CN112040491A - Charger deployment comprehensive cost optimization method for multi-hop wireless charging - Google Patents

Charger deployment comprehensive cost optimization method for multi-hop wireless charging Download PDF

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CN112040491A
CN112040491A CN202010842583.8A CN202010842583A CN112040491A CN 112040491 A CN112040491 A CN 112040491A CN 202010842583 A CN202010842583 A CN 202010842583A CN 112040491 A CN112040491 A CN 112040491A
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徐佳
吴思徐
靳勇
胡苏怡
周凯军
张佳垒
徐力杰
胡洋
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Nanjing University of Posts and Telecommunications
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Abstract

The invention discloses a charger deployment comprehensive cost optimization method facing multi-hop wireless charging, which is characterized by firstly formalizing the problem of comprehensive cost minimization of charger node deployment aiming at the characteristics of a multi-hop charging technology, wherein the comprehensive cost is jointly composed of charger deployment cost and energy cost; and further provides a scheme for optimizing the comprehensive cost, and the scheme is divided into two stages: initializing a charging forest at a first stage to minimize the deployment cost of a charger; the second stage maximizes overall cost savings by adding a charger on the basis of the first stage. The invention solves the problem of the deployment of the multi-hop wireless charging charger and reduces the total comprehensive cost.

Description

Charger deployment comprehensive cost optimization method for multi-hop wireless charging
Technical Field
The invention belongs to the technical field of wireless chargeable sensor networks and optimization algorithms, and particularly relates to a comprehensive cost optimization method for deployment of a charger facing multi-hop wireless charging.
Background
With the development of wireless charging technology, wireless rechargeable sensing networks are widely applied in various fields, such as military target tracking and monitoring, natural disaster rescue, biomedical health monitoring, dangerous environment exploration and the like. Wireless charging technologies can be generally classified into two categories according to their charging characteristics: one is single-hop charging, i.e. the sensor receives power only from the charger; the other is multi-hop charging, that is, the sensor can receive electric energy from the charger and can also forward the electric energy to charge other sensors. By using the multi-hop charging technology, the maximum charging distance can be prolonged, and more sensors can be charged under the condition of the same number of chargers. In practice, for a given sensor network, how many chargers are set to meet the charging requirements of the entire network, and where these chargers should be deployed. The total cost of the charger deployment and subsequent charging includes both the cost of initially deploying the charger nodes and the cost of energy consumed during charging. Based on such practical problems, the present invention proposes to measure the actual economic cost with the combined cost, and proposes a charger deployment scheme to minimize the combined cost. The scheme of the invention can greatly reduce the comprehensive cost.
Disclosure of Invention
The purpose of the invention is as follows: the invention provides a comprehensive cost optimization method for charger deployment facing multi-hop wireless charging, which can solve the problem of charger deployment and reduce comprehensive cost.
The invention content is as follows: a comprehensive cost optimization method for deployment of a charger facing multi-hop wireless charging comprises the following steps:
(1) establishing a charging model, and formalizing a comprehensive cost optimization problem;
(2) an initialization charging forest algorithm is adopted to find out the minimum deployment number and deployment position of the chargers capable of meeting the charging requirements of all the sensor nodes;
(3) and executing a comprehensive cost saving algorithm according to a result obtained by initializing the forest charging algorithm, obtaining an optimized deployment scheme, and calculating comprehensive cost.
Further, the step (1) is realized as follows:
the multi-hop wireless sensor network comprises n sensor nodes, wherein the set of deployable charger positions is V ═ 1,2, …, n }, each deployable charger position is provided with a sensor node i, and the sensor nodes have energy requirements DiThe charger deployment is regarded as that a high-capacity battery is installed on the sensor node, the charger is homogeneous, and the upper limit of the battery capacity is DMAXThe unit energy cost is alpha and represents the cost required by one unit of energy, each charger has a deployment cost beta, which is regarded as lease cost, depreciation cost or installation cost, only the charger can become an energy source, but when the energy requirement of each sensor node is met and redundant energy is obtained, the energy is transmitted to other sensor nodes within the maximum charging distance of the sensor node in a magnetic resonance mode; the sensor nodes have the same maximum charging distance, once the distance between the sensor nodes exceeds the maximum charging distance, the sensor nodes cannot transmit energy, the energy loss ratio is set to be infinite, and the energy loss ratio of multi-hop transmission is the product of the energy loss ratio of each transmission, namely
Figure BDA0002641978610000021
πabIs the energy loss ratio between (a, b), PijIs a path from i to j, (a, b) represents an edge on the path, i, j ∈ V is a source point and an end point on the multi-hop transmission path respectively;
the comprehensive cost optimization problem of charger deployment in formalized multi-hop charging is as follows:
Figure BDA0002641978610000022
and (3) constraint:
Figure BDA0002641978610000023
Figure BDA0002641978610000024
Figure BDA0002641978610000025
Figure BDA0002641978610000026
Figure BDA0002641978610000027
wherein x isijIndicates whether the sensor at location j is being charged by the charger at location i, and if so, xij1, otherwise xij＝0，yiIndicating whether a charger is deployed at position i, and if so, yi1, otherwise y i0, the objective function F in the formula (1) indicates that the composite cost is the sum of the actual energy consumption cost and the charger deployment cost, α is the unit energy cost and indicates the cost required by one unit energy, β is the charger deployment cost and can be the leasing cost, the depreciation cost or the installation cost, the constraint (2) ensures that one sensor node is only provided with energy by one charger, the constraint (3) ensures that the total energy consumption on the charging tree does not exceed the upper limit of the battery capacity, the constraint (4) ensures that the tree is generated, and the constraints (5) and (6) ensure that x is the sum of the actual energy consumption cost and the charger deployment costij，yiIs a boolean value.
