by the following technical solutions:

An analytical method for dynamic analysis of a natural gas network is provided, including:

In some embodiments, the establishing a dynamic model of natural gas transmission according to the conservation equations includes the following steps:

In some embodiments, the reconstructing the dynamic gas flow model into the equations in a heat conduction equation form includes the following steps:

deriving a partial derivative for x from a second equation in Eq. before substituting into a first equation in Eq. (1) to obtain a heat conduction form equation as follows:

In some embodiments, the transforming the heat conduction equation into a partial differential equation set with homogeneous boundary conditions using a method of variation of parameters includes the following steps:

In some embodiments, the constructing a general analytical expression for the partial differential equation set under constant pressure boundaries includes the following steps:

In some embodiments, the constructing, based on superposition principle, practical analytical expressions for the dynamic model of natural gas transmission includes the following steps:

In some embodiments, the constructing, according to system topologies and time-invariant, and causal properties of a natural gas system, a natural gas network equivalent model based on the practical analytical expressions includes the following steps:

This method directly establishes an analytical solution for the dynamic model of natural gas system. Compared to traditional numerical methods based on discretization, this method completely avoids approximation errors, numerical dispersion, and dissipation. Additionally, by avoiding the discretization process during the solution, this method significantly improves the computational efficiency and solution accuracy of dynamic analysis of the natural gas system.

DETAILED DESCRIPTION

The technical solutions in the embodiments of the present disclosure will be described clearly and completely below in combination with the drawings in the embodiments of the present disclosure. Clearly, the described embodiment is not all but only part of embodiments of the present disclosure. All other embodiments obtained by those ordinarily skilled in the art based on the embodiments in the present disclosure without creative work shall fall within the scope of protection of the present disclosure.

In the description of this specification, references to the terms “one embodiment”, “an example”, “a specific example”, and the like indicate that a specific feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the present disclosure. In this specification, schematic representations of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific feature, structure, material, or characteristic described may be combined in any suitable manner in any one or more embodiments or examples.

Embodiment: Taking the natural gas system shown in FIG. 2 as an example, the state distribution is studied under given initial conditions and boundary conditions.

As shown in FIG. 1, an embodiment of the present disclosure provides an analytical method for dynamic analysis of a natural gas system, including the following steps:

At step 10, a dynamic model of natural gas transmission is established according to the conservation equations and reconstructed into the equations in a heat conduction equation form.

At step 101, a dynamic model of natural gas transmission is established in a pipeline, including a mass conservation equation and a momentum conservation equation as below:

At step 102, a partial derivative for x is derived from a second equation in Eq. (1) before substituting into a first equation in Eq. (1) to obtain a heat conduction form equation as follows:

Eq. (1) may be converted to as follows:

At step 20, the reconstructed dynamic model of natural gas transmission is transformed into a partial differential equation set with homogeneous boundary conditions using a method of variation of parameters, followed by deriving a general analytical expression for the dynamic model for natural gas transmission when the pressure boundary is constant.

At step 201, p0,t is taken constant with the value of ψp,1 in response to the pressure boundary being constant, followed by introducing an intermediate variable ux,t and reconstructing px,t as follows:

A partial derivative for x is derived on both sides of Eq. (6) at x=L to obtain a further set of boundary conditions for ux,t, expressed as follows:

A second-order partial derivative for x and a first-order partial derivative for t are derived on both sides of Eq. (6), respectively:

Eq. (9) is substituted into Eq. (5), with Eq. (5) re-expressing a partial differential equation set with homogeneous boundary conditions as follows:

At step 202, ux,t* is defined as a solution of a homogeneous problem of Eq. (10), and the homogeneous problem is solved as follows:

Eq. (17) is substituted into Eq. (6) to obtain an analytical expression for px,t at constant pressure:

Eq. (18) is substituted into a second equation in Eq. to obtain an analytical expression for qx,t at constant pressure:

Eq. (18) and Eq. (19) are general analytical expressions for the partial differential equation set of the dynamic model for natural gas transmission under constant pressure boundaries.

