CN113890029A - Multi-objective optimization method for power distribution network with openable capacity improvement - Google Patents

Multi-objective optimization method for power distribution network with openable capacity improvement Download PDF

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CN113890029A
CN113890029A CN202111327215.0A CN202111327215A CN113890029A CN 113890029 A CN113890029 A CN 113890029A CN 202111327215 A CN202111327215 A CN 202111327215A CN 113890029 A CN113890029 A CN 113890029A
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CN113890029B (en
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汪超群
陈懿
迟长云
李晓波
蒋雪冬
乔辉
史立勤
陶媛
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Zhejiang Zheda Energy Technology Co ltd
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    • H—ELECTRICITY
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Abstract

The invention discloses a multi-objective optimization method for a power distribution network with openable capacity improvement, which comprises the following steps: s1, collecting basic data required by multi-objective optimization of the power distribution network; s2, constructing a multi-objective function taking the openable capacity of the power distribution network, the network loss, the operation cost and the transformation cost as optimization targets; and S3, solving the multi-objective function by using a layered optimization method with the constraints of transformer replacement, line reconstruction, distributed resource location and capacity determination, network radiation operation, power balance, node voltage, branch loss and branch capacity to obtain the optimal solution of the multi-objective function. According to the invention, various resources and behaviors related in the process of modifying the power distribution system are reasonably modeled, so that the comprehensive optimality of a modification scheme is effectively improved, and the method has a certain application value for improving the power supply capacity of the system, reducing the operation and modification cost and delaying the construction investment.

Description

Multi-objective optimization method for power distribution network with openable capacity improvement
Technical Field
The invention relates to the technical field of power grids, in particular to a multi-objective optimization method for a power distribution network for improving the openable capacity.
Background
The power distribution system is an important link for connecting a power grid and a user, and has an irreplaceable effect on meeting the electricity utilization requirements of residents and guaranteeing normal production of enterprises. In recent years, with the rapid development of economy, the demand of various users on electric energy continuously rises, so that the openable capacity of the conventional power distribution system is in rapid shortage, and the further development of the economic society is severely restricted. In order to improve the load carrying capacity of the system and meet the social power consumption requirements, upgrading and modifying the power distribution network on the original basis is urgent. At present, most of methods related to upgrading and reconstruction of a power distribution network adopt a simple capacity-to-load ratio method, the method combines subjective experience of designers, and an upgrading and reconstruction scheme is determined through simple capacity setting calculation, so that the method has the advantages of large subjective randomness, extensive calculation process and difficulty in ensuring the optimality of the scheme, and resource waste or unqualified reconstruction is caused.
Disclosure of Invention
The invention provides a multi-objective optimization method for a power distribution network with an openable capacity improvement function, which is characterized in that various resources and behaviors related in the process of modifying the power distribution system are reasonably modeled, so that the comprehensive optimality of a modification scheme is effectively improved, and the method has certain application value for improving the power supply capacity of the system, reducing the operation and modification cost and delaying the construction investment.
In order to achieve the purpose, the invention adopts the following technical scheme:
the method for optimizing the multiple targets of the power distribution network facing the improvement of the openable capacity comprises the following steps:
s1, collecting basic data required by multi-objective optimization of the power distribution network;
s2, constructing a multi-objective function taking the openable capacity of the power distribution network, the network loss, the operation cost and the transformation cost as optimization targets;
and S3, solving the multi-objective function by using a hierarchical optimization method with the constraints of transformer replacement, line reconstruction, distributed resource location and capacity determination, network radiation operation, power balance, node voltage, branch loss and branch capacity to obtain an optimal solution of the multi-objective function.
Further, the basic data in step S1 includes active load and reactive load of the power distribution network; resistance, reactance, capacity of the transformer and distribution lines; the length of the distribution line; hourly wind speed and illumination of the area where the power distribution system is located; the technical and economic parameters of the alternative equipment and the importance ranking of the technical experts, operating personnel and involved personnel on the category 4 optimization objectives of the openable capacity, the network loss, the operating costs, the retrofit costs.
Further, the multi-objective function constructed in step S2 is expressed by the following formula (1):
Figure BDA0003347614460000021
in the formula (1), the first and second groups,
Figure BDA0003347614460000022
representing the a-th objective function f of the constructed multi-objective functionsaMaking a target function expression after dimensionless processing;
wato represent
Figure BDA0003347614460000023
The corresponding weight;
a＝1，2，3，4。
further, the air conditioner is provided with a fan,
Figure BDA0003347614460000024
calculated by the following formula (2):
Figure BDA0003347614460000025
in the formula (2), the first and second groups, af、
Figure BDA0003347614460000026
respectively representing the a-th objective function faMinimum and maximum values of.
Further, calculating
Figure BDA0003347614460000027
Corresponding weight waThe method comprises the following steps:
l1, forming an importance comparison matrix A for pairwise comparison of 4 types of optimization targets of the openable capacity, the network loss, the operation cost and the transformation cost of the power distribution network(u)U-1, 2,3, u-1, u-2, and u-3 respectively represent the relative ranking of importance of technical experts, operators, and designers for the 4 types of optimization targets, a(u)Expressed by the following formula (3):
Figure BDA0003347614460000028
l2, calculating the importance comparison matrix A by the following formula (4)(u)The product of each of the rows in the column,
Figure BDA0003347614460000029
in the formula (4), the first and second groups,
Figure BDA00033476144600000210
representing the comparison matrix of importance A(u)The product of the g-th row;
Figure BDA00033476144600000211
representing the comparison matrix of importance A(u)Row g and column h of (1);
h represents the importance comparison matrix A(u)H is 1,2,3, 4;
l3, calculating the product by the following formula (5)
Figure BDA00033476144600000212
4 th order root of (1)
Figure BDA00033476144600000213
Figure BDA00033476144600000214
L4, by the following equation (6) pair
Figure BDA00033476144600000215
Normalization is carried out to obtain a normalization result
Figure BDA00033476144600000216
Figure BDA0003347614460000031
In the formula (6), the first and second groups,
Figure BDA0003347614460000032
expressed in equation (5)
Figure BDA0003347614460000033
h, taking 1,2,3 and 4;
l5 calculated by the following formula (7)
Figure BDA0003347614460000034
Corresponding weight wa：
Figure BDA0003347614460000035
In formula (7), a is g.
Further, when a is 1, the objective function fa＝f1Representing an objective function for optimizing the openability of the distribution network, f1Calculated by the following equation (8):
f1becoming-xi formula (8)
In the formula (8), ξ represents the load increase factor variable of the node i in the power distribution network, and is used for indicating the increase factor relative to the original ground state load.
Further, when a is 2, the objective function fa＝f2Expressing an objective function for optimizing the network loss of the distribution network, f2Calculated by the following equation (9):
Figure BDA0003347614460000036
in formula (9), SBRepresenting sets of branches in a distribution network, SB＝SL∪SV；SLRepresenting a set of feeder branches in the distribution network; sVRepresenting a set of transformer branches in a power distribution network;
STrepresents the set of all optimization periods, S, of each dayT＝{1，2，…，24}；
k represents a k-type transformer branch or a k-type feeder branch in the power distribution network;
Figure BDA0003347614460000037
and the active loss of a k-type transformer branch or a k-type feeder branch connected with the nodes i and j in a t period is represented.
Further, when a is 3, the objective function fa＝f3An objective function representing the cost of operating an optimized distribution network, f3Calculated by the following equation (10):
Figure BDA0003347614460000038
in the formula (10), ptRepresenting the electricity purchase price of the t period;
Figure BDA0003347614460000039
representing the active output of a transformer of a node i of the power distribution network in a period t;
STrepresents the set of all optimization periods, S, of each dayT＝{1，2，…，24}。
Further, when a is 4, the objective function fa＝f4An objective function representing the cost of modification to optimize the distribution network, f4Calculated by the following equation (11):
Figure BDA0003347614460000041
in the formula (11), the reaction mixture,
r represents loan rate;
y represents a return on investment period;
Figure BDA0003347614460000042
representing the actual cost of transformation of a transformer branch (i, j) in the distribution network;
the branch (i, j) of the transformer belongs to SV，SVRepresenting a set of transformer branches in a power distribution network;
Figure BDA0003347614460000043
representing the actual cost of transformation of the feeder branch (i, j) in the distribution network;
the feeder branch (i, j) belongs to SL，SLRepresenting a set of feeder branches in the distribution network;
Figure BDA0003347614460000044
representing the actual installation cost of the static var compensator svc at node i of the power distribution network;
Figure BDA0003347614460000045
representing the total investment cost of the fan at the node i;
Figure BDA0003347614460000046
representing the actual installation cost of the photovoltaic at the node i;
SNrepresenting the set of all nodes.
Further, the transformer replacement constraint described in step S3 is expressed by the following equations (12) to (13):
Figure BDA0003347614460000047
Figure BDA0003347614460000048
in equations (12) to (13), K represents the number of types of alternative transformers;
k represents a kth type transformer;
Figure BDA0003347614460000049
indicating whether the transformer branch (i, j) connecting the node i and the node j of the distribution network is replaced by a kth type transformer,
Figure BDA00033476144600000410
or a combination of the values of 0,
Figure BDA00033476144600000411
indicates that no replacement is to be performed on the transformer in the transformer branch (i, j);
Figure BDA00033476144600000412
represents the replacement of the transformers in the transformer branch (i, j) with a kth type transformer;
formula (12) shows that at most one of K transformers is selected to replace the transformer in the transformer branch (i, j);
Figure BDA00033476144600000413
representing the replacement cost of the kth type transformer;
Figure BDA00033476144600000414
the capacity of a kth type transformer is shown;
Figure BDA0003347614460000051
representing the actual cost of retrofitting of the transformer branch (i, j);
the branch (i, j) of the transformer belongs to SV，SVRepresenting a set of transformer legs in a power distribution network.