Further, the step (2) comprises the steps of:
(2.1) obtaining a charging network graph G (V, E) according to an energy loss ratio between nodes in a sensor network, wherein V is a set of positions where the sensor nodes are located, E is a set of edges, if the loss ratio between two sensor nodes is not infinite, one edge exists, otherwise, no edge exists between two points, and the value on the edge is the energy loss ratio, formally initializing the charging forest problem as follows, wherein a formula (7) represents the minimum number of chargers:
Figure BDA0002641978610000031
and (3) constraint: formula (2) -formula (6)
(2.2) for each deployable charger position i ∈ V, deploy the charger at i, let the charging tree rooted at i be Ti＝(Vi,Ei) In which V isiFor charging tree TiSet of intermediate sensor node positions, EiFor charging tree TiThe first two sets are both empty sets;
(2.3) initializing the uncovered position set VuCharging forest ═ V
Figure BDA0002641978610000032
(2.4) if
Figure BDA0002641978610000033
Executing the step (2.5) to the step (2.7), otherwise executing the step (2.8);
(2.5) for each i ∈ VuFinding the root position with total energy consumption not exceeding DMAXContains the charging tree T with the largest number of sensor nodesi'；
(2.6) finding the tree containing the most sensor nodes from all the charging trees formed in the step (2.5), and marking the tree as T'i；
(2.7) delete sensor nodes in the tree from the set of uncovered nodes, i.e. Vu＝Vu\ViUpdating the charging Tree into the charging forest, i.e. Ti＝T'i；
(2.8) Return-Charge SensenForest (forest)
Figure BDA0002641978610000034
The root positions of all charging trees in the charging forest form a charger set C1Indicating that the charger is set at these positions.
Further, the step (25) comprises the steps of:
(251) initializing remaining energy D of the chargerr＝DMAX；
(252) Judging whether the residual energy is enough to charge the sensor node at the position i, and if so, enabling the Vi＝Vi∪{i}；Vu＝Vu\{i}；Dr＝Dr-Di(ii) a Otherwise, returning to the charging tree Ti；
(253) If D isrIf > 0, repeating the steps (1.1.5.4) to (1.1.5.5); otherwise, returning to the charging tree Ti；
(254) Finding out the sensor node with the minimum energy consumption without setting the node at joSo as to minimize the energy consumed, i.e., piijiπjijoDjoHas the smallest value of (a), wherein jiCharging the current tree TiThe position of the sensor node in (j)oIs the current charging tree TiThe positions of the outer sensor nodes;
(255) if the remaining energy of the charger is sufficient to charge the battery at joCharging of sensor nodes, i.e. Dr-πijiπjijoDj0If not less than 0, let Dr＝Dr-πijiπjijoDj0，πijo＝πijiπjijo，Vi＝Vi∪{jo}，Vu＝Vu\{jo}，Ei＝Ei∪(ji,jo) (ii) a Otherwise, return to charging tree Ti。
Further, the step (3) includes the steps of:
(31) defining the integrated cost saving function Δ F as:
Figure BDA0002641978610000041
adding collections
Figure BDA0002641978610000047
The sensor node in the system is a charger, after a new charger is added to a certain charging tree, the new charger and the child nodes thereof in the charging tree are added together to obtain a new charging tree, the saved comprehensive cost function is the cost saving obtained by subtracting the total energy cost of the new forest from the total energy cost of the old forest, and the added value of the deployment cost caused by adding the charger is subtracted, the aim is to maximize the comprehensive cost saving function, and in order to ensure that the function value is non-negative, the method defines
Figure BDA0002641978610000042
Since β n is a constant, maximizing Δ F is equivalent to maximizing
Figure BDA0002641978610000043
(32) Initializing a newly added charger set
Figure BDA0002641978610000044
Newly-added alternative charger set Y ═ V \ C1；
(33) For each position w ∈ V \ C1Repeatedly executing the step (34) to the step (35);
(34) calculate out
Figure BDA0002641978610000045
And
Figure BDA0002641978610000046
(35) if a is equal to b, adding the node into a newly added charger set, namely X is equal to X, U { w }, and updating the charging forest; otherwise, two different operations are performed with a certain probability:
performing an addition operation with a/(a + b) probability, that is, X ═ X utou { w }, and updating the charged forest;
② do not add with b/(a + b) probability, i.e. Y ═ Y \ w }.
