CN107689890A - A kind of power distribution method of the WiMAX alignment system based on genetic algorithm - Google Patents
A kind of power distribution method of the WiMAX alignment system based on genetic algorithm Download PDFInfo
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04W—WIRELESS COMMUNICATION NETWORKS
- H04W52/00—Power management, e.g. TPC [Transmission Power Control], power saving or power classes
- H04W52/04—TPC
- H04W52/06—TPC algorithms
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- H04L41/00—Arrangements for maintenance, administration or management of data switching networks, e.g. of packet switching networks
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04W—WIRELESS COMMUNICATION NETWORKS
- H04W52/00—Power management, e.g. TPC [Transmission Power Control], power saving or power classes
- H04W52/04—TPC
- H04W52/18—TPC being performed according to specific parameters
- H04W52/28—TPC being performed according to specific parameters using user profile, e.g. mobile speed, priority or network state, e.g. standby, idle or non transmission
- H04W52/283—Power depending on the position of the mobile
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04W—WIRELESS COMMUNICATION NETWORKS
- H04W64/00—Locating users or terminals or network equipment for network management purposes, e.g. mobility management
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Abstract
The invention discloses a kind of power distribution method of the WiMAX alignment system based on genetic algorithm.First, using the object function that square position error circle of node to be positioned is power distribution, the actual restrictive condition of power distribution is constraints, establishes the mathematical model of optimization based on power distribution.Under multi-target condition, multi-objective optimization question is changed into single-object problem using beeline ideal point method.Finally using the genetic algorithm direct solution of the belt restraining Optimized model, so as to obtain power allocation scheme.This algorithm can reduce the computation complexity of algorithm without any system priori.Obtain specific power mean allocation algorithm preferably theoretical square error circle.
Description
Technical field
The present invention relates to the power distribution method of WiMAX alignment system, belongs to navigator fix field.
Background technology
In recent years, as the development of wireless communication technology, wireless location technology play in daily life
The location-based applications such as more and more important effect, target following, Emergency Assistance, logistics transport are constantly closed by industry
Note.
In WiMAX alignment system, location algorithm, parameter measurement algorithm improvement and network topology structure can be passed through
The precision of alignment system [1-2] is improved etc. technology.Current research shows：The power resource reasonable distribution of system equally can
Improve positioning precision [3-5].Therefore researcher has carried out the research work in alignment system on power distribution both at home and abroad.
Power assignment of doctor Shen Yuan of Massachusetts Polytechnics in development alignment system in 2008, according to etc.
The form of valency Fisher's information matrix (Equivalent Fisher Information Matrix, EFIM) gives succinctly
Square position error lower bound SPEB of form definition, and according to ranging localization information strength (Ranging
Information Intensity, RII) functional relation of SPEB and power parameter is established, establish power optimization point
Theoretical foundation [4] is provided with positioning precision can be improved.Target letter in research afterwards using SPEB as Optimized model
Number, it is proposed that the power optimization scheme of the geometric interpretation based on EFIM characteristic values, but the algorithm is only suitable for special scene, and
Without generality, due to SPEB expression formula complexity, so needing to carry out certain mathematic(al) manipulation, and become
Change and be converted into SOCP forms or be converted into Semidefinite Programming (Semi-Definite Programming, SDP) problem, apply
The Optimization Toolbox that matlab is carried carries out the solution [6-10] of power optimization.Pass through above-mentioned research work, it has been found that by
More complicated in SPEB calculation formula, the emphasis of research work is deployed on the basis of the optimized algorithm of routine.And mesh
Preceding power distribution mathematical modeling is all based under single goal localizing environment mostly, is studied the problem of under multiple target less.
The algorithm that tradition improves positioning precision refers to：
[1]Boushaba M,Hafid A,Benslimane A.High accuracy localization method
using AoA in sensor networks[J].Computer Networks,2009,53(18):3076-3088.
[2]Yu K,Guo Y J.Statistical NLOS identification based on AOA,TOA,and
signal strength[J]. IEEE Transactions on Vehicular Technology,2009,58(1):274-
286.
The algorithm that power distribution is carried out using traditional optimization refers to：
[3]Tinqting Z,Qinyu Z,Naiionq Z,et al.Performance analysis of an
indoor UWB ranging system[J].Journal of Systems Engineering and Electronics,
2009,20(3):450-456.
