A method for establishing optimal power flow model for three-phase unbalanced distribution network

Abstract

The present invention relates to a method for establishing an optimal power flow model of a three-phase unbalanced distribution network, and belongs to the technical field of power flow optimization of a three-phase unbalanced distribution network. With the large-scale access of electric vehicles and photovoltaic power generation systems to the power grid, the single-phase power supply within the distribution network has increased, the load has increased, and the severity of its single-phase operation problem has been aggravated. In order to ensure the power supply quality of the power grid, a large number of adjustable single- and three-phase compensation and regulation equipment have been used in the power grid. Taking into account the above power grid operation conditions, the present invention first establishes a multi-period distribution network optimal power flow model containing reactive compensation devices and energy storage; through second-order cone programming, the original non-convex nonlinear optimal power flow model is transformed into a linear model, which reduces the difficulty of solving without losing accuracy and improves the solution efficiency. It provides a good reference for subsequent research on the three-phase imbalance problem in the optimal power flow process of the distribution network.

Description

Technical Field

[0001] The present invention relates to a method for establishing an optimal power flow model of a three-phase unbalanced distribution network, and in particular to a method for transforming an original non-convex nonlinear optimal power flow model into a linear model in a three-phase unbalanced distribution network based on second-order cone programming, and optimizing the optimal power flow model of the distribution network to ensure its accuracy and efficiency. The method belongs to the technical field of power flow optimization of three-phase unbalanced distribution networks. Background Art

[0002] In recent years, with the development of the social economy, governments and all sectors of society have gradually begun to pay attention to environmental and energy issues. Green energy, represented by photovoltaics, has been widely promoted for consumer use. The integration of a large number of household photovoltaic power generation systems into distribution networks has led to widespread asymmetric loads and asymmetric line parameters within the distribution network. The non-full-phase operation of the photovoltaic systems exacerbates the three-phase imbalance of the distribution network. Several research results have been published on the operation of household photovoltaic power generation systems after they are connected to the distribution network. These studies generally analyze the power flow of a single phase under the condition of balanced three-phase operation, without considering the characteristics of three-phase imbalance in the distribution network. As three-phase imbalance in distribution networks intensifies, this analysis clearly fails to accurately reflect the complex conditions of each phase in the current system. Three-phase unbalance in low-voltage distribution networks increases system losses, and heavily loaded phases reduce bus voltage, thereby affecting user power quality. The installation of reactive power compensation and relay protection devices to address abnormal operating conditions of unbalanced phases depends on the system's power flow parameters. Therefore, it is urgent to establish a platform for rapidly solving optimal power flow (OPF) for low-voltage distribution networks with three-phase imbalance.

[0003] With the emergence of numerous intelligent algorithms, some researchers have adopted them directly to solve various linear and nonlinear models. However, intelligent algorithms can easily fall into local optimal solutions during iterative solutions within the feasible domain, and repeated iterations near the optimal solution also reduce solution efficiency. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a method for establishing an optimal power flow model of a three-phase unbalanced distribution network, which solves the problem of poor power quality for users caused by the three-phase imbalance of the distribution network when a large number of household photovoltaic power generation systems are connected to the distribution network.

[0005] The technical solution of the present invention is: a method for establishing an optimal power flow model of a three-phase unbalanced distribution network, which can effectively optimize the optimal power flow model of the distribution network to ensure its accuracy and efficiency under the condition of three-phase imbalance of the distribution network.

[0006] The specific steps are:

[0007] Step 1: Construct a three-phase, four-wire distribution network model and incorporate constraints on reactive power compensation equipment and energy storage devices. This results in a multi-period OPF model for the three-phase distribution network, including reactive power compensation and energy storage. This model, based on the three-phase imbalance of the distribution network, uses a 4x4 impedance matrix to represent the relationship between phases A, B, and C and the neutral line. It also considers the compensation provided by control units such as distributed power sources, energy storage, and discrete and continuous reactive power compensation devices for each phase of the unbalanced grid.

[0008] Step 2: Based on the OPF model, the objective function is to minimize the active network loss in the distribution network. The constraints of the distribution network operation are added to obtain the OPF optimization model of the three-phase distribution network. The OPF optimization model is a non-convex nonlinear model with mixed integer variables.

[0009] Step 3: According to the second-order cone relaxation theory, based on the non-convex nonlinear OPF optimization model with mixed integer variables established in Step 2, by converting complex variables into real variables in amplitude form, the original non-convex nonlinear model is transformed into a linear model through phase relaxation. Finally, the optimal power flow model of the three-phase unbalanced distribution network based on second-order cone relaxation is obtained.

[0010] The Step 1 is specifically as follows:

[0011] Step 1.1: Construct a three-phase four-wire model of the distribution network.

[0012] First, we use the 4*4 impedance matrix Z l,m To express the relationship between the three phases A, B, and C and the neutral line:

[0013]

[0014] Where Z gg , g takes a, b, c, n as the diagonal elements of the series impedance matrix as the self-impedance of the three-phase line and the neutral line, Z gh , h takes a, b, c, n as the non-diagonal elements in the series impedance matrix, g≠h, as the mutual impedance of the three-phase line and the neutral line.