Has the advantages that: compared with the prior art, the invention has the beneficial effects that: 1. the invention formalizes the problem of charger deployment in multi-hop wireless charging and provides a comprehensive cost calculation formula; an initialized charging forest algorithm and a comprehensive cost saving algorithm are provided, the problem of charger deployment is solved, and the total comprehensive cost is reduced; 2. the time complexity of the initialized charging forest algorithm is O (n)5) Approximate ratio is lnn + 1; the overall cost saving algorithm time complexity is O (n)2) The approximate ratio to the equivalence problem of maximizing the composite cost saving function is 1/2.
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Fig. 1 is a schematic diagram of a deployment scenario of a charger facing multi-hop charging according to the present invention;
FIG. 2 is a flow chart of an initialized forest charging algorithm according to the present invention;
FIG. 3 is a flowchart of solving a maximum charging tree in the present invention;
FIG. 4 is a flow chart of the integrated cost savings algorithm of the present invention.
Detailed Description
In order to more clearly illustrate the technical solution of the present invention, the present invention will be further described below; obviously, the following description is only a part of the embodiments, and it is obvious for a person skilled in the art to apply the technical solutions of the present invention to other similar situations without creative efforts; in order to more clearly illustrate the technical solution of the present invention, the technical solution of the present invention is further described in detail below with reference to the accompanying drawings.
First a set of concepts is defined:
set coverage: given a full set U, a set S containing n elements, and a subset of the full set U. The set coverage problem is to find a minimum subset of S such that their union is equal to the full set U.
Sub-model functions: a submodular function is a set function that increases a single element as the number of elements in the input set increasesThe difference in function increments caused to the set of inputs is reduced. The formalization is described as: given a finite set U, a set of real-valued functions
Figure BDA0002641978610000051
If f (A ≦ W }) -f (A) ≧ f (B ≦ W }) -f (B) for all
Figure BDA0002641978610000052
And w ∈ V \ B are both true, then f is called the submodular function.
The invention relates to a comprehensive cost optimization method for deploying a charger facing multi-hop wireless charging, which specifically comprises the following steps:
step 1: and establishing a charging model, and obtaining a network graph used as algorithm input according to the energy loss ratio between any two multi-hop sensor nodes and the limit of the capacity of a rechargeable battery, thereby formalizing the comprehensive cost optimization problem.
The invention considers a multi-hop wireless sensing network with n sensor nodes, the position set of the deployable chargers is set as V ═ 1,2, …, n }, each position of the deployable chargers is provided with a sensor node i, and the sensor nodes have an energy requirement DiIs more than or equal to 0. In the multi-hop infinite sensor network, because the sensor nodes have the function of energy transmission, the charger deployment can be regarded as that high-capacity batteries are installed on the sensor nodes, the charger is regarded as homogeneous, and the upper limit of the battery capacity is DMAXThe unit energy cost is alpha, which represents the cost required by one unit of energy, each charger has a deployment cost beta, which can be regarded as lease cost, depreciation cost or installation cost, only the charger can become an energy source, but when the energy requirement of each sensor node is met and surplus energy is obtained, the energy can be transmitted to other sensor nodes within the maximum charging distance of the sensor node in a magnetic resonance mode.
When the distance between any two sensor nodes a, b and V at the position of the deployable charger is less than or equal to the maximum charging distance, the energy loss ratio pi isabNot less than 1, the energy loss ratio depends on the distance between two sensor nodesThe invention considers that the energy loss ratio of energy transmission between any two sensor nodes a and b is a constant and the energy loss ratio piab≧ 1 means that if the sensor node b has an energy demand of 1, energy is transmitted from a to b, and pi is actually required to be transmittedabAmong them is (pi)ab-1) energy is lost in transmission, while energy is transmitted from b to a with the same transmission loss ratio, i.e.. piab＝πbaIn particular, the energy loss ratio of the sensor node a to itself is piaaThe invention considers that the sensor nodes have the same maximum charging distance, and once the distance between the sensor nodes exceeds the maximum charging distance, the energy transmission between the sensor nodes can not be carried out, the energy loss ratio is set to be infinite, because the storage forwarding mode is adopted, each energy transmission is independent, and therefore, the energy loss ratio of the multi-hop transmission is the product of the energy loss ratio of each transmission, namely the product of the energy loss ratios of each transmission
Figure BDA0002641978610000061
PijFor a path from i to j, (a, b) denotes an edge on the path, i, j e V being the source and destination points on the multi-hop transmission path, respectively.
