CN105225016B - Based on the energy distributing method of cooperative game in the cloud computing system of renewable energy supply - Google Patents
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Abstract
The invention discloses a kind of in the cloud computing system of renewable energy supply based on the energy distributing method of cooperative game, mainly comprises the steps that the time interval for calculating the distribution of this energy；Predict the utilisable energy in this time interval；Judge whether energy is sufficient；Energy assignment problem is modeled based on the theory of cooperative game, and is converted to the optimization problem of a belt restraining；The optimization problem is converted to its dual problem；With the gradient projection method solution dual problem.In the present invention, role is broadly divided into user and cloud service provider, and unlike traditional cloud computing, service provider will be in view of the particularity of energy supply while the service of offer.By acquire the electric energy that converts of outside energy be it is unstable, the present invention consider when the electric energy deficiency of conversion how by energy reasonable distribution to each user.In this case, cloud service provider needs to take into account the demand of each user, while being accounted for the cost of oneself, and the present invention models this with cooperative game, and it is converted into the optimization problem of a belt restraining, the solution of the optimization problem is corresponding energy allocation plan.
Description
Technical field
The present invention relates to cloud computing, game theory relevant knowledge more particularly to a kind of energy for comprehensively considering justice with efficiency
Measure allocation plan；Specifically a kind of energy distribution side in the cloud computing system of renewable energy supply based on cooperative game
Method.
Background technique
Renewable energy can be used as the power supply source of computer system, but can generate a series of problems, including electricity in this way
It is sufficient whether, whether the stabilization of electric energy, whether the operation of computer system impacted etc., and domestic and international expert opens with regard to these problems
In-depth study is opened up.Jing Chen devise one by renewable energy energize real-time system, comprehensively consider task and
The characteristic of processor, author propose a static state and dynamic algorithm respectively come when reducing the cut-off of the energy consumption of system and task
Between miss rate.Longbo Huang proposes the network energized by renewable energy, extraneous collected energy stores to one
In a limited battery of capacity, author proposes on a line algorithm to manage collected energy and be reasonably allocated to energy
On each node in network.Shaobo Liu proposes the energy pipe adjustment based on dynamic voltage frequency selection (DVFS)
Method, it is intended to rationally improve the service quality of system simultaneously using extraneous collected energy.Jing Yang studies one by can be more
The communication system of new energy energy supply, due to needing the speed in view of information transmission in communication, author designed one calculation
Method minimizes call duration time under the premise of finite energy.
A novel business prototype of the cloud computing as computer field has become and changes the one of computer usage mode
A important means, major IT giant is in the Cloud Server for building oneself.Cloud computing model includes the service of following level:
Infrastructure services (IaaS), and platform services (PaaS) and software services (SaaS).IaaS(Infrastructure-
As-a-Service): infrastructure services, and consumer can be obtained by Internet from perfect Basis of Computer Engineering facility
Service；PaaS (Platform-as-a-Service): platform services, and PaaS, which actually refers to, makees the platform of research and development of software
For a kind of service, user is submitted to the mode of SaaS, therefore, PaaS is also a kind of application of SaaS mode；SaaS
(Software-as-a-Service): software services, it is a kind of mode by Internet offer software, Yong Huwu
Software need to be bought, but rents the software based on Web, Lai Guanli business operation to provider.
Summary of the invention
The purpose of the present invention is propose for the energy assignment problem in the cloud computing system energized by renewable energy
Comprehensively consider the demand of user and the cost of service provider when carrying out energy distribution, make every effort to one it is not only fair but also effectively
The allocation plan of rate, i.e., a kind of energy allocation plan in the cloud computing system of renewable energy supply based on cooperative game.It is visiting
During the rope program, energy distribution is modeled with cooperative game, and the optimization for being converted into a belt restraining is asked
Topic, the solution of the optimization problem is corresponding energy allocation plan.
