METHOD FOR CALCULATING OPTIMAL ENERGY FLOW OF INTEGRATED ELECTRICITY-GAS SYSTEM BASED ON SEQUENTIAL CONVEX PROGRAMMING

Abstract

A method for calculating an optimal energy flow of an integrated electricity-gas system comprises: a convex optimization part of an optimal energy flow model is established according to a fuel cost of each thermal power unit node and a gas supply cost of each nature gas source node, and a convex relaxation formula of a quadratic constraint of the optimal energy flow model is established, and a convex function is expanded by a first order Taylor expansion at a relaxation solution to form an expansion formula; a precision requirement threshold value of a non-convex constraint is given, the threshold value is compared with an unbalance magnitude of a non-convex constraint in the expansion formula; when the unbalance magnitude is greater than the threshold value, the expansion is iterated into a solution model of the integrated electricity-gas hybrid energy system until the unbalance magnitude is not greater than the threshold value.

Description

TECHNICAL FIELD

The present disclosure relates to the technical field of an optimal energy flow of an integrated electricity-gas system, and in particular to a method for calculating an optimal energy flow of an integrated electricity-gas system based on sequential convex programming.

BACKGROUND

With the rapid development of energy revolution, the hybrid energy system constructed by the coupling of the electrical power system and the nature gas system has become an important development direction on the optimization of energy structure in China. The multi-energy interlinking hybrid energy system has broken the operation barrier of the traditional energy system, has realized the mutual coupling, substitution and supplement of the multi-energy form, and has promoted the diversified utilization of energy. The calculation of optimal energy flow of the integrated electricity-gas system is one of the important theoretical bases for researching the planning and operation of the system.

At present, the greatest difficulty in solving this problem is the non-convex constraint in the optimal energy flow model of the integrated electricity-gas system. Although the traditional method for calculating the optimal energy flow based on a convex relaxation strategy has relatively high calculation speed, it frequently fails to satisfy the requirements of calculation accuracy, and the obtained solutions cannot guarantee the feasibility requirements for non-convex constraints. Therefore, it is extremely important for the development of the integrated electricity-gas system to provide an algorithm that can efficiently solve the non-convex optimization problem of the optimal energy flow of the integrated electricity-gas system.

SUMMARY

The objectives of the present disclosure are to provide a method for calculating an optimal energy flow of an integrated electricity-gas system based on sequential convex programming, by adding the unbalance magnitude of the non-convex constraints into the optimization objective as a penalty factor, thereby guaranteeing the feasibility and tightness of the solutions and guaranteeing the feasibility of the non-convex constraints.

In order to solve the above problems, the present disclosure is achieved by the following technical solutions.

Provided in the present disclosure is a method for calculating an optimal energy flow of an integrated electricity-gas system based on sequential convex programming. The method comprises the following steps.

A convex relaxation formula of a quadratic constraint of an optimal energy flow model is established in an integrated electricity-gas hybrid energy system, and a convex function is expanded by a first order Taylor expansion at a relaxation solution to form an expansion.

A precision requirement threshold value of a non-convex constraint is given, the threshold value is compared with an unbalance magnitude of the non-convex constraint in the expansion formula.

When the unbalance magnitude is greater than the threshold value, the expansion formula is iterated into a solution model of the integrated electricity-gas hybrid energy system until the unbalance magnitude is not greater than the threshold value, to obtain a relaxation solution in the solution model.

Further, iterating the expansion formula into a solution model of the integrated electricity-gas hybrid energy system includes as follows.

The precision requirement threshold value of the non-convex constraint is given, an unbalance magnitude obtained from a difference of the convex relaxation formula of the quadratic constraint is compared with the threshold value.

When the unbalance magnitude is not greater than a set precision requirement threshold value of the non-convex constraint, the expansion formula is solved according to an energy flow model.

When the unbalance magnitude is greater than the set precision requirement threshold value of the non-convex constraint, the first order Taylor expansion at the relaxation solution of the convex function is introduced into the model as a penalty term, a relaxation solution is recalculated with the penalty term of the model;

Before the relaxation solution with the penalty term of the model is calculated, the unbalance magnitude of an solved expansion formula of the relaxation formula after the penalty term is introduced is compared with the threshold value, when the unbalance magnitude is greater than the set precision requirement threshold value of the non-convex constraint, the penalty term is iterated until the unbalance magnitude is not greater than the set precision requirement threshold value of the non-convex constraint.

Further, before establishing the convex relaxation formula of the quadratic constraint of the optimal energy flow model, a convex optimization part of the optimal energy flow model is established according to a fuel cost of each thermal power unit node and a gas supply cost of each nature gas source node in the integrated electricity-gas hybrid energy system;

The constraints in the electricity-gas interlinking hybrid energy system include a power flow constraint of an electrical power system and a power flow constraint of a gas system, a power flow model of the electrical power system and a power flow model of the gas system are both quadratic nonlinear models.

Further, an objective function of the convex optimization part of the optimal energy flow model in the integrated electricity-gas system is established as:

$\min \sum _{i\in {\Omega }_c}\left[{C_i^a\left(p_i^c\right)}^2+C_i^b{p}_i^c+C_i^c\right]+\sum _{i\in {\Omega }_s}{C}_m^f{f}_m^s,$

where sets Ω_c and Ω_s represent a thermal power unit node set and a nature gas source node set respectively, a variable p_i^c represents an active power of the thermal power unit, a variable f_m^s represents a gas supply rate of the nature gas source, parameters C_i^a, C_i^b and C_i^c represent a second-order term coefficient, a first-order term coefficient and a zero-order term coefficient of the fuel cost in the thermal power unit respectively, and C_m^f represents a gas supply cost coefficient of the nature gas source.