[0015] Correspondingly, the inverse matrix of the impedance matrix Y=Z -1 As the admittance matrix Y of this line l,m .

[0016] The node admittance matrix Y of a distribution network with m nodes is expressed as:

[0017]

[0018] In the formula, s(m) is the set of nodes adjacent to node m, is the inverse matrix of the series impedance between two adjacent nodes, is the sum of the inverse matrices of all series impedance matrices connected to node m.

[0019] In order to obtain the voltage value of each phase in each node, the voltage vector equation of each phase in each node of the distribution network is added:

[0020] V(t)=Y -1 *I inj (t) (3)

[0021] Where: V(t) is the Nth-order vector formed by the voltage values of each phase at time t. -1 It is the inverse matrix of the distribution network node admittance matrix Y. inj (t) is the N-order vector formed by the current injected into the node by each phase at time t.

[0022] In order to ensure that the results of the power flow calculation can accurately reflect the actual situation in the system, the polynomial ZIP model that considers the static characteristics of the load is finally used as the load model (Z is a constant impedance model, I is a constant current model, and P is a constant power model) to calculate the voltage characteristics or static characteristics of the user. The injected current of the s phase on the node can be obtained as:

[0023]

[0024] Where, is the current injected into phase s at time t, is the active and reactive power generated by the household photovoltaic power generation at the s phase on the node l, Active power and reactive power purchased from the upper main grid, is the ZIP component of the s-phase active power demand of the user load at node l, is the ZIP component of the s-phase reactive power demand of the user load at node l, V norml (t) is the standard voltage of the load on node l, V l s (t) is the actual voltage of the s-phase load on node l, and * represents the conjugate.

[0025] At this point, the three-phase four-wire model of the distribution network has been constructed.

[0026] Step 1.2: Add the constraints of reactive compensation equipment and energy storage devices to the constructed three-phase four-wire distribution network model.

[0027] Static VAR compensator (SVC) is a continuous reactive power compensation device with the following constraints:

[0028]

[0029] The group switching capacitor (CB) group is used as a discrete reactive power compensation device, and its constraints are as follows:

[0030]

[0031]

[0032]

[0033]

[0034] In the above formula, are the upper and lower limits of the reactive power input by the static VAR compensation device at node m, are the reactive power of the static VAR compensation device and the group switching capacitor connected to node m at time t, are the number of switching groups and the maximum number of switching groups of grouped switching capacitors on node m, respectively. Indicates whether the number of capacitor switching groups on node m changes within the scheduling period. If it changes, it is 1; if it does not change, it is 0.

[0035] Formula (5) is the power limit of the static VAR compensator (SVC), Formula (6) is the constraint on the switching capacity and the number of switching groups of the grouped switching capacitors (CB), and Formulas (7) and (8) are the constraints on the number of switching groups and the number of switching times of the grouped switching capacitors (CB), respectively.

[0036] The constraints of the on-load tap-changing transformer (OLTC) are:

[0037]

[0038] Where K lm,max is the maximum adjustment position of OLTC contacts, n lm is the ratio between nodes l and m, K lm,t Indicates the position of the OLTC contact connected to node j at time t, n lm,t , n lm,0 are the transformation ratios at time t and time 0, respectively. It is a 0-1 variable. When the variable is 1, it means the tap position has changed, and when it is 0, it means the tap position has not changed.

[0039] Formula (10) is the relationship between the high-side and low-side voltages and the transformation ratio, formula (11) is the relationship between the transformation ratio and the position of the OLTC tap and the transformation ratio, formula (12) is the maximum position limit of the OLTC tap, and formula (13) is the adjustment limit of the OLTC tap within the dispatch period. The value of is 0 or 1.

[0040] The inverter capacity of the photovoltaic system must satisfy the following relationship:

[0041]

[0042] In the formula, s is a, b, c. S is the maximum limit of reactive power generated by the household photovoltaic inverter installed on phase S. PV,s is the rated capacity of the household photovoltaic inverter installed on phase S, P PV,s is the active power of household photovoltaic installed in phase s.

[0043] The energy storage system (ESS) constraints are:

[0044]

[0045] Where, and are the minimum and maximum limits of energy storage SOC, S SOC,i (t0) and S SOC,i (t n ) are the energy storage SOC value at the beginning of the day and the energy storage SOC value at the end of the day, and are the minimum and maximum charging power of the energy storage system, respectively. and are the minimum and maximum discharge power of the energy storage system, D char,i (t) and D disc,i (t) is a binary 0-1 variable.

[0046] At this point, a multi-period three-phase distribution network OPF model with reactive compensation devices and energy storage is obtained.