Further, the route modification constraint described in step S3 is expressed by the following equations (14) to (15):
Figure BDA0003347614460000052
Figure BDA0003347614460000053
in equations (14) - (15), K represents the number of alternative feeder branch types;
k represents a k-th type feeder branch;
Figure BDA0003347614460000054
representing whether a feeder branch connecting a node i and a node j of the power distribution network is replaced by a k-th type feeder branch，
Figure BDA0003347614460000055
Or a combination of the values of 0,
Figure BDA0003347614460000056
indicates that no replacement is to be made for feeder leg (i, j);
Figure BDA0003347614460000057
representing the replacement of feeder branch (i, j) with a k-th type feeder branch;
formula (14) represents that at most one of K feeder branches is selected to replace the feeder branch (i, j);
Figure BDA0003347614460000058
representing the replacement cost per unit length of the k-th type feeder branch;
Figure BDA0003347614460000059
representing a length of the feeder branch (i, j) e SL，SLThe feeder branch sets in the power distribution network are provided;
Figure BDA00033476144600000510
representing the actual cost of retrofitting of the feeder branch (i, j).
Further, the distributed resource location and volume constraints described in step S3 include static var compensator location and volume constraints, distributed fan location and volume constraints, and distributed photovoltaic location and volume constraints, where the static var compensator location and volume constraints are expressed by the following equations (16) - (17):
Figure BDA00033476144600000511
Figure BDA00033476144600000512
in the formulae (16) to (17),
Figure BDA00033476144600000513
indicating whether a static var compensator svc is installed at the node i of the power distribution network,
Figure BDA00033476144600000514
or 1 of the number of the groups in the group,
Figure BDA00033476144600000515
indicating that the static var compensator svc is not installed at the node i;
Figure BDA00033476144600000516
represents the installation of the static var compensator svc at the node i;
Figure BDA00033476144600000517
respectively representing the minimum installation capacity and the maximum installation capacity of the static var compensator svc installed at the node i;
Figure BDA0003347614460000061
representing the actual installation capacity of the static var compensator svc at the node i;
Csvcrepresenting the unit capacity manufacturing cost of the static var compensator svc;
Figure BDA0003347614460000062
represents the actual installation cost of the static var compensator svc at the node i.
Further, the distributed fan siting volume constraint is expressed by the following equations (18) - (20):
Figure BDA0003347614460000063
Figure BDA0003347614460000064
Figure BDA0003347614460000065
in the formulae (18) to (20),
Figure BDA0003347614460000066
or 1 of the number of the groups in the group,
Figure BDA0003347614460000067
indicating that no fan wt is installed at the node i;
Figure BDA0003347614460000068
indicating that a fan wt is installed at the node i; the formula (19) shows that only 1 type fan in the K type fans is allowed to be installed at most;
Figure BDA0003347614460000069
respectively representing the minimum number and the maximum number of the installed k-type fans at the node i;
Figure BDA00033476144600000610
the rated capacity of the k-type fan is represented;
Figure BDA00033476144600000611
representing an actual installed capacity of the k-type fan at the node i;
Figure BDA00033476144600000612
expressing the unit capacity cost of the k-type fan;
Figure BDA00033476144600000613
representing the total investment cost of the fan at the node i;
k represents the model number of the fan wt;
k represents the k-type fan.
Further, the distributed photovoltaic siting constraint is expressed by the following equations (21) - (22):
Figure BDA00033476144600000614
Figure BDA00033476144600000615
in the formulae (21) to (22),
Figure BDA00033476144600000616
or 1 of the number of the groups in the group,
Figure BDA00033476144600000617
means that no photovoltaic is connected at said node i;
Figure BDA00033476144600000618
represents the photovoltaic access at the node i;
Figure BDA00033476144600000619
respectively representing the minimum installation capacity and the maximum installation capacity of the photovoltaic at the node i;
Figure BDA00033476144600000620
the actual photovoltaic installation capacity at the node i is obtained;
Cpvthe unit capacity cost for photovoltaic;
Figure BDA00033476144600000621
representing the actual installation cost of the photovoltaic at the node i.
Further, the network radiation operation constraint described in step S3 is expressed by the following equation (23):
Figure BDA0003347614460000071
in the formula (23), the first and second groups,
Figure BDA0003347614460000072
representing the switching state of a branch ij connecting nodes i, j in the distribution network,
Figure BDA0003347614460000073
or 1 of the number of the groups in the group,
Figure BDA0003347614460000074
it means that said branch ij is open,
Figure BDA0003347614460000075
when indicates that the branch ij is closed;
Nnoderepresenting the number of nodes in the power distribution network;
SBrepresenting sets of branches in a distribution network, SB＝SL∪SV；SLRepresenting a set of feeder branches in the distribution network; sVRepresenting a set of transformer branches in a power distribution network;
i. j denotes a tributary node.
Further, the power balance constraint described in step S3 is expressed by the following equations (24) - (25):
Figure BDA0003347614460000076
Figure BDA0003347614460000077
in the formulae (24) to (25),
Figure BDA0003347614460000078
respectively representing a transformer active output variable, a fan active output variable, a photovoltaic active output variable and an active load measured value of a node i of a power distribution network in a period t;
ξ represents the load increase magnification variable of the node i;
Figure BDA0003347614460000079
respectively representing a transformer reactive power output variable of the node i, a static reactive power compensator reactive power output variable and a reactive load measured value of the node i in a period t;
Figure BDA00033476144600000710
respectively representing the injected active and reactive variables of the node i;
Figure BDA00033476144600000711
subject to constraints shown by the following equations (26) to (27):
Figure BDA00033476144600000712
Figure BDA00033476144600000713
in formulae (26) to (27), Pi，j，k，t、Qi，j，k，tRespectively representing the active power and the reactive power of a k-type feeder branch circuit where the nodes i and j are located in a t period;
Figure BDA00033476144600000714
respectively representing the active loss and the reactive loss of the k-type feeder branch circuit where the nodes i and j are located in the period t;
Figure BDA0003347614460000081
subject to the constraint shown in equation (28) below:
Figure BDA0003347614460000082
in the formula (28), the first and second groups,
Figure BDA0003347614460000083
representing the actual installed capacity of the static var compensator svc at said node i.
Further, the air conditioner is provided with a fan,
Figure BDA0003347614460000084
calculated by the following equation (29):
Figure BDA0003347614460000085
in the formula (29), the reaction is carried out,
Figure BDA0003347614460000086
representing the output active power of a k-type fan accessed to the node i in a t period;
Figure BDA0003347614460000087
calculated by the following equation (30):
Figure BDA0003347614460000088
in the formula (30), vtThe average wind speed is obtained by measuring the average wind speed in the t period;
vk，ci、vk，r、vk，cothe cut-in wind speed, the rated wind speed and the cut-out wind speed of the k-type fan are respectively.
Further, the air conditioner is provided with a fan,
Figure BDA0003347614460000089
calculated by the following equation (31):
Figure BDA00033476144600000810
in formula (31), It、θtRespectively representing the average illumination intensity and the illumination incidence angle of the photovoltaic in the t period;
Figure BDA00033476144600000811
representing the actual installation capacity of the photovoltaic at the node i;
μpvrepresenting the power temperature coefficient of the photovoltaic module;
Ttthe average working temperature of the photovoltaic is t period;
TSTCis the photovoltaic module temperature under standard test conditions.
Further, the node voltage constraint described in step S3 is expressed by the following equations (32) - (34):
Figure BDA00033476144600000812
Figure BDA00033476144600000813
Figure BDA00033476144600000814
in the formulae (32) to (34), Ri，j，k、Xi，j，kRespectively representing the resistance and reactance of a k-type feeder branch circuit connected with nodes i and j of the power distribution network;
Pi，j，k，t、Qi，j，k，trespectively represents the flow through the connection node at the time ti, j, the active and reactive power of the k-type feeder branch;
Vi，trepresenting the voltage amplitude of the node i in a period t;
iV、
Figure BDA0003347614460000091
respectively representing the minimum voltage amplitude value and the maximum voltage amplitude value of the node i;
Vref，trepresenting a voltage per unit value of a first node, wherein the first node is a direct connection node of a main network and a distribution network;
V0representing a default value of the voltage of the head node.
Further, the branch loss constraint described in step S3 is expressed by the following equations (35) to (42):
Figure BDA0003347614460000092
Figure BDA0003347614460000093
Figure BDA0003347614460000094
Figure BDA0003347614460000095
Figure BDA0003347614460000096
Figure BDA0003347614460000097
Figure BDA0003347614460000098
Figure BDA0003347614460000099
in the formulae (35) to (42),
Figure BDA00033476144600000910
respectively representing the active loss and the reactive loss of a k-type feeder branch circuit where the nodes i and j are located in a t period;
Ri，j，k、Xi，j，krespectively representing the resistance and reactance of the k-type feeder branch circuit where the nodes i and j are located;
Pi，j，k，t、Qi，j，k，trespectively representing the active power and the reactive power of the k-type feeder branch circuit where the nodes i and j are located in a t period;
Figure BDA00033476144600000911
respectively representing the maximum allowable active power and the maximum allowable reactive power of the k-type feeder branch circuit where the nodes i and j are located;
Nvindicates a section of
Figure BDA00033476144600000912
Or
Figure BDA00033476144600000913
Number of segments to be linearly equally divided;
v denotes a linear segment number, v is 1,2, … Nv。
Further, the branch capacity constraint described in step S3 is expressed by the following equations (43) to (44):
Figure BDA0003347614460000101
Figure BDA0003347614460000102
in formulae (43) to (44), Pi，j，k，t、Qi，j，k，tRespectively representing the active power and the reactive power of a k-type feeder branch circuit where nodes i and j in the power distribution network are located in a t period;
Figure BDA0003347614460000103
representing the switching state of the k-type feeder branch in which the node i, j is located in the distribution network,
Figure BDA0003347614460000104
or 1 of the number of the groups in the group,
Figure BDA0003347614460000105
when the k-type feeder branch is disconnected,
Figure BDA0003347614460000106
when the k-type feeder branch is closed, the k-type feeder branch is closed;
Figure BDA0003347614460000107
representing the capacity of a k-type transformer or the k-type feeder branch in the power distribution network;
Figure BDA0003347614460000108
is a variable from 0 to 1, indicates whether a k-type transformer connecting node i and node j or the k-type feeder branch is replaced,
Figure BDA0003347614460000109
indicating that either the k-type transformer or the k-type feeder leg is replaced,
Figure BDA00033476144600001010
indicating that the k-type transformer or the k-type feeder leg is not replaced.