The object of the present invention is achieved like this:
Energy distributing method based on cooperative game in a kind of cloud computing system of renewable energy supply, feature is this method packet
Include following steps:
Step 1: the time interval of this energy distribution is determined；
Step 2: predicting the utilisable energy in this time interval, energy source of the utilisable energy as system；
Step 3: judging whether energy is sufficient, four gone to step when inadequate, turns distribution according to need energy when sufficient, and turn step
Rapid six；
Step 4: energy distribution is modeled with game theory, and is converted to the optimization problem of belt restraining, then converted
At dual problem；
Step 5: dual problem is solved with gradient projection method, and verifies the Global Optimality for seeking solution；
Step 6: distribution terminates.
The step 1 specifically includes:
Step A1: minimum energy required for computing system:
Emin(Δ t)=M*min { δisi 2c*Ni|, i=1,2 ..., M }
Wherein: M is the number of server, siFor the frequency of server, NiFor the number of task on server, c is task
Execute frequency, δiFor circuit efficiency factor；
Step A2: most energy required for computing system:
Emax(Δ t)=M*max { δisi 2c*Ni|, i=1,2 ..., M }
Wherein: M is the number of server, siFor the frequency of server, NiFor the number of task on server, c is task
Execute frequency, δiFor circuit efficiency factor；
Step A3: energy distribution time interval Δ t is defined as Eharv(Δ t) is in Emin(Δ t) and Emax(between Δ t)
Time:
Δ t=min Δ t | Eharv(t+Δt)∈[Emin(Δ t), Emax(Δt)]}。
The step 2 specifically includes:
Step B1: the utilisable energy in this time interval predicted are as follows:
Wherein, PharvIt (t) is energy acquisition power.
The step 3 specifically includes:
Step C1: the energy that computing system needs in total:
ECons, i=Ni*Econs(τ, si)
Wherein, Edemand(Δ t) is gross energy required for M server, ECons, iThe energy needed for i-th of server
Amount, Econs(τ, si) it is the energy that each task consumes, when task is s in frequencyiServer on when running；NiIt is taken for i-th
The number of task on business device；
Step C2: judge whether energy is sufficient
Work as Eharv(Δ t) > Edemand(when Δ t), system capacity is abundance, and otherwise system capacity is inadequate.
The step 4 specifically includes:
Step D1: modeling energy assignment problem with game theory, and be converted into the optimization problem of a belt restraining,
That is:
S.t: ∑I=1,2 ..., MEAloc, i=Eavl(Δt)
Wherein, EAloc, iFor the energy distributed on i-th of server, μi 0It handles up for what i-th of server subsistence level met
Amount, M are the number of server, siFor the frequency of server, NiFor the number of task on server, c is the execution frequency of task,
δiFor circuit efficiency factor；
Step D2: the corresponding Lagrangian of the optimization problem are as follows:
Local derviation is asked to Lagrangian, and local derviation is made to be equal to 0, obtains allocation plan are as follows:
Know that allocation plan is Lagrange multiplier (α, βi, γi) function, these variables are sought by its dual problem；
Step D3: willLagrangian is substituted into, its dual problem is obtained are as follows:
s.t.βi>=0, i=1,2 ..., M
The step 5 specifically includes:
Step E1: dual problem is solved with gradient projection method.The initial value of algorithm is provided first, can be obtained after algorithmic statement
To the solution of dual problem, α*, βi *, γi *, substituted intoObtain final energy point
With scheme；Step E2: the Global Optimality of solution is sought in verifying
Firstly, finding out the Hessian matrix of dual problem objective function, i.e., Hessian matrix；Each single item calculates as follows in Hessian matrix:
As it can be seen that each single item is both greater than equal to 0 in Hessian matrix, Hessian matrix positive semidefinite, objective function is in dual problem
Convex function, therefore, the solution α asked by gradient projection*, βi *, γi *It is global optimum, therefore final energy allocation planIt is also global optimum.
The present invention comprehensively considers the situation that energy is sufficient or supplement is sufficient.When energy abundance, according to each server
Actual needs distributes corresponding energy；When energy is inadequate, energy distribution, final energy distribution are carried out using cooperative game
Scheme had not only considered user's physical examination but also had looked after the cost of cloud service provider, met user and clothes simultaneously to a certain extent
The demand of business quotient both sides.
