CN113394771B - Method for solving feasible domain of nonlinear power system tie line - Google Patents

Method for solving feasible domain of nonlinear power system tie line Download PDF

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CN113394771B
CN113394771B CN202110630299.9A CN202110630299A CN113394771B CN 113394771 B CN113394771 B CN 113394771B CN 202110630299 A CN202110630299 A CN 202110630299A CN 113394771 B CN113394771 B CN 113394771B
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代伟
简江艺
王帅
赵静怡
石博臣
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    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
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Abstract

The invention discloses a method for solving a feasible domain of a nonlinear power system tie line, which comprises the following steps: 1) Establishing constraint conditions for calculating the feasible domain of the tie line; 2) Establishing a boundary set V; 3) Establishing a polyhedron R based on the boundary set; 4) At least one face of the polyhedron R is translated to obtain a plurality of new boundary points meeting constraint conditions, and the new boundaries are written into the boundary set V to obtain a new boundary set V new (ii) a 5) Based on new boundary set V new Establishing a new polyhedron R new (ii) a 6) Judging polyhedron R and new polyhedron R new Whether the difference between the two is less than a preset condition or not, if so, obtaining a feasible region R of the power system tie line new Otherwise, updating the boundary set V = V new Update polyhedron R = R new And returns to step 4). The method can effectively approximate the prior tie line feasible region, approximate the prior nonlinear feasible region into the linear feasible region, and has small calculated amount and higher precision.

Description

Method for solving feasible domain of nonlinear power system tie line
Technical Field
The invention relates to the field of nonlinear power system calculation, in particular to a method for solving a feasible domain of a nonlinear power system tie line.
Background
With the ever increasing demand for electricity and renewable energy, the efficient use of electricity resources in regional power grids has increasingly relied on the exchange of electricity by junctors in interconnected power grids. The feasible region is a space where the system can stably operate under operation constraint and safety constraint, and the optimality and safety of power dispatching are guaranteed through accurate description of the feasible region of the tie line. However, the crossline feasible domain is often complex due to the presence of non-linear constraints. The tie-line power determination is not properly going to violate grid operating constraints, which include the crew contribution constraint and the voltage magnitude constraint. Therefore, in order to enable the tie to normally perform power exchange under the constraint, it is necessary to determine the feasible range of the tie. Although many methods are available for establishing a feasible domain solution model, the method has more or less limitations. For example, a network flow method ignores the load flow balance constraint of a regional power grid; the breadth-first progressive vertex algorithm, while delineating the boundaries of the feasible region, is limited by ignoring reactive and voltage.
Disclosure of Invention
The invention aims to provide a method for solving a feasible domain of a nonlinear power system tie line, which comprises the following steps:
1) Constraints for tie-line feasible domain calculations are established.
The constraints include power balance constraints, transmission constraints, generator power constraints, tie-line power thermal limits.
The power balance constraint comprises active balance and reactive balance regional power system power flow constraint, namely:
Figure BDA0003103172160000011
Figure BDA0003103172160000012
in the formula (I), the compound is shown in the specification,
Figure BDA0003103172160000013
is a collection of buses in the electrical grid. P is L，i 、P B，i And P G，i Respectively the active load power, the active tie line power and the active power generation level at the bus i. Q L，i 、Q B，i And Q G，i Respectively the reactive load power, the reactive tie line power and the reactive power generation level at the bus i. V i And V j The voltage values at the bus i and the bus j are respectively. G ij And B ij Are respectively provided withThe conductance and admittance of the bus i-j branch. Theta ij Is the difference in voltage phase angle at the bus i and j.
The transmission constraints are as follows:
Figure BDA0003103172160000014
Figure BDA0003103172160000021
Figure BDA0003103172160000022
in the formula, P ij And Q ij The active power and the reactive power of the branch circuits connected with the i bus and the j bus are respectively. S ij Is the apparent power of the corresponding branch.