Further, a linear part of the power flow constraints of the gas system is as follows:

$\sum _{{\mathrm{ji}}^{\in {\beta }_i}}\left(p_{\mathrm{ji}}-l_{\mathrm{ji}}{R}_{\mathrm{ji}}\right)-\sum _{\mathrm{ik}\in {\alpha }_i}{p}_{\mathrm{ik}}+p_i^c+p_i^g=D_i^p,\forall i\in {\Omega }_b$ $\sum _{{\mathrm{ji}\in {\beta }_i}^{}}\left(q_{\mathrm{ji}}-l_{\mathrm{ji}}{X}_{\mathrm{ji}}\right)-\sum _{\mathrm{ik}\in {\alpha }_i}{q}_{\mathrm{ik}}+q_i^c+q_i^g=D_i^q,\forall i\in {\Omega }_b$ $v_i-2\left(p_{\mathrm{ij}}{R}_{\mathrm{ij}}+q_{\mathrm{ij}}{X}_{\mathrm{ij}}\right)+l_{\mathrm{ij}},\left(R_{\mathrm{ij}}^2+X_{\mathrm{ij}}^2\right)=v_j,\forall \mathrm{ij}\in {\Omega }_l$ ${\left(V_i^{\min }\right)}^2\le v_i\le {\left(V_i^{\max }\right)}^2,\forall i\in {\Omega }_b$ $P_{c,i}^{\min }\le p_i^c\le P_{c,i}^{\max },Q_{c,i}^{\min }\le q_i^c\le Q_{c,i}^{\max },\forall i\in {\Omega }_c$ $P_{g,i}^{\min }\le p_i^g\le P_{g,i}^{\max },Q_{g,i}^{\min }\le q_i^g\le Q_{g,i}^{\max },\forall i\in {\Omega }_g$ $P_{\mathrm{ij}}^{\min }\le p_{\mathrm{ij}}\le P_{\mathrm{ij}}^{\max },\forall ij\in {\Omega }_l$ $Q_{\mathrm{ij}}^{\min }\le q_{\mathrm{ij}}\le Q_{\mathrm{ij}}^{\max },\forall ij\in {\Omega }_l$ $0\le l_{\mathrm{ij}}\le {\left(I_{\mathrm{ij}}^{\max }\right)}^2,\forall ij\in {\Omega }_l,$

where, sets Ω_b, Ω_l and Ω_g represent an electrical grid node set, an electrical transmission lines set and a gas generator node set respectively, sets α_i and β_i represent an electrical transmission line set with a node i as a head-terminal node and an electrical transmission line set with the node i as a tail-terminal node respectively, variables p_ij, p_ji and p_ik represent active powers flowing over lines ij, ji and ik, q_ij, q_ji and q_ik represent reactive powers flowing over the lines ij, ji and ik, l_ij and l_ji represent squares of currents flowing over the lines ij and ji, p_i^c and q_i^c represent an active power and a reactive power output by the thermal power unit respectively, p_i^g and q_i^g represent an active power and a reactive power output by the gas generator respectively, v_i and v_j represent squares of voltage amplitudes of the node i and a node j, parameters R_ij and R_ji represent resistances on the line ij and the line ji respectively, X_ij and X_ji represent reactances on the line ij and the line ji respectively, D_i^p and D_i^q represent an active load and a reactive load of the node i respectively, V_i^min and V_i^max represent a lower limit and an upper limit of the voltage amplitude of the node i respectively, P_c,i^min and P_c,i^max represent a lower limit and an upper limit of the active power output by a thermal power unit i respectively, Q_c,i^min and Q_c,i^max represent a lower limit and an upper limit of the reactive power output by the thermal power unit i respectively, P_g,i^min and P_g,i^max represent a lower limit and an upper limit of the active power output by a gas power unit i respectively, Q_g,i^min and Q_g,i^max represent a lower limit and an upper limit of the reactive power output by the gas power unit i respectively, P_ij^min and P_ij^max represent a transmission lower limit and a transmission upper limit of the active power on an electrical transmission line ij respectively, Q_ij^min and Q_ij^max represent a transmission lower limit and a transmission upper limit of the reactive power on the electrical transmission line ij respectively, I_ij^max represents a heat stable current value for the line ij.

Further, a linear part of the power flow constraints of the gas system is as follows:

$\sum _{\mathrm{im}\in {\delta }_m}\left(e_{\mathrm{im}}-{\tau }_{\mathrm{im}}\right)-\sum _{\mathrm{mn}\in {\gamma }_m}{e}_{\mathrm{mn}}-\sum _{\mathrm{mn}\in {\Xi }_m}{f}_{\mathrm{mn}}+f_m^s-f_m^g=D_m^g,\forall m\in {\Omega }_n$ ${\tau }_{\mathrm{mn}}=K_{\mathrm{mn}}{e}_{\mathrm{mn}},\forall \mathrm{mn}\in {\Omega }_k$ ${\left(H_{\mathrm{mn}}^{\min }\right)}^2{\pi }_m\le {\pi }_n\le {\left(H_{\mathrm{mn}}^{\max }\right)}^2{\pi }_m,\forall \mathrm{mn}\in {\Omega }_k$ $p_i^g=T_m{f}_m^g,\forall i,m\in {\Omega }_g$ ${\left({\prod }_m^{\min }\right)}^2\le {\pi }_m\le {\left({\prod }_m^{\max }\right)}^2,\forall m\in {\Omega }_n$ $-F_{\mathrm{mn}}^{\max }\le f_{\mathrm{mn}}\le F_{\mathrm{mn}}^{\max },\forall \mathrm{mn}\in {\Omega }_p$ $0\le e_{\mathrm{mn}}\le E_{\mathrm{mn}}^{\max },\forall \mathrm{mn}\in {\Omega }_k$ $0\le f_m^g\le F_{g,m}^{\max },\forall m\in {\Omega }_g$ $F_{s,m}^{\min }\le f_m^s\le F_{s,m}^{\max },\forall m\in {\Omega }_s,$