Further, the method for solving the multi-objective function by using the hierarchical optimization method in step S3 includes the steps of:
s31, initializing iterative computation parameters, wherein the iterative computation parameters comprise iteration times n and computation errors epsilon;
s32, dividing the multi-objective function expressed by the formula (1) and the solving constraint conditions expressed by the formulas (12) to (44) into a first sub-problem and a second sub-problem for solving the multi-objective function;
s33, solving the first subproblem to obtain a solution
Figure BDA00033476144600001011
α(n)And
Figure BDA00033476144600001012
x(n)is the whole of the variable stated in parentheses, α(n)For the value of the nth iteration of the introduced continuous variable,
Figure BDA00033476144600001013
representing the value of the 4 th objective function after the nth iteration;
s34, solving the second subproblem to obtain continuous variables in the solution of the multi-objective function, wherein the continuous variables comprise multipliers gamma(n)And
Figure BDA00033476144600001014
γ(n)for the equality constraint x ═ x in the subproblem after the nth iterationnThe corresponding value of the lagrange multiplier,
Figure BDA00033476144600001015
is the value of the objective function a after the nth iteration;
s35, the convergence check is performed on the solution results of the steps S33 and S34,
if the check is passed, outputting the solved optimal solution;
if the check fails, a cut plane constraint of the variable α is constructed, 1 is added to the iteration number, and then the step S33 is returned to.
Further, the first subproblem is expressed by the following formula (45):
Figure BDA0003347614460000111
in the formula (45), γ(m)Representing the second sub-problem coupling equation constraint x ═ x for the mth iteration(m)Lagrange multiplier of (a);
Figure BDA0003347614460000112
representing the value of the objective function a after the mth iteration;
x(m)representing the variables after the m-th iteration
Figure BDA0003347614460000113
The value of (c).
Further, the second subproblem is expressed by the following formula (46):
Figure BDA0003347614460000114
Figure BDA0003347614460000115
further, in step S35, the method for checking convergence of the solution results of step S33 and step S34 includes:
s351, calculating the upper bound of the function values of the multi-target function
Figure BDA0003347614460000116
And lower bound
Figure BDA0003347614460000117
S352, calculating the upper bound
Figure BDA0003347614460000118
And said lower bound
Figure BDA0003347614460000119
A difference of (d);
s353, judging whether the difference value is less than or equal to the calculation error epsilon,
if yes, judging that the convergence check is passed, and outputting the optimal solution of the multi-objective function;
if not, constructing the cut plane constraint of the variable alpha, accumulating 1 for the iteration times, and returning to the step S33 to continue iterative computation.
Further, the upper bound
Figure BDA00033476144600001110
Calculated by the following equation (47):
Figure BDA00033476144600001111
in the formula (47), the first and second groups,
Figure BDA00033476144600001112
representing the value of the objective function a after the nth iteration.
Further, the lower bound
Figure BDA00033476144600001113
Calculated by the following equation (48):
Figure BDA00033476144600001114
in the formula (48), the reaction mixture is,
Figure BDA00033476144600001115
representing the value of the 4 th objective function term after the nth iteration;
α(n)expressed as the value of the nth iteration of the introduced continuous variable.
Further, the method of constructing the cut plane constraint of the variable α is expressed by the following formula (49):
Figure BDA0003347614460000121
in the formula (49), the first and second groups of the compound,
Figure BDA0003347614460000122
representing the value of the objective function a after the nth iteration;
γ(n)representing an equal constraint x ═ x in the second class of subproblems after the nth iterationnA corresponding Lagrangian multiplier value;
x represents a variable
Figure BDA0003347614460000123
x(n)Representing the nth iteration variable
Figure BDA0003347614460000124
The value of (c).
The invention has the following beneficial effects:
1. when a multi-objective function taking the open capacity, the network loss, the operation cost and the transformation cost of the power distribution system as optimization targets is constructed, accurate modeling is carried out on various constraints of transformer replacement, line transformation, distributed resource location and capacity determination, network radiation operation and the like solved by the objective function, and the optimality of the made transformation scheme is effectively improved;
2. the multi-objective function is solved by using a hierarchical optimization method, the constructed large-scale mixed integer linear programming problem is divided into two sub-problems with smaller scales, and the two sub-problems are coordinated by constructing a cutting plane constraint, so that the original problem is accurately and quickly solved. .
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings required to be used in the embodiments of the present invention will be briefly described below. It is obvious that the drawings described below are only some embodiments of the invention, and that for a person skilled in the art, other drawings can be derived from them without inventive effort.
Fig. 1 is a flowchart illustrating an implementation of a multi-objective optimization method for a power distribution network with scalable capacity improvement according to an embodiment of the present invention;
FIG. 2 is a flow chart of a method for solving a multi-objective function in a hierarchical optimization method;
FIG. 3 is a network topology diagram of a power distribution system;
FIG. 4 is a graph of the variation trend of the upper and lower bound errors of a target value during an iterative process;
fig. 5 is a network structure topological diagram of the modified power distribution system and schematic diagrams of installation positions of the static var compensator, the photovoltaic and the wind turbine.
Detailed Description
The technical scheme of the invention is further explained by the specific implementation mode in combination with the attached drawings.
Wherein the showings are for the purpose of illustration only and are shown by way of illustration only and not in actual form, and are not to be construed as limiting the present patent; to better illustrate the embodiments of the present invention, some parts of the drawings may be omitted, enlarged or reduced, and do not represent the size of an actual product; it will be understood by those skilled in the art that certain well-known structures in the drawings and descriptions thereof may be omitted.
The same or similar reference numerals in the drawings of the embodiments of the present invention correspond to the same or similar components; in the description of the present invention, it should be understood that if the terms "upper", "lower", "left", "right", "inner", "outer", etc. are used for indicating the orientation or positional relationship based on the orientation or positional relationship shown in the drawings, it is only for convenience of description and simplification of description, but it is not indicated or implied that the referred device or element must have a specific orientation, be constructed in a specific orientation and be operated, and therefore, the terms describing the positional relationship in the drawings are only used for illustrative purposes and are not to be construed as limitations of the present patent, and the specific meanings of the terms may be understood by those skilled in the art according to specific situations.
In the description of the present invention, unless otherwise explicitly specified or limited, the term "connected" or the like, if appearing to indicate a connection relationship between the components, is to be understood broadly, for example, as being fixed or detachable or integral; can be mechanically or electrically connected; they may be directly connected or indirectly connected through intervening media, or may be connected through one or more other components or may be in an interactive relationship with one another. The specific meanings of the above terms in the present invention can be understood in specific cases to those skilled in the art.
The multi-objective optimization method for the power distribution network capable of improving the open capacity, provided by the embodiment of the invention, is shown in fig. 1, and specifically comprises the following four steps:
1) basic data collection
Respectively acquiring a distribution network model section (such as branch connection relation) and load measurement data (such as node load active power and reactive power) from a distribution network system (such as a distribution network automation master station D5200 system) and a marketing management system; acquiring data (such as information of a transformer, resistance, reactance, length and the like of a line) such as topological connection relation, geographical position parameters, equipment ledger information and the like from a distribution network GIS system; local hourly wind speed and illumination data are obtained from a meteorological database; acquiring economic and technical parameters of equipment such as alternative transformers, overhead lines, cables, switches, fans, photovoltaics, static reactive compensators and the like from an equipment library, wherein the main parameters comprise but are not limited to equipment models, unit manufacturing cost, rated capacity and the like; and issuing questionnaires to operators, planners and technical experts according to the table 1 to obtain the importance ranking of the optimization targets of the three groups of people on the openable capacity, the network loss, the operation cost and the transformation cost.
Figure BDA0003347614460000131
Figure BDA0003347614460000141
TABLE 1
2) Constraint introduction
The method considers 8 constraint conditions for solving the multi-objective function, and respectively comprises transformer replacement constraint, line transformation constraint, distributed resource location and volume constraint, network radiation operation constraint, power balance constraint, node voltage constraint, branch circuit loss monthly flood and branch circuit capacity constraint.
(1) Transformer replacement constraints
Provided that a 1-K type transformer is available for replacement, wherein the capacity and replacement cost of the kth type transformer are respectively
Figure BDA0003347614460000142
Figure BDA0003347614460000143
For the transformer branch (i, j) connecting the node i and the node j of the power distribution network, whether the transformer branch (i, j) is replaced by the k type transformer or not is replaced by a variable of 0-1
Figure BDA0003347614460000144
It is shown that,
Figure BDA0003347614460000145
the method comprises the following steps that (1) the transformer in the transformer branch (i, j) is not replaced, and the original type transformer is still adopted to continue operation;
Figure BDA0003347614460000146
representing the replacement of a transformer in a transformer leg (i, j) with a kth type transformer, the transformer replacement constraint can be expressed by the following equations (1) - (2):
Figure BDA0003347614460000147
Figure BDA0003347614460000148
in the formulae (1) to (2),
Figure BDA0003347614460000149
representing the actual cost of transformation of the transformer branch (i, j);
transformer branch (i, j) is belonged to SV，SVRepresenting a set of transformer legs in a power distribution network.
Formula (1) shows that at most one of K transformers is selected to replace the transformers in the transformer branches (i, j); k is 0 to represent the original type of transformer, and the unit replacement cost of the transformer
Figure BDA00033476144600001410
Is 0.
(2) Line reconstruction constraints
If 1-K feeder branches are provided for replacement, wherein the replacement cost per unit length of the K-th feeder branch is
Figure BDA00033476144600001411
For the feeder branch connecting the node i and the node j, a variable of 0-1 is available for replacing the feeder branch of the k-th type
Figure BDA00033476144600001412
It is shown that,
Figure BDA0003347614460000151
indicates that no replacement is to be made for feeder leg (i, j);
Figure BDA0003347614460000152
representing the replacement of feeder leg (i, j) with a type k feeder leg, the line modification constraint may be expressed by the following equations (3) - (4):
Figure BDA0003347614460000153
Figure BDA0003347614460000154
in formulas (3) to (4), K represents the number of alternative feeder branch types;
k represents a k-th type feeder branch;
Figure BDA0003347614460000155
representing the length of the feeder branch (i, j) e.g. SL，SLThe feeder branch sets in the power distribution network are provided;
Figure BDA0003347614460000156
representing the actual retrofit cost of the feeder branch (i, j).
