CN112861315B - One-dimensional descent search method for non-convex single-target optimal power flow global solution of power system - Google Patents

One-dimensional descent search method for non-convex single-target optimal power flow global solution of power system Download PDF

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CN112861315B
CN112861315B CN202110030468.5A CN202110030468A CN112861315B CN 112861315 B CN112861315 B CN 112861315B CN 202110030468 A CN202110030468 A CN 202110030468A CN 112861315 B CN112861315 B CN 112861315B
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杨柳林
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    • G06F30/20—Design optimisation, verification or simulation
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    • G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
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    • G06Q50/06—Electricity, gas or water supply
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J3/00—Circuit arrangements for ac mains or ac distribution networks
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J3/00—Circuit arrangements for ac mains or ac distribution networks
    • H02J3/04—Circuit arrangements for ac mains or ac distribution networks for connecting networks of the same frequency but supplied from different sources
    • H02J3/06—Controlling transfer of power between connected networks; Controlling sharing of load between connected networks
    • G—PHYSICS
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J2203/00—Indexing scheme relating to details of circuit arrangements for AC mains or AC distribution networks
    • H02J2203/10—Power transmission or distribution systems management focussing at grid-level, e.g. load flow analysis, node profile computation, meshed network optimisation, active network management or spinning reserve management
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J2203/00—Indexing scheme relating to details of circuit arrangements for AC mains or AC distribution networks
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Abstract

The invention discloses a one-dimensional descent search method for a non-convex single-target optimal power flow global solution of an electric power system, which comprises the following steps of: 1) Determining an objective function; 2) Determining a constraint condition; 3) Performing primary power flow calculation to obtain a primary power flow solution and an initial objective function value; 4) Subtracting a positive small deviation value from the initial objective function value, and substituting the initial objective function value into the original objective function to obtain an equation, and further forming an equation set with the constraint condition; 5) Solving the formed equation set by adopting an interior point method to obtain a new trend solution and an objective function value; 6) And (5) repeatedly executing the step 4) and the step 5), when the condition that the equation set has no solution occurs, ending the loop, and obtaining the set value and the solution of the equation set in the last iteration step, namely the required global optimal value and the required global optimal solution. The method can solve the problem of trapping in the local extreme point and ensure continuous search in the direction of the global extreme point; meanwhile, the global extreme point can be correctly identified, and the reliability of the global extreme value is ensured.

Description

One-dimensional descent search method for non-convex single-target optimal power flow global solution of power system
Technical Field
The invention belongs to the technical field of intelligent control of an electric power system, and particularly relates to a one-dimensional descent search method for a non-convex single-target optimal power flow global solution of the electric power system.
Background
The Optimal Power Flow (OPF) is to find a load that satisfies all specified system operation and safety constraints and makes a predetermined system performance index reach Optimal Power Flow distribution by adjusting control variables when structural parameters and load conditions of the Power system are given. The research of the optimal power flow problem is mainly divided into the following two aspects: on one hand, constraint conditions and optimization targets are added in the optimal power flow model, such as unit combination problems, dynamic reactive power constraints and the like, and the large-scale power system engineering problems are analyzed and solved. On the other hand, the optimal power flow problem is often an NP-hard (Non-deterministic polymeric) problem, for example, optimization methods such as a convex programming method and an uncertainty algorithm are introduced to ensure that the global extreme value of the optimal power flow of the power system is accurately and reliably solved.
The optimization algorithm is an important basis for solving the optimal power flow problem, and the quality of the algorithm directly determines the quality of the optimal power flow solution. The existing global optimization algorithm, whether a probabilistic algorithm or a deterministic algorithm, has respective advantages and disadvantages and needs to be further perfected. The probabilistic algorithm has simple form and strong robustness, but the characteristics of black-boxed optimization problem and random search cause the solving efficiency to be low, and only approximate solution is often obtained when the search time is limited; the deterministic search algorithm utilizes the analytic information of the optimization problem, has a definite search direction and high efficiency, but when the analytic information of the optimization problem does not have strict global property, the search is trapped in a local extreme point. Therefore, if the traditional optimization algorithm is adopted, only local extremum solutions can be obtained, and the global optimal solution cannot be reliably solved.
Disclosure of Invention
The invention aims to provide a one-dimensional descent search method for a non-convex single-target optimal power flow global solution of an electric power system, aiming at solving the problem of non-convex power flow optimization of the electric power system in the prior art. The invention relates to a single-target optimization algorithm, which can effectively solve the problems of extreme point jumping and judgment in the global extreme value optimizing process and is applied to the conventional optimal power flow problem.