where sets Ω_n, Ω_p and Ω_k represent a gas node set, a gas pipeline set and a nature gas compressor set respectively, sets δ_m and γ_m represent a gas pipeline set with a node m as head-terminal node and a gas pipeline set with the node m as a tail-terminal node respectively, Ξ_m represents a nature gas compressor set with node m as a gas-intake node, e_lm and e_mn represent a gas amount flowing through a nature gas compressor im and a gas amount flowing through a nature gas compressor mn, τ_im and τ_mn represent a gas amount consumed by the nature gas compressor im and a gas amount consumed by the nature gas compressor mn respectively, f_mn represents a gas amount flowing through a gas pipeline mn, f_m^s represents a nature gas amount injected into the node m by the nature gas source per unit time, f_m^g represents a gas consumption amount by a gas generator connected to the node m per unit time, π_m and π_n represent squares of gas pressure values at the node m and a node n respectively, a parameter D_m^g represents a gas load at the node m, W_mn represents a Weymouth coefficient of a gas transmission pipeline mn, K_mn represents a ratio coefficient of a gas compression amount and a gas consumption amount per unit time of the nature gas compressor, H_mn^min and H_mn^max represent a lower limit and an upper limit of a compression ratio of the nature gas compressor, T_m represents a ratio coefficient of the gas consumption amount and a power generation amount of the gas generator, Π_m^min and Π_m^max represent a lower limit and an upper limit of a nodal pressure, F_mn^min and F_mn^max represent a lower limit and an upper limit of a gas transmission amount of the gas pipeline mn per unit time, E_mn^max represents an upper limit of a compression rate of the nature gas compressor, F_g,m^max represents an upper limit of a gas consumption rate of the gas generator, F_s,m^min and F_s,m^max represent a lower limit and an upper limit of a gas supply amount of the nature gas source per unit time.

Further, the model is expressed in a matrix form:

$\min x^{T}Qx+cx+d$ $s.t.\mathrm{Ax}\le b,$

where matrices Q, c, d represent a second-order term coefficient, a first-order term coefficient and a constant term matrix in the objective function respectively, a matrix A represents a coefficient matrix in the linear constraint, and b represents a constant term coefficient matrix in the linear constraint.

Further, the convex relaxation formula of the quadratic constraint of the electricity-gas interlinking hybrid energy system is:

(p_ij)²+(q_ij)²≤v_il_ij, ∀ij∈Ω_l

W _mn(f_mn)²≤π_m−π_n, ∀mn∈Ω_p,

the first order Taylor expansion is:

(l_ij+v_i)²−8p*_ij−8q*_ijq_ij+4(p*_ij)²+4(q*_ij)²+(l*_ij−v*_i)²−2(l*_ij−v*_i)≤μ_ij^l, ∀ij∈Ω_l

(π_m−π_n−2W_minf_mnf_mn+W_mn(f*_mn)²≤μ_mn^p, ∀mn∈Ω_p,

where p*_ij, q*_ij, l*_ij, v*_i and f*_mn respectively represents a given value for a corresponding variable, namely, a value for a line active power, a value for a line reactive power, a square value for a line current, a square value for a node voltage, and a value for a pipeline gas flow optimized and obtained in a previous iteration, variables μ_ij^l and μ_mn^p respectively represent an unbalance magnitude of a non-convex constraint of a corresponding line ij and the gas transmission pipeline mn.

The present disclosure has the following beneficial effects.

The existing researches on the method for calculating an optimal energy flow of an integrated electricity-gas system lacks the effective means to deal with non-convex constraints. The traditional convex relaxation techniques are usually difficult to ensure the relaxation tightness, which leads to infeasible solutions. The present disclosure adopts the solution idea of sequential convex programming, and adds the unbalance of non-convex constraint into the optimization objective as a penalty factor to ensure the feasibility and tightness of the solution.

In order to ensure the feasibility of the solution, the existing researches on the method for calculating an optimal energy flow of an integrated electricity-gas system frequently need to adopt a more complex global optimization algorithm to solve, which leads to a low efficiency of the solution and cannot be applied to the power flow calculation of super large-scale energy system. The power flow calculation method based on sequential convex programming provided in the present disclosure is to solve based on the convex optimization problem in each iteration, therefore on the basis of ensuring the feasibility of the solution, the high efficiency of the solution is taken into account.

Certainly, the implementation of any product of the present disclosure is not necessary to achieve all the advantages described above at the same time.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to illustrate the technical solutions of the embodiments in the present disclosure more clearly, the following will give a brief introduction to the drawings needed for the description of the embodiments. It will be apparent that the drawings described below are only some embodiments of the present disclosure, for ordinary who skilled in the art, other drawings can be obtained according to these drawings without any creative effort.

The FIGURE illustrates a schematic diagram of solving the relaxation solution of the model.

DETAILED DESCRIPTION OF THE EMBODIMENTS

The following will clearly and completely describe the technical solutions in the embodiments of the present disclosure in combination with the drawings of the embodiments in the present disclosure. It will be apparent that the described embodiments are only some of the embodiments in the present disclosure, but not all of the embodiments. Based on the embodiments in the present disclosure, all other embodiments obtained by those skilled in the art without creative effort are within the protection scope of the present disclosure.

With reference to the FIGURE, provided in the present disclosure is a method for calculating an optimal energy flow of an integrated electricity-gas system based on sequential convex programming. The method is implemented through the following steps.

In Step 1, a convex optimization part of an optimal energy flow model in the integrated electricity-gas system is established.

In Step 101, an objective function is:

$\min \sum _{i\in {\Omega }_c}\left[{C_i^a\left(p_i^c\right)}^2+C_i^b{p}_i^c+C_i^c\right]+\sum _{i\in {\Omega }_s}{C}_m^f{f}_m^s$

where sets Ω_c and Ω_s represent a thermal power unit node set and a nature gas source node set respectively, a variable p_i^c represents an active power of the thermal power unit, a variable f_m^s represents a gas supply rate of the nature gas source, parameters C_i^a, C_i^b and C_i^c represent a second-order term coefficient, a first-order term coefficient and a zero-order term coefficient of the fuel cost in the thermal power unit respectively, and C_m^f represents a gas supply cost coefficient of the nature gas source.

