Non-isothermal power-natural gas optimal power flow calculation method based on convex relaxation method

Abstract

The present application relates to a kind of non-isothermal electric power-gas optimal flow calculation method based on convex relaxation method, comprising the following steps: step S1: non-isothermal natural gas system model is built;Step S2: based on convex envelope, cubic and bilinear constraint of non-isothermal natural gas system model is relaxed into a set of linear constraints, obtain natural gas system convex model;Step S3: the direct current flow model of power system is coupled with the natural gas system convex model established, and non-isothermal electric power-gas optimal flow convex model is established, and based on non-isothermal electric power-gas optimal flow convex model carries out flow calculation.The present application effectively improves the efficiency and reliability of flow calculation.

Description

Technical Field

[0001] This invention relates to a non-isothermal power-natural gas optimal power flow calculation method based on convex relaxation. Background Technology

[0002] Gas temperature has a significant impact on the assessment of nodal pressures and pipeline mass flow rates in natural gas systems. Under isothermal conditions, it is difficult to obtain accurate optimal operating strategies for power-gas power flow. Existing research has explored modeling and solution methods for non-isothermal power-gas optimal power flow, but the models obtained by traditional methods are still highly nonconvex and nonlinear, making it difficult to find the global optimal solution and resulting in low computational efficiency. Summary of the Invention

[0003] In view of this, the purpose of this invention is to provide a non-isothermal power-natural gas optimal power flow calculation method based on the convex relaxation method, which effectively improves the efficiency and reliability of power flow calculation.

[0004] To achieve the above objectives, the present invention adopts the following technical solution:

[0005] A non-isothermal power-natural gas optimal power flow calculation method based on convex relaxation includes the following steps:

[0006] Step S1: Construct a model of a non-isothermal natural gas system;

[0007] Step S2: Based on the convex enveloping, relax the cubic sum bilinear constraints of the non-isothermal natural gas system model into a set of linear constraints to obtain the convex model of the natural gas system;

[0008] Step S3: Couple the DC power flow model of the power system with the established convex model of the natural gas system to establish a non-isothermal power-natural gas optimal power flow convex model, and perform power flow calculation based on the non-isothermal power-natural gas optimal power flow convex model.

[0009] Furthermore, the non-isothermal natural gas system model is represented as follows:

[0010]

[0011]

[0012]

[0013] In the formula, K c It is the heat transfer coefficient; c p T represents the specific heat under constant pressure; eA represents the ambient temperature; T represents the gas temperature; A represents the cross-sectional area of ​​the pipe; ρ is the gas density; f is the gas mass flow rate; d is the pipe diameter; R is the gas constant; Z is the gas compressibility factor; l is the pipe length; p i Here, represents the pressure at node i; j refers to the junction of the pipes. This represents the outlet temperature of pipe (i,j).

[0014] Furthermore, step S2 specifically includes:

[0015] By using the cubic terms that appear in the momentum equation Rewritten as f ij (T i ·f ij Let a convex hull be defined to linearize the cubic term, expressed as:

[0016]

[0017] The upper and lower bounds of the auxiliary variable are represented as

[0018] ω i ≥f ij min (f ij min (T i -2T i min )+2f ij ·T i min )

[0019] ω i ≥f ij max (f ij min (T i -2T i max )+2f ij ·T i max )

[0020] ω i ≥f ij min (-f ij min T i min +f ij max (T i -T i max )+f ij (T i min +T imax ))

[0021] ω i ≥f ij max (f ij min (T i -T i min )-f ij max T i max +f ij (T i min +T i max ))

[0022] ω i ≤(f ij max ) 2 T i +(f ij -f ij max (f) ij min +f ij max )T i min

[0023] ω i ≤(f ij min ) 2 T i +(f ij -f ij min (f) ij min +f ij max )T i max

[0024]

[0025] Substitute the auxiliary variable ω i The momentum equation becomes:

[0026]

[0027] Using a convex hull to relax bilinear constraints, it can be expressed as:

[0028] χ i =f ij ·T i

[0029]

[0030] auxiliary variable χ i The upper and lower bounds are determined by the following equations:

[0031]

[0032]

[0033]

[0034]

[0035] In the formula, and These refer to the minimum and maximum mass flow rates of pipe (i,j), respectively; T i min and T i max These represent the lowest and highest gas temperatures at node i, respectively;