The generator power constraint is as follows:
Figure BDA0003103172160000023
Figure BDA0003103172160000024
in the formula (I), the compound is shown in the specification,
Figure BDA0003103172160000025
and
Figure BDA0003103172160000026
respectively active power generation level P G Upper and lower bounds.
Figure BDA0003103172160000027
And
Figure BDA0003103172160000028
respectively a reactive power generation level Q G The upper and lower limits of (2).
The tie-line power thermal limit conditions are as follows:
Figure BDA0003103172160000029
in the formula (I), the compound is shown in the specification,
Figure BDA00031031721600000210
and
Figure BDA00031031721600000211
respectively a tie line power P B Upper and lower bounds.
2) A set of boundaries V is established. In an initial condition, the elements within the boundary set V include active tie line power P B，i The extreme points of which satisfy the constraint condition.
3) A polyhedron R is built based on the set of boundaries.
The step of building the polyhedron R comprises:
3.1 Determine the ith tie-line power by using the equations (9) and (10)
Figure BDA00031031721600000220
Of (2) an optimal solution
Figure BDA00031031721600000212
And an optimal solution
Figure BDA00031031721600000213
The formula (9) and the formula (10) are respectively as follows:
Figure BDA00031031721600000214
Figure BDA00031031721600000215
3.2 ) optimal solution
Figure BDA00031031721600000216
And an optimal solution
Figure BDA00031031721600000217
Written as boundary points in the boundary set V. Construction of a bounded set polyhedron with a set of boundaries V
Figure BDA00031031721600000218
A and B are the face coefficient vectors of R. The expression of the kth face of the polyhedron R is A k P B ≤B k Wherein A is k Is the kth row sub-matrix of vector A, B k Is the kth element of the column vector B.
4) At least one face of the polyhedron R is translated to obtain a plurality of new boundary points meeting constraint conditions, and the new boundaries are written into the boundary set V to obtain a new boundary set V new 。
Preferably, each face of the polyhedron R is translated to obtain a plurality of new boundary points meeting constraint conditions, and the new boundaries are written into the boundary set V to obtain a new boundary set V new 。
5) Based on new boundary set V new Establishing a new polyhedron R new 。
Establishing a new polyhedron R new Comprises the following steps:
5.1 To establish an optimization equation, namely:
Figure BDA0003103172160000031
wherein the optimal solution of optimization equation (11) is denoted as P B|k 。
5.2 Establish a new set of boundaries
Figure BDA0003103172160000032
Based on the new boundary set V new Building new polyhedrons
Figure BDA0003103172160000033
A new And B new Is a novel polyhedron R new The surface coefficient vector of (2).
6) Judging polyhedron R and new polyhedron R new Whether the difference between the two is smaller than a preset condition or not, if yes, obtaining a feasible region R of the power system tie line new Otherwise, updating the boundary set V = V new Update polyhedron R = R new And returns to step 4).
Judging polyhedron R and new polyhedron R new The method of whether the difference between them is smaller than the preset condition is: calculating polyhedron R and New polyhedron R new The volume difference between the two polyhedrons is delta R, whether the delta R is less than the epsilon is judged, if yes, the polyhedron R and the new polyhedron R are judged new The difference therebetween is smaller than a preset condition. Epsilon is a preset volume difference threshold.
It is worth noting that the present invention first determines the transmission capability of a tie to construct a tie feasible domain, which can also be represented by a polyhedron R. Secondly, each surface of the feasible region is used as an objective function to solve an optimization problem, which can be regarded as a process of moving each surface of R outwards to find the boundary point of the original feasible region Ω. All searched boundary points are then connected so that R will become a new polyhedron R new I.e. a new boundary of the tie-line feasible domain is found. Finally, this process is repeated several times until R new The difference from Ω is below a preset value. This then completes the approximation of the original feasible domain omega.