In Step 102, a linear part of the power flow constraints of the electrical power system is as follows:

$\sum _{{\mathrm{ji}}^{\in {\beta }_i}}\left(p_{\mathrm{ji}}-l_{\mathrm{ji}}{R}_{\mathrm{ji}}\right)-\sum _{\mathrm{ik}\in {\alpha }_i}{p}_{\mathrm{ik}}+p_i^c+p_i^g=D_i^p,\forall i\in {\Omega }_b$ $\sum _{{\mathrm{ji}\in {\beta }_i}^{}}\left(q_{\mathrm{ji}}-l_{\mathrm{ji}}{X}_{\mathrm{ji}}\right)-\sum _{\mathrm{ik}\in {\alpha }_i}{q}_{\mathrm{ik}}+q_i^c+q_i^g=D_i^q,\forall i\in {\Omega }_b$ $v_i-2\left(p_{\mathrm{ij}}{R}_{\mathrm{ij}}+q_{\mathrm{ij}}{X}_{\mathrm{ij}}\right)+l_{\mathrm{ij}},\left(R_{\mathrm{ij}}^2+X_{\mathrm{ij}}^2\right)=v_j,\forall \mathrm{ij}\in {\Omega }_l$ ${\left(V_i^{\min }\right)}^2\le v_i\le {\left(V_i^{\max }\right)}^2,\forall i\in {\Omega }_b$ $P_{c,i}^{\min }\le p_i^c\le P_{c,i}^{\max },Q_{c,i}^{\min }\le q_i^c\le Q_{c,i}^{\max },\forall i\in {\Omega }_c$ $P_{g,i}^{\min }\le p_i^g\le P_{g,i}^{\max },Q_{g,i}^{\min }\le q_i^g\le Q_{g,i}^{\max },\forall i\in {\Omega }_g$ $P_{\mathrm{ij}}^{\min }\le p_{\mathrm{ij}}\le P_{\mathrm{ij}}^{\max },\forall ij\in {\Omega }_l$ $Q_{\mathrm{ij}}^{\min }\le q_{\mathrm{ij}}\le Q_{\mathrm{ij}}^{\max },\forall ij\in {\Omega }_l$ $0\le l_{\mathrm{ij}}\le {\left(I_{\mathrm{ij}}^{\max }\right)}^2,\forall ij\in {\Omega }_l,$

where, sets Ω_b, Ω_l and Ω_g represent an electrical grid node set, an electrical transmission lines set and a gas generator node set respectively, sets α_i and β_i represent an electrical transmission line set with a node i as a head-terminal node and an electrical transmission lines set with the node i as a tail-terminal node respectively, variables p_ij, p_ji and p_ik represent active powers flowing over lines ij, ji and ik, q_ij, q_ji and q_ik represent reactive powers flowing over the lines ij, ji and ik, l_ij and l_ji represent squares of currents flowing over the lines ij and ji, p_i^c and q_i^c represent an active power and a reactive power output by the thermal power unit respectively, p_i^g and q_i^g represent an active power and a reactive power output by the gas generator respectively, v_i and v_j represent squares of voltage amplitudes of the node i and a node j, parameters R_ij and R_ji represent resistances on the lines ij and the line ji respectively, X_ij and X_ji represent reactances on the line ij and the line ji respectively, D_i^p and D_i^q represent an active load and a reactive load of the node i respectively, V_i^min and V_i^max represent a lower limit and an upper limit of the voltage amplitude of the node i respectively, P_c,i^min and P_c,i^max represent a lower limit and an upper limit of the active power output by a thermal power unit i respectively, Q_c,i^min and Q_c,i^max represent a lower limit and an upper limit of the reactive power output by the thermal power unit i respectively, P_g,i^min and P_g,i^max represent a lower limit and an upper limit of the active power output by a gas power unit i respectively, Q_g,i^min and Q_g,i^max represent a lower limit and an upper limit of the reactive power output by the gas power unit i respectively, P_ij^min and P_ij^max represent a transmission lower limit and a transmission upper limit of the active power on an electrical transmission line ij respectively, Q_ij^min and Q_ij^max represent a transmission lower limit and a transmission upper limit of the reactive power on the electrical transmission line ij respectively, I_ij^max represents a heat stable current value for the line ij.

In Step 103, a linear part of the power flow constraints of the gas system is as follows:

$\sum _{\mathrm{im}\in {\delta }_m}\left(e_{\mathrm{im}}-{\tau }_{\mathrm{im}}\right)-\sum _{\mathrm{mn}\in {\gamma }_m}{e}_{\mathrm{mn}}-\sum _{\mathrm{mn}\in {\Xi }_m}{f}_{\mathrm{mn}}+f_m^s-f_m^g=D_m^g,\forall m\in {\Omega }_n$ ${\tau }_{\mathrm{mn}}=K_{\mathrm{mn}}{e}_{\mathrm{mn}},\forall \mathrm{mn}\in {\Omega }_k$ ${\left(H_{\mathrm{mn}}^{\min }\right)}^2{\pi }_m\le {\pi }_n\le {\left(H_{\mathrm{mn}}^{\max }\right)}^2{\pi }_m,\forall \mathrm{mn}\in {\Omega }_k$ $p_i^g=T_m{f}_m^g,\forall i,m\in {\Omega }_g$ ${\left({\prod }_m^{\min }\right)}^2\le {\pi }_m\le {\left({\prod }_m^{\max }\right)}^2,\forall m\in {\Omega }_n$ $-F_{\mathrm{mn}}^{\max }\le f_{\mathrm{mn}}\le F_{\mathrm{mn}}^{\max },\forall \mathrm{mn}\in {\Omega }_p$ $0\le e_{\mathrm{mn}}\le E_{\mathrm{mn}}^{\max },\forall \mathrm{mn}\in {\Omega }_k$ $0\le f_m^g\le F_{g,m}^{\max },\forall m\in {\Omega }_g$ $F_{s,m}^{\min }\le f_m^s\le F_{s,m}^{\max },\forall m\in {\Omega }_s,$