[4]Shen Y,Win M Z.Fundamental limits of wideband localization—Part
I:A general framework[J].IEEE Transactions on Information Theory,2010,56(10):
4956-49
[5]Gharehshiran O N,Krishnamurthy V.Coalition formation for bearings-
only localization in sensor networks—a cooperative game approach[J].IEEE
Transactions on Signal Processing,2010, 58(8):4322-4338.
[6]Shen Y,Win M Z.Energy efficient location-aware networks[C]//IEEE
International Conference on.IEEE,2008:2995-3001.
[7]Li W W L,Shen Y,Zhang Y J,et al.Efficient anchor power allocation
for location-aware networks[C]//IEEE International Conference on.IEEE,2011:1-
6.
[8]Shen Y,Dai W,Win M Z.Power optimization for network localization
[J].IEEE/ACM Transactions on Networking(TON),2014,22(4):1337-1350.
[9]Shen Y,Dai W,Win M Z.Power optimization for network localization
[J].IEEE/ACM Transactions on Networking(TON),2014,22(4):1337-1350.
[10]Dai W N,Shen Y,Win M Z.Energy efficient cooperative network
localization[C]//IEEE International Conference on.IEEE,2014:4969-4974.
The content of the invention
For problem above, the invention provides a kind of power distribution of the WiMAX alignment system based on genetic algorithm
Method.The present invention can further optimize the result of power distribution, drop using the power allocation scheme based on intelligent optimization algorithm
Low computation complexity.On the other hand, it is preferable first with the beeline without any priori under Multi-target position
Multi-objective optimization question is changed into single-object problem by point method.Then with the Optimization Solution thinking of single goal alignment system
It is similar, the mathematical modeling of definition is solved using genetic algorithm, and then the power allocation scheme under Multi-target position is obtained, lead to
Overpower distributes optimisation technique, improves system accuracy.
In order to solve problem above, the present invention adopts the following technical scheme that：A kind of WiMAX based on genetic algorithm is determined
The power distribution method of position system, it is characterised in that comprise the following steps：
Step 1：The system model of WiMAX positioning network is established, and theory analysis is carried out to system model；
Step 2：Abbreviation represents the mathematic(al) representation of square position error lower bound of unknown node positioning precision；With unknown section
Object function of square position error lower bound of point as optimization, the restrictive condition of actual anchors node transmitting power is constraint bar
The mathematical problem of power distribution is abstracted as function optimization problem by part, each anchor node power to be allocated as variable；
Step 3：The Optimized model of the power distribution of single goal alignment system is established, is solved using the genetic algorithm of belt restraining
The Optimized model；
Step 4：Establish the Optimized model of the power distribution of multi-target positioning system；To be more using beeline ideal point method
Objective optimisation problems are changed into single-object problem；Then the mathematical model of optimization of definition is solved using genetic algorithm, is obtained
Power allocation scheme under to Multi-target position.
The step 1 includes herein below：If the position coordinates of unknown node is p=[x, y]T, anchor node j coordinate is
Pj=[xj,yj]T(j∈Nb), wherein NbThe set of anchor node is represented, P is position coordinates to be asked, PjFor known location coordinate；Not
Know that node receives the signal that j-th of anchor node is sent and is expressed as：
Wherein s (t) is the transmission signal of anchor node,WithRepresent respectively jth road reception signal range value and when
Prolong, LjFor the multipath number of transmission signal, zj(t) represent that bilateral power spectral density is N0/ 2 additive white Gaussian noise；
Information inequality principle must is fulfilled for according to Square Error matrix, we draw the position estimation error of unknown node
Performance
Wherein p0For the location estimation of unknown node, JeIt is the Fischer matrix EFIM of equal value on unknown node, can be with table
It is shown as
WhereinRepresent k-th of anchor node to the propagation angle between unknown node；Represent location information intensity, wherein εk=1000, which are one, represents the normal more than zero of propagation channel performance
Number；Represent the distance between k-th of anchor node and unknown node；pkFor k-th of anchor section
The transmission power of point；
According to above-mentioned content, square position error circle SPEB in WiMAX positioning network can be obtained：
The step 2 includes herein below：
Further abbreviation processing is done to formula (4) square position error；Can be in the hope of Fischer square of equal value by formula (3)
The inverse J of battle arraye -1For：
Wherein | | Je| | it is matrix JeDeterminant,
Because the mark of matrix is the sum of all elements on matrix leading diagonal, square position error lower bound SPEB takes to be of equal value
Xie Er inverse of a matrix matrix Jse -1Leading diagonal on all elements sum；
So SPEB expression formula can be further represented as：
Wherein
The step 3 includes herein below：Under single goal location condition, the Optimized model of power distribution can define
For：
WhereinFor the transmission power of each anchor node to be optimized, restrictive condition is：1st, Ge Gemao
The transmission power of node is more than or equal to zero and maximum is2nd, the transmission power sum of all anchor nodes is equal to p0；It can see
Go out the single objective programming mathematical modeling that above-mentioned model is belt restraining condition.