The optimal power flow mathematical model of the power system is as follows:
Figure BDA0002891907230000011
in formula (1): f is an objective function, h is an equality constraint, g is an inequality constraint,
Figure BDA0002891907230000021
and u is the upper and lower limits of the continuous variable u, the symbols m, p and n represent equality constraint, inequality constraint and the number of variables, respectively, and the symbol RnRepresenting an n-dimensional real vector space; the OPF problem is to find a solution u from the feasible domain such that f is minimized, under the constraint of h and g.
In order to achieve the purpose, the invention adopts the following technical scheme:
a one-dimensional descent search method for a non-convex single-target optimal power flow global solution of an electric power system is a deterministic global optimal value search method established according to analysis information of global properties of an optimization problem, has clear global property analysis information, and can solve the problem of trapping in a local extreme point and ensure continuous search in the direction of the global extreme point, wherein the search direction is a one-dimensional descent direction; meanwhile, a judgment mechanism of search termination is provided, the global extreme point can be correctly identified, and the reliability of the sought global extreme value is ensured.
The one-dimensional descent search method comprises the following steps:
1) Determining an objective function;
2) Determining a constraint condition;
3) Performing primary power flow calculation to obtain a primary power flow solution and an initial objective function value;
4) Subtracting a positive small deviation value from the initial objective function value, and substituting the initial objective function value into the original objective function to obtain an equation, and further forming an equation set with the constraint condition;
5) Solving the formed equation set by adopting an interior point method to obtain a new power flow solution and an objective function value;
6) And (5) repeatedly executing the step 4) and the step 5), when the condition that the equation set has no solution occurs, ending the loop, and obtaining the set value and the solution of the equation set in the last iteration step, namely the required global optimal value and the required global optimal solution.
The termination criterion is when at the local minimum point GiIf a feasible solution X can be foundiThe setpoint is decremented cyclically in the direction of the decline of the setpoint of the objective function. When the set value G isvLess than global minimum G*And when the situation that no solution exists occurs, the search is stopped at the moment, and the solutions of the set values and the equation sets obtained in the last iteration step are the required global optimal values and the required global optimal solutions. Whereby the set value G can be decremented as a loop in the descending direction of the sequence SvTo search for the global optimum G*The termination criterion of (1).
The invention further illustrates that the objective function is:
Min F(x) (1a)
wherein F is an objective function and x is a continuous variable.
The invention further explains that the constraint conditions are as follows:
Figure BDA0002891907230000031
in the formula, h is equality constraint, g is inequality constraint, and symbols m and p respectively represent the number of equality constraint and inequality constraint.
According to a further aspect of the invention, the continuous variable x is:
x＝(x1,x2,...,xn)T∈Rn
in the formula, RnA real vector space representing the dimension, n representing the number of variables;
the symbol I is used to represent the feasible field of Problem1, i.e., I = { x | h (x) =0,g (x) ≦ 0,x ∈ Rn}; problem1 contains an objective function F, an equality constraint h and an inequality constraint g;
let it be assumed that the calculated value of the objective function F is identified by the symbol G, i.e. F (x) = G; according to the definition of the global minimum, there must be a sequence of values:
S＝{F(x1)＝G1,...,F(xv)＝Gv,...,F(x*)＝G*}；G1>...>Gv>...>G*
wherein G is the global optimum value of F; when x of Problem1 is continuous in the feasible region, S is a continuous interval; in the case of a discrete interval, at a certain solution x of the optimization problemvWith a certain value G of the objective function FvThere is a correspondence between, i.e. xv→Gv。
The invention further explains that in the step 3), the primary power flow calculation is carried out to obtain a primary power flow solution and an initial objective function value G0To form Problem1.
The invention further explains that the specific process of the step 4) is as follows:
setting a precision parameter sigma>0; small deviation amount delta>0; the initial objective function value F (x)0)＝G0Subtracting a positive small deviation amount delta, and substituting Problem1 into a substitution formula (1 a), thereby obtaining a new Problem form Problem 2; setting an iteration count value v =1;
Figure BDA0002891907230000032
in the formula, Δ is a small positive deviation amount, and may be set to 0.001, for example.
The invention further explains that the step 5) is specifically as follows:
with xv-1As an initial point, solving Problem 2 by using an interior point method to obtain GvAnd a corresponding solution x thereofv。
The invention further explains that the step 6) is specifically as follows:
if Problem 2 has no solution, G obtained from the last iterationv-1And a corresponding solution xv-1The loop stops for the optimal solution. Otherwise, let Gv＝Gv-1-Δ；xv＝xv-1(ii) a v = v +1, forming a new equation set, and continuing iteration until no solution exists; and after the solution is finished, outputting a global optimal value and a global optimal solution.