[0047] Equations (1) to (15) are the final established multi-period three-phase distribution network OPF model with reactive power compensation devices and energy storage. In this model, the energy storage device (ESS), distributed generation (DG) inverter represented by photovoltaics, group switching capacitors (CB), static VAR compensators (SVCs), and on-load tap changers (OLTCs) serve as adjustable active and reactive sources. On the basis of ensuring the balance of supply and demand in the distribution network, the voltage on the line is changed by adjusting the reactive power input and output. During the day, photovoltaic power generation is at its peak and user electricity consumption is at its low point. By absorbing reactive power, the grid overvoltage is reduced. At night, when photovoltaic power generation is at its low point and user electricity consumption is at its peak, reactive power is generated to increase the grid voltage.

[0048] In Step 2, the objective function is to minimize the active power loss in the distribution network:

[0049] Network loss is an important reference for judging the economic indicators of the distribution network. Taking a 24-hour period, the objective function is to minimize the active network loss in the distribution network:

[0050]

[0051] Where, P loss is the sum of the active power losses of each branch of the distribution system in 24 hours, E is the set of low-voltage distribution network branches, T is the total number of each time period throughout the day, r lm is the resistance of the branch 1m, I lm,t is the branch current during the period t.

[0052] In Step 2, the constraints added to the distribution network operation are specifically:

[0053] (1) Branch flow constraints:

[0054]

[0055] Where i and j are the node numbers, P j,t , Q j,t are the active injection power and reactive injection power of node j at time t, P ij,t , Q ij,t are respectively the active and reactive power at the head end of branch ij at time t, P jk,t , Q jk,t They are respectively the active and reactive power of the head end of branch jk at time t, k is the set of all nodes with node j as the parent node, U i,t 、U j,t are the voltages of nodes i and j at time t respectively.

[0056] (2) Branch current constraints:

[0057]

[0058] Where, is the current of the s-phase branch ij at time t, and s is the three phases a, b, and c. is the maximum allowable value of branch current.

[0059] (3) Voltage constraint:

[0060]

[0061] Where, |V i s (t)| is the absolute value of the voltage amplitude of node i in phase s at time t, is the minimum voltage of phase s on node i, is the maximum voltage of phase s at node i.

[0062] After adding the distribution network operation constraints of Equations (17) to (22), the resulting OPF optimization model is a nonconvex nonlinear model containing mixed integer variables, and the convergence of the solution cannot be guaranteed. This form of the model is difficult to find the optimal solution. To obtain the optimal solution efficiently and quickly, the present invention uses the second-order cone method to perform a relaxation transformation on the model.

[0063] In Step 3, the non-convex nonlinear OPF optimization model with mixed integer variables established in Step 2 is transformed into a linear model according to the second-order cone relaxation theory. The second-order cone transformation process of the non-convex model is as follows:

[0064] The standard form of a second-order cone is:

[0065]

[0066] Where x∈R n is an n-order vector, A i ∈R m*n , b i ∈R m , c i ∈R n , d i ∈R are all known constants.

[0067] For branch (i, j)∈E, the branch flow constraints are as follows:

[0068]

[0069]

[0070] Substituting equation (25) into equation (24) yields:

[0071]

[0072] By converting complex variables into real variables in amplitude form, the original model is subjected to phase relaxation to linearize the branch power flow equation.

[0073] make Taking the square of the modulus on both sides of the equation, equation (26) can be rewritten as:

[0074]

[0075] Rewrite equations (18) and (19) as follows:

[0076]

[0077]

[0078] The standard second-order cone form can be obtained by transforming the nonlinear inequality constraint (20) through the second-order cone transformation:

[0079]

[0080] The specific transformation process of formula (20) is formula (31) to formula (37):

[0081]

[0082]

[0083]

[0084]

[0085]

[0086]

[0087]

[0088]

[0089] After the above series of relaxation transformations, the original non-convex, nonlinear, and NP-hard model is transformed into the following classic second-order cone programming model:

[0090]

[0091] In order to improve the calculation speed and simplify the solution difficulty, the second-order cone relaxation relaxes the constraints. The relaxation process will inevitably produce errors. The relaxation error is defined as follows using Equation (39):

[0092]

[0093] The model has global optimality and can be solved with the help of the mature Gurobi algorithm package.

[0094] The optimal power flow model of three-phase unbalanced distribution network based on second-order cone relaxation has been established.

[0095] The beneficial effects of the present invention are: based on the three-phase unbalanced operation of the low-voltage distribution network, while taking into account controllable equipment such as distributed power supplies and single-phase reactive power compensation devices, the optimal power flow model of the distribution network is effectively optimized and its accuracy and efficiency are guaranteed, providing a good reference for subsequent research on the scheduling and planning problems of the three-phase unbalanced distribution network. BRIEF DESCRIPTION OF THE DRAWINGS

[0096] Figure 1 is a flow chart of the steps of the present invention;

[0097] Figure 2 This is a three-phase four-wire circuit model diagram of the power distribution network of the present invention;

[0098] Figure 3 It is an IEEE 33 node in an embodiment of the present invention;

[0099] Figure 4 1 is a diagram of the three-phase voltages of A, B, and C at node 6 under the IEEE 33-node distribution network control condition in an embodiment of the present invention;

[0100] Figure 5 This is a comparison of active power loss before and after optimization in an embodiment of the present invention;