Formula (3) represents that at most one of K feeder branches is selected to replace the feeder branch (i, j), K is 0 and represents the original feeder line, and the unit replacement cost is low
Figure BDA0003347614460000157
(3) Distributed resource site selection and volume fixing constraint
In this embodiment, the distributed resource location and volume constraints include a static reactive power compensator location and volume constraint, a distributed fan location and volume constraint, and a distributed photovoltaic location and volume constraint, which are explained below.
Location and volume fixing constraint of static reactive compensator
Giving the minimum and maximum access capacity of the static var compensator svc at the node i of the power distribution network, and introducing a variable of 0-1
Figure BDA0003347614460000158
For indicating whether the node i is installed with a static var compensator,
Figure BDA0003347614460000159
indicating that no static var compensator svc is installed at node i;
Figure BDA00033476144600001510
representing the installation of the static var compensator svc at node i, the static var compensator siting capacity constraint can be expressed by the following equations (5) - (6):
Figure BDA00033476144600001511
Figure BDA00033476144600001512
in the formulae (5) to (6),
Figure BDA00033476144600001513
respectively representing the minimum installation capacity and the maximum installation capacity of the static var compensator svc installed at the node i;
Figure BDA00033476144600001514
representing the actual installation capacity of the static var compensator svc at node i;
Csvcthe manufacturing cost of the unit capacity of the static reactive power compensator svc is shown;
Figure BDA00033476144600001515
representing the actual installation cost of the static var compensator svc at node i.
Second, the distributed fan is restricted in location and volume
For the node i, K types of fans can be installed optionally, and when the minimum and maximum installation numbers of the fans at the node i are respectively equal
Figure BDA0003347614460000161
Then, the constraints shown by the following equations (7) to (9) need to be satisfied:
Figure BDA0003347614460000162
Figure BDA0003347614460000163
Figure BDA0003347614460000164
in the formulae (7) to (9),
Figure BDA0003347614460000165
is a variable from 0 to 1, and is,
Figure BDA0003347614460000166
indicating that no fan is installed at node i (fan is indicated by wt);
Figure BDA0003347614460000167
indicating that a fan is installed at the node i; the formula (8) shows that only 1 type fan is allowed to be installed in the K type fans at most so as to avoid the maintenance difficulty caused by the simultaneous installation of multiple types of fans;
Figure BDA0003347614460000168
the rated capacity of the k-type fan is represented;
Figure BDA0003347614460000169
representing the actual installation capacity of the k-type fan at the node i;
Figure BDA00033476144600001610
the unit capacity cost of the k-type fan is represented;
Figure BDA00033476144600001611
representing the total investment cost of the fan at the node i;
k represents the model number of the fan wt;
k represents a k-type fan.
Distributed photovoltaic addressing constant volume constraint
Given minimum and maximum access capacity of the photovoltaic (photovoltaic module) at node i, a 0-1 variable is introduced
Figure BDA00033476144600001612
Is used for indicating whether the node is connected with the photovoltaic,
Figure BDA00033476144600001613
indicating that no photovoltaic is connected at node i;
Figure BDA00033476144600001614
representing the access of photovoltaic at node i, the distributed photovoltaic siting constraint can be expressed by the following equations (10) - (11):
Figure BDA00033476144600001615
Figure BDA00033476144600001616
in the formulae (10) to (11),
Figure BDA00033476144600001617
respectively representing the minimum installation capacity and the maximum installation capacity of the photovoltaic at the node i;
Figure BDA00033476144600001618
the actual photovoltaic installation capacity at the node i;
Cpvthe unit capacity cost for photovoltaic;
Figure BDA00033476144600001619
representing the actual installation cost of the photovoltaic at node i.
(4) Network radiation operation constraints
In order to ensure that the power distribution network always operates in an open loop mode, network radiation operation constraint needs to be introduced, and is expressed by the following formula (12):
Figure BDA0003347614460000171
in the formula (12), the first and second groups,
Figure BDA0003347614460000172
representing the switching state of a branch ij connecting nodes i, j in the distribution network,
Figure BDA0003347614460000173
is a variable from 0 to 1, and is,
Figure BDA0003347614460000174
indicating that the branch ij is open,
Figure BDA0003347614460000175
when it indicates that branch ij is closed;
Nnoderepresenting the number of nodes in the power distribution network;
SBrepresenting sets of branches in a distribution network, SB＝SL∪SV；SLRepresenting a set of feeder branches in the distribution network; sVRepresenting a set of transformer branches in a power distribution network;
formula (12) shows that closed switch quantity equals to distribution network node number and subtracts 1 in whole distribution network, and this restraint can restrict the action of switch in the distribution network, avoids appearing the looped netowrk (the looped netowrk can lead to earth fault electric current too big, causes equipment damage, enlarges the accident range) to the reliability of guarantee distribution system operation.
(5) Power balance constraint
And modeling the power flow distribution of the power distribution network by adopting a linear power flow model. The model can take the influence of factors such as voltage, network loss and line capacity into account, and has the advantages of complete constraint, accurate result and the like while keeping the linearization characteristic of the model. The power balance constraint is expressed by the following equations (13) to (14):
Figure BDA0003347614460000176
Figure BDA0003347614460000177
in the formulae (13) to (14),
Figure BDA0003347614460000178
respectively representing a transformer active output variable, a fan active output variable, a photovoltaic active output variable and an actual measurement value of an active load of a node i of the power distribution network in a time period t;
Figure BDA0003347614460000179
all are constants, and are obtained through actual measurement;
xi represents a load increase multiplying factor variable of the node i, and the variable xi is the ratio of the maximum load to the basic load which can be supplied by the power distribution system and is used for measuring the power supply capacity of the power distribution system, and the larger the variable xi is, the larger the variable xi represents that the openable capacity of the power distribution system is more abundant.
Figure BDA00033476144600001710
Respectively representing the transformer reactive power output variable of the node i, the static reactive power compensator reactive power output variable and the measured value of the reactive load of the node i in the period t;
Figure BDA00033476144600001711
respectively representing the injected active and reactive variables of the node i;
Figure BDA0003347614460000181
subject to constraints shown by the following equations (15) to (16):
Figure BDA0003347614460000182
Figure BDA0003347614460000183
in formulae (15) to (16), Pi，j，k，t、Qi，j，k，tRespectively representing the active power and the reactive power of a k-type feeder branch circuit where the nodes i and j are located in a t period;
Figure BDA0003347614460000184
respectively representing the active loss and the reactive loss of a k-type feeder branch circuit where the nodes i and j are located in a t period;
Figure BDA0003347614460000185
subject to the constraint shown in the following equation (17):
Figure BDA0003347614460000186
in the formula (17), the reaction is carried out,
Figure BDA0003347614460000187
representing the actual installed capacity of the static var compensator svc at node i.
Figure BDA0003347614460000188
Calculated by the following equation (18):
Figure BDA0003347614460000189
in the formula (18), the first and second groups,
Figure BDA00033476144600001810
k type fan representing access node i in t periodThe output active power of (a);
Figure BDA00033476144600001811
calculated by the following equation (19):
Figure BDA00033476144600001812
in the formula (19), vtThe average wind speed is obtained by measuring the average wind speed in the t period;
vk，ci、vk，r、vk，cothe cut-in wind speed, the rated wind speed and the cut-out wind speed of the k-type fan are respectively, the cut-in wind speed represents the minimum wind speed required by the fan for generating power, the rated wind speed represents the corresponding wind speed when the fan generates rated power, and the cut-out wind speed represents that the fan needs to be stopped after the wind speed reaches the value so as to avoid damage to the fan structure caused by overlarge wind speed.
Figure BDA00033476144600001813
Calculated by the following equation (20):
Figure BDA00033476144600001814
in the formula (20), It、θtRespectively representing the average illumination intensity and the illumination incidence angle of the photovoltaic in the t period;
Figure BDA0003347614460000191
representing the actual installation capacity of the photovoltaic at node i;
μpvrepresenting the power temperature coefficient of the photovoltaic module;
Ttthe average working temperature of the photovoltaic at the time t can be replaced by the ambient temperature at the time t;
TSTCis light under standard test conditionsThe module temperature, 25 ℃ in this example, was measured.
(6) Node voltage constraint
The node voltage constraints to be satisfied for solving the multi-objective function are expressed by the following equations (21) to (23):
Figure BDA0003347614460000192
Figure BDA0003347614460000193
Figure BDA0003347614460000194
in the formulae (21) to (23), Ri，j，k、Xi，j，kRespectively representing the resistance and reactance of a k-type feeder branch circuit connected with nodes i and j of the power distribution network;
Pi，j，k，t、Qi，j，k，trespectively representing the active power and the reactive power of a k-type feeder line branch flowing through the connection node i, j at the moment t;
Vi，trepresenting the voltage amplitude of the node i in the period t;
iV、
Figure BDA0003347614460000195
respectively representing the minimum value and the maximum value of the voltage amplitude of the node i;
Vref，texpressing a voltage per unit value of a first node, wherein the first node is a direct connection node of a main network and a distribution network;
V0indicates the default voltage of the head node, in this embodiment, V0The value is 1.0.
It should be noted here that the above-mentioned,
Figure BDA0003347614460000196
as a quadratic term, to ensure that the constraint is linear, it can be treated as a wholeTreatment of variables, e.g. with the symbol V'i，tReplacement of
Figure BDA0003347614460000197
The corresponding constraint is still linear.