The technical effect of the invention is undoubted, the invention can effectively approximate the prior connecting line feasible domain, approximate the prior nonlinear feasible domain into the linear feasible domain, and has small calculation amount and higher precision.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is an IEEE-30 bus system;
FIG. 3 is a diagram of a feasible domain of links characterized by Monte Carlo sampling in an IEEE 30-bus system;
FIG. 4 is a cross-tie feasible domain characterized by a multi-segment boundary approximation in an IEEE 30-bus system.
FIG. 5 is a cross-tie feasible region characterized by the maximum transfer capability method in an IEEE 30-bus system.
Detailed Description
The present invention is further illustrated by the following examples, but it should not be construed that the scope of the above-described subject matter is limited to the following examples. Various substitutions and alterations can be made without departing from the technical idea of the invention and the scope of the invention is covered by the present invention according to the common technical knowledge and the conventional means in the field.
Example 1:
referring to fig. 1, a method for solving a feasible domain of a nonlinear power system tie line includes the following steps:
1) Constraints for tie-line feasible domain calculations are established.
The constraints include power balance constraints, transmission constraints, generator power constraints, and tie line power thermal limits.
The power balance constraint comprises active balance and reactive balance regional power system power flow constraint, namely:
Figure BDA0003103172160000041
Figure BDA0003103172160000042
in the formula (I), the compound is shown in the specification,
Figure BDA0003103172160000043
is a set of buses in the grid. P L，i 、P B，i And P G，i Respectively the active load power, the active tie line power and the active power generation level at the bus i. Q L，i 、Q B，i And Q G，i Respectively the reactive load power, the reactive tie line power and the reactive power generation level at the bus i. V i And V j The voltage values at the bus i and the bus j are respectively. G ij And B ij Respectively the conductance and admittance of the bus i-j branch. Theta ij Is the difference in voltage phase angle at the bus i and j.
The transmission constraints are as follows:
Figure BDA0003103172160000044
Figure BDA0003103172160000045
Figure BDA0003103172160000046
in the formula, P ij And Q ij The active power and the reactive power of the branch circuits connecting the i bus and the j bus are respectively. S ij Is the apparent power of the corresponding branch.
The generator power constraint is as follows:
Figure BDA0003103172160000047
Figure BDA0003103172160000048
in the formula (I), the compound is shown in the specification,
Figure BDA0003103172160000049
and
Figure BDA00031031721600000410
respectively active power generation level P G Upper and lower bounds.
Figure BDA00031031721600000411
And
Figure BDA0003103172160000051
respectively a reactive power generation level Q G The upper and lower limits of (2).
The tie-line power thermal limit conditions are as follows:
Figure BDA0003103172160000052
in the formula (I), the compound is shown in the specification,
Figure BDA0003103172160000053
and
Figure BDA0003103172160000054
respectively a tie line power P B Upper and lower bounds.
2) A set of boundaries V is established. In an initial condition, the elements within the boundary set V include active tie line power P B，i The extreme points of which satisfy the constraint condition.
3) A polyhedron R is built based on the set of boundaries.
The step of building the polyhedron R comprises:
3.1 Determine the ith tie line power by formula (9) and formula (10)
Figure BDA0003103172160000055
Of (2) an optimal solution
Figure BDA0003103172160000056
And optimal solution
Figure BDA0003103172160000057
The formula (9) and the formula (10) are respectively as follows:
Figure BDA0003103172160000058
Figure BDA0003103172160000059
3.2 ) optimal solution
Figure BDA00031031721600000510
And an optimal solution
Figure BDA00031031721600000511
Written as boundary points in the boundary set V. Construction of a bounded set polyhedron with a set of boundaries V
Figure BDA00031031721600000512
A and B are the face coefficient vectors of R. The expression of the kth face of the polyhedron R is A k P B ≤B k Wherein A is k Is the kth row sub-matrix of vector A, B k Is the kth element of the column vector B.