[0036] Bilinear constraints introduce auxiliary variables and Transformed into a set of linear constraints, it can be represented as:

[0037]

[0038]

[0039]

[0040] in, and

[0041] Furthermore, step S3 specifically includes:

[0042] The objective of the electricity-natural gas optimal power flow model is to minimize operating costs, expressed as:

[0043]

[0044] Where, α j and β i This represents the cost coefficient between natural gas suppliers and coal-fired power units. and This indicates the output of natural gas suppliers and coal-fired power units;

[0045] Power system constraints:

[0046]

[0047]

[0048]

[0049] θ i min ≤θ i,t ≤θ i max

[0050] in, For electrical load, For the output of the gas turbine, B f For the susceptance of the transmission line, θ i,t It is the phase angle of node i;

[0051] Natural gas system constraints:

[0052]

[0053]

[0054]

[0055]

[0056] In the formula, This refers to the gas mass flow rate at the gas source node; It is the gas load of node n; It is the gas consumption of the gas turbine; f p,t It is the mass flow rate of pipe p;

[0057] The compressor ensures that the node pressure of the natural gas system is maintained within a certain range, which is represented as:

[0058]

[0059]

[0060]

[0061]

[0062] In the formula, It is the horsepower of the compressor; κ ij and r n HV represents the compressor horsepower constant; γ represents the total calorific value of natural gas; ij This refers to the energy conversion coefficient of the compressor; and These are the minimum and maximum compression ratios of the compressor, respectively. and These are the minimum and maximum mass flow rates of the compressor, respectively.

[0063] In addition, the compressor's discharge temperature is obtained by the following formula.

[0064]

[0065] The nonlinear constraints of the compressor are simplified to

[0066]

[0067]

[0068] In the formula, τ ij and ν ij All are constants.

[0069] Coupling constraints:

[0070]

[0071]

[0072] Where, η g This refers to the energy conversion coefficient of the gas turbine. and These represent the minimum and maximum output of the gas turbine.

[0073] A non-isothermal power-natural gas optimal power flow calculation system based on a convex relaxation method includes a processor, a memory, and a computer program stored in the memory. When the processor executes the computer program, it specifically performs the steps in the non-isothermal power-natural gas optimal power flow calculation method based on a convex relaxation method as described above.

[0074] Compared with the prior art, the present invention has the following advantages:

[0075] This invention transforms the strongly nonconvex and nonlinear non-isothermal power-natural gas optimal power flow model into a convex model, solving the problem of the difficulty in efficiently solving nonconvex models, greatly improving the accuracy of the solutions obtained by the model, and significantly improving the solution efficiency of the model, thus effectively improving the efficiency and reliability of power flow calculation. Attached Figure Description

[0076] Figure 1 This is a 6-node power system model in one embodiment of the present invention;

[0077] Figure 2 This is a 6-node natural gas system in one embodiment of the present invention. Detailed Implementation

[0078] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0079] This invention provides a non-isothermal power-natural gas optimal power flow calculation method based on convex relaxation, comprising the following steps:

[0080] Step S1: Construct a model of a non-isothermal natural gas system, represented as follows:

[0081]

[0082]

[0083]

[0084] In the formula, K c It is the heat transfer coefficient; c p T represents the specific heat under constant pressure; e A represents the ambient temperature; T represents the gas temperature; A represents the cross-sectional area of ​​the pipe; ρ is the gas density; f is the gas mass flow rate; d is the pipe diameter; R is the gas constant; Z is the gas compressibility factor; l is the pipe length; p i Here, represents the pressure at node i; j refers to the junction of the pipes. This represents the outlet temperature of pipe (i,j).

[0085] Step S2: Based on the convex enveloping, relax the cubic sum bilinear constraints of the non-isothermal natural gas system model into a set of linear constraints to obtain the convex model of the natural gas system;

[0086] In this embodiment, the nonconvexity of the non-isothermal natural gas system model is mainly caused by the cubic constraints (momentum equation) and bilinear constraints (pipeline thermal model and temperature mixing equation). To eliminate the nonconvexity of the momentum equation, a relaxation method based on the McCormick linear approximation is proposed to linearize the cubic term.