(7) Branch loss constraint
The branch loss constraint is expressed by the following equations (24) - (31):
Figure BDA0003347614460000198
Figure BDA0003347614460000199
second order terms in formulas (24) to (25)
Figure BDA00033476144600001910
Can be represented by a linear constraint set of a set of branch flows, namely:
Figure BDA00033476144600001911
Figure BDA0003347614460000201
Figure BDA0003347614460000202
Figure BDA0003347614460000203
Figure BDA0003347614460000204
Figure BDA0003347614460000205
in the formulae (24) to (31),
Figure BDA0003347614460000206
respectively representing the active power loss and the reactive power loss of a k-type feeder branch circuit where the nodes i and j are located in a t period;
Ri，j，k、Xi，j，krespectively representing the resistance and reactance of a k-type feeder branch circuit where the nodes i and j are positioned;
Pi，j，k，t、Qi，j，k，trespectively representing the active power and the reactive power of a k-type feeder branch circuit where the nodes i and j are located in a t period;
Figure BDA0003347614460000207
respectively representing the maximum allowable active power and the maximum allowable reactive power of a k-type feeder branch where the nodes i and j are located;
Nvindicates a section of
Figure BDA0003347614460000208
Or
Figure BDA0003347614460000209
Number of segments linearly equally divided, N in the present embodimentvTaking the value as 5, after segmentation, the quadratic term function
Figure BDA00033476144600002010
Respectively by linear functions Pi，j，k，t、Qi，j，k，tRepresents;
v denotes a linear segment number, v is 1,2, … Nv。
It should be noted here that the above-mentioned,
Figure BDA00033476144600002011
for quadratic terms, to ensure that the constraints are linear, one can combine
Figure BDA00033476144600002012
Or
Figure BDA00033476144600002013
The whole being treated as a variable, e.g. by the symbol P'i，j，k，tReplacement of
Figure BDA00033476144600002014
The corresponding constraint is still linear.
(8) Branch capacity constraint
The branch capacity constraint may be represented by the following equations (32) - (33):
Figure BDA00033476144600002015
Figure BDA00033476144600002016
in formulae (32) to (33), Pi，j，k，t、Qi，j，k，tRespectively representing the active power and the reactive power of a k-type feeder branch circuit where nodes i and j in the power distribution network are located in a t period;
Figure BDA00033476144600002017
representing the switch state of the k-type feeder branch circuit of the node i, j in the distribution network,
Figure BDA00033476144600002018
or 1 of the number of the groups in the group,
Figure BDA00033476144600002019
when the time is short, the k-type feeder branch is disconnected,
Figure BDA0003347614460000211
the time indicates that the k-type feeder branch is closed;
Figure BDA0003347614460000212
indicating the capacity of k-transformers or k-type feeder branches in the distribution network, i.e.
Figure BDA00033476144600002119
Is composed of
Figure BDA0003347614460000214
Or
Figure BDA0003347614460000215
The capacity of the k-transformer is represented,
Figure BDA0003347614460000216
representing the branch capacity of the k-type feeder line;
Figure BDA0003347614460000217
whether a k-type transformer or a k-type feeder branch connecting the node i and the node j is replaced or not is represented;
Figure BDA0003347614460000218
Included
Figure BDA0003347614460000219
and
Figure BDA00033476144600002110
Figure BDA00033476144600002111
indicates whether or not to replace a variable of the k-type transformer,
Figure BDA00033476144600002112
indicating whether a variable is replaced by a type k feeder leg,
Figure BDA00033476144600002113
indicating that a k-type transformer or k-type feeder leg is replaced,
Figure BDA00033476144600002114
indicating that either the k-type transformer or the k-type feeder leg is not replaced.
3) Multi-objective function construction
The goal of upgrading and modifying the distribution network is to maximize the power supply capacity of the distribution system with minimal modification costs, while minimizing network losses and electricity purchase costs. The multi-objective function constructed by the invention is expressed by a formula (34) as follows:
Figure BDA00033476144600002115
in the formula (1), the first and second groups,
Figure BDA00033476144600002116
representing the a-th objective function f in the constructed multi-objective functionaMaking a target function expression after dimensionless processing;
wato represent
Figure BDA00033476144600002117
The corresponding weight;
a＝1，2，3，4。
when a is 1, the objective function fa＝f1Representing an objective function for optimizing the openability of the distribution network, f1Calculated by the following equation (35):
f1becoming-xi formula, 35)
In the formula (35), a variable ξ represents a load increase rate variable of a node i in the power distribution network to be solved, the variable represents the ratio of the maximum load to the base load which can be supplied by the power distribution system and is used for measuring the power supply capacity of the power distribution system, and the larger the variable value is, the more abundant the openable capacity of the power distribution system is.
When a is 2, the objective function fa＝f2Expressing an objective function for optimizing the network loss of the distribution network, f2Calculated by the following equation (36):
Figure BDA00033476144600002118
in the formula (36), SBRepresenting sets of branches in a distribution network, SB＝SL∪SV；SLRepresenting a set of feeder branches in the distribution network; sVRepresenting a set of transformer branches in a power distribution network;
STrepresents a set of optimized periods of each day, ST＝{1，2，…，24}；
k represents a k-type transformer branch or a k-type feeder branch in the power distribution network;
Figure BDA0003347614460000221
and the active loss of a k-type transformer branch or a k-type feeder branch connected with the nodes i and j in a t period is represented.
When a is 3, the objective function fa＝f3An objective function representing the cost of operating an optimized distribution network, f3Calculated by the following equation (37):
Figure BDA0003347614460000222
in the formula (37), ptRepresenting the electricity purchase price of the t period;
Figure BDA0003347614460000223
representing the active output of a transformer of a node i of the power distribution network in a period t;
STrepresenting the set of all periods of each day, STIs 1 hour, i.e., S in the present embodimentTThe number of periods in (1) is 24.
When a is 4, the objective function fa＝f4An objective function representing the cost of modification to optimize the distribution network, f4Calculated by the following equation (38):
Figure BDA0003347614460000224
in the formula (38), the first and second groups,
r represents loan rate;
y represents a return on investment period;
Figure BDA0003347614460000225
representing the actual cost of transformation of a transformer branch (i, j) in the distribution network;
transformer branch (i, j) is belonged to SV，SVRepresenting a set of transformer branches in a power distribution network;
Figure BDA0003347614460000226
representing the actual cost of transformation of the feeder branch (i, j) in the distribution network;
feeder branch (i, j) is belonged to SL，SLRepresenting a set of feeder branches in the distribution network;
Figure BDA0003347614460000227
representing the actual installation cost of the static var compensator svc at node i of the power distribution network;
Figure BDA0003347614460000228
representing the total investment cost of the fan at the node i;
Figure BDA0003347614460000229
representing the actual installation cost of the photovoltaic at node i;
SNrepresenting the set of all nodes.
Due to f1、f2、f3、f4The four target functions are not uniform in unit, and have large order difference, so that the weights of all the target items are difficult to determine, and linear addition cannot be directly carried out. For this purpose, the invention adopts a normalization method, which is toThe four objective functions are mapped to the interval of 0-1, and then the quantitative calculation of the weight is realized in a matrix transformation mode by combining the experiences of technical experts, operators and designers.
(1) Object function mapping
First, using equations (1) to (33) in the embodiment as constraint conditions, and f is used as1、f2、f3、f4Performing single-target optimization on the target to obtain the minimum value and the maximum value of each target item;
secondly, carrying out normalization processing on the formulas (35) to (38) through the following formula (39) to obtain a dimensionless objective function
Figure BDA0003347614460000231
Figure BDA0003347614460000232
In the formula (39), the reaction mixture, af、
Figure BDA0003347614460000233
respectively representing the a-th objective function faMinimum and maximum values of.
(2) Weight quantitative calculation
Respectively forming an importance comparison matrix A for comparing 4 types of optimization targets of technical experts, operators and designers for the openable capacity, the network loss, the operation cost and the transformation cost of the power distribution network in pairs according to the investigation result in the step 1)(u)U-1, 2,3, u-1, u-2, and u-3 respectively represent the relative ranking of importance of technical experts, operators, and designers for the 4 types of optimization targets, a(u)Expressed by the following formula (40):
Figure BDA0003347614460000234
② an importance comparison matrix A is calculated by the following formula (41)(u)The product of each row in (1), namely:
Figure BDA0003347614460000235
in the formula (41), the first and second groups,
Figure BDA0003347614460000236
representing the comparison matrix of importance A(u)The product of the g-th row;
Figure BDA0003347614460000237
representing the comparison matrix of importance A(u)Row g and column h of (1);
h represents the importance comparison matrix A(u)H is 1,2,3, 4;
calculating product by formula (42)
Figure BDA0003347614460000238
4 th order root of (1)
Figure BDA0003347614460000239
Figure BDA00033476144600002310
Fourthly, by the formula (43)
Figure BDA00033476144600002311
Normalization is carried out to obtain a normalization result
Figure BDA00033476144600002312
Figure BDA0003347614460000241
In the formula (43), the first and second groups,
Figure BDA0003347614460000242
expressed in formula (42)
Figure BDA0003347614460000243
h, taking 1,2,3 and 4;
calculated by the formula (44)
Figure BDA0003347614460000244
Corresponding weight wa，
Figure BDA0003347614460000245
In the formula (44), a is g.
4) Hierarchical optimization solution
The equations (1) - (33) and the equation (40) jointly form a multi-objective optimization transformation model of the power distribution network, and the direct solving difficulty is high in consideration of the fact that the model has more constraints and variables, so that the invention adopts a layered optimization technology to solve. The basic idea of the hierarchical optimization solution is to separately calculate 0-1 variables and continuous variables in a multi-objective optimization transformation model, correspondingly obtain a 0-1 planning sub-problem (a first sub-problem) and a linear planning sub-problem (a second sub-problem), then introduce secant plane constraints to coordinate the two sub-problems, and gradually reduce feasible domains in an alternating iteration mode until an optimal solution is obtained, wherein the calculation flow is shown in fig. 2:
(1) initializing iterative computation parameters, wherein the iterative computation parameters comprise iteration times n and a computation error epsilon;
(2) dividing the multi-objective function represented by the formula (34) and the solution constraint conditions represented by the formulas (1) to (33) into a first sub-problem and a second sub-problem for solving the multi-objective function, and then calculating the first sub-problem to obtain a solution
Figure BDA0003347614460000246
α(n)And
Figure BDA00033476144600002411
x(n)is an integer of the variable stated in parenthesesBody, alpha(n)For the value of the nth iteration of the continuous variable introduced (to construct the cut plane constraint),
Figure BDA0003347614460000248
representing the value of the 4 th objective function after the nth iteration;
the calculation process of the first subproblem can be expressed by the following formula (45):
Figure BDA0003347614460000249
in the formula (45), alpha is an introduced variable, the constraint corresponding to alpha is a secant plane constraint, and gamma is(m)Coupling equation constraint x ═ x representing the second subproblem of the mth iteration(m)Lagrange multiplier of (a);
(3) the second subproblem is computed, resulting in the constraint x-x on the equationnMultiplier gamma of(n)And
Figure BDA00033476144600002410
and the calculation process of the second sub-problem is expressed by the following formula (46):
Figure BDA0003347614460000251
Figure BDA0003347614460000252
γ(n)representing the constraint x-x obtained after solving the second subproblem in the nth iterationnLagrange multiplier values of;
Figure BDA0003347614460000253
is the value of the objective function a after the nth iteration.