[0101] Figure 6 is the output of the phase A reactive power compensation device at node 6 in each time period in the embodiment of the present invention;

[0102] Figure 7 is the output of the B-phase reactive power compensation device at each time period at node 6 in an embodiment of the present invention;

[0103] Figure 8 is the output of the C-phase reactive power compensation device at each time period at node 6 in an embodiment of the present invention;

[0104] Figure 9 is the relaxation error of each branch under multiple time periods in the embodiment of the present invention. DETAILED DESCRIPTION

[0105] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0106] Example 1: Figure 1As shown in the figure, a method for establishing an optimal power flow model for a three-phase unbalanced distribution network is proposed. First, a three-phase four-wire mathematical model of the distribution network is constructed, and the constraints of reactive compensation equipment and energy storage devices are added to obtain a multi-period three-phase distribution network OPF model containing reactive compensation equipment and energy storage. The model is based on the three-phase imbalance of the distribution network and uses a 4*4 impedance matrix to express the relationship between the three phases A, B, and C and the neutral line. It also takes into account the compensation of each phase of the unbalanced power grid by control units including distributed power sources, energy storage, discrete and continuous reactive compensation devices, making the OPF model of the entire three-phase distribution network more accurate. Secondly, the reactive compensation device and energy storage are added to the OPF model. The multi-period three-phase distribution network OPF model is used as the basis, with the maximization of economic benefits, that is, minimization of active network losses in the distribution network as the objective function, and the constraints on the operation of the distribution network are added. The resulting three-phase distribution network OPF optimization model is a non-convex nonlinear model containing mixed integer variables; then, according to the second-order cone relaxation theory, based on the non-convex nonlinear OPF optimization model containing mixed integer variables established in the previous step, by converting complex variables into real variables in amplitude form, the model, which was originally non-convex nonlinear in nature, is transformed into a linear model through phase angle relaxation; finally, the optimal power flow model of the three-phase unbalanced distribution network based on the second-order cone relaxation is obtained.

[0107] The specific steps are:

[0108] Step 1: Construct a three-phase, four-wire distribution network model and incorporate constraints on reactive power compensation equipment and energy storage devices. This results in a multi-period OPF model for the three-phase distribution network, including reactive power compensation and energy storage. This model, based on the three-phase imbalance of the distribution network, uses a 4x4 impedance matrix to represent the relationship between phases A, B, and C and the neutral line. It also considers the compensation provided by control units such as distributed power sources, energy storage, and discrete and continuous reactive power compensation devices for each phase of the unbalanced grid.

[0109] Step 2: Based on the OPF model, the objective function is to minimize the active network loss in the distribution network. The constraints of the distribution network operation are added to obtain the OPF optimization model of the three-phase distribution network. The OPF optimization model is a non-convex nonlinear model containing mixed integer variables.

[0110] Step 3: According to the second-order cone relaxation theory, based on the non-convex nonlinear OPF optimization model with mixed integer variables established in Step 2, by converting complex variables into real variables in amplitude form, the original non-convex nonlinear model is transformed into a linear model through phase relaxation. Finally, the optimal power flow model of the three-phase unbalanced distribution network based on second-order cone relaxation is obtained.

[0111] The Step 1 is specifically as follows:

[0112] Step 1.1: Construct a three-phase four-wire model of the distribution network.

[0113] Most existing power distribution systems use a three-phase four-wire topology. The power distribution line between node L and node M is as follows: Figure 2 As shown, all nodes use a star connection with a grounded neutral point. The first neutral line of the line serves as the sole reference node for this model. Each phase line has its own self-impedance, and the coupling relationship between lines is represented by mutual impedance. The lines are connected to user equipment to form a closed loop.

[0114] According to the above three-phase four-wire line model of the distribution network, considering the three-phase imbalance of the distribution network, the 4*4 impedance matrix Z is first used. l,m To express the relationship between the three phases A, B, and C and the neutral line:

[0115]

[0116] Where Z gg , g takes a, b, c, n as the diagonal elements of the series impedance matrix as the self-impedance of the three-phase line and the neutral line, Z gh , h takes a, b, c, n as the non-diagonal elements in the series impedance matrix, g≠h, as the mutual impedance of the three-phase line and the neutral line.

[0117] Correspondingly, the inverse matrix of the impedance matrix Y=Z -1 As the admittance matrix Y of this line l,m .

[0118] The node admittance matrix Y of a distribution network with m nodes is expressed as:

[0119]

[0120] In the formula, s(m) is the set of nodes adjacent to node m, is the inverse matrix of the series impedance between two adjacent nodes, is the sum of the inverse matrices of all series impedance matrices connected to node m.

[0121] In order to obtain the voltage value of each phase in each node, the voltage vector equation of each phase in each node of the distribution network is added:

[0122] V(t)=Y -1 *I inj (t) (3)

[0123] Where: V(t) is the Nth-order vector formed by the voltage values of each phase at time t. -1 It is the inverse matrix of the admittance matrix Y of the distribution network node. inj (t) is the N-order vector formed by the current injected into the node by each phase at time t.