The invention has the advantages that:
the one-dimensional descent search method provided by the invention has clear global property analysis information, the search direction is a one-dimensional descent direction, the problem of trapping in a local extreme point can be solved, and continuous search in the direction of the global extreme point is ensured; meanwhile, a judgment mechanism of search termination is provided, the global extreme point can be correctly identified, and the reliability of the sought global extreme value is ensured.
Drawings
FIG. 1 is a graph of local extrema and global extrema.
Fig. 2 is a calculation flow chart.
Fig. 3 is a five node system diagram.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
Example (b):
a one-dimensional descent search method for a non-convex single-target optimal power flow global solution of an electric power system is a deterministic global optimal value search method established according to analysis information of global properties of an optimization problem, has clear global property analysis information, and can solve the problem of trapping in a local extreme point and ensure continuous search in the direction of the global extreme point, wherein the search direction is a one-dimensional descent direction; meanwhile, a judgment mechanism of search termination is provided, the global extreme point can be correctly identified, and the reliability of the sought global extreme value is ensured. The method comprises the following steps:
1) Determining an objective function:
Min F(x)
wherein F is an objective function and x is a continuous variable;
2) Determining a constraint condition:
Figure BDA0002891907230000051
in the formula, h is equality constraint, g is inequality constraint, and symbols m and p respectively represent the number of equality constraint and inequality constraint;
the continuous variable x is as follows:
x＝(x1,x2,...,xn)T∈Rn
in the formula, RnA real vector space representing the dimension, n representing the number of variables;
the symbol I is used to represent the feasible field of Problem1, i.e., I = { x | h (x) =0,g (x) ≦ 0,x ∈ Rn}; problem1 contains an objective function F, an equality constraint h and an inequality constraint g;
let it be assumed that the calculated value of the objective function F is identified by the symbol G, i.e. F (x) = G; according to the definition of the global minimum, there must be a sequence of values:
S＝{F(x1)＝G1,...,F(xv)＝Gv,...,F(x*)＝G*}；G1>...>Gv>...>G*
wherein G is the global optimum value of F; when the temperature is higher than the set temperatureWhen x of Problem1 is continuous in the feasible region, S is a continuous interval; in the case of a discrete interval, at a certain solution x of the optimization problemvWith a certain value G of the objective function FvThere is a corresponding relationship between them, i.e. xv→Gv；
3) Performing primary power flow calculation to obtain a primary power flow solution and an initial objective function value G0Forming Problem1;
4) Subtracting a positive small deviation value from the initial objective function value, and substituting the initial objective function value into the original objective function to obtain an equation, and further forming an equation set with the constraint condition; the specific process is as follows:
setting a precision parameter sigma>0; small deviation amount delta>0; the initial objective function value F (x)0)＝G0Subtracting a positive small deviation amount delta, and substituting Problem1 into a substitution formula (1 a), thereby obtaining a new Problem form Problem 2; setting an iteration count value v =1;
Figure BDA0002891907230000052
where Δ is a small positive deviation;
5) Solving the formed equation set by adopting an interior point method to obtain a new trend solution and an objective function value; the method comprises the following specific steps:
with xv-1For the initial point, solving Problem 2 by using an interior point method to obtain GvAnd a corresponding solution xv；
6) Repeatedly executing the step 4) and the step 5), when the condition that the equation set has no solution appears, the loop is terminated, and the iteration of the previous step obtains the set value and the solution of the equation set, namely the required global optimal value and the global optimal solution; the method specifically comprises the following steps:
if Problem 2 has no solution, G obtained in last iterationv-1And a corresponding solution xv-1The loop is stopped for an optimal solution. Otherwise, let Gv＝Gv-1-Δ；xv＝xv-1(ii) a v = v +1, forming a new equation set, and continuing iteration until no solution exists; and after the solution is finished, outputting a global optimal value and a global optimal solution.
One-dimensional descent search method optimization power flow problem example analysis based on the embodiment
Step 1, determining a target function, converting the original problem target function into an equality constraint condition, converting the original optimization power flow problem into an operation power flow problem, converting the solution of the global optimal power flow into a circular solution of a power flow equation in the descending direction of the target function value for the current example, and calculating the flow as shown in figure 2.
The test system adopts a 5-node standard test system, and a system single line diagram and impedance parameters are shown in figure 3. Wherein nodes 4 and 5 are generator nodes, the balance node is node 5, and the remaining nodes are PQ nodes. Node voltage data and node power data of the system are respectively shown in tables 1 and 2, limited line transmission power data are shown in table 3, and fuel consumption coefficients of the generator set are shown in table 4. All data are in per unit value form unless otherwise specified.