The comprehensive cost optimization problem of charger deployment in formalized multi-hop charging is as follows:
Figure BDA0002641978610000062
and (3) constraint:
Figure BDA0002641978610000063
Figure BDA0002641978610000064
Figure BDA0002641978610000065
Figure BDA0002641978610000066
Figure BDA0002641978610000067
wherein x isijIndicates whether the sensor at location j is being charged by the charger at location i, and if so, xij1, otherwise xij＝0，yiIndicating whether a charger is deployed at position i, and if so, yi1, otherwise y i0, the objective function F in the formula (1) indicates that the composite cost is the sum of the actual energy consumption cost and the charger deployment cost, α is the unit energy cost and indicates the cost required by one unit energy, β is the charger deployment cost and can be the leasing cost, the depreciation cost or the installation cost, the constraint (2) ensures that one sensor node is only provided with energy by one charger, the constraint (3) ensures that the total energy consumption on the charging tree does not exceed the upper limit of the battery capacity, the constraint (4) ensures that the tree is generated, and the constraints (5) and (6) ensure that x is the sum of the actual energy consumption cost and the charger deployment costij，yiIs a boolean value.
Step 2: and finding out the minimum deployment number and deployment position of the chargers capable of meeting the charging requirements of all the sensor nodes by adopting an initialized charging forest algorithm. The charging forest is a set consisting of charging trees, and the charging trees are nodes in the trees, wherein one charger is used as an energy source, and energy is transmitted to the nodes in the trees according to edges on the trees.
And (2.1) obtaining a charging network graph G (V, E) according to the energy loss ratio among the nodes in the sensor network. Wherein V is a set of positions where sensor nodes are located, E is a set of edges, if a loss ratio between two sensor nodes is not infinite, there is an edge, otherwise, there is no edge between two points, a value on the edge is an energy loss ratio, the formalized initialization charging forest problem is as follows, where formula (7) represents a minimum number of chargers:
Figure BDA0002641978610000071
and (3) constraint: formula (2) -formula (6)
(2.2) for each deployable charger position i ∈ V, deploy the charger at i, let the charging tree rooted at i be Ti＝(Vi,E i) In which V isiFor charging tree TiSet of intermediate sensor node positions, EiFor charging tree TiThe first two sets are empty sets.
(2.3) initializing the uncovered position set VuCharging forest ═ V
Figure BDA0002641978610000072
(2.4) if
Figure BDA0002641978610000073
And (5) executing the step (2.5) to the step (2.7), otherwise, executing the step (2.8).
(2.5) for each i ∈ VuFinding the root position with total energy consumption not exceeding DMAXContains the charging tree T with the largest number of sensor nodesi'。
1) Initializing remaining energy D of the chargerr＝DMAX；
2) Judging whether the residual energy is enough to charge the sensor node at the position i, and if so, enabling the Vi＝Vi∪{i}；Vu＝Vu\{i}；Dr＝Dr-Di(ii) a Otherwise, returning to the charging tree Ti；
3) If D isrIf > 0, repeatedly executing the step 4) to the step 5); otherwise, returning to the charging tree Ti；
4) Finding out the sensor node with the minimum energy consumption without setting the node at joSo as to minimize the energy consumed, i.e., piijiπjijoDjoHas the smallest value of (a), wherein jiCharging the current tree TiThe position of the sensor node in (j)oIs the current charging tree TiThe positions of the outer sensor nodes;
5) if the remaining energy of the charger is sufficient to charge the battery at joCharging of sensor nodes, i.e. Dr-πijiπjijoDj0If not less than 0, let Dr＝Dr-πijiπjijoDj0，πijo＝πijiπjijo，Vi＝Vi∪{jo}，Vu＝Vu\{jo}，Ei＝Ei∪(ji,jo) (ii) a Otherwise, return to charging tree Ti。
(2.6) finding the tree containing the most sensor nodes from all the charging trees formed in the step (2.5), and marking the tree as T'i。
(2.7) delete sensor nodes in the tree from the set of uncovered nodes, i.e. Vu＝Vu\ViUpdating the charging Tree into the charging forest, i.e. Ti＝T'i。
(2.8) Return to charging forest
Figure BDA0002641978610000081
The root positions of all charging trees in the charging forest form a charger set C1Indicating that the charger is set at these positions.
And step 3: and executing a comprehensive cost saving algorithm according to a result obtained by initializing the forest charging algorithm, obtaining an optimized deployment scheme, and calculating comprehensive cost.
(3.1) defining the integrated cost saving function Δ F as:
Figure BDA0002641978610000082
adding collections
Figure BDA0002641978610000083
The sensor node in (1) is a chargerAfter the charger is added into a certain charging tree, a new charger and child nodes thereof in the charging tree are added together to obtain a new charging tree, the saved comprehensive cost function is the cost saving obtained by subtracting the total energy cost of the new forest from the total energy cost of the old forest, and the added value of the deployment cost caused by adding the charger is subtracted, the aim is to maximize the comprehensive cost saving function, and in order to ensure that the function value is not negative, the comprehensive cost saving function is defined
Figure BDA0002641978610000084
Since β n is a constant, maximizing Δ F is equivalent to maximizing
Figure BDA0002641978610000085
(3.2) initializing the newly added charger set
Figure BDA0002641978610000086
Newly-added alternative charger set Y ═ V \ C1；
(3.3) for each position w ∈ V \ C1Repeatedly executing the step (3.4) to the step (3.5);
(3.4) calculating
Figure BDA0002641978610000087
And
Figure BDA0002641978610000088
(3.5) if a is equal to b, adding the node into a newly added charger set, namely, X is equal to X, U, w, and updating the charging forest; otherwise, two different operations are performed with a certain probability:
performing an addition operation with a/(a + b) probability, that is, X ═ X utou { w }, and updating the charged forest;
② do not add with b/(a + b) probability, i.e. Y ═ Y \ w }.