[0124] In order to ensure that the results of the power flow calculation can accurately reflect the actual situation in the system, the polynomial ZIP model that considers the static characteristics of the load is finally used as the load model (Z is a constant impedance model, I is a constant current model, and P is a constant power model) to calculate the voltage characteristics or static characteristics of the user. The injected current of the s phase on the node can be obtained as:

[0125]

[0126] Where, is the current injected into phase s at time t, is the active and reactive power generated by the household photovoltaic power generation at the s phase on the node l, Active power and reactive power purchased from the upper main grid, is the ZIP component of the s-phase active power demand of the user load at node l, is the ZIP component of the s-phase reactive power demand of the user load at node l, V norml (t) is the standard voltage of the load on node l, V l s (t) is the actual voltage of the s-phase load on node l, and * represents the conjugate.

[0127] At this point, the three-phase four-wire model of the distribution network has been constructed.

[0128] Step 1.2: Add the constraints of reactive compensation equipment and energy storage devices to the constructed three-phase four-wire distribution network model.

[0129] Static VAR compensator (SVC) is a continuous reactive power compensation device with the following constraints:

[0130]

[0131] The group switching capacitor (CB) group is used as a discrete reactive power compensation device, and its constraints are as follows:

[0132]

[0133]

[0134]

[0135]

[0136] In the above formula, are the upper and lower limits of the reactive power input by the static VAR compensation device at node m, are the reactive power of the static VAR compensation device and the group switching capacitor connected to node m at time t, are the number of switching groups and the maximum number of switching groups of grouped switching capacitors on node m, respectively. Indicates whether the number of capacitor switching groups on node m changes within the scheduling period. If it changes, it is 1; if it does not change, it is 0.

[0137] Formula (5) is the power limit of the static VAR compensator (SVC), Formula (6) is the constraint on the switching capacity and the number of switching groups of the grouped switching capacitors (CB), and Formulas (7) and (8) are the constraints on the number of switching groups and the number of switching times of the grouped switching capacitors (CB), respectively.

[0138] The constraints of the on-load tap-changing transformer (OLTC) are:

[0139]

[0140] Where K lm,max is the maximum adjustment position of OLTC contacts, n lm is the ratio between nodes l and m, K lm,t Indicates the position of the OLTC contact connected to node j at time t, n lm,t , n lm,0 are the transformation ratios at time t and time 0, respectively. It is a 0-1 variable. When the variable is 1, it means the tap position has changed, and when it is 0, it means the tap position has not changed.

[0141] Formula (10) is the relationship between the high-side and low-side voltages and the transformation ratio, formula (11) is the relationship between the transformation ratio and the position of the OLTC tap and the transformation ratio, formula (12) is the maximum position limit of the OLTC tap, and formula (13) is the adjustment limit of the OLTC tap within the dispatch period. The value of is 0 or 1.

[0142] The inverter capacity of the photovoltaic system must satisfy the following relationship:

[0143]

[0144] In the formula, s is a, b, c. S is the maximum limit of reactive power generated by the household photovoltaic inverter installed on phase S. PV , s is the rated capacity of the household photovoltaic inverter installed in phase s, P PV,s is the active power of household photovoltaic installed in phase s.

[0145] The energy storage system (ESS) constraints are:

[0146]

[0147] Where, and are the minimum and maximum limits of energy storage SOC, S SOC,i (t0) and S SOC,i (t n ) are the energy storage SOC value at the beginning of the day and the energy storage SOC value at the end of the day, and are the minimum and maximum charging power of the energy storage system, respectively. and are the minimum and maximum discharge power of the energy storage system, D char,i (t) and D disc,i (t) is a binary 0-1 variable.

[0148] At this point, a multi-period three-phase distribution network OPF model with reactive compensation devices and energy storage is obtained.

[0149] Equations (1) to (15) are the final established multi-period three-phase distribution network OPF model with reactive power compensation devices and energy storage. In this model, the energy storage device (ESS), distributed generation (DG) inverter represented by photovoltaics, group switching capacitors (CB), static VAR compensators (SVCs), and on-load tap changers (OLTCs) serve as adjustable active and reactive sources. On the basis of ensuring the balance of supply and demand in the distribution network, the voltage on the line is changed by adjusting the reactive power input and output. During the day, photovoltaic power generation is at its peak and user electricity consumption is at its low point. By absorbing reactive power, the grid overvoltage is reduced. At night, when photovoltaic power generation is at its low point and user electricity consumption is at its peak, reactive power is generated to increase the grid voltage.

[0150] In Step 2, the objective function is to minimize the active power loss in the distribution network:

[0151] Network loss is an important reference for judging the economic indicators of the distribution network. Taking a 24-hour period, the objective function is to minimize the active network loss in the distribution network:

[0152]

[0153] Where, P loss is the sum of the active power losses of each branch of the distribution system in 24 hours, E is the set of low-voltage distribution network branches, T is the total number of each time period throughout the day, r lm is the resistance of the branch 1m, I lm,t is the branch current during the period t.