4) Translating each face of the polyhedron R to obtain a plurality of new boundary points meeting constraint conditions, writing the new boundaries into a boundary set V to obtain a new boundary set V new 。
5) Based on the new boundary set V new Establishment of a novel polyhedron R new 。
Establishment of a novel polyhedron R new Comprises the following steps:
5.1 To establish an optimization equation, namely:
Figure BDA00031031721600000513
wherein the optimal solution of optimization equation (11) is denoted as P B|k 。
5.2 Establish a new set of boundaries
Figure BDA00031031721600000514
Based on new boundary set V new Building new polyhedron
Figure BDA00031031721600000515
A new And B new Is a novel polyhedron R new The surface coefficient vector of (1).
6) Judgment of polyhedron R and New polyhedron R new Whether the difference between the two is smaller than a preset condition or not, if yes, obtaining a feasible region R of the power system tie line new Otherwise, updating the boundary set V = V new Update polyhedron R = R new And returns to step 4).
Judging polyhedron R and new polyhedron R new The method for determining whether the difference is smaller than the preset condition includes: calculating polyhedron R and New polyhedron R new The volume difference between the polyhedron R and the polyhedron R is judged, whether the size is smaller than the epsilon is judged, and if the size is larger than the epsilon, the polyhedron R and the new polyhedron R are judged new The difference therebetween is smaller than a preset condition. Epsilon is a preset volume difference threshold.
Example 2:
a method of solving nonlinear power system tie line feasible regions, comprising the steps of:
1) Establishing constraint conditions
a) Constraint of power balance
Regional power system power flow constraints, including active and reactive balance, can be represented by the following matrices and vectors:
Figure BDA0003103172160000061
Figure BDA0003103172160000062
in the formula
Figure BDA0003103172160000063
Is a bus set in the grid; p is L，i ，P B，i And P is G，i The active load power, the active tie line power and the active power generation level at the bus i are respectively; q L，i ，Q B，i And Q G，i Respectively representing the reactive load power, the reactive interconnection line power and the reactive power generation level at the position of a bus i; v i And V j The voltage values of the buses i and j are respectively; g ij And B ij Respectively the conductance and admittance of the bus i-j branch; theta ij Is the difference in voltage phase angle at bus i and j.
b) Transmission constraints
The transmission flow of each branch should be limited within a given range, specifically as follows:
Figure BDA0003103172160000064
Figure BDA0003103172160000065
Figure BDA0003103172160000066
in the formula P ij And Q ij Active power and reactive power of branches connected with the i bus and the j bus are respectively; s ij Is the apparent power of the corresponding branch.
c) Power constraint of generator
During the output period of the unit, the power generation level meets the capacity requirement of the generator:
Figure BDA0003103172160000067
Figure BDA0003103172160000068
in the formula
Figure BDA0003103172160000069
And
Figure BDA00031031721600000610
respectively active power generation level P G Upper and lower bounds of (a);
Figure BDA00031031721600000611
and
Figure BDA00031031721600000612
respectively as reactive power generation level Q G The upper and lower limits of (2).
d) Junctor power thermal limit
When the tie between regional grids is exchanging power, the tie power should not exceed upper and lower limits:
Figure BDA0003103172160000071
in the formula
Figure BDA0003103172160000072
And
Figure BDA0003103172160000073
respectively, the tie line power P B Upper and lower bounds.
2) Initialization of polyhedrons R
By searching
Figure BDA0003103172160000074
The maximum value and the minimum value of each variable in the range are used for determining the upper limit and the lower limit of each link line, so that an initial feasible domain is obtained. Of the ith tie-line
Figure BDA0003103172160000075
Can be expressed as the following two optimization problems:
Figure BDA0003103172160000076
Figure BDA0003103172160000077
the above equations (9) and (10) relate to P B Is expressed as
Figure BDA0003103172160000078
And
Figure BDA0003103172160000079
then we can use
Figure BDA00031031721600000710
And
Figure BDA00031031721600000711
construction of a closed set polyhedron as boundary points V
Figure BDA00031031721600000712
Where A and B are the surface coefficients of R.