Detailed description of the invention
Fig. 1 is flow chart of the present invention；
Energy when Fig. 2 is 3 servers distributes schematic diagram；
Energy when Fig. 3 is 5 servers distributes schematic diagram；
Energy when Fig. 4 is 8 servers distributes schematic diagram；
When Fig. 5 is 3 servers and changes utilisable energy, pair of the present invention and naive method in terms of fair and efficiency
Compare schematic diagram；
When Fig. 6 is 5 servers and changes utilisable energy, pair of the present invention and naive method in terms of fair and efficiency
Compare schematic diagram；
When Fig. 7 is 8 servers and changes utilisable energy, pair of the present invention and naive method in terms of fair and efficiency
Compare schematic diagram；
When Fig. 8 is 3 servers and changes load, the present invention shows with comparison of the naive method in terms of justice is with efficiency
It is intended to；
When Fig. 9 is 5 servers and changes load, the present invention shows with comparison of the naive method in terms of justice is with efficiency
It is intended to；
When Figure 10 is 8 servers and changes load, the present invention shows with comparison of the naive method in terms of justice is with efficiency
It is intended to；
Figure 11 is the contrast schematic diagram of the present invention and MT scheme in handling capacity；
Figure 12 is the present invention and contrast schematic diagram of the MT scheme in terms of fairness.
Specific embodiment
Below in conjunction with drawings and the specific embodiments, the present invention is described in further detail.
For the present invention using solar energy as energy source, following formula presents the spy that solar panels capacitation changes over time
Property:
Wherein, N (t) is the stochastic variable of 01 Gaussian Profile of obedience.
Task in the present invention, which reaches, has randomness, unpredictability, therefore the load on each server is random
It generates, also constantly changes the validity that the load on each server carrys out proof scheme in test, experiment shows regardless of service
How load on device changes, and the allocation plan mentioned can preferably take into account cloud service provider and user.In addition to this, this hair
In the task averagely execution period having the same in bright, in test to carry out having for verification algorithm there is no the task execution period is changed
Effect property because the execution period of all tasks be it is the same, it is very necessary for changing the parameter not, nevertheless, task
Executing the period can be set to meet actual any value.
Server is also random in the present invention, here at random be embodied in two aspect.First, the number of server is
Random, cloud service provider determines the quantity for the virtual server that can externally provide according to one's own hardware resource situation.
Second, the frequency of server be it is random, in practical cloud service system, expense that the frequency of virtual server is paid by user
It determines, therefore the frequency of server can be handled as a random value.Constantly change the quantity and frequency of server in test
To verify the validity of allocation plan, the results showed that, when number of servers and frequency shift, the allocation plan that is proposed in article
Expected effect is preferably met.
The distribution mechanism proposed in the present invention is intended to consider simultaneously the handling capacity of cloud entirety and the clothes of each user oneself
The handling capacity being engaged on device, a good allocation plan are to take into account the scheme of performance and fairness, how to indicate that a scheme is that have " effect
Rate ", how to indicate that a scheme is " justice ", how to measure the degree that a scheme takes into account performance and fairness.
One scheme is that have the handling capacity of the expression cloud whole system of " efficiency " relatively high, that is, Servers-all gulps down
The summation for the amount of spitting is bigger, and under the premise of limited utilisable energy, system entire throughput is higher, indicates selected scheme
More efficiently, also more meet the wish of service provider.
" fairness " of one scheme should be showed by the handling capacity of all users, if the handling capacity between user
Differ larger, it will cause the discontented of user, this also illustrates that the fairness of the program is poor.Therefore, " fairness " can use use
The variance of family handling capacity indicates that variance is smaller to illustrate that fairness is better, variance is bigger to illustrate that fairness is poorer.
Indicate that a scheme takes into account degree to fair and efficiency in the present invention with fairness/efficiency, the value is smaller, then
It is better to illustrate to take into account, and the value is bigger to illustrate that scheme does not take into account service provider and user well.