where sets Ω_n, Ω_p and Ω_k represent a gas node set, a gas pipeline set and a nature gas compressor set respectively, sets δ_m and γ_m represent a gas pipeline set with a node m as a head-terminal node and a gas pipeline set with the node m as a tail-terminal node respectively, Ξ_m represents a nature gas compressor set with node m as a gas-intake node, e_lm and e_mn represent a gas amount flowing through a nature gas compressor im and a gas amount flowing through a nature gas compressor mn, τ_im and τ_mn represent a gas amount consumed by the nature gas compressor im and a gas amount consumed by the nature gas compressor mn respectively, f_mn represents a gas amount flowing through a gas pipeline mn, f_m^s represents a nature gas amount injected into the node m by the nature gas source per unit time, f_m^g represents a gas consumption amount by a gas generator connected to the node m per unit time, π_m and π_n represent squares of gas pressure values at the node m and a node n respectively, a parameter D_m^g represents a gas load at the node m, W_mn represents a Weymouth coefficient of a gas transmission pipeline mn, K_mn represents a ratio coefficient of a gas compression amount and a gas consumption amount per unit time of the nature gas compressor, H_mn^min and H_mn^max represent a lower limit and an upper limit of a compression ratio of the nature gas compressor, T_m represents a ratio coefficient of the gas consumption amount and a power generation amount of the gas generator, Π_m^min and Π_m^max represent a lower limit and an upper limit of a nodal pressure, F_mn^min and F_mn^max represent a lower limit and an upper limit of a gas transmission amount of the gas pipeline mn per unit time, E_mn^max represents an upper limit of a compression rate of the nature gas compressor, F_g,m^max represents an upper limit of a gas consumption rate of the gas generator, F_s,m^min and F_s,m^max represent a lower limit and an upper limit of a gas supply amount of the nature gas source per unit time.

In Step 104, the above model is expressed in a matrix form as follows:

$\min x^{T}Qx+cx+d$ $s.t.\mathrm{Ax}\le b,$

where matrices Q c, d represent a second-order term coefficient, a first-order term coefficient and a constant term matrix in the objective function respectively, a matrix A represents a coefficient matrix in the linear constraint, and b represents a constant term coefficient matrix in the linear constraint.

In Step 2, a convex relaxation formula of a quadratic constraint of the optimal energy flow model and its first order Taylor expansion are established.

Specifically, the convex relaxation formula of the quadratic constraint of the optimal energy flow model is established in an integrated electricity-gas hybrid energy system, and a convex function is expanded by the first order Taylor expansion at a relaxation solution to form expansion formulas, the expansion formulas described below are Taylor expansion formulas. In other words. That is, linear constraints are established by the optimal energy flow model based on the first-order Taylor expansion instead of the quadratic constraints, and solutions of the model at this time are approximate solutions.

A precision requirement threshold value of a non-convex constraint is given, then the set threshold value is compared with an unbalance magnitude of the non-convex constraint in the expansion formula, and whether to solve the model is determined according to the comparison results.

When the unbalance magnitude is greater than the threshold value, the expansion formula is iterated into a solution model of the integrated electricity-gas hybrid energy system until the unbalance magnitude is not greater than the threshold value, to obtain a relaxation solution in the solution model.

For the unbalance magnitude mentioned above, the approximate solutions may not satisfy an original quadratic constraint, and the shortest distance between the approximate solutions and a feasible domain is the unbalance magnitude.

Specifically, iterating the expansion formula into a solution model of the integrated electricity-gas hybrid energy system includes as follows.

The precision requirement threshold value of the non-convex constraint is given, an unbalance magnitude obtained from a difference of the convex relaxation formula of the quadratic constraint is compared with the threshold value.

When the unbalance magnitude is not greater than a set precision requirement threshold value of the non-convex constraint, the expansion formula is solved according to an energy flow model.

When the unbalance magnitude is greater than the set precision requirement threshold value of the non-convex constraint, the first order Taylor expansion at the relaxation solution of the convex function is introduced into the model as a penalty term, a relaxation solution is recalculated with the penalty term of the model.

Before the relaxation solution with the penalty term of the model is calculated, the unbalance magnitude of an solved expansion formula of the relaxation formula after the penalty term is introduced is compared with the threshold value, when the unbalance magnitude is greater than the set precision requirement threshold value of the non-convex constraint, the penalty term is iterated until the unbalance magnitude is not greater than the set precision requirement threshold value of the non-convex constraint.

The actual calculation processes are as follows.

In Step 201, the convex relaxation formula of the quadratic constraint of the electricity-gas interconnected hybrid energy system is:

(p_ij)²+(q_ij)²≤v_il_ij, ∀ij∈Ω_l

W _mn(f_mn)²≤π_m−π_n, ∀mn∈Ω_p,

In Step 202, a first order Taylor expansion formula of the above constraint is:

(l_ij+v_i)²−8p*_ij−8q*_ijq_ij+4(p*_ij)²+4(q*_ij)²+(l*_ij−v*_i)²−2(l*_ij−v*_i)≤μ_ij^l, ∀ij∈Ω_l

(π_m−π_n−2W_minf_mnf_mn+W_mn(f*_mn)²≤μ_mn^p, ∀mn∈Ω_p,

where p*_ij, q*_ij, l*_ij, v*_i and f*_mn respectively represents a given value for a corresponding variable, namely, a value for a line active power, a value for a line reactive power, a square value for a line current, a square value for a node voltage, and a value for a pipeline gas flow optimized and obtained in a previous iteration, variables μ_ij^l and μ_mn^p respectively represents an unbalance magnitude of a non-convex constraint of a corresponding line ij and the gas transmission pipeline mn.