At step 30, a practical analytical expression for the dynamic model for natural gas transmission is constructed combining the superposition principle of natural gas transmission and according to the discrete form of the initial conditions and boundary conditions.

At step 301, first, the pressure boundary is reconstructed, expressed as:

At step 302, according to superposition principle of natural gas transmission, the pressures px,t is decomposed as follows:

When k=1, a form of χk,x,t is consistent with Eq. (18), namely:

When k>1, a form of χk,x,t is as follows:

Eq. (23) and Eq. (24) are substituted into Eq. (21), it may be found that at any pressure boundary, the analytical expressions for px,t and qx,t are still as shown in Eq. (18) and Eq. (19) with the difference that p0,t is transformed from a constant to a set of discrete sequences.

At step 303, the initial conditions are calculated according to steady-state energy flow in practice, expressed as follows:

An integral formula in Eq. (27) may be modified as follows:

Eq. (30) is substituted into Eq. (18) and Eq. (19), to obtain the practical analytical expressions for the dynamic model for natural gas transmission as follows:

At step 40, the dynamic model of natural gas transmission is reconstructed; and the natural gas network equivalent model is derived based on the analytical method, as well as the simplified analysis method thereof combining topological characteristics, the time-invariant and causal properties of natural gas transmission.

At step 401, x=L and x=0 are substituted into Eq. (32) and Eq. (33), respectively, to obtain pressure at the outlet and flow at the inlet of the pipeline at arbitrary tj as follows:

At step 402, the number of nodes and pipelines in the natural gas system are defined as Ng and Nb, respectively, and an identity matrix and a zero matrix are defined as 1 and 0, respectively; incidence matrices are introduced to describe topological properties in the natural gas system.

(1) Ain includes Ng×Nb sub-matrices for associating flow at end of a branch with flow at a node, and in response to a sub-matrix of an i-th row and j-th column being 1, the flow flows from the end of the branch j into the node i, otherwise, the sub-matrix is 0.

(2) Aout includes Ng×Nb sub-matrices for associating flow at head end of a branch with flow at a node, and in response to a sub-matrix of an i-th row and j-th column being 1, the flow flows from the node i into the head end of the branch j, otherwise, the sub-matrix is 0.

(3) Anb1 includes Nb×Ng sub-matrices for associating pressure at end of a branch with pressure at a node, and in response to a sub-matrix of an i-th row and j-th column being 1, pressure at the node j is equal to pressure at the end of the branch i, otherwise, the sub-matrix is 0.

(4) Anb2 includes Nb×Ng sub-matrices for associating pressure at a head end of a branch with pressure at a node, and in response to a sub-matrix of an i-th row and j-th column being 1, pressure at the node j is equal to pressure at the head end of the branch i, otherwise, the sub-matrix is 0; particularly, in response to the node j being a compressor node with a pressure ratio of Kcp and the node j being connected to the head end of the branch i, the sub-matrix of the i-th row and jth column is Kcp×1.

At step 403, equations of mass conservation at nodes and pressure continuity between nodes and pipelines are constructed based on the incidence matrices, expressed as follows:

Eq. (46) is substituted into a first row of Eq. (36) as follows:

The pressures at load nodes and mass flows at source nodes are calculated using the analytical functions as shown in FIG. 3.

The present disclosure proposes an analytical method for dynamic analysis of a natural gas system, which is based on a dynamic model of the natural gas system. Furthermore, based on the superposition principle, time-invariant and causal properties of natural gas transmission, practical analytical forms of the dynamic model under different initial conditions and boundary conditions, as well as the equivalent network model, are constructed, reducing the computational complexity and greatly improving the computational accuracy.

The foregoing has shown and described the basic principles, principal features, and advantages of the present disclosure. It is to be understood by those skilled in the art that the present disclosure is not limited to the above embodiments. The above embodiments and description in the present disclosure are merely illustrative of the principles of the present disclosure. Without deviating from the spirit and scope of the present disclosure, the present disclosure is subject to various changes and improvements, and these changes and improvements fall within the scope of the present disclosure that requires protection.