3) Each plane is translated
The expression for the kth face of the bounded polyhedron R is A k P B ≤B k Wherein A is k Is the k-th row sub-matrix of A, B k Is the kth element of the column vector B. To find new boundary points, the following optimization problem needs to be solved:
Figure BDA00031031721600000713
expressing the optimal solution of the above problem as P B|k Then the new set of boundary points for the feasible region is
Figure BDA00031031721600000714
From the new boundary points and the boundary points in the previous polyhedron, a new polyhedron of the feasible region of tie line power can be obtained, i.e.
Figure BDA00031031721600000715
Figure BDA00031031721600000716
4) Accuracy determination
The greater the difference between the new polyhedron and the previous polyhedron, the more significant the change in shape. When the difference is small, the boundary point search may then stop because a small difference indicates that the new polyhedron is similar to the feasible domain of the original tie. The expression of the precision judgment is that iteration is repeatedly carried out by a multi-segment boundary approximation method until the volume difference between a new polyhedron and a previous polyhedron is smaller than a preset threshold value.
Example 3:
an accuracy verification test of a method for solving a nonlinear power system tie line feasible region comprises the following processes:
1) An IEEE-30 bus system is established, see fig. 2.
2) The following three methods are used to describe the feasible region:
m0: monte carlo sampling method.
M1: multi-segment boundary approximation.
M2: tie-line maximum transmission power method.
As shown in fig. 3 to 5, the feasible domains of the IEEE-30 node system are characterized by the above three methods, respectively. PB1 and PB2 are transmission powers of boundary links. Wherein, a positive value represents the junctor power injected to the border node of the regional network from other adjacent regional networks, and a negative value represents the junctor power transmitted to other adjacent regional networks.
As shown in fig. 3, 5000 points are randomly sampled, and 2358 points are totally arranged in the feasible region Ω, so as to obtain a group of feasible solutions x = [ P ] satisfying the constraints (1) - (8) ij Q ij P G Q G ]。
For the other 2642 operating points that fall outside the feasible region Ω, no feasible solution can be found that satisfies the constraints. From these points within the feasible region, the original feasible region Ω can be outlined. The approximate feasible region characterized by the method M1 of the invention is shown in FIG. 4, the whole calculation process goes through 3 rounds of translation, 16 boundary points are found, and the points are connected to obtain a linear feasible region. The feasible regions, which only take into account the maximum and minimum power that can be transmitted by each link, are depicted in fig. 5. Table one gives specific accuracy and computation time comparisons.
TABLE 1 comparison of time and area of calculation of feasible fields
Figure BDA0003103172160000081
From the experimental results, it can be seen that:
from the analysis of the calculation time and accuracy of M0, M1 and M2 in table 1, the maximum feasible region area of M0 is 4.0871, which means that the accuracy is highest and is close to the original feasible region, but the calculation process of M0 requires 24159.01 seconds and is very time-consuming. In contrast, M2 only needs 13.51 seconds, the calculation time is minimum, but the area of the feasible region 2.7857 is far smaller than M0, and the error is large. Finally, the multi-segment boundary approximation method M1 provided by the invention makes a good compromise between accuracy and time, the feasible region area is 4.0378, which is close to the feasible region area M0, but the calculation time is only 87.20 seconds. Therefore, M1 has the best computational performance among the three methods.