As shown in fig. 1, the charger deployment cost and the energy cost constitute a comprehensive cost of a charging scenario, and a problem of charger deployment in multi-hop charging is solved with the goal of minimizing the comprehensive cost of the entire charging scenario; taking a multi-hop charging sensor network as an example, the comprehensive cost optimization method for charger deployment in multi-hop wireless charging specifically comprises the following steps:
according to the energy loss ratio between any two multi-hop sensors and the limit of the capacity of a rechargeable battery, an initialized charging forest algorithm is adopted to find out the minimum deployment number and the deployment position of the charger nodes which can meet the charging requirements of all the sensors, wherein the charging forest is a set formed by a charging tree, the charging tree is formed by taking one charger as an energy source and transmitting energy in a tree-shaped mode, and the energy is transmitted according to the edges of the tree;
assuming that a sensor node set V is {1,2,3,4,5,6,7}, energy loss ratios between sensor nodes are: pi12＝1.1，π14＝1.5，π17＝1.2，π23＝1.2，π45＝1.3，π47＝1.3，π562.2, the energy loss ratio between the same positions is 1, and the quantity loss ratio between other sensor nodes is infinite;
the energy requirement of each sensor node is as follows: d10.2 kWh, D20.3 kWh, D30.3 kWh, D40.5 kWh, D50.2 kWh, D60.3 kWh, D7When the kilowatt hour is 0.5 kilowatt hour, set DMAX1 kilowatt hour, setting alpha to be 0.5 yuan/kilowatt hour, and setting beta to be 1 yuan/unit;
and executing the comprehensive cost saving algorithm according to the result obtained by initializing the forest charging algorithm to obtain an optimized deployment scheme.
As shown in fig. 2, the specific steps of initializing the forest charging algorithm are as follows:
obtaining a charging network graph G (V, E) according to the energy loss ratio among the nodes in the sensor network, wherein the graph is shown in a table 1;
TABLE 1 charging network diagram G (V, E)
Figure BDA0002641978610000091
Figure BDA0002641978610000101
For each i ∈ V, let the charging tree with it as the charger be
Figure BDA0002641978610000102
Initializing uncovered node set VuInitializing a charging forest as shown in table 2, i.e., {1,2,3,4,5,6,7 };
TABLE 2 charging forest
Figure BDA0002641978610000103
If it is
Figure BDA0002641978610000104
For each i ∈ VuFinding the charging tree T 'with the highest number of sensor nodes and with the position as the energy source and total energy consumption not more than 1 kilowatt-hour'i(ii) a Finding out one tree containing most sensor nodes from all charging trees, and marking the tree as T'i. In this example, T'1And T'3All have 3 sensors contained, are the most in the optimal charging tree. One of the charging trees is randomly selected, and T 'is selected in the embodiment'3(ii) a Deleting sensor nodes in the tree from the uncovered node set, i.e. updating VuUpdate the charging tree into charging forest, i.e. T4, 5,6,73＝T'3(ii) a Cycling this operation until
Figure BDA0002641978610000105
Until now.
If it is
Figure BDA0002641978610000106
Returning to the charged forest to obtain the charged forest as shown in the table 4;
as shown in fig. 3, taking i ═ 1 as an example, the charging tree T ″, which includes the largest number of sensor nodes and has total energy consumption of not more than 1 kilowatt hour, of the energy source with position 1 is calculated'1Is the process of step 1) Go to step 5):
1) initializing remaining energy D of the chargerr1 kilowatt-hour;
2) determine if the remaining energy is sufficient to charge the sensor node at location 1, in this embodiment Dr＞D1Then order V1＝{1}；Vu＝{2,3,4,5,6,7}；Dr＝Dr＞D10.8 kilowatt-hour;
3) if D isrIf the value is more than 0, repeatedly executing the steps 4) to 5) to meet the execution condition;
4) finding out sensor node j with minimum energy consumptionoSo as to minimize the energy consumed, i.e.
Figure BDA0002641978610000107
Has the smallest value of (a), wherein jiCharging the current tree TiSensor node j inoIs the current charging tree TiAn outer sensor node;
the sensor node found in this embodiment is jo＝2，
Figure BDA0002641978610000111
When kilowatt hour is reached, the other sensor nodes meeting the conditions are 4 and 7, and the consumed energy ratio is 0.75 kilowatt hour and 0.6 kilowatt hour;
5) if the remaining energy of the charger is sufficient for the sensor node joCharging, i.e.