In Step 3, the sequential convex programming strategy is used to solve the optimal energy flow model iteratively.

In Step 301, the precision requirement threshold value ε of the non-convex constraint is set, and the counter is set to be k=0.

In Step 302, the model is solved

$\min x^{T}Qx+cx+d$ $s.t.\mathrm{Ax}\le b$

to obtain the current solutions p*_ij, q*_ij, l*_ij, v*_i and f*_mn of the variables p_ij, q_ij, l_ij, v_i and f_mn.

In Step 303, whether the unbalance magnitude of the non-convex constraint is satisfied is determined

v* _i l* _ij−(p*_ij)²−(q*_ij)²≤ε, ∀ij∈Ω_l

π*_m−π*_n−W_min(f*_mn)²≤ε, ∀mnεΩ_p

when the accuracy requirement is satisfied, the calculation results are output; if not, Step 304 is executed.

In Step 304, the model is solved

$\min x^{T}Qx+cx+d+2^k\left(\sum _{\mathrm{ij}\in {\Omega }_l}{\mu }_{\mathrm{ij}}^l+\sum _{\mathrm{mn}\in {\Omega }_p}{\mu }_{\mathrm{mn}}^p\right)$ $s.t.\mathrm{Ax}\le b$ ${\left(l_{\mathrm{ij}}+v_i\right)}^2-8p_{\mathrm{ij}}^{*}{p}_{\mathrm{ij}}-8q_{\mathrm{ij}}^{*}{q}_{\mathrm{ij}}+4{\left(p_{\mathrm{ij}}^{*}\right)}^2+4{\left(q_{\mathrm{ij}}^{*}\right)}^2+{\left(l_{\mathrm{ij}}^{*}-v_i^{*}\right)}^2-2\left(l_{\mathrm{ij}}^{*}-v_i^{*}\right)\left(l_{\mathrm{ij}}-v_i\right)\le {\mu }_{\mathrm{ij}}^l,\forall \mathrm{ij}\in {\Omega }_l$ $\left({\pi }_m-{\pi }_n\right)-2W_{\mathrm{mn}}{f}_{\mathrm{mn}}^{*}{f}_{\mathrm{mn}}+{W_{\mathrm{mn}}\left(f_{\mathrm{mn}}^{*}\right)}^2\le {\mu }_{\mathrm{mn}}^p,\forall \mathrm{mn}\in {\Omega }_p.$

In Step 305, whether the unbalance magnitude of the non-convex constraint is satisfied is determined

v* _i l* _ij−(p*_ij)²−(q*_ij)²≤ε, ∀ij∈Ω_l

π*_m−π*_n−W_min(f*_mn)²≤ε, ∀mnεΩ_p

when accuracy requirement is satisfied, the calculation results are output; if not, the current solutions p*_ij, q*_ij, l*_ij, v*_i and f*_mn of the variables p_ij, q_ij, l_ij, v_i and f_mn are obtained, and Step 304 is executed.

The following is an example of testing the integrated electricity-gas hybrid energy system, whose system parameters are as shown in Table 1 to Table 9.

TABLE 1
Node parameters for an electrical power system
			Lower	Upper		Load
	Active	Reactive	voltage	voltage	Voltage	rejection
Serial	load	load	limit	limit	reference	cost
number	(MW)	(MVAr)	(p.u.)	(p.u.)	(kV)	($/MWh)
0	97.6	44.2	0.94	1.06	345	1000
1	0	0	0.94	1.06	345	1000
2	322	2.4	0.94	1.06	345	1000
3	500	184	0.94	1.06	345	1000
4	0	0	0.94	1.06	345	1000
5	0	0	0.94	1.06	345	1000
6	233.8	84	0.94	1.06	345	1000
7	522	176.6	0.94	1.06	345	1000
8	6.5	−66.6	0.94	1.06	345	1000
9	0	0	0.94	1.06	345	1000
10	0	0	0.94	1.06	345	1000
11	8.53	88	0.94	1.06	345	1000
12	0	0	0.94	1.06	345	1000
13	0	0	0.94	1.06	345	1000
14	320	153	0.94	1.06	345	1000
15	329	32.3	0.94	1.06	345	1000
16	0	0	0.94	1.06	345	1000
17	158	30	0.94	1.06	345	1000
18	0	0	0.94	1.06	345	1000
19	680	103	0.94	1.06	345	1000
20	274	115	0.94	1.06	345	1000
21	0	0	0.94	1.06	345	1000
22	247.5	84.6	0.94	1.06	345	1000
23	308.6	−92.2	0.94	1.06	345	1000
24	224	47.2	0.94	1.06	345	1000
25	139	17	0.94	1.06	345	1000
26	281	75.5	0.94	1.06	345	1000
27	206	27.6	0.94	1.06	345	1000
28	283.5	26.9	0.94	1.06	345	1000
29	0	0	0.94	1.06	345	1000
30	9.2	4.6	0.94	1.06	345	1000
31	0	0	0.94	1.06	345	1000
32	0	0	0.94	1.06	345	1000
33	0	0	0.94	1.06	345	1000
34	0	0	0.94	1.06	345	1000
35	0	0	0.94	1.06	345	1000
36	0	0	0.94	1.06	345	1000
37	0	0	0.94	1.06	345	1000
38	1104	250	0.94	1.06	345	1000

TABLE 2
Node parameters for a gas system
		Lower	Upper
Serial	Gas load	pressure	pressure	Load rejection
number	(1000 m3/h)	limit (bar)	limit (bar)	cost ($/1000 m3)
0	0	40	70	1000
1	0	40	70	1000
2	0	40	70	1000
3	100	40	70	1000
4	120	40	60	1000
5	80	40	60	1000
6	0	40	70	1000
7	0	40	70	1000
8	0	40	70	1000
9	0	40	70	1000
10	0	40	70	1000