TABLE 1 System node Voltage data
Figure BDA0002891907230000061
Note: means do not set
TABLE 2 System node Power data
Figure BDA0002891907230000062
Figure BDA0002891907230000071
TABLE 3 line Transmission Power Limit data
Figure BDA0002891907230000072
TABLE 4 System Generator set Fuel consumption coefficient
Figure BDA0002891907230000073
System admittance matrix formation
The admittance matrix for test system I can be formed from the data of fig. 3 as follows:
Figure BDA0002891907230000074
(II) System variable Condition
In the test system I, 5 nodes are provided, 2 generator sets are provided, and no other reactive power source exists. According to the node voltage and power data, the number of variables to be solved of the system is 13:
X＝(V1,θ1,V2,θ2,V3,θ3,V4,θ4,V5,PG4,QG4,PG5,QG5)
wherein the state variable is (V)1,θ1,V2,θ2,V3,θ3,V4,θ4,V5) (ii) a The control variable is (P)G4,QG4,PG5,QG5)。
(III) System optimal Power flow model
With the generator operating cost as the objective function, the data according to table 4 can be obtained:
Figure BDA0002891907230000081
the equality constraint of the optimal power flow model of the test system is 10 node power equations in total; there are 14 inequality constraints and variable constraints, including: 5 node voltage constraints, 2 power supply active power constraints, 2 power supply reactive power constraints, and 5 branch transmission power constraints.
Further, the analysis is integrated to obtain an optimal power flow model of the test system I:
Figure BDA0002891907230000082
Figure BDA0002891907230000083
Figure BDA0002891907230000084
Vimin≤Vi≤Vimax；(i＝1,...,5)
PGimin≤PGi≤PGimax；(i＝4,5)
QGimin≤QGi≤QGimax；(i＝4,5)
Figure BDA0002891907230000085
wherein, Pij＝ViVj(Gijcosθij+Bijsinθij)-ViGij。
And 2, constructing an objective function equation constraint. When the objective function value is set to be G, an objective function equation constraint is formed:
Figure BDA0002891907230000086
this, together with other constraints in the optimal power flow model, forms a system of operational power flow equations.
Step 3, performing primary power flow calculation to obtain a primary power flow solution and an initial objective function value G0Entering the optimizing process.
Firstly, initialization setting is carried out: the convergence condition is 1e-6; the initial value of the system variable is V1＝1.0，θ1＝0.0， V2＝1.0，θ2＝0.0，V3＝1.0，θ3＝0.0，V4＝1.0，θ4＝0.0，V5＝1.0，PG4＝4.0，QG4＝0.1，PG5＝3.0，QG5=0.1, it is noted that node 5 is a balanced node, the phase angle of which is a fixed value θ5=0.0, but the voltage amplitude of the node 5 is adjustable. The small deviation for the objective function is set to Δ =0.1; the remaining known parameters are: pD1＝-1.6，QD1＝-0.8，PD2＝-2，QD2＝-1，PD3＝-3.7，QD3And (4) = -1.3. And obtaining an initial set value G =7973.8 of the objective function through primary power flow calculation.
And 4, subtracting a positive small deviation amount delta from the initial set value G =7973.8 of the target function, and substituting the formula Min F (x) = G into Problem1 to obtain a Problem 2.
And 5, solving the formed equation set by adopting an interior point method to obtain a new trend solution and an objective function value.
And 6, repeatedly executing the step sequence of the step 4 and the step 5. If no solution exists in the 507 circulation equation set, the circulation is terminated, the global optimal solution is obtained by searching, and the intermediate calculation process is shown as follows. Table 5 is a calculated value of the voltage parameter of each node in the search process, and table 6 is a calculated value of the power parameter of the generator set.
TABLE 5 variation of the Voltage parameters of the nodes
Figure BDA0002891907230000091
TABLE 6 variation of generator power
Figure BDA0002891907230000101
The global solutions searched by the one-dimensional descent search method are shown in tables 7, 8 and 9. Wherein, table 7 is the optimal solution for each node voltage; table 8 is the calculated value of the branch active power; table 9 is the optimal solution for active and reactive power, and the global optimal value for the objective function.
TABLE 7 optimal solution for each node voltage
Figure BDA0002891907230000102
Table 8 branch active power
Figure BDA0002891907230000103
TABLE 9 active, reactive, and objective function optima
Figure BDA0002891907230000111
It should be understood that the above-described embodiments are merely examples for clearly illustrating the present invention and are not intended to limit the practice of the present invention. Other variations and modifications will be apparent to persons skilled in the art in light of the above description; it is not necessary or exhaustive for all embodiments to be presented herein; and obvious variations or modifications are intended to be within the scope of the present invention.