(4) And (6) checking convergence. The upper bound of the objective function value is calculated by the following equations (47) to (48)
Figure BDA0003347614460000254
And lower bound
Figure BDA0003347614460000255
If it is
Figure BDA0003347614460000256
If yes, calculating convergence and outputting an optimal solution; otherwise, constructing a cut plane constraint of the variable alpha, accumulating 1 for the iteration number (n is n +1), and returning to the step (2) to continue the calculation.
Figure BDA0003347614460000257
In the formula (47), the first and second groups,
Figure BDA0003347614460000258
representing the value of the objective function a after the nth iteration.
Figure BDA0003347614460000259
In the formula (48), the reaction mixture is,
Figure BDA00033476144600002510
representing the value of the 4 th objective function term after the nth iteration;
α(n)expressed as the value of the nth iteration of the introduced continuous variable.
The cut plane constraint construction method of the variable α is expressed by the following formula (49):
Figure BDA00033476144600002511
in the formula (49), the first and second groups of the compound,
Figure BDA00033476144600002512
representing the value of the objective function a after the nth iteration;
γ(n)representing second after the nth iterationEquality constraint x ═ x in class problemnA corresponding Lagrangian multiplier value;
x represents a variable
Figure BDA00033476144600002513
x(n)Representing the nth iteration variable
Figure BDA00033476144600002514
The value of (c).
The multi-objective optimization method for the power distribution network facing the openable capacity improvement, provided by the invention, is further explained in combination with a specific application scene.
Fig. 3 is a network topology diagram of a power distribution system. The numerical value "0-33" in fig. 3 is the node number in the power distribution system, the node 0 is the first node (the connection node between the main network and the distribution network), the solid line between the nodes represents the line branch, the dotted line represents the interconnection switch, and the double-circle symbol between the node 0 and the node 1 is the transformer. Aiming at the power distribution system, the invention provides a power distribution network multi-objective optimization method for improving the openable capacity, and the implementation process comprises the following steps:
1) collecting reactive load of a power distribution system, technical and economic parameters of a transformer and a line ledger (resistance, reactance, capacity, line length and the like), hourly wind speed of an area where the power distribution system is located, illumination, alternative equipment (an alternative transformer and the like), and importance ranking results of technical experts, operators and designers for four types of optimization targets;
2) constructing constraint conditions according to the collected data and equations (1) - (33) in the detailed description section of the specification, including transformer replacement constraint, line modification constraint, distributed resource location capacity constraint, network radiated operation constraint, power balance constraint, node voltage constraint, branch loss and capacity constraint;
3) respectively taking equations (35), (36), (37) and (38) in the detailed embodiment of the specification as single targets, solving the optimization problem containing the constraint equations (1) to (33) to obtain the upper limit and the lower limit of each objective function, and then performing linear mapping on each objective function according to an equation (39) to obtain the expression of each mapped objective function as follows:
Figure BDA0003347614460000261
Figure BDA0003347614460000262
Figure BDA0003347614460000263
Figure BDA0003347614460000264
according to the investigation result in 1), the importance ranking matrixes of technical experts, operators and designers are respectively obtained as follows:
Figure BDA0003347614460000265
Figure BDA0003347614460000266
Figure BDA0003347614460000267
according to equations (41) - (44), we obtain:
Figure BDA0003347614460000268
Figure BDA0003347614460000271
Figure BDA0003347614460000272
Figure BDA0003347614460000273
Figure BDA0003347614460000274
from equation (34), a solution for the multi-objective function is obtained:
min f＝0.404427f1+1.607043×10-7f2+1.803653×10-7f3+3.654271×10-7f4
4) and (3) dividing the constraint conditions and the multi-objective functions in the above 2) and 3) according to formulas (45) - (46), namely substituting variables, constraint conditions and objective functions belonging to the formula (45) into formula (45), substituting variables, constraint conditions and objective functions belonging to the formula (46) into formula (46) to obtain two corresponding sub-problems, and respectively solving by adopting a branch-and-bound algorithm and an interior point method. After the calculation is finished, solving the upper bound of the objective function value according to the formulas (47) - (48)
Figure BDA0003347614460000275
And lower bound
Figure BDA0003347614460000276
And judging convergence conditions
Figure BDA0003347614460000277
If yes, outputting an optimal solution, and quitting the calculation process; if not, forming a cut plane constraint of the variable alpha, and recalculating the two subproblems. Fig. 4 shows the variation trend of the upper and lower bound errors of the target value in the iterative process, and as can be seen from fig. 4, the upper and lower bound errors become smaller and smaller with the increase of the iterative computation times, which indicates that the solution method for dividing the constraint condition and the multi-objective function into two sub-problems is effective.
Fig. 5 shows a schematic diagram of the modified power distribution network topology structure and the installation positions of the static var compensator, the photovoltaic and the wind turbine. The capacities of the static var compensators of the access nodes 15, 30 in fig. 5 are 342kVar and 716kVar, respectively, the photovoltaic capacity of the access node 24 is 650kW, and the fan capacity of the access node 31 is 420 kW. The dotted line in fig. 5 shows the replacement and modification of the original transformer and lines. The capacity of the transformer is increased to 7500kVA from 5000kVA, and the capacity of the line is increased to 375A from 265A. After capacity expansion modification, the value of a load increase factor (namely a load increase rate variable) ξ of the power distribution system is increased from 1.0653 to 1.6029, and the annual modification cost is converted to 50.78 ten thousand yuan.
To sum up, the method is based on strict mathematical optimization theory and various collected basic data, accurate modeling is carried out on transformer replacement, line transformation, distributed resource location and sizing and other various operation constraints, and meanwhile, the influence of factors such as voltage, network loss and line capacity is considered; then, by combining collective wisdom of technical experts, operators and designers, the construction of a multi-target function with the openable capacity, the network loss, the operation cost and the transformation cost as optimization targets is realized by using a mode of function mapping and matrix transformation, and the problem of high difficulty in weight calculation caused by the inconsistency of various dimensions and magnitude orders of the targets is effectively solved; and finally, dividing the established large-scale mixed integer linear programming problem into two sub-problems with smaller scale by adopting a layered solving and interactive coordination mode, and coordinating the two sub-problems by constructing a secant plane constraint, thereby realizing the accurate and fast solving of the original problem.
It should be understood that the above-described embodiments are merely preferred embodiments of the invention and the technical principles applied thereto. It will be understood by those skilled in the art that various modifications, equivalents, changes, and the like can be made to the present invention. However, such variations are within the scope of the invention as long as they do not depart from the spirit of the invention. In addition, certain terms used in the specification and claims of the present application are not limiting, but are used merely for convenience of description.

Claims (28)

1. A multi-objective optimization method for a power distribution network with an openable capacity improvement function is characterized by comprising the following steps:
s1, collecting basic data required by multi-objective optimization of the power distribution network;
s2, constructing a multi-objective function taking the openable capacity of the power distribution network, the network loss, the operation cost and the transformation cost as optimization targets;
and S3, solving the multi-objective function by using a hierarchical optimization method with the constraints of transformer replacement, line reconstruction, distributed resource location and capacity determination, network radiation operation, power balance, node voltage, branch loss and branch capacity to obtain an optimal solution of the multi-objective function.
2. The multi-objective optimization method for the power distribution network capable of realizing the capacity increase according to claim 1, wherein the basic data in the step S1 comprises active load and reactive load of the power distribution network; resistance, reactance, capacity of the transformer and distribution lines; the length of the distribution line; hourly wind speed and illumination of the area where the power distribution system is located; the technical and economic parameters of the alternative equipment and the importance ranking of the technical experts, operating personnel and involved personnel on the category 4 optimization objectives of the openable capacity, the network loss, the operating costs, the retrofit costs.
3. The multi-objective optimization method for the power distribution network capable of realizing capacity increase according to claim 1, wherein the multi-objective function constructed in the step S2 is expressed by the following formula (1):
Figure FDA0003347614450000011
in the formula (1), the first and second groups,
Figure FDA0003347614450000012
representation pair constructedThe a-th objective function f in the multi-objective functionaMaking a target function expression after dimensionless processing;
wato represent
Figure FDA0003347614450000013
The corresponding weight;
a＝1,2,3,4。
4. the multi-objective optimization method for the power distribution network with the openable capacity being improved according to claim 3,
Figure FDA0003347614450000014
calculated by the following formula (2):
Figure FDA0003347614450000015
in the formula (2), the first and second groups,f a、
Figure FDA0003347614450000016
respectively representing the a-th objective function faMinimum and maximum values of.
5. The multi-objective optimization method for distribution network capable of achieving capacity improvement according to claim 3, wherein calculation is carried out
Figure FDA0003347614450000017
Corresponding weight waThe method comprises the following steps:
l1, forming an importance comparison matrix A for pairwise comparison of 4 types of optimization targets of the openable capacity, the network loss, the operation cost and the transformation cost of the power distribution network(u)U-1, 2,3, u-1, u-2, and u-3 respectively represent the relative ranking of importance of technical experts, operators, and designers for the 4 types of optimization targets, a(u)Expressed by the following formula (3):
Figure FDA0003347614450000021
l2, calculating the importance comparison matrix A by the following formula (4)(u)The product of each of the rows in the column,
Figure FDA0003347614450000022
in the formula (4), the first and second groups,
Figure FDA0003347614450000023
representing the comparison matrix of importance A(u)The product of the g-th row;
Figure FDA0003347614450000024
representing the comparison matrix of importance A(u)Row g and column h of (1);
h represents the importance comparison matrix A(u)H is 1,2,3, 4;
l3, calculating the product by the following formula (5)
Figure FDA0003347614450000025
4 th order root of (1)
Figure FDA0003347614450000026
Figure FDA0003347614450000027
L4, by the following equation (6) pair
Figure FDA0003347614450000028
Normalization is carried out to obtain a normalization result
Figure FDA0003347614450000029
Figure FDA00033476144500000210
In the formula (6), the first and second groups,
Figure FDA00033476144500000211
expressed in equation (5)
Figure FDA00033476144500000212
h, taking 1,2,3 and 4;
l5 calculated by the following formula (7)
Figure FDA00033476144500000213
Corresponding weight wa：
Figure FDA00033476144500000214
In formula (7), a is g.