Figure BDA0002641978610000112
Then order
Figure BDA0002641978610000113
Vi＝Vi∪{jo}，Vu＝Vu\{jo}，Ei＝Ei∪(ji,jo) (ii) a Otherwise, return to charging tree Ti；
0.8 in this example>0.33, the remaining energy of the charger is enough for the sensor node 2 to update Dr0.8-0.33-0.47 kWh, pi12＝1×1.1＝1.1，V1＝{1,2}，Vu＝{3,4,5,6,7}，E1＝{(1,2)}；
Continuing to repeat steps 4) to 5) until DrLess than or equal to 0 to obtain charging tree T'1Of which is V'1＝{1,2,3}。
V was calculated in the same manneruThe charging trees of other nodes in the table 3 show that all the charging trees are obtained;
TABLE 3 charging tree node set
Charging tree Node set
T'1 V'1＝{1,2,3}
T'2 V'2＝{2,1}
T'3 V'3＝{3,2,1}
T'4 V'4＝{4,5}
T'5 V'5＝{5,4}
T'6 V'6＝{6,5}
T'7 V'7＝{7,1}
TABLE 4 charging forest after initialization
Figure BDA0002641978610000114
Figure BDA0002641978610000121
At this time, the charger set C13,4,6,7, and graph G (V, E) also changed, as shown in table 5,
table 5 updated charging network diagram G (V, E)
1 2 3 4 5 6 7
1 1 1.1 1.32 1.5 ∞ ∞ 1.2
2 1.1 1 1.2 ∞ ∞ ∞ ∞
3 1.32 1.2 1 ∞ ∞ ∞ ∞
4 1.5 ∞ ∞ 1 1.3 ∞ 3.7
5 ∞ ∞ ∞ 1.3 1 2.2 ∞
6 ∞ ∞ ∞ ∞ 2.2 1 ∞
7 1.2 ∞ ∞ 3.7 ∞ ∞ 1
The overall cost saving function Δ F at this time is:
Figure BDA0002641978610000122
adding set C2After a new charger is added to a certain charging tree, the new charger and the child nodes thereof in the charging tree together obtain a new charging tree, and the saved comprehensive cost function is the cost saving obtained by subtracting the total energy cost of the new forest from the total energy cost of the old forest and then subtracting the added value of the deployment cost caused by adding the charger.
As shown in fig. 4, the newly added charger set is initialized
Figure BDA0002641978610000123
The set of the alternative newly-added chargers Y is {1,2,5 }; for each position w ∈ V \ C1Calculate out
Figure BDA0002641978610000124
And
Figure BDA0002641978610000125
wherein
Figure BDA0002641978610000126
With respect to the sensor node 1, it is,
Figure BDA0002641978610000127
wherein
Figure BDA0002641978610000128
And the sensor node 1 originally belongs to T3Therefore, the charging tree T1And a charging tree T3The sensor nodes in the set are divided into {3,2,1}, the edges directly upstream of the added chargers are deleted, the edges are (1,2) for the sensor node 1, the original charging tree is divided into two charging trees, and the sensor node 2 is distributed to T3(ii) a Calculating the actual total energy demand, T3Actual energy demand is D3+π32D20.3+1.2 × 0.3-0.66 kwh, wherein D3Is the electric quantity, pi, required by the sensor node 3 itself32D2In order to transmit energy from the sensor node 3 to the electric quantity actually required to be transmitted by the sensor node 2, it is possible to determine T in the same way1Has an actual energy consumption of D1At 0.2 kwh, the sum gives a total demand of 0.66+0.2 at 0.86 kwh; similarly, when the sensor node 1 is not used as a charger, the actual energy consumption of other parts is not changed, and only the original T needs to be obtained3Actual energy consumption of pi13D1+π23D2+D3When the actual energy is saved by 0.924 to 0.86 to 0.064 kwh when the actual energy is 1.32 × 0.2+1.2 × 0.3+0.3 to 0.924 kwh, the equation (1) is used to obtain the actual energy saving
Figure BDA0002641978610000131
Because of the fact that
Figure BDA0002641978610000132
Without any improvement, so
Figure BDA0002641978610000133
Then a is 0 and similarly b is 0.99.
If a is equal to b, adding the node into a newly added charger set, namely X is equal to X, U { w }, and updating the charging forest; otherwise, two different operations are performed with a certain probability:
performing an addition operation with a/(a + b) probability, that is, X ═ X utou { w }, and updating the charged forest;
② do not add with b/(a + b) probability, i.e. Y ═ Y \ w }.
Here, with probability 1 not added, Y ═ 2,5, until all w have been traversed; the final forest charging of this example is still as shown in table 4, since the result obtained by initializing the forest charging algorithm in this embodiment is already good enough; if it is an instance that there is also an optimization space, the end result may change.
The time complexity of the initialized charging forest algorithm is O (n)5) The approximate ratio is lnn + 1: the complexity of the process time for circularly constructing the charging tree by initializing the charging forest algorithm is O (n)2) And the time complexity of each time of constructing a charging tree is O (n)3) So the time complexity of constructing all possible charging trees is O (n)5). According to the definition of the initialized charging forest problem, the initialized charging forest problem is equivalent to the set coverage problem, the initialized charging forest algorithm is designed based on a set coverage greedy algorithm of lnn +1 approximate ratio, and if the maximum charging tree accurately corresponding to each sensor node can be obtained, the initialized charging forest algorithm approximate ratio is also lnn +1. Since step (2.5) can find the charging tree containing the largest number of sensor nodes with each sensor node as a charger, the approximate ratio of the initialized charging forest algorithm is lnn +1.