The step 4 includes herein below：Under the conditions of Multi-target position, the Optimized model of power distribution can define
For：
……
Wherein It is the constant for being more than zero of an expression propagation channel performance；Represent k-th of anchor node and unknown node NaThe distance between；Represent k-th of anchor node to unknown node NaBetween propagation angle；
Then objective optimization model can be expressed as：
Represent unknown node NaSquare position error circle SPEB；Above-mentioned model is more mesh of belt restraining condition
Mark mathematical model of optimization；
In the solution procedure of Model for Multi-Objective Optimization, first with beeline ideal point method by multi-objective optimization question
It is changed into single-object problem；Then single objective programming mathematical modeling is solved using genetic algorithm, and then obtained in more mesh
The power allocation scheme demarcated under position；
Beeline ideal point method comprises the following steps that：
Step 1：To each component in object functionThe optimal solution under constraints is obtained respectively
Wherein i=1,2 ... NaUnknown node is represented, therefore above-mentioned solutionReferred to as ideal point；
Step 2：In order that object function and ideal point are close to i.e. object functionAnd ideal point (p)It
Between " distance " it is minimum, construct following single object optimization function, the Model for Multi-Objective Optimization of formula (8) be converted into monocular
Mark Optimized model：
Beneficial effect：1. the present invention proposes the power distribution side in the WiMAX alignment system based on genetic algorithm
Case.With average power allocation algorithm ratio, system accuracy can be effectively improved, and run time is more excellent than existing routine
The power distribution algorithm of change is short.2. multiple target mathematical optimization problem is converted into list by the present invention using beeline ideal point method
The mathematical optimization problem of target, so as to realize the optimized power allocation method under Multi-target position demand.
Brief description of the drawings
Fig. 1 detail flowcharts of the present invention；
Fig. 2 WiMAX positioning system models figures；
The SPEB changed during the mono- unknown nodes of Fig. 3 with anchor node number；
With the run time of each algorithm of system anchor node number of variations during the mono- unknown nodes of Fig. 4；
The SPEB sums changed during Fig. 5 multiple target nodes with anchor node；
With the run time of each algorithm of system anchor node number of variations during the more unknown nodes of Fig. 6.
Embodiment
Further detailed description is done to the present invention below in conjunction with the accompanying drawings.
As shown in figure 1, the invention provides a kind of power distribution side of the WiMAX alignment system based on genetic algorithm
Method, comprise the following steps：
Step 1：Establish the system model of WiMAX positioning network；And calculate square position error lower bound of unknown node
Mathematic(al) representation.
Step 2：Object function using square position error lower bound of unknown node as optimization, actual anchor node launch work(
The restrictive condition of rate is constraints, and each anchor node power to be allocated establishes the mathematical optimization mould of power distribution as variable
Type.
Step 3：The Optimized model of the power distribution of single goal alignment system is established, is solved using the genetic algorithm of belt restraining
The Optimized model.
Step 4：Establish the Optimized model of the power distribution of multi-target positioning system；To be more using beeline ideal point method
Objective optimisation problems are changed into single-object problem；Then the mathematical model of optimization of definition is solved using genetic algorithm, is obtained
Power allocation scheme under to Multi-target position.
As shown in Fig. 2 describing one contains NbThe anchor node and N of individual known locationaThe nothing of individual Location-Unknown node to be asked
Line width band positions network.Anchor node is communicated with unknown node, and unknown node is believed by receiving the transmission that anchor node is sent
Number estimate self-position.Below by taking single unknown node as an example, illustrate position estimation procedure.