Claims (3)

1. A method for solving a feasible domain of a nonlinear power system tie line, comprising the steps of:
1) Establishing constraint conditions for calculating the feasible domain of the tie line;
the constraint conditions comprise power balance constraint, transmission constraint, generator power constraint and tie line power thermal limit condition;
the power balance constraint comprises active balance and reactive balance regional power system power flow constraint, namely:
Figure FDA0003860766140000011
Figure FDA0003860766140000012
in the formula (I), the compound is shown in the specification,
Figure FDA0003860766140000013
is a bus set in the grid; p L,i 、P B,i And P G,i Respectively the active load power, the active tie line power and the active power generation level at the bus i; q L,i 、Q B,i And Q G,i Respectively representing the reactive load power, the reactive tie line power and the reactive power generation level at the position of a bus i; v i And V j Are respectively provided withThe voltage values of the bus i and the bus j are obtained; g ij And B ij Respectively the conductance and admittance of the bus i-j branch; theta.theta. ij Is the difference of the phase angles of the voltages at the bus i and j;
the transmission constraints are as follows:
Figure FDA0003860766140000014
Figure FDA0003860766140000015
Figure FDA0003860766140000016
in the formula, P ij And Q ij The active power and the reactive power of a branch circuit connected with the i bus and the j bus are respectively; s ij Is the apparent power of the corresponding branch;
the generator power constraint is as follows:
Figure FDA0003860766140000017
Figure FDA0003860766140000018
in the formula (I), the compound is shown in the specification,
Figure FDA0003860766140000019
and
Figure FDA00038607661400000110
respectively active power generation level P G Upper and lower bounds of (a);
Figure FDA00038607661400000111
and
Figure FDA00038607661400000112
respectively as reactive power generation level Q G The upper and lower limits of (d);
the tie-line power thermal limit conditions are as follows:
Figure FDA00038607661400000113
in the formula (I), the compound is shown in the specification,
Figure FDA00038607661400000114
and
Figure FDA00038607661400000115
respectively a tie line power P B Upper and lower bounds of (a);
2) Establishing a boundary set V; in an initial condition, the elements within the boundary set V include active tie power P B，i The extreme points meeting the constraint conditions;
3) Establishing a polyhedron R based on the boundary set;
the step of building the polyhedron R comprises:
3.1 Determine the ith tie line power by formula (9) and formula (10)
Figure FDA0003860766140000021
Of (2) an optimal solution
Figure FDA0003860766140000022
And an optimal solution
Figure FDA0003860766140000023
The formula (9) and the formula (10) are respectively as follows:
Figure FDA0003860766140000024
Figure FDA0003860766140000025
3.2 ) optimal solution
Figure FDA0003860766140000026
And optimal solution
Figure FDA0003860766140000027
Writing the boundary point into the boundary set V; construction of a bounded set polyhedron with a set of boundaries V
Figure FDA0003860766140000028
A and B are the face coefficient vectors of R; the expression of the kth face of the polyhedron R is A k P B ≤B k Wherein A is k Is the kth row sub-matrix of vector A, B k Is the kth element of column vector B;
4) At least one face of the polyhedron R is translated to obtain a plurality of new boundary points meeting constraint conditions, and the new boundaries are written into the boundary set V to obtain a new boundary set V new ；
5) Based on the new boundary set V new Establishment of a novel polyhedron R new ；
Establishment of a novel polyhedron R new Comprises the following steps:
5.1 ) establish an optimization equation, i.e.:
Figure FDA0003860766140000029
wherein the optimal solution of the optimization equation (11) is denoted as P B|k ；
5.2 Establish a new set of boundaries
Figure FDA00038607661400000210
Based on the new boundary set V new Building new polyhedrons
Figure FDA00038607661400000211
A new And B new Is a novel polyhedron R new A surface coefficient vector of (a);
6) Judgment of polyhedron R and New polyhedron R new Whether the difference between the two is smaller than a preset condition or not, if yes, obtaining a feasible region R of the power system tie line new Otherwise, updating the boundary set V = V new Update polyhedron R = R new And returns to step 4).
2. The method according to claim 1, wherein each face of the polyhedron R is translated to obtain a plurality of new boundary points satisfying constraint conditions, and the new boundaries are written into the boundary set V to obtain a new boundary set V new 。
3. The method for solving the feasible region of the nonlinear power system tie line according to claim 1, characterized in that a judgment polyhedron R and a new polyhedron R new The method for determining whether the difference is smaller than the preset condition includes: calculating the polyhedron R and the new polyhedron R new The volume difference between them is DeltaR, and DeltaR is judged<If epsilon is true, judging whether the polyhedron R and the new polyhedron R are true or not, if yes, judging that the polyhedron R is a new polyhedron R new The difference between the two is smaller than a preset condition; epsilon is a preset volume difference threshold.
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