Claims (8)

1. A one-dimensional descent search method for a non-convex single-target optimal power flow global solution of an electric power system is characterized by comprising the following steps of: the method is a deterministic global optimum value searching method established according to the analysis information of the global property of the optimization problem, has definite global property analysis information, and can solve the problem of trapping in a local extreme point and ensure continuous searching in the direction of the global extreme point, wherein the searching direction is a one-dimensional descending direction; meanwhile, a judgment mechanism of search termination is provided, so that the global extreme point can be correctly identified, and the reliability of the sought global extreme value is ensured;
the one-dimensional descent search method comprises the following steps:
1) Determining an objective function;
2) Determining a constraint condition;
3) Performing primary power flow calculation to obtain a primary power flow solution and an initial objective function value;
4) Subtracting a positive small deviation value from the initial objective function value, and substituting the initial objective function value into the original objective function to obtain an equation, and further forming an equation set with the constraint condition;
5) Solving the formed equation set by adopting an interior point method to obtain a new trend solution and an objective function value;
6) And (5) repeatedly executing the step 4) and the step 5), when the condition that the equation set has no solution occurs, ending the loop, and obtaining the set value and the solution of the equation set in the last iteration step, namely the required global optimal value and the required global optimal solution.
2. The one-dimensional descent search method for the non-convex single-target optimal power flow global solution of the power system as claimed in claim 1, wherein the objective function is:
Min F(x) (1a)
wherein F is an objective function and x is a continuous variable.
3. The one-dimensional descent search method for the non-convex single-target optimal power flow global solution of the power system as claimed in claim 2, wherein the constraint conditions are as follows:
Figure FDA0003844590230000011
in the formula, h is equality constraint, g is inequality constraint, and symbols m and p respectively represent the number of equality constraint and inequality constraint.
4. The one-dimensional descent search method for the non-convex single-target optimal power flow global solution of the power system as claimed in claim 3, wherein the continuous variable x is:
x＝(x1,x2,...,xn)T∈Rn
in the formula, RnA real vector space representing the dimension, n representing the number of variables;
the symbol I is used to represent the feasible field of Problem1, i.e., I = { x | h (x) =0,g (x) ≦ 0,x ∈ Rn}; problem1 contains an objective function F, an equality constraint h and an inequality constraint g;
let it be assumed that the calculated value of the objective function F is identified by the symbol G, i.e. F (x) = G; according to the definition of the global minimum, there must be a sequence of values:
S＝{F(x1)＝G1,...,F(xv)＝Gv,...,F(x*)＝G*}；G1>...>Gv>...>G*
wherein G is the global optimum value of F; when x of Problem1 is continuous in the feasible region, S is a continuous interval; in the case of a discrete interval, at a certain solution x of the optimization problemvWith a certain value G of the objective function FvThere is a corresponding relationship between them, i.e. xv→Gv。
5. The one-dimensional descent search method for the non-convex single-target optimal power flow global solution of the power system as claimed in claim 4, wherein in the step 3), a primary power flow calculation is performed to obtain a primary power flow solution and an initial objective function value G0To form Problem1.
6. The one-dimensional descent search method for the non-convex single-target optimal power flow global solution of the power system according to claim 5, wherein the specific process of the step 4) is as follows:
setting a precision parameter sigma>0; small deviation Δ>0; the initial objective function value F (x)0)＝G0Subtracting a positive small deviation amount delta, and substituting Problem1 into a Problem1 substitution formula (1 a) so as to obtain a new Problem form Problem 2; setting an iteration count value v =1;
Figure FDA0003844590230000021
in the formula, Δ is a small positive deviation amount.
7. The one-dimensional descent search method for the non-convex single-target optimal power flow global solution of the power system according to claim 6, wherein the step 5) is specifically as follows:
with xv-1As an initial point, solving Problem 2 by using an interior point method to obtain GvAnd a corresponding solution xv。
8. The one-dimensional descent search method for the non-convex single-target optimal power flow global solution of the power system according to claim 7, wherein the step 6) is specifically as follows:
if Problem 2 has no solution, G obtained in last iterationv-1And a corresponding solution x thereofv-1For the optimal solution, the cycle is stopped; otherwise, let Gv＝Gv-1-△；xv＝xv-1(ii) a v = v +1, forming a new equation set, and continuing iteration until no solution exists; and after the solution is finished, outputting a global optimal value and a global optimal solution.
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