Comprehensive cost saving algorithm time complexityIs O (n)2) The approximate ratio of the equivalence problem to maximize the composite cost saving function is 1/2: traversing all w e V \ C1The time complexity of (a) is O (n), and the time complexity of each calculation sum is O (n), so the time complexity of the comprehensive cost saving algorithm is O (n)2). According to the comprehensive cost saving function in step (3.1), there are:
Figure BDA0002641978610000141
similarly, there are:
Figure BDA0002641978610000142
depending on the position of w, consider the following two cases:
(1)w∈Vi,i∈C1and U.A. That is, w is not located on the charging tree rooted at the sensor node in B \ A. Thus, at C1U.S. A or C1The same division is found in U.B. Because the change in energy cost only occurs on the tree where w is located, there are:
Figure BDA0002641978610000143
the combinations (9), (10) and (11) include: Δ F (a utou { w }) - Δ F (a) ═ Δ F (B utou { w }) - Δ F (B).
(2)w∈ViI belongs to B \ A. That is, w is located on the charging tree rooted at the sensor node in B \ A. Without loss of generality, consider TiIs at Ti'And dividing a previous subtree. Comprises the following steps:
Figure BDA0002641978610000144
Figure BDA0002641978610000145
for each j ∈ VwDue to PijIs P ofi'jA sub-path having:
πi'j＞πij (14)
the combinations (9), (10), (12), (13) and (14) include:
Δ F (A { U { w }) - Δ F (A) > Δ F (B { U { w }) - Δ F (B). In summary, Δ F (-) is a sub-model function.
And because it is known that there is a submodel maximization approximation algorithm with random linear time, if a submodel function is non-negative, its approximation ratio is 1/2. However, Δ F (-) is likely negative. To solve this problem, let
Figure BDA0002641978610000151
Obviously for any
Figure BDA0002641978610000152
Since β n is a constant, so
Figure BDA0002641978610000153
And is also a sub-mold function. Furthermore, maximizing Δ F (-) is equivalent to maximizing
Figure BDA0002641978610000154
The approximate ratio of the overall cost saving algorithm is 1/2.
The above description is only an example of the present invention, but the protection scope of the present invention is not limited thereto, and any person skilled in the art can substitute or change the technical solution of the present invention and the inventive concept thereof within the scope of the present invention.

Claims (5)

1. A comprehensive cost optimization method for charger deployment in multi-hop wireless charging is characterized by comprising the following steps:
(1) establishing a charging model, and formalizing a comprehensive cost optimization problem;
(2) an initialization charging forest algorithm is adopted to find out the minimum deployment number and deployment position of the chargers capable of meeting the charging requirements of all the sensor nodes;
(3) and executing a comprehensive cost saving algorithm according to a result obtained by initializing the forest charging algorithm, obtaining an optimized deployment scheme, and calculating comprehensive cost.
2. The comprehensive cost optimization method for deployment of chargers in multi-hop wireless charging according to claim 1, wherein the step (1) is implemented as follows:
the multi-hop wireless sensor network comprises n sensor nodes, wherein the set of deployable charger positions is V ═ 1,2, …, n }, each deployable charger position is provided with a sensor node i, and the sensor nodes have energy requirements DiThe charger deployment is regarded as that a high-capacity battery is installed on the sensor node, the charger is homogeneous, and the upper limit of the battery capacity is DMAXThe unit energy cost is alpha and represents the cost required by one unit of energy, each charger has a deployment cost beta, which is regarded as lease cost, depreciation cost or installation cost, only the charger can become an energy source, but when the energy requirement of each sensor node is met and redundant energy is obtained, the energy is transmitted to other sensor nodes within the maximum charging distance of the sensor node in a magnetic resonance mode; the sensor nodes have the same maximum charging distance, once the distance between the sensor nodes exceeds the maximum charging distance, the sensor nodes cannot transmit energy, the energy loss ratio is set to be infinite, and the energy loss ratio of multi-hop transmission is the product of the energy loss ratio of each transmission, namely
Figure FDA0002641978600000011
πabIs the energy loss ratio between (a, b), PijIs a path from i to j, (a, b) represents an edge on the path, i, j ∈ V is a source point and an end point on the multi-hop transmission path respectively;
the comprehensive cost optimization problem of charger deployment in formalized multi-hop charging is as follows:
Figure FDA0002641978600000012
and (3) constraint:
Figure FDA0002641978600000013
Figure FDA0002641978600000014
Figure FDA0002641978600000015
Figure FDA0002641978600000016
Figure FDA0002641978600000021
wherein x isijIndicates whether the sensor at location j is being charged by the charger at location i, and if so, xij1, otherwise xij＝0，yiIndicating whether a charger is deployed at position i, and if so, yi1, otherwise yi0, the objective function F in the formula (1) indicates that the composite cost is the sum of the actual energy consumption cost and the charger deployment cost, α is the unit energy cost and indicates the cost required by one unit energy, β is the charger deployment cost and can be the leasing cost, the depreciation cost or the installation cost, the constraint (2) ensures that one sensor node is only provided with energy by one charger, the constraint (3) ensures that the total energy consumption on the charging tree does not exceed the upper limit of the battery capacity, the constraint (4) ensures that the tree is generated, and the constraints (5) and (6) ensure that x is the sum of the actual energy consumption cost and the charger deployment costij，yiIs a boolean value.