6. The multi-objective optimization method for power distribution network with openable capacity being improved according to claim 3, wherein when a is 1, the objective function f isa＝f1Representing an objective function for optimizing the openability of the distribution network, f1Calculated by the following equation (8):
f1becoming-xi formula (8)
In the formula (8), ξ represents the load increase factor variable of the node i in the power distribution network, and is used for indicating the increase factor relative to the original ground state load.
7. The multi-objective optimization method for power distribution network with openable capacity being improved according to claim 3, wherein when a is 2, the objective function f isa＝f2Expressing an objective function for optimizing the network loss of the distribution network, f2Calculated by the following equation (9):
Figure FDA0003347614450000031
in formula (9), SBRepresenting sets of branches in a distribution network, SB＝SL∪SV；SLRepresenting a set of feeder branches in the distribution network; sVRepresenting a set of transformer branches in a power distribution network;
STrepresents the set of all optimization periods, S, of each dayT＝{1，2，…，24}；
k represents a k-type transformer branch or a k-type feeder branch in the power distribution network;
Figure FDA0003347614450000032
and the active loss of a k-type transformer branch or a k-type feeder branch connected with the nodes i and j in a t period is represented.
8. The multi-objective optimization method for power distribution network with openable capacity being improved according to claim 3, wherein when a is 3, the objective function f isa＝f3An objective function representing the cost of operating an optimized distribution network, f3Calculated by the following equation (10):
Figure FDA0003347614450000033
in the formula (10), ptRepresenting the electricity purchase price of the t period;
Figure FDA0003347614450000034
representing the active output of a transformer of a node i of the power distribution network in a period t;
STeach representsSet of all optimization periods of the day, ST＝{1，2，…，24}。
9. The multi-objective optimization method for power distribution network with openable capacity being improved according to claim 3, wherein when a is 4, the objective function f isa＝f4An objective function representing the cost of modification to optimize the distribution network, f4Calculated by the following equation (11):
Figure FDA0003347614450000035
in the formula (11), the reaction mixture,
r represents loan rate;
y represents a return on investment period;
Figure FDA0003347614450000036
representing the actual cost of transformation of a transformer branch (i, j) in the distribution network;
the branch (i, j) of the transformer belongs to SV，SVRepresenting a set of transformer branches in a power distribution network;
Figure FDA0003347614450000037
representing the actual cost of transformation of the feeder branch (i, j) in the distribution network;
the feeder branch (i, j) belongs to SL，SLRepresenting a set of feeder branches in the distribution network;
Figure FDA0003347614450000041
representing the actual installation cost of the static var compensator svc at node i of the power distribution network;
Figure FDA0003347614450000042
presentation instrumentThe total investment cost of the fan at the node i is reduced;
Figure FDA0003347614450000043
representing the actual installation cost of the photovoltaic at the node i;
SNrepresenting the set of all nodes.
10. The multi-objective optimization method for distribution network with scalable capacity boost as claimed in claim 3, wherein the transformer replacement constraint in step S3 is expressed by the following equations (12) - (13):
Figure FDA0003347614450000044
Figure FDA0003347614450000045
in equations (12) to (13), K represents the number of types of alternative transformers;
k represents a kth type transformer;
Figure FDA0003347614450000046
indicating whether the transformer branch (i, j) connecting the node i and the node j of the distribution network is replaced by a kth type transformer,
Figure FDA0003347614450000047
or a combination of the values of 0,
Figure FDA0003347614450000048
indicates that no replacement is to be performed on the transformer in the transformer branch (i, j);
Figure FDA0003347614450000049
representing the transformers in the transformer branches (i, j)Replacing with a k-th type transformer;
formula (12) shows that at most one of K transformers is selected to replace the transformer in the transformer branch (i, j);
Figure FDA00033476144500000410
representing the replacement cost of the kth type transformer;
Figure FDA00033476144500000411
the capacity of a kth type transformer is shown;
Figure FDA00033476144500000412
representing the actual cost of retrofitting of the transformer branch (i, j);
the branch (i, j) of the transformer belongs to SV，SVRepresenting a set of transformer legs in a power distribution network.
11. The method for multi-objective optimization of power distribution network oriented to openable capacity boost according to claim 10, wherein the line transformation constraints in step S3 are expressed by the following equations (14) - (15):
Figure FDA00033476144500000413
Figure FDA00033476144500000414
in equations (14) - (15), K represents the number of alternative feeder branch types;
k represents a k-th type feeder branch;
Figure FDA0003347614450000051
indicating whether the feeder branch connecting distribution network node i and node j is replaced by a type k feeder branch,
Figure FDA0003347614450000052
Figure FDA0003347614450000053
indicates that no replacement is to be made for feeder leg (i, j);
Figure FDA0003347614450000054
representing the replacement of feeder leg (i, j) with a type k feeder leg:
formula (14) represents that at most one of K feeder branches is selected to replace the feeder branch (i, j);
Figure FDA0003347614450000055
representing the replacement cost per unit length of the k-th type feeder branch;
Figure FDA0003347614450000056
representing a length of the feeder branch (i, j) e SL，SLThe feeder branch sets in the power distribution network are provided;
Figure FDA0003347614450000057
representing the actual cost of retrofitting of the feeder branch (i, j).
12. The multi-objective optimization method for the power distribution network capable of increasing the open capacity according to claim 11, wherein the distributed resource siting constraints in step S3 include static reactive power compensator siting constraints, distributed fan siting constraints, and distributed photovoltaic siting constraints, and the static reactive power compensator siting constraints are expressed by the following equations (16) to (17):
Figure FDA0003347614450000058
Figure FDA0003347614450000059
in the formulae (16) to (17),
Figure FDA00033476144500000510
indicating whether a static var compensator svc is installed at the node i of the power distribution network,
Figure FDA00033476144500000511
Figure FDA00033476144500000512
Figure FDA00033476144500000513
indicating that the static var compensator svc is not installed at the node f;
Figure FDA00033476144500000514
represents the installation of the static var compensator svc at the node i;
Figure FDA00033476144500000515
respectively representing the minimum installation capacity and the maximum installation capacity of the static var compensator svc installed at the node i;
Figure FDA00033476144500000516
representing the actual installation capacity of the static var compensator svc at the node i;
Csvcindicating said static var compensatorCost per unit volume of svc;
Figure FDA00033476144500000517
represents the actual installation cost of the static var compensator svc at the node i.
13. The openably capacity-scalable-boost-oriented power distribution network multiobjective optimization method according to claim 12, wherein the distributed fan siting capacity constraint is expressed by the following equations (18) to (20):
Figure FDA00033476144500000518
Figure FDA00033476144500000519
Figure FDA0003347614450000061
in the formulae (18) to (20),
Figure FDA0003347614450000062
Figure FDA0003347614450000063
indicating that no fan wt is installed at the node i;
Figure FDA0003347614450000064
indicating that a fan wt is installed at the node i; the formula (19) shows that only 1 type fan in the K type fans is allowed to be installed at most;
Figure FDA0003347614450000065
are respectively shown inThe minimum number and the maximum number of the k-type fans are installed at the node i;
Figure FDA0003347614450000066
the rated capacity of the k-type fan is represented;
Figure FDA0003347614450000067
representing an actual installed capacity of the k-type fan at the node i;
Figure FDA0003347614450000068
expressing the unit capacity cost of the k-type fan;
Figure FDA0003347614450000069
representing the total investment cost of the fan at the node i;
k represents the model number of the fan wt;
k represents the k-type fan.
14. The openably capacity-scalable-boost-oriented power distribution network multiobjective optimization method according to claim 13, wherein the distributed photovoltaic siting constraint is expressed by the following equations (21) to (22):
Figure FDA00033476144500000610
Figure FDA00033476144500000611
in the formulae (21) to (22),
Figure FDA00033476144500000612
Figure FDA00033476144500000613
means that no photovoltaic is connected at said node i;
Figure FDA00033476144500000614
represents the photovoltaic access at the node i;
Figure FDA00033476144500000615
respectively representing the minimum installation capacity and the maximum installation capacity of the photovoltaic at the node i;
Figure FDA00033476144500000616
the actual photovoltaic installation capacity at the node i is obtained;
Cpvthe unit capacity cost for photovoltaic;
Figure FDA00033476144500000617
representing the actual installation cost of the photovoltaic at the node i.
15. The multi-objective optimization method for distribution network with scalable capacity boost as claimed in claim 14, wherein the network radiated operation constraint in step S3 is expressed by the following equation (23):
Figure FDA00033476144500000618
in the formula (23), the first and second groups,
Figure FDA00033476144500000619
representing the switching state of a branch ij connecting nodes i, j in the distribution network,
Figure FDA00033476144500000620
Figure FDA00033476144500000621
it means that said branch ij is open,
Figure FDA00033476144500000622
when indicates that the branch ij is closed;
Nnoderepresenting the number of nodes in the power distribution network;
SBrepresenting sets of branches in a distribution network, SB＝SL∪SV；SLRepresenting a set of feeder branches in the distribution network; sVRepresenting a set of transformer branches in a power distribution network;
i. j denotes a tributary node.