TABLE 3
Line parameters for an electrical power system
					Suscep-	Transmission
Serial	Head	Tail	Resistance	Reactance	tance	power
number	node	node	(p.u.)	(p.u.)	(p.u.)	(MVA)
0	0	1	0.0035	0.0411	0.6987	600
1	0	38	0.001	0.025	0.75	1000
2	1	2	0.0013	0.0151	0.2572	500
3	1	24	0.007	0.0086	0.146	500
4	1	29	0	0.0181	0	900
5	2	3	0.0013	0.0213	0.2214	500
6	2	17	0.0011	0.0133	0.2138	500
7	3	4	0.0008	0.0128	0.1342	600
8	3	13	0.0008	0.0129	0.1382	500
9	4	5	0.0002	0.0026	0.0434	1200
10	4	7	0.0008	0.0112	0.1476	900
11	5	6	0.0006	0.0092	0.113	900
12	5	10	0.0007	0.0082	0.1389	480
13	5	30	0	0.025	0	1800
14	6	7	0.0004	0.0046	0.078	900
15	7	8	0.0023	0.0363	0.3804	900
16	8	38	0.001	0.025	1.2	900
17	9	10	0.0004	0.0043	0.0729	600
18	9	12	0.0004	0.0043	0.0729	600
19	9	31	0	0.02	0	900
20	11	10	0.0016	0.0435	0	500
21	11	12	0.0016	0.0435	0	500
22	12	13	0.0009	0.0101	0.1723	600
23	13	14	0.0018	0.0217	0.366	600
24	14	15	0.0009	0.0094	0.171	600
25	15	16	0.0007	0.0089	0.1342	600
26	15	18	0.0016	0.0195	0.304	600
27	15	20	0.0008	0.0135	0.2548	600
28	15	23	0.0003	0.0059	0.068	600
29	16	17	0.0007	0.0082	0.1319	600
30	16	26	0.0013	0.0173	0.3216	600
31	18	19	0.0007	0.0138	0	900
32	18	32	0.0007	0.0142	0	900
33	19	33	0.0009	0.018	0	900
34	20	21	0.0008	0.014	0.2565	900
35	21	22	0.0006	0.0096	0.1846	600
36	21	34	0	0.0143	0	900
37	22	23	0.0022	0.035	0.361	600
38	22	35	0.0005	0.0272	0	900
39	24	25	0.0032	0.0323	0.531	600
40	24	36	0.0006	0.0232	0	900
41	25	26	0.0014	0.0147	0.2396	600
42	25	27	0.0043	0.0474	0.7802	600
43	25	28	0.0057	0.0625	1.029	600
44	27	28	0.0014	0.0151	0.249	600
45	28	37	0.0008	0.0156	0	1200

TABLE 4
Pipeline parameters for a gas system
			Minimum	Maximum	Weymouth
			transmission	transmission	coefficient
Serial	Head	Tail	rate	rate	((bar/(1000
number	node	node	(1000 m3/h)	(1000 m3/h)	m3/h)){circumflex over ( )}2)
0	0	1	−1100	1100	0.058913
1	6	7	−1100	1100	0.058913
2	2	8	−1100	1100	0.061036
3	7	3	−1100	1100	0.066929
4	7	9	−1100	1100	0.077628
5	8	9	−1100	1100	0.061036
6	10	4	−1100	1100	0.063635
7	10	5	−1100	1100	0.071318

TABLE 5
Parameters for a gas valve
			Upper limit	Minimum	Maximum
			of pressure	transmission	transmission
Serial	Head	Tail	difference	rate	rate
number	node	node	(bar)	(1000 m3/h)	(1000 m3/h)
0	6	8	120	−1100	1100

TABLE 6
Parameters for a nature gas compressor
					Minimum	Maximum
			Minimum	Maximum	transmission	transmission	Gas-
Serial	Head	Tail	compression	compression	rate	rate	driven
number	node	node	ratio	ratio	(1000 m3/h)	(1000 m3/h)	ratio
0	1	6	1.2	2.8	0	1100	0.035
1	9	10	1.2	2.8	0	1100	0.035

TABLE 7
Parameters for a thermal power unit
		Minimum	Maximum	Minimum	Maximum
		active	active	reactive	reactive
Serial		power	power	power	power	c2	c1	c0
number	Node	(MW)	(MW)	(MVAr)	(MVAr)	($/MWh2)	($/MWh)	($/h)
0	29	0	1040	140	400	0.01	0.3	0.2
1	30	0	646	−100	300	0.01	0.3	0.2
2	32	0	652	0	250	0.01	0.3	0.2
3	33	0	508	0	167	0.01	0.3	0.2
4	34	0	687	−100	300	0.01	0.3	0.2
5	35	0	580	0	240	0.01	0.3	0.2
6	36	0	564	0	250	0.01	0.3	0.2
7	38	0	1100	−100	300	0.01	0.3	0.2

TABLE 8
Parameters for nature gas source
		Minimum gas	Maximum gas
Serial		supply rate	supply rate	Gas supply cost
number	Node	(1000 m3/h)	(1000 m3/h)	($/1000 m3)
0	0	50	750	88.80555
1	1	0	500	88.80555
2	2	100	500	88.80555

TABLE 9
Parameters for a gas generator
			Minimum	Maximum	Minimum	Maximum
	Electric		active	active	reactive	reactive	Conversion
Serial	power	Gas	power	power	power	power	ratio
number	node	node	(MW)	(MW)	(MVAr)	(MVAr)	(1000 m3/MW)
0	31	3	0	725	150	300	0.25
1	37	1	0	865	−150	300	0.25

Taking the above electricity-gas interlinking hybrid energy system as an example, a calculation for optimal energy flow is carried out for a unit time section, a time scale is set to one hour, and a threshold value for the unbalance magnitude of the non-convex constraint is set to 0.1%. The main technical indexes of the algorithm are as shown in Table 10.