[0154] In Step 2, the constraints added to the distribution network operation are specifically:

[0155] (1) Branch flow constraints:

[0156]

[0157] Where i and j are the node numbers, P j,t , Q j,t are the active injection power and reactive injection power of node j at time t, P ij,t , Q ij,t are respectively the active and reactive power at the head end of branch ij at time t, P jk,t , Q jk,t They are respectively the active and reactive power of the head end of branch jk at time t, k is the set of all nodes with node j as the parent node, U i,t 、U j,t are the voltages of nodes i and j at time t respectively.

[0158] (2) Branch current constraints:

[0159]

[0160] Where, is the current of the s-phase branch ij at time t, and s is the three phases a, b, and c. is the maximum allowable value of branch current.

[0161] (3) Voltage constraint:

[0162]

[0163] Where, |V i s (t)| is the absolute value of the voltage amplitude of node i in phase s at time t, is the minimum voltage of phase s on node i, is the maximum voltage of phase s at node i.

[0164] After adding the distribution network operation constraints of Equations (17) to (22), the resulting OPF optimization model is a nonconvex nonlinear model containing mixed integer variables, and the convergence of the solution cannot be guaranteed. This form of the model is difficult to find the optimal solution. To obtain the optimal solution efficiently and quickly, the present invention uses the second-order cone method to perform a relaxation transformation on the model.

[0165] In Step 3, the non-convex nonlinear OPF optimization model with mixed integer variables established in Step 2 is transformed into a linear model according to the second-order cone relaxation theory. The second-order cone transformation process of the non-convex model is as follows:

[0166] The standard form of a second-order cone is:

[0167]

[0168] Where x∈R n is an n-order vector, Ai ∈R m*n , b i ∈R m , c i ∈R n , d i ∈R are all known constants.

[0169] For branch (i, j)∈E, the branch flow constraints are as follows:

[0170]

[0171]

[0172] Substituting equation (25) into equation (24) yields:

[0173]

[0174] By converting complex variables into real variables in amplitude form, the original model is subjected to phase relaxation to linearize the branch power flow equation.

[0175] make Taking the square of the modulus on both sides of the equation, equation (26) can be rewritten as:

[0176]

[0177] Rewrite equations (18) and (19) as follows:

[0178]

[0179]

[0180] The standard second-order cone form can be obtained by transforming the nonlinear inequality constraint (20) through the second-order cone transformation:

[0181]

[0182] The specific transformation process of formula (20) is formula (31) to formula (37):

[0183]

[0184]

[0185]

[0186]

[0187]

[0188]

[0189]

[0190]

[0191] After the above series of relaxation transformations, the original non-convex, nonlinear, and NP-hard model is transformed into the following classic second-order cone programming model:

[0192]

[0193] In order to improve the calculation speed and simplify the solution difficulty, the second-order cone relaxation relaxes the constraints. The relaxation process will inevitably produce errors. The relaxation error is defined as follows using Equation (39):

[0194]

[0195] The model has global optimality and can be solved with the help of the mature Gurobi algorithm package.

[0196] The optimal power flow model of three-phase unbalanced distribution network based on second-order cone relaxation has been established.

[0197] Use Figure 3 The IEEE 33-node network shown in the figure is simulated and analyzed. To reflect the unbalanced load characteristics of each phase in the example, the number of users configured on phase C is significantly greater than the number of users on phases A and B. Energy storage devices are installed at nodes 15 and 32, in conjunction with DGs, to achieve peak load shifting. Single-phase reactive power compensation devices (CBs) are installed on the three phases of nodes 6 and 16, with each CB group having a capacity of 50 kvar, for a total of 10 groups. Single-phase reactive power compensation devices (SVCs) are installed on each phase of nodes 6, 16, and 32, with a compensation range of -0.1 to 0.3 Mvar. An on-load tap-changing transformer (OLTC) is installed at node 33, with a tap adjustment step size of 0.01 and a maximum daily adjustment of 5 times. The node voltage operating range is 0.98 to 1.13 pu. The proposed model is used to optimize the 24-hour operating conditions of the IEEE 33-node distribution system.

[0198] The Gurobi solver is called to solve the relaxed model, and the sum of the active power loss in each period is 691.25kW. The active power loss before system optimization is 1036.88kW. The active power loss after optimization is reduced to 66.7% before optimization. The voltage amplitude of each phase on node 6 of the system and the loss comparison of the entire distribution system before and after optimization in each period are shown in the figure below. Figure 4 and Figure 5 shown.

[0199] according to Figure 5It is not difficult to see that the active power loss curve of the system in each period is similar to the load demand trend, and the active power loss of the system is proportional to the load. The distribution system model can optimize the three-phase unbalanced system in operation by controlling and coordinating controllable units represented by single-phase reactive power compensation devices, thereby reducing active power loss and improving the economic efficiency of distribution network operation. The compensation status of each phase reactive power compensation device on node 6 is shown in Figure 2. Figure 6-8 As shown in the figure, each controllable device adjusts the compensation amount according to the load change within the operating conditions.