16. The method for multi-objective optimization of power distribution network oriented to openable capacity boost according to claim 15, wherein the power balance constraint in step S3 is expressed by the following equations (24) - (25):
Figure FDA0003347614450000071
Figure FDA0003347614450000072
in the formulae (24) to (25),
Figure FDA0003347614450000073
respectively representing a transformer active output variable, a fan active output variable, a photovoltaic active output variable and an active load measured value of a node i of a power distribution network in a period t;
ξ represents the load increase magnification variable of the node i;
Figure FDA0003347614450000074
respectively representing a transformer reactive power output variable of the node i, a static reactive power compensator reactive power output variable and a reactive load measured value of the node i in a period t;
Figure FDA0003347614450000075
respectively representing the injected active and reactive variables of the node i;
Figure FDA0003347614450000076
subject to constraints shown by the following equations (26) to (27):
Figure FDA0003347614450000077
Figure FDA0003347614450000078
in formulae (26) to (27), Pi，j，k，t、Qi，j，k，tRespectively representing the active power and the reactive power of a k-type feeder branch circuit where the nodes i and j are located in a t period;
Figure FDA0003347614450000079
respectively representing the active loss and the reactive loss of the k-type feeder branch circuit where the nodes i and j are located in the period t;
Figure FDA00033476144500000710
subject to the constraint shown in equation (28) below:
Figure FDA00033476144500000711
in the formula (28), the first and second groups,
Figure FDA00033476144500000712
representing the actual installed capacity of the static var compensator svc at said node i.
17. The multi-objective optimization method for distribution network with scalable capacity boost as claimed in claim 16,
Figure FDA0003347614450000081
calculated by the following equation (29):
Figure FDA0003347614450000082
in the formula (29), the reaction is carried out,
Figure FDA0003347614450000083
representing the output active power of a k-type fan accessed to the node i in a t period;
Figure FDA0003347614450000084
calculated by the following equation (30):
Figure FDA0003347614450000085
in the formula (30), vtThe average wind speed is obtained by measuring the average wind speed in the t period;
vk，ci、vk，r、vk，cothe cut-in wind speed, the rated wind speed and the cut-out wind speed of the k-type fan are respectively.
18. The multi-objective optimization method for distribution network with scalable capacity boost as claimed in claim 17,
Figure FDA0003347614450000086
calculated by the following equation (31):
Figure FDA0003347614450000087
in formula (31), It、θtRespectively representing the average illumination intensity and the illumination incidence angle of the photovoltaic in the t period;
Figure FDA0003347614450000088
representing the actual installation capacity of the photovoltaic at the node i;
μpvrepresenting the power temperature coefficient of the photovoltaic module;
Ttthe average working temperature of the photovoltaic is t period;
TSTCis the photovoltaic module temperature under standard test conditions.
19. The multi-objective optimization method for distribution network with scalable capacity boost as claimed in claim 18, wherein the node voltage constraint in step S3 is expressed by the following equations (32) - (34):
Figure FDA0003347614450000089
Figure FDA00033476144500000810
Figure FDA00033476144500000811
in the formulae (32) to (34), Ri，j，k、Xi，j，kRespectively representing connected distribution network sectionsThe resistance and reactance of the k-type feeder branch at the point i, j;
Pi，j，k，t、Qi，j，k，trespectively representing the active power and the reactive power of the k-type feeder line branch flowing through the connection node i, j at the moment t;
Vi，trepresenting the voltage amplitude of the node i in a period t;
iV、
Figure FDA0003347614450000091
respectively representing the minimum voltage amplitude value and the maximum voltage amplitude value of the node i;
Vref，trepresenting a voltage per unit value of a first node, wherein the first node is a direct connection node of a main network and a distribution network;
V0representing a default value of the voltage of the head node.
20. The method for multi-objective optimization of power distribution network oriented to openable capacity boost according to claim 19, wherein the branch loss constraint in step S3 is expressed by the following equations (35) - (42):
Figure FDA0003347614450000092
Figure FDA0003347614450000093
Figure FDA0003347614450000094
Figure FDA0003347614450000095
Figure FDA0003347614450000096
Figure FDA0003347614450000097
Figure FDA0003347614450000098
Figure FDA0003347614450000099
in the formulae (35) to (42),
Figure FDA00033476144500000910
respectively representing the active loss and the reactive loss of a k-type feeder branch circuit where the nodes i and j are located in a t period;
Ri，j，k、Xi，j，krespectively representing the resistance and reactance of the k-type feeder branch circuit where the nodes i and j are located;
Pi，j，k，t、Qi，j，k，trespectively representing the active power and the reactive power of the k-type feeder branch circuit where the nodes i and j are located in a t period;
Figure FDA00033476144500000911
respectively representing the maximum allowable active power and the maximum allowable reactive power of the k-type feeder branch circuit where the nodes i and j are located;
Nvindicates a section of
Figure FDA00033476144500000912
Or
Figure FDA00033476144500000913
Number of segments to be linearly equally divided;
v denotes a linear segment number, v is 1,2, … Nv。
21. The method for multi-objective optimization of power distribution network oriented to openable capacity boost according to claim 20, wherein the branch capacity constraint in step S3 is expressed by the following equations (43) - (44):
Figure FDA0003347614450000101
Figure FDA0003347614450000102
in formulae (43) to (44), Pi，j，k，t、Qi，j，k，tRespectively representing the active power and the reactive power of a k-type feeder branch circuit where nodes i and j in the power distribution network are located in a t period;
Figure FDA0003347614450000103
representing the switching state of the k-type feeder branch in which the node i, j is located in the distribution network,
Figure FDA0003347614450000104
Figure FDA0003347614450000105
for a line indicating a disconnection of the k-type feeder branch,
Figure FDA0003347614450000106
when the k-type feeder branch is closed, the k-type feeder branch is closed;
Figure FDA0003347614450000107
representing the capacity of a k-type transformer or the k-type feeder branch in the power distribution network;
Figure FDA0003347614450000108
is a variable from 0 to 1, indicates whether a k-type transformer connecting node i and node j or the k-type feeder branch is replaced,
Figure FDA0003347614450000109
indicating that either the k-type transformer or the k-type feeder leg is replaced,
Figure FDA00033476144500001010
indicating that the k-type transformer or the k-type feeder leg is not replaced.
22. The multi-objective optimization method for distribution networks with scalable capacity boost according to claim 21, wherein the method for solving the multi-objective function by using the hierarchical optimization method in step S3 comprises the steps of:
s31, initializing iterative computation parameters, wherein the iterative computation parameters comprise iteration times n and computation errors epsilon;
s32, dividing the multi-objective function expressed by the formula (1) and the solving constraint conditions expressed by the formulas (12) to (44) into a first sub-problem and a second sub-problem for solving the multi-objective function;
s33, solving the first subproblem to obtain a solution
Figure FDA00033476144500001011
α(n)And
Figure FDA00033476144500001012
x(n)is the whole of the variable stated in parentheses, α(n)For the value of the nth iteration of the introduced continuous variable,
Figure FDA00033476144500001013
representing the value of the 4 th objective function after the nth iteration;
s34, solving the second subproblem to obtain continuous variables in the solution of the multi-objective function, wherein the continuous variables comprise multipliers gamma(n)And
Figure FDA00033476144500001014
γ(n)for the equality constraint x ═ x in the subproblem after the nth iterationnThe corresponding value of the lagrange multiplier,
Figure FDA00033476144500001015
is the value of the objective function a after the nth iteration;
s35, the convergence check is performed on the solution results of the steps S33 and S34,
if the check is passed, outputting the solved optimal solution;
if the check fails, a cut plane constraint of the variable α is constructed, 1 is added to the iteration number, and then the step S33 is returned to.
23. The multi-objective optimization method for distribution network with scalable capacity boost capability according to claim 22, wherein the first sub-problem is expressed by the following formula (45):
Figure FDA0003347614450000111
in the formula (45), γ(m)Representing the second sub-problem coupling equation constraint x ═ x for the mth iteration(m)Lagrange multiplier of (a);
Figure FDA0003347614450000112
representing the value of the objective function a after the mth iteration;
x(m)representing the variables after the m-th iteration
Figure FDA0003347614450000113
The value of (c).
24. The multi-objective optimization method for distribution network with scalable capacity boost capability according to claim 22, wherein the second sub-problem is expressed by the following formula (46):
Figure FDA0003347614450000114
25. the multi-objective optimization method for distribution networks with scalable capacity improvement according to claim 22, wherein in step S35, the method for checking convergence of the solution results of steps S33 and S34 comprises:
s351, calculating the upper bound of the function values of the multi-target function
Figure FDA0003347614450000115
And lower bound
Figure FDA0003347614450000116
S352, calculating the upper bound
Figure FDA0003347614450000117
And said lower bound
Figure FDA0003347614450000118
A difference of (d);
s353, judging whether the difference value is less than or equal to the calculation error epsilon,
if yes, judging that the convergence check is passed, and outputting the optimal solution of the multi-objective function;
if not, constructing the cut plane constraint of the variable alpha, accumulating 1 for the iteration times, and returning to the step S33 to continue iterative computation.
26. Multi-objective optimization method for power distribution network capable of achieving openable capacity improvement according to claim 25Method, characterized in that said upper bound
Figure FDA0003347614450000119
Calculated by the following equation (47):
Figure FDA00033476144500001110
in the formula (47), the first and second groups,
Figure FDA0003347614450000121
representing the value of the objective function a after the nth iteration.
27. The multi-objective optimization method for distribution network with scalable capacity boost as claimed in claim 25 or 26, wherein the lower bound is
Figure FDA0003347614450000122
Calculated by the following equation (48):
Figure FDA0003347614450000123
in the formula (48), the reaction mixture is,
Figure FDA0003347614450000124
representing the value of the 4 th objective function term after the nth iteration;
α(n)expressed as the value of the nth iteration of the introduced continuous variable.
28. The openably capacity-scalable-boost-oriented power distribution network multi-objective optimization method according to claim 25 or 26, wherein the cut plane constraint construction method of the variable α is expressed by the following formula (49):
Figure FDA0003347614450000125
in the formula (49), the first and second groups of the compound,
Figure FDA0003347614450000126
representing the value of the objective function a after the nth iteration;
γ(n)representing an equal constraint x ═ x in the second class of subproblems after the nth iterationnA corresponding Lagrangian multiplier value;
x represents a variable
Figure FDA0003347614450000127
x(n)Representing the nth iteration variable
Figure FDA0003347614450000128
The value of (c).
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