TABLE 10
The main technical indexes
Objective function value	Unbalance magnitude	Calculation time
17647.89 RMB	0.06945%	0.194031 seconds

The results show that the method for calculating an optimal energy flow of an integrated electricity-gas system based on sequential convex programming provided by the present disclosure can effectively obtain the objective function value of the problem, and the calculation time is only 0.194 seconds on the premise of ensuring that the maximum value for the unbalance magnitude of the non-convex constraint does not exceed 0.1% of the threshold value, and the feasibility and efficiency of the optimal energy flow algorithm are considered.

In the description of this specification, the reference terms “an embodiment”, “example”, “specific example”, etc., are used to refer to the specific features, structures, materials or characteristics described in conjunction with the embodiments or examples, which are 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 identical embodiments or examples. Furthermore, the specific features, structures, materials or characteristics described may be combined in an appropriate manner in any one or more embodiments or examples.

The above disclosed preferred embodiments of the present disclosure are only intended to help explain the present disclosure. The preferred embodiments do not describe all of the details and do not limit the present disclosure to the specific embodiments described. It will be apparent that many modifications and changes can be made can be made according to the contents of this specification. These embodiments are selected and described in details in this specification to better explain the principle and practical application of the present disclosure, so that those skilled in the art would better understand and utilize the present disclosure. The present disclosure is only limited by the claims and its full scope and equivalents.

Claims

1. A method for calculating an optimal energy flow of an integrated electricity-gas system based on sequential convex programming, wherein the method comprises following steps: establishing, in an integrated electricity-gas hybrid energy system, a convex relaxation formula of a quadratic constraint of an optimal energy flow model, and expanding, by a first order Taylor expansion, a convex function at a relaxation solution to form an expansion formula;giving a precision requirement threshold value of a non-convex constraint, comparing the threshold value with an unbalance magnitude of the non-convex constraint in the expansion formula;iterating, when the unbalance magnitude is greater than the threshold value, the expansion formula into a solution model of the integrated electricity-gas hybrid energy system until the unbalance magnitude is not greater than the threshold value, to obtain a relaxation solution in the solution model.

2. The method for calculating the optimal energy flow of the integrated electricity-gas system based on sequential convex programming according to claim 1, wherein the iterating the expansion formula into a solution model of the integrated electricity-gas hybrid energy system includes: giving, the precision requirement threshold value of the non-convex constraint, comparing an unbalance magnitude obtained from a difference of the convex relaxation formula of the quadratic constraint with the threshold value;solving, when the unbalance magnitude is not greater than a set precision requirement threshold value of the non-convex constraint, the expansion formula according to an energy flow model;introducing, when the unbalance magnitude is greater than the set precision requirement threshold value of the non-convex constraint, the first order Taylor expansion at the relaxation solution of the convex function into the model as a penalty term, recalculating, the model relaxation solution with the penalty term;comparing, before calculating the model relaxation solution with the penalty term, an unbalance magnitude of an solved expansion formula of the relaxation formula with the penalty term with the threshold value, iterating, when the unbalance magnitude is greater than the set precision requirement threshold value of the non-convex constraint, the penalty term until the unbalance magnitude is not greater than the set precision requirement threshold value of the non-convex constraint.

3. The method for calculating the optimal energy flow of the integrated electricity-gas system based on sequential convex programming according to claim 1, wherein, establishing, before establishing the convex relaxation formula of the quadratic constraint of the optimal energy flow model, a convex optimization part of the optimal energy flow model according to a fuel cost of each thermal power unit node and a gas supply cost of each nature gas source node in the integrated electricity-gas hybrid energy system;wherein, constraints in the electricity-gas interlinking hybrid energy system include a power flow constraint of an electrical power system and a power flow constraint of a gas system, a power flow model of the electrical power system and a power flow model of the gas system are both quadratic nonlinear models.

4. The method for calculating the optimal energy flow of the integrated electricity-gas system based on sequential convex programming according to claim 3, wherein, establishing, an objective function of the convex optimization part of the optimal energy flow model in the integrated electricity-gas system:

5. The method for calculating the optimal energy flow of the integrated electricity-gas system based on sequential convex programming according to claim 4, wherein a linear part of the power flow constraint of the electrical power system is as follows:

6. The method for calculating the optimal energy flow of the integrated electricity-gas system based on sequential convex programming according to claim 5, wherein a linear part of the power flow constraint of the gas system is as follows:

7. The method for calculating the optimal energy flow of the integrated electricity-gas system based on sequential convex programming according to claim 6, wherein the model is expressed in a matrix form:

8. The method for calculating the optimal energy flow of the integrated electricity-gas system based on sequential convex programming according to claim 4, wherein, the convex relaxation formula of the quadratic constraint of the electricity-gas interlinking hybrid energy system is: (pij)2+(qij)2≤vilij, ∀ij∈Ωl Wmn(fmn)2≤πm−πn, ∀mn∈Ωp,the first order Taylor expansion is: (lij+vi)2−8p*ij−8q*ijqij+4(p*ij)2+4(q*ij)2+(l*ij−v*i)2−2(l*ij−v*i)≤μijl, ∀ij∈Ωl (πm−πn−2Wminfmnfmn+Wmn(f*mn)2≤μmnp, ∀mn∈Ωp,where p*ij, q*ij, l*ij, v*i and f*mn respectively represents a given value for a corresponding variable, namely, a value for a line active power, a value for a line reactive power, a square value for a line current, a square value for a node voltage, and a value for a pipeline gas flow optimized and obtained in a previous iteration, variables μijl and μmnp respectively represent an unbalance magnitude of a non-convex constraint of a corresponding line ij and the gas transmission pipeline mn.