The advantages of in order to illustrate the allocation plan proposed in the present invention, experimental section will carry out according to the following steps:
1. verifying, in the case of different utilisable energy, different loads, the method for proposition can converge to one entirely
Office's optimal solution, the i.e. allocation plan based on cooperative game can obtain a unique solution.
2. the load in fixed server changes utilisable energy, and in different processor number, the feelings of different task scale
Under shape, verifying proposes the validity of distribution method.
3. fixed utilisable energy, changes server load, and in different processor number, the situation of different task scale
Under, verify the validity of proposition method scheme.
4. the method for proposition and the scheme with maximum throughput are compared.
Embodiment
Step 1: determine that this energy distributes time interval.In implementation process of the present invention, using 8 servers, 50-150
A task carrys out presentation process.
Frequency on server are as follows:
| Server1 | Server2 | Server3 | Server4 | Server5 | Server6 | Server7 | Server8 |
| 2.2 | 2.3 | 2.1 | 2.5 | 2.4 | 2.5 | 2.6 | 2.8 |
Circuit factor delta on serveriAre as follows:
| δ1 | δ2 | δ3 | δ4 | δ5 | δ6 | δ7 | δ8 |
| 13*10-9 | 12*10-9 | 11*10-9 | 11*10-9 | 14*10-9 | 15*10-9 | 11*10-9 | 13*10-9 |
Load in service, i.e. task number are as follows:
| Load1 | Load2 | Load3 | Load4 | Load5 | Load6 | Load7 | Load8 |
| 70 | 80 | 100 | 120 | 130 | 110 | 90 | 80 |
By Emin(Δ t)=M*min { δisi 2c*Ni|, i=1,2 ..., M } required minimum energy is calculated, by Emax
(Δ t)=M*max { δisi 2c*Ni|, i=1,2 ..., M } the required most energy of calculating, and by
Δ t=min Δ t | Eharv(t+Δt)∈[Emin(Δ t), Emax(Δ t)] } calculate this energy distribution the time between
It is divided into 130 minutes.
Step 2: formula is usedThe utilisable energy for predicting this is 14580J.
Step 3: formula is usedCalculating energy actually required is 18500J, therefore,
Utilisable energy is inadequate, is allocated using the allocation plan based on game theory.
Step 4: dual problem and basis are solved by gradient projection methodIt calculates
Final allocation plan out are as follows: 1250J, 2500J, 2100J, 2500J, 1000J, 2500J, 1750J, 990J.
Step 5: the uniqueness of allocation plan is verified
Firstly, generating the number of server at random, the load and utilisable energy on each server also give at random, not
It is disconnected to change these parameters, constantly change the initial value of algorithm, whether verification algorithm converges to an optimal solution.
Respectively in the case of difference: three servers, task scale are 0-20；Five servers, task scale are 20-
50；Eight servers, task scale are 50-150, observe the relationship of different initial value and gradient projection the number of iterations.
Table 1
As shown in table 1, when aerver is 3, changing the initial value of gradient projection, the number of iterations is changed,
But final energy allocation plan is consistent, as shown in Figure 2.
When server number is 5,8, change the initial value of gradient projection, final energy allocation plan is also unique
, it is as shown in Figure 3 and Figure 4 respectively.
As it can be seen that the allocation plan proposed in the present invention is the scheme of a global optimum.
Step 6: change energy estimate methods load, the validity of the allocation plan proposed in the verifying present invention
Firstly, generating the load on server number and server at random, and the load in fixed server, change outer
Boundary's utilisable energy, and be compared with the naive method of distribution according to need, verify the validity to propose a plan.It is compared each time
When, the number of server is fixed as 3,5,8 respectively, and the load on each server is kept not change, changed simultaneously
Become extraneous utilisable energy.
Fig. 5 is three servers, and when task scale is 0-20, extraneous utilisable energy is respectively 100J, 110J,
Energy is allocated in the case where 120J, 130J, 140J, 150J, ordinate indicates fairness/efficiency ratio.It distributes
At justice/efficiency ratio later, is calculated, light column is naive method in figure, and dark bars column is distribution side of the invention
Case, it can be seen that the scheme proposed is strictly better than naive method.