If the position coordinates of unknown node is p=[x, y]T, anchor node j coordinate is Pj=[xj,yj]T(j∈Nb), wherein
P is position coordinates to be asked, PjFor known location coordinate.It is unknown due to being influenceed by noise, decline, shade, Multipath Transmission
Node receives the signal that j-th of anchor node is sent and can be expressed as：
Wherein s (t) is the transmission signal of anchor node,Represent respectively jth road reception signal range value and when
Prolong, LjFor the multipath number of transmission signal, zj(t) represent that bilateral power spectral density is N0/ 2 additive white Gaussian noise.
Information inequality principle must is fulfilled for according to Square Error matrix, we draw the position estimation error of unknown node
Performance
Wherein p0For the location estimation of unknown node, JeIt is the Fischer matrix (EFIM) of equal value on unknown node, can be with
It is expressed as
WhereinRepresent k-th of anchor node to the propagation angle between unknown node；Location information intensity (Ranging inform Intensity, RII) is represented, wherein wherein εk=1000 are
The constant for being more than zero of one expression propagation channel performance；Represent k-th anchor node with
The distance between unknown node；pkFor the transmission power of k-th of anchor node.
According to above-mentioned content, square position error circle (SPEB) in WiMAX positioning network can be obtained：
A. description of the formula (4) to square position error circle is passed through, it has been found that：It is unknown under given localizing environment
The positioning performance of node is closely bound up with the installation position and transmission power of anchor node.If in the anchor node bar of given position
Under part, transmission power assignment problem will directly determine positioning performance.
B. further abbreviation processing can be done to formula (4) square position error.It can be taken by formula (3) in the hope of of equal value
Xie Er inverses of a matrix Je -1For：
Wherein | | Je| | it is matrix JeDeterminant,
Because the mark of matrix is the sum of all elements on matrix leading diagonal, square position error lower bound SPEB takes to be of equal value
Xie Er inverse of a matrix matrix Jse -1Leading diagonal on all elements sum.
So SPEB expression formula can be further represented as
Wherein
Under single goal location condition, the Optimized model of power distribution can be defined as：
WhereinFor the transmission power of each anchor node to be optimized, restrictive condition is (1) each anchor
The transmission power of node is more than or equal to zero and maximum is(2) the transmission power sum of all anchor nodes is equal to p0.Can be with
Find out the single objective programming mathematical modeling that above-mentioned model is belt restraining condition.
Under the conditions of Multi-target position, the Optimized model of power distribution can be defined as：
……
WhereinRepresent k-th of anchor node to unknown node NaBetween propagation angle, It is the constant for being more than zero of an expression propagation channel performance；Represent k-th of anchor node and unknown node NaThe distance between；
Then objective optimization model can be expressed as：
Represent unknown node NaSquare position error circle SPEB.Above-mentioned model is more mesh of belt restraining condition
Mark mathematical model of optimization.
In the solution procedure of Model for Multi-Objective Optimization, first with beeline ideal point method by multi-objective optimization question
It is changed into single-object problem.Then single objective programming mathematical modeling is solved using genetic algorithm, and then obtained in more mesh
The power allocation scheme demarcated under position.
Beeline ideal point method comprises the following steps that：
Step 1：To each component in object functionThe optimal solution under constraints is obtained respectively
Wherein i=1,2 ... Na, therefore above-mentioned solutionReferred to as ideal point.
Step 2：In order that object function and ideal point are close to (object functionWith ideal pointBetween
" distance " it is minimum), construct following single object optimization function, the Model for Multi-Objective Optimization of formula (8) be converted into single goal
Optimized model：
For the power distribution Optimized model under above-mentioned single goal and multiple target, genetic algorithm will be used herein to determining
The model of justice optimizes the solution of problem.Genetic algorithm is a kind of intelligent search algorithm of solving-optimizing problem.Can be effective
Prevent search procedure from converging on locally optimal solution.
Simulation result explanation
If anchor node and unknown node are distributed in the square area of [- 10 × 10] × [- 10 × 10], anchor node coordinate is such as
Shown in table 1；In emulation experiment, taking N number of anchor node to represent to take anchor node marked as 1,2 ..., N anchor node combines.Launch work(
The parameter of rate is as follows：Normalized system total power p0Equal to 1, the transmission power of anchor nodeMeetPass
Broadcast the constant ε of channel performancekFor 1000.Termination genetic algebra M in genetic algorithm is 200, and initialization colony number N is
100。
The combination distribution of the anchor node of table 1 and coordinate
Computer configuration is as follows：Processor is Intel (R) Core (TM) i7-4770CPU@3.4GHz 3.40GHz, in operation
4GB is saved as, operating system is Windows 7 (64) Legend computer, and Matlab software versions are R2013a.