3. The comprehensive cost optimization method for deployment of chargers in multi-hop wireless charging according to claim 1, wherein the step (2) comprises the steps of:
(21) obtaining a charging network graph G (V, E) according to an energy loss ratio between nodes in a sensor network, wherein V is a set of positions where the sensor nodes are located, E is a set of edges, if the loss ratio between two sensor nodes is not infinite, an edge exists, otherwise, an edge does not exist between two points, the value on the edge is the energy loss ratio, formally initializing the charging forest problem as follows, wherein a formula (7) represents the minimum number of chargers:
Figure FDA0002641978600000022
and (3) constraint: formula (2) -formula (6)
(22) For each deployable charger position i ∈ V, deploying the charger at i, and setting a charging tree with i as a root as Ti＝(Vi,Ei) In which V isiFor charging tree TiSet of intermediate sensor node positions, EiFor charging tree TiThe first two sets are both empty sets;
(23) initializing a set of uncovered locations VuCharging forest ═ V
Figure FDA0002641978600000023
(24) If it is
Figure FDA0002641978600000024
Performing steps (25) to (27), otherwise performing step (28);
(25) for each i ∈ VuFinding the root position with total energy consumption not exceeding DMAXContains the charging tree T with the largest number of sensor nodesi'；
(26) Finding the tree containing the most sensor nodes from all the charging trees formed in the step (25),marking this tree as T'i；
(27) Deleting sensor nodes in the tree from the set of uncovered nodes, i.e., Vu＝Vu\ViUpdating the charging Tree into the charging forest, i.e. Ti＝T'i；
(28) Return charging forest
Figure FDA0002641978600000031
The root positions of all charging trees in the charging forest form a charger set C1Indicating that the charger is set at these positions.
4. The comprehensive cost optimization method for charger deployment in multi-hop wireless charging according to claim 3, wherein the step (25) comprises the steps of:
(251) initializing remaining energy D of the chargerr＝DMAX；
(252) Judging whether the residual energy is enough to charge the sensor node at the position i, and if so, enabling the Vi＝ViU{i}；Vu＝Vu\{i}；Dr＝Dr-Di(ii) a Otherwise, returning to the charging tree Ti；
(253) If D isrIf > 0, repeating the steps (1.1.5.4) to (1.1.5.5); otherwise, returning to the charging tree Ti；
(254) Finding out the sensor node with the minimum energy consumption without setting the node at joSo as to minimize the energy consumed, i.e.
Figure FDA0002641978600000032
Has the smallest value of (a), wherein jiCharging the current tree TiThe position of the sensor node in (j)oIs the current charging tree TiThe positions of the outer sensor nodes;
(255) if the remaining energy of the charger is sufficient to charge the battery at joCharging of sensor nodes, i.e.
Figure FDA0002641978600000033
Then order
Figure FDA0002641978600000034
Vi＝Vi∪{jo}，Vu＝Vu\{jo}，Ei＝Ei∪(ji,jo) (ii) a Otherwise, return to charging tree Ti。
5. The comprehensive cost optimization method for deployment of chargers in multi-hop wireless charging according to claim 1, wherein the step (3) comprises the steps of:
(31) defining the integrated cost saving function Δ F as:
Figure FDA0002641978600000035
adding collections
Figure FDA0002641978600000036
The sensor node in the system is a charger, after a new charger is added to a certain charging tree, the new charger and the child nodes thereof in the charging tree are added together to obtain a new charging tree, the saved comprehensive cost function is the cost saving obtained by subtracting the total energy cost of the new forest from the total energy cost of the old forest, and the added value of the deployment cost caused by adding the charger is subtracted, the aim is to maximize the comprehensive cost saving function, and in order to ensure that the function value is non-negative, the method defines
Figure FDA0002641978600000037
Since β n is a constant, maximizing Δ F is equivalent to maximizing
Figure FDA0002641978600000038
(32) Initializing a newly added charger set
Figure FDA0002641978600000039
Newly-added alternative charger set Y ═ V \ C1；
(33) For each position w ∈ V \ C1Repeatedly executing the step (34) to the step (35);
(34) calculate out
Figure FDA0002641978600000041
And
Figure FDA0002641978600000042
(35) if a is equal to b, adding the node into a newly added charger set, namely X is equal to X, U { w }, and updating the charging forest; otherwise, two different operations are performed with a certain probability:
performing an addition operation with a/(a + b) probability, that is, X ═ X utou { w }, and updating the charged forest;
② do not add with b/(a + b) probability, i.e. Y ═ Y \ w }.
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