[0087] By using the cubic terms that appear in the momentum equation Rewritten as f ij (T i ·f ij Let a convex hull be defined to linearize the cubic term, expressed as:

[0088]

[0089] The upper and lower bounds of the auxiliary variable are represented as

[0090] ω i ≥f ij min (f ij min (T i -2T i min )+2f ij ·Ti min )

[0091] ω i ≥f ij max (f ij min (T i -2T i max )+2f ij ·T i max )

[0092] ω i ≥f ij min (-f ij min T i min +f ij max (T i -T i max )+f ij (T i min +T i max ))

[0093] ω i ≥f ij max (f ij min (T i -T i min )-f ij max T i max +f ij (T i min +T i max ))

[0094] ω i ≤(f ij max ) 2 T i +(f ij -f ij max )(f ij min +f ij max )T i min

[0095] ωi ≤(f ij min ) 2 T i +(f ij -f ij min (f) ij min +f ij max )T i max

[0096]

[0097] Substitute the auxiliary variable ω i The momentum equation becomes:

[0098]

[0099] Using a convex hull to relax bilinear constraints, it can be expressed as:

[0100] χ i =f ij ·T i

[0101]

[0102] auxiliary variable χ i The upper and lower bounds are determined by the following equations:

[0103]

[0104]

[0105]

[0106]

[0107] In the formula, and These refer to the minimum and maximum mass flow rates of pipe (i,j), respectively; T i min and T i max These represent the lowest and highest gas temperatures at node i, respectively;

[0108] Bilinear constraints introduce auxiliary variables and Transformed into a set of linear constraints, it can be represented as:

[0109]

[0110]

[0111]

[0112] in, and

[0113] Step S3: Couple the DC power flow model of the power system with the established convex model of the natural gas system to establish a non-isothermal power-natural gas optimal power flow convex model, and perform power flow calculation based on the non-isothermal power-natural gas optimal power flow convex model.

[0114] The objective of the electricity-natural gas optimal power flow model is to minimize operating costs, expressed as:

[0115]

[0116] Where, α j and β i This represents the cost coefficient between natural gas suppliers and coal-fired power units. and This indicates the output of natural gas suppliers and coal-fired power units;

[0117] Power system constraints:

[0118]

[0119]

[0120]

[0121]

[0122] in, For electrical load, For the output of the gas turbine, B f For the susceptance of the transmission line, θ i,t It is the phase angle of node i;

[0123] Natural gas system constraints:

[0124]

[0125]

[0126]

[0127]

[0128] In the formula, This refers to the gas mass flow rate at the gas source node; It is the gas load of node n; It is the gas consumption of the gas turbine; f p,t It is the mass flow rate of pipe p;

[0129] The compressor ensures that the node pressure of the natural gas system is maintained within a certain range, which is represented as:

[0130]

[0131]

[0132]

[0133]

[0134] In the formula, It is the horsepower of the compressor; κ ij and r n HV represents the compressor horsepower constant; γ represents the total calorific value of natural gas; ij This refers to the energy conversion coefficient of the compressor; and These are the minimum and maximum compression ratios of the compressor, respectively. and These are the minimum and maximum mass flow rates of the compressor, respectively.

[0135] In addition, the compressor's discharge temperature is obtained by the following formula.

[0136]

[0137] The nonlinear constraints of the compressor are simplified to

[0138]

[0139] T j =T i +ν ij T i

[0140] In the formula, τ ij and ν ij All are constants.

[0141] Coupling constraints:

[0142]

[0143]

[0144] Where, η g P is the energy conversion coefficient of the gas turbine. i min and P i max These represent the minimum and maximum output of the gas turbine.

[0145] Example 1:

[0146] In this embodiment, a test system consisting of a 6-node power system and a 6-node natural gas system is used to verify the effectiveness of the proposed convex relaxation method for the non-isothermal power-natural gas optimal power flow model. Figure 1 and Figure 2 The topology of the test system is presented. The non-convex model is solved using the interior-point method (IP) and the spatial branch-and-bound method (sBB). The proposed non-isothermal power-natural gas optimal power flow convex model is solved using Cplex-Matlab.

[0147] Table 1 shows a comparison of the computational efficiency of the convex and non-convex models. As can be seen, the convex model proposed in this invention can complete the solution in 0.2 seconds, which is significantly better than the solution speed of the non-convex model, proving the effectiveness of the convex relaxation method of the proposed non-isothermal power-natural gas optimal power flow model.

[0148] Table 1 Comparison between convex and non-convex models

[0149]

[0150] The above description is only a preferred embodiment of the present invention. All equivalent changes and modifications made within the scope of the claims of the present invention should be included in the scope of the present invention.