The present invention program is in emulation experiment, by the power allocation scheme based on optimization routine algorithm, power averaging distribution
Scheme is compared with the present invention program is in positioning performance theoretical value and time complexity.Wherein it is based on optimization routine algorithm
Power allocation scheme constrained optimization problem is converted into by unconstrained problem by method of inner penalty function algorithm, then pass through newton
Method is met former constraints, tends to the optimal solution of former problem, that is, the power allocation scheme needed and side of the present invention
Case is compared.
When Fig. 3 is comprising the single unknown node that coordinate is (0,0), curve map that the SPEB of target changes with anchor node.
It can be drawn by curve in figure, as anchor node number increases within the specific limits, the SPEB of the unknown node of three kinds of algorithms is
It is gradually reduced, but the SPEB that the present invention program is smaller than what mean allocation can obtain.In addition, the present invention program and routine
Optimized algorithm has similar power distribution result.
Fig. 4 is in the alignment system of (0,0), the power of the present invention program and optimization routine divides in single unknown node coordinate
Compare with Riming time of algorithm.The influence that anchor node number increases to inventive algorithm run time is smaller, and conventional algorithm is transported
The row time but increases and increased dramatically with anchor node number, and it is because the search procedure of genetic algorithm is to be based on this result occur
The evaluation information of target function value, basic thought is simple, and fitness function information is only used in search procedure, and without leading
Number or other auxiliary informations, conventional algorithm try to achieve optimal solution with penalty function method, tend to the optimal solution of former problem, while meet again
Original constraints, but because it is to converge on optimum point, newton by solving a series of extreme point of unconstrained problems
Method needs to calculate Hession inverse of a matrix matrixes, and calculating process is complicated, calculating speed is affected, especially with optimization
Problem scale and complexity gradually increase, and calculating speed is decreased obviously.Therefore the calculating speed of the present invention program is substantially than normal
It is fast to advise algorithm, especially when anchor node number is more, this advantage is more obvious.
Fig. 5 describes two unknown node coordinates in the case of (0,0) and (- 4,2), the present invention program, conventional calculation
Method and average distribution system these three schemes carry out power distribution, system unknown node SPEB sums with anchor node number change
Change.It can be seen that it is proposed by the present invention based on the power allocation scheme of multiple target than the system obtained by average power allocation algorithm
SPEB sums it is substantially much smaller, obtained power allocation scheme significantly improves the positioning precision of system unknown node.
Fig. 6 describes two unknown node coordinates in the case of (0,0) and (- 4,2), the present invention program and conventional calculation
The comparison of method run time.It is similar with the situation of change of run time required for algorithm in the alignment system of single unknown node,
Gradually increase with the model and complexity of optimization problem, conventional algorithm due to being related to the computings such as Hession matrix inversions,
It is greatly affected calculating speed, run time is as anchor node number increases and quickly increases.The evaluation of the present invention program
Information is only relevant with target function value, is not related to other auxiliary operations such as derivation, so run time is by anchor node number of variations
Influence it is smaller.Therefore, in the alignment system of multiple target, the calculating speed of the present invention program faster, and by anchor node number shadow
Sound is smaller.
To those skilled in the art, according to above-mentioned implementation type can be easy to association other the advantages of and deformation.
Therefore, the present invention is not limited to above example, and it carries out detailed, exemplary as just example to a kind of form of the present invention
Explanation.In the range of without departing substantially from present inventive concept, those skilled in the art are according to above-mentioned instantiation, by various etc.
With the technical scheme obtained by replacing, should be included within scope of the presently claimed invention and its equivalency range.