[0200] from Figure 6-8 As can be seen, the reactive power compensation devices on each phase at node 6 provide reactive power compensation to prevent a drop in bus voltage due to insufficient reactive power. Each phase optimizes the system based on single-phase power flow parameters. Because the load on phase C is greater than on phases A and B, its reactive power compensation is also relatively large. Furthermore, compensation is concentrated between 3:00 PM and 8:00 PM. This is because in the evening, household photovoltaic power generation in the system is at its lowest point, while residential electricity consumption is at its peak. This mismatch between photovoltaic power generation and load usage can easily cause the voltage to exceed the lower limit, necessitating that the reactive power compensation devices generate sufficient reactive power to ensure node voltage stability.

[0201] The charge and discharge power and SOC changes of the energy storage device ESS in each period are shown in Table 1:

[0202]

[0203] Table 1 ESS power output and remaining power in each time period

[0204] Table 1 Power output and remaining power of ESS in each period

[0205] When the ESS output power is greater than 0, it is in the charging state, and less than 0, it is in the discharging state. The data in the table shows that the ESS charges during the daytime low load period and discharges during the evening peak load period, which has a good peak-shaving and valley-filling effect.

[0206] In order to verify the accuracy of the second-order cone relaxation processing model, the relaxation error of each branch in multiple time periods is calculated according to the calculation formula (33), as follows: Figure 9 As shown in the figure, it can be seen that the relaxation error is 10 -6 The magnitude of the error is sufficient to meet the operational requirements in this case, proving that this method has high feasibility.

[0207] Based on the three-phase unbalanced operation of low-voltage distribution networks, this paper takes into account controllable devices such as distributed power sources and single-phase reactive power compensation devices, and aims to minimize the active power loss of the distribution network. It establishes an optimal power flow model for the multi-period distribution network. A second-order cone relaxation method is used to convert the mixed integer non-convex nonlinear model into an easy-to-solve linear model. The feasibility and effectiveness of this method are verified through a case study of an IEEE 33-node distribution system, laying a certain foundation for further research on the scheduling and planning of three-phase unbalanced distribution networks.

[0208] The above describes the specific embodiments of the present invention in detail with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. Various changes can be made within the knowledge of ordinary technicians in this field without departing from the scope of the present invention.

Claims

1. A method for establishing an optimal power flow model for a three-phase unbalanced distribution network, characterized by: Step 1: Construct a three-phase four-wire distribution network model and add constraints on reactive power compensation equipment and energy storage devices to the model, thereby obtaining a multi-period three-phase distribution network OPF model including reactive power compensation equipment and energy storage. Step 2: Based on the OPF model, the objective function is to minimize the active network loss in the distribution network. The constraints of the distribution network operation are added to obtain the OPF optimization model of the three-phase distribution network. The OPF optimization model is a non-convex nonlinear model with mixed integer variables. Step 3: Based on the non-convex nonlinear OPF optimization model with mixed integer variables established in Step 2, the original non-convex nonlinear model is transformed into a linear model by performing phase relaxation according to the second-order cone relaxation theory. Finally, the optimal power flow model of the three-phase unbalanced distribution network based on the second-order cone relaxation is obtained. Step 1.1: Construct a three-phase four-wire model of the distribution network; First, we use the 4*4 impedance matrix Z l,m To express the relationship between the three phases A, B, and C and the neutral line: Where Z gg , g takes a, b, c, n as the diagonal elements of the series impedance matrix as the self-impedance of the three-phase line and the neutral line, Z gh , h takes a, b, c, n as the non-diagonal elements in the series impedance matrix, g ≠ h, as the mutual impedance of the three-phase line and the neutral line; Correspondingly, the inverse matrix of the impedance matrix Y=Z -1 As the admittance matrix Y of this line l,m ; The node admittance matrix Y of a distribution network with m nodes is expressed as: In the formula, s(m) is the set of nodes adjacent to node m, is the inverse matrix of the series impedance between two adjacent nodes, is the sum of the inverse matrices of all series impedance matrices connected to node m; Then add the voltage vector equation of each phase of each node in the distribution network: V(t)=Y -1 *I inj (t) (3) Where: V(t) is the Nth-order vector formed by the voltage values of each phase at time t; Y -1 is the inverse matrix of the distribution network node admittance matrix Y; I inj (t) is the N-order vector formed by the current injected into the node by each phase at time t; Finally, the polynomial ZIP model considering the static characteristics of the load is used as the load model to calculate the voltage characteristics or static characteristics of the user, and the injected current of the s phase on the node can be obtained as: Where, is the current injected into phase s at time t, is the active and reactive power generated by the household photovoltaic power generation at the s phase on the node l, Active power and reactive power purchased from the upper main grid, is the ZIP component of the s-phase active power demand of the user load at node l, is the ZIP component of the s-phase reactive power demand of the user load at node l, is the standard voltage of the load on node l, V l s (t) is the actual voltage of the s-phase load on node l, * indicates conjugate; At this point, the three-phase four-wire model of the distribution network has been constructed.