Claims

1. A non-isothermal power-natural gas optimal power flow calculation method based on convex relaxation, characterized in that, Includes the following steps: Step S1: Construct a model of a non-isothermal natural gas system; Step S2: Based on the convex enveloping, relax the cubic sum bilinear constraints of the non-isothermal natural gas system model into a set of linear constraints to obtain the convex model of the natural gas system; Step S3: Couple the DC power flow model of the power system with the established convex model of the natural gas system to establish a non-isothermal power-natural gas optimal power flow convex model, and perform power flow calculation based on the non-isothermal power-natural gas optimal power flow convex model; The non-isothermal natural gas system model is represented as follows: In the formula, K c It is the heat transfer coefficient; c p T represents the specific heat under constant pressure; e The ambient temperature is represented by T; the gas temperature by A; the cross-sectional area of ​​the pipe by A; the mass flow rate of the gas by f; the diameter of the pipe by d; the gas constant by R; the compressibility factor of the gas by Z; and the pipe length by l. i Here, represents the pressure at node i; j refers to the junction of the pipes. This represents the outlet temperature of pipe (i,j); Step S2 specifically involves: By using the cubic terms that appear in the momentum equation Rewritten as f ij (T i ·f ij Let a convex hull be defined to linearize the cubic term, expressed as: Auxiliary variable ω i The upper and lower bounds are represented as ω i ≥f ij min (f ij min (T i -2T i min )+2f ij ·T i min ) ω i ≥f ij max (f ij min (T i -2T i max )+2f ij ·T i max ) ω i ≥f ij min (-f ij min T i min +f ij max (T i -T i max )+f ij (T i min +T i max )) ω i ≥f ij max (f ij min (T i -T i min )-f ij max T i max +f ij (T i min +T i max )) ω i ≤(f ij max ) 2 T i +(f ij -f ij max )(f ij min +f ij max )T i min ω i ≤(f ij min ) 2 T i +(f ij -f ij min )(f ij min +f ij max )T i max Substitute the auxiliary variable ω i The momentum equation becomes: Using a convex hull to relax bilinear constraints, it can be expressed as: x i =f ij ·T i auxiliary variable χ i The upper and lower bounds are determined by the following equations: In the formula, and These refer to the minimum and maximum mass flow rates of pipe (i,j), respectively. and These represent the lowest and highest gas temperatures at node i, respectively; Bilinear constraints introduce auxiliary variables and Transformed into a set of linear constraints, it can be represented as:

2. The non-isothermal power-natural gas optimal power flow calculation method based on convex relaxation method according to claim 1, characterized in that, Step S3 specifically involves: The objective of the electricity-natural gas optimal power flow model is to minimize operating costs, expressed as: Where, α j and β i This represents the cost coefficient between natural gas suppliers and coal-fired power units. and This indicates the output of natural gas suppliers and coal-fired power units; Power system constraints: in, For electrical load, For the output of the gas turbine, B f For the susceptance of the transmission line, θ i,t It is the phase angle of node i; Natural gas system constraints: In the formula, This refers to the gas mass flow rate at the gas source node; It is the gas load of node n; It is the gas consumption of the gas turbine; f p,t It is the mass flow rate of pipe p; The compressor ensures that the node pressure of the natural gas system is maintained within a certain range, which is represented as: In the formula, It is the horsepower of the compressor; κ ij and r n HV represents the compressor horsepower constant; γ represents the total calorific value of natural gas; ij This refers to the energy conversion coefficient of the compressor; and These are the minimum and maximum compression ratios of the compressor, respectively. and These are the minimum and maximum mass flow rates of the compressor, respectively. In addition, the compressor's discharge temperature is obtained by the following formula. The nonlinear constraints of the compressor are simplified to T j =T i +ν ij T i In the formula, τ ij and ν ij All are constants; Coupling constraints: Where, η g P is the energy conversion coefficient of the gas turbine. i min and P i max These represent the minimum and maximum output of the gas turbine.

3. A non-isothermal power-natural gas optimal power flow calculation system based on the convex relaxation method, characterized in that, The computing system includes a processor, a memory, and a computer program stored in the memory. When the processor executes the computer program, it specifically performs the steps in the non-isothermal power-natural gas optimal power flow calculation method based on the convex relaxation method as described in any one of claims 1-2.

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