Claims (5)
1. a kind of power distribution method of the WiMAX alignment system based on genetic algorithm, it is characterised in that including following step
Suddenly：
Step 1：The system model of WiMAX positioning network is established, and theory analysis is carried out to system model；
Step 2：Abbreviation represents the mathematic(al) representation of square position error lower bound of unknown node positioning precision；With unknown node
The object function of square position error lower bound as optimization, the restrictive condition of actual anchors node transmitting power is constraints, often
The mathematical problem of power distribution is abstracted as function optimization problem by individual anchor node power to be allocated as variable；
Step 3：The Optimized model of the power distribution of single goal alignment system is established, it is excellent to solve this using the genetic algorithm of belt restraining
Change model；
Step 4：Establish the Optimized model of the power distribution of multi-target positioning system；Using beeline ideal point method by multiple target
Optimization problem is changed into single-object problem；Then the mathematical model of optimization of definition is solved using genetic algorithm, is obtained more
Power allocation scheme under target positioning.
2. a kind of power distribution method of WiMAX alignment system based on genetic algorithm according to claim 1, its
It is characterised by, the step 1 includes herein below：If the position coordinates of unknown node is p=[x, y]T, anchor node j coordinate
For Pj=[xj,yj]T(j∈Nb), wherein NbThe set of anchor node is represented, P is position coordinates to be asked, PjFor known location coordinate；
Unknown node receives the signal that j-th of anchor node is sent and is expressed as：
Wherein s (t) is the transmission signal of anchor node,WithThe range value and time delay of jth road reception signal, L are represented respectivelyj
For the multipath number of transmission signal, zj(t) represent that bilateral power spectral density is N0/ 2 additive white Gaussian noise；
Information inequality principle must is fulfilled for according to Square Error matrix, we draw the position estimation error performance of unknown node
Wherein p0For the location estimation of unknown node, JeIt is the Fischer matrix EFIM of equal value on unknown node, can be expressed as
WhereinRepresent k-th of anchor node to the propagation angle between unknown node；Represent location information intensity, wherein εk=1000 be the constant for being more than zero of an expression propagation channel performance；Represent the distance between k-th of anchor node and unknown node；pkFor k-th anchor node
Transmission power；
According to above-mentioned content, square position error circle SPEB in WiMAX positioning network can be obtained：
3. a kind of power distribution method of WiMAX alignment system based on genetic algorithm according to claim 2, its
It is characterised by, the step 2 includes herein below：
Further abbreviation processing is done to formula (4) square position error；Can be in the hope of Fischer matrix of equal value by formula (3)
Inverse Je -1For：
Wherein | | Je| | it is matrix JeDeterminant,
Because the mark of matrix is the sum of all elements on matrix leading diagonal, square position error lower bound SPEB is Fischer of equal value
Inverse of a matrix matrix Je -1Leading diagonal on all elements sum；
So SPEB expression formula can be further represented as：
Wherein
4. a kind of power distribution method of WiMAX alignment system based on genetic algorithm according to claim 3, its
It is characterised by, the step 3 includes herein below：Under single goal location condition, the Optimized model of power distribution can define
For：
WhereinFor the transmission power of each anchor node to be optimized, restrictive condition is：1st, each anchor node
Transmission power be more than or equal to zero and maximum be2nd, the transmission power sum of all anchor nodes is equal to p0；On it can be seen that
State the single objective programming mathematical modeling that model is belt restraining condition.
5. a kind of power distribution method of WiMAX alignment system based on genetic algorithm according to claim 4, its
It is characterised by, the step 4 includes herein below：Under the conditions of Multi-target position, the Optimized model of power distribution can define
For：
……
Wherein It is the constant for being more than zero of an expression propagation channel performance；Represent k-th of anchor node and unknown node NaThe distance between；Represent k-th of anchor node to unknown node NaBetween propagation angle；
Then objective optimization model can be expressed as：
Represent unknown node NaSquare position error circle SPEB；Above-mentioned model be the multiple target of belt restraining condition most
Optimized mathematical model；
In the solution procedure of Model for Multi-Objective Optimization, multi-objective optimization question is changed first with beeline ideal point method
For single-object problem；Then single objective programming mathematical modeling is solved using genetic algorithm, and then obtains determining in multiple target
Power allocation scheme under position；
Beeline ideal point method comprises the following steps that：
Step1：To each component in object functionThe optimal solution under constraints is obtained respectively
Wherein i=1,2 ... NaUnknown node is represented, therefore above-mentioned solutionReferred to as ideal point；
Step2：In order that object function and ideal point are close to i.e. object functionWith ideal pointBetween
" distance " is minimum, constructs following single object optimization function, the Model for Multi-Objective Optimization of formula (8) is converted into single object optimization
Model：
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