2. The method for establishing an optimal power flow model for a three-phase unbalanced distribution network according to claim 1, wherein: The Step 1 is specifically as follows: Step 1.2: Add the constraints of reactive power compensation equipment and energy storage devices to the constructed three-phase four-wire distribution network model; As a continuous reactive power compensation device, the static VAR compensator has the following constraints: The group switching capacitor bank is used as a discrete reactive power compensation device, and its constraints are as follows: In the above formula, are the upper and lower limits of the reactive power input by the static VAR compensation device at node m, are the reactive power of the static VAR compensation device and the group switching capacitor connected to node m at time t, are the number of switching groups and the maximum number of switching groups of grouped switching capacitors on node m, respectively. Whether the number of capacitor switching groups on node m changes within the scheduling period, if it changes, it is 1, if it does not change, it is 0; Formula (5) is the power limit of the static VAR compensator, formula (6) is the constraint of the switching capacity and the number of switching groups of the group switching capacitor, and formulas (7) and (8) are the constraints of the number of switching groups and the number of switching times of the group switching capacitor respectively. The constraints of the on-load tap-changing transformer are: Where K lm,max is the maximum adjustment position of OLTC contacts, n lm is the ratio between nodes l and m, K lm,t Indicates the position of the OLTC contact connected to node j at time t, n lm,t , n lm,0 are the transformation ratios at time t and time 0, respectively. It is a 0-1 variable. When the variable is 1, it means the tap position has changed, and when it is 0, it means the tap position has not changed. Formula (10) is the relationship between the high-side and low-side voltages and the transformation ratio, formula (11) is the relationship between the transformation ratio and the position of the tap of the on-load tap-changing transformer and the transformation ratio, formula (12) is the maximum position limit of the tap of the on-load tap-changing transformer, and formula (13) is the adjustment limit of the tap of the on-load tap-changing transformer within the dispatching period. The value of is 0 or 1; The inverter capacity of the photovoltaic system must satisfy the following relationship: In the formula, s is a, b, c; S is the maximum limit of reactive power generated by the household photovoltaic inverter installed on phase S. PV,s is the rated capacity of the household photovoltaic inverter installed on phase S, P PV,s is the active power of household photovoltaic installed in phase s; The energy storage device constraints are: Where, and are the minimum and maximum limits of energy storage SOC, S SOC,i (t0) and S SOC,i (t n ) are the energy storage SOC value at the beginning of the day and the energy storage SOC value at the end of the day, and are the minimum and maximum charging power of the energy storage system, respectively. and are the minimum and maximum discharge power of the energy storage system, D char,i (t) and D disc,i (t) is a binary 0-1 variable; At this point, a multi-period three-phase distribution network OPF model with reactive compensation devices and energy storage is obtained.

3. The method for establishing an optimal power flow model for a three-phase unbalanced distribution network according to claim 1, wherein: In Step 2, the objective function is to minimize the active power loss in the distribution network: Where, P loss is the sum of the active power losses of each branch of the distribution system in 24 hours, E is the set of low-voltage distribution network branches, T is the total number of each time period throughout the day, r lm is the resistance of the branch 1m, I lm,t is the branch current during the period t.

4. The method for establishing an optimal power flow model for a three-phase unbalanced distribution network according to claim 1, wherein: In Step 2, the constraints added to the distribution network operation are specifically: (1) Branch flow constraints: Where i and j are the node numbers, P j,t , Q j,t are the active injection power and reactive injection power of node j at time t, P ij,t , Q ij,t are respectively the active and reactive power at the head end of branch ij at time t, P jk,t , Q jk,t They are respectively the active and reactive power of the head end of branch jk at time t, k is the set of all nodes with node j as the parent node, U i,t 、U j,t are the voltages of nodes i and j at time t respectively; (2) Branch current constraints: Where, is the current of the s-phase branch ij at time t, where s is the three phases a, b, and c; is the maximum allowable value of branch current; (3) Voltage constraint: Where, |V i s (t)| is the absolute value of the voltage amplitude of node i in phase s at time t, is the minimum voltage of phase s on node i, is the maximum voltage of phase s at node i.

5. The method for establishing an optimal power flow model for a three-phase unbalanced distribution network according to claim 2, wherein: In Step 3, the non-convex nonlinear OPF optimization model with mixed integer variables established in Step 2 is transformed into a linear model according to the second-order cone relaxation theory. The second-order cone transformation process of the non-convex model is as follows: The standard form of a second-order cone is: Where x∈R n is an n-order vector, A i ∈R m*n , b i ∈R m , c i ∈R n , d i ∈R are all known constants; For branch (i, j)∈E, the branch flow constraints are as follows: Substituting equation (25) into equation (24) yields: By converting complex variables into real variables in amplitude form, the original model is relaxed in phase angle to linearize the branch power flow equation; make Taking the square of the modulus on both sides of the equation, equation (26) can be rewritten as: After the relaxation transformation, the original non-convex, nonlinear, and NP-hard model is transformed into the following classic second-order cone programming model: The relaxation error is defined by equation (29): The optimal power flow model of three-phase unbalanced distribution network based on second-order cone relaxation has been established.

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