Fig. 6 is five servers, and task scale is 20-50, and outside energy is respectively 650J, 700J, 750J, 800J,
Distribute energy in the case of 850J, 900J with naive method and the present invention respectively, ordinate indicates fairness/efficiency ratio
Value, can significantly find out, scheme proposed by the present invention is to fair taking into account degree and to be strictly better than naive method with efficiency.
Fig. 7 is eight servers, and task scale is 50150, and extraneous utilisable energy is respectively 3500J, 4000J, 4500J,
In the case of 5000J, 5500J, 6000J, energy distribution is carried out with naive method and scheme proposed by the present invention respectively, indulges and sits
Mark indicates fairness/efficiency ratio, it is not difficult to find out that, energy allocation plan of the invention has preferable effect.
It can be seen from Fig. 5, Fig. 6, Fig. 7 when number of servers and task scale are fixed, and outside energy changes, it is based on
The allocation algorithm of cooperative game can take into account performance and fairness well.
Step 7: change load fixed energies, the validity of the allocation plan proposed in the verifying present invention
Firstly, generating server number and utilisable energy at random, constantly change the load on server, and have in system
In the case of the server of different number, " validity " and " fairness " of proof scheme.
Fig. 8 is 3 servers, respectively (load is as shown in the figure), the comparison feelings of two schemes in different loads
Condition, You Tuzhong is it is found that allocation plan is strictly better than naive method in the present invention.
Fig. 9 is 5 servers, and the comparison under different loads situation cannot be shown in figure since load data is more
Come, therefore load data is enumerated in table 2.By in figure it is found that allocation plan in the present invention is better than naive method.
25 servers of table, different loads
| Load1 | Load2 | Load3 | Load4 | Load5 | Load6 |
| 20,25,30,33,40 | 20,28,32,25,45 | 23,30,35,38,50 | 20,32,40,45,50 | 22,34,42,41,48 | 24,33,39,40,45 |
Figure 10 is 8 servers, and the comparison under different loads situation, load data sieve is listed in Table 3 below.By in figure it is found that
Allocation plan based on game theory is better than naive method.
38 servers of table, different loads
Step 8: comparing with the scheme (MT) with maximum throughput, and the validity of allocation plan will in the verifying present invention
Scheme in the present invention is carried out with a scheme (maximum throughput strategy, MT) with maximum throughput
Comparison；It is square directly proportional due to the energy consumption of individual task and the length of task and frequency, and task all in the present invention
Length be it is identical, therefore, if energy priority is assigned on the server of low frequency, system can obtain higher handle up
Amount, because same energy will support more task runs.Two schemes will carry out in validity and fairness respectively
Comparison.
Figure 11 is the comparison diagram of the corresponding handling capacity of two schemes (comparison of validity), and abscissa is utilisable energy,
Ordinate indicates handling capacity, EneedTo complete the minimum energy that all required by task are wanted.As can be seen from Figure, work as utilisable energy
Less than EneedWhen, handling capacity outline of the invention is lower than MT scheme, and reason is provided in the preceding paragraph.When utilisable energy gradually increases
Added-time, throughput of system also will increase, and when utilisable energy is more than or equal to EneedWhen, two kinds of energy allocation plans are corresponding to handle up
It measures equal.
Figure 12 is comparison diagram of two kinds of allocation plans in terms of fairness, indicates fair with standard deviation in comparison
Property, because the numerical value of variance is excessive, ordinate indicates standard deviation.Compared with MT scheme, the scheme in the present invention has preferable
Fairness, this is because every time carry out energy distribution when, all more fair partitions energy on each server, and the side MT
Case is only merely that will measure preferentially to be assigned to above the server of low frequency, and therefore, MT scheme does not ensure that fairness.When available
Energy is more than or equal to EneedWhen, the two fairness having the same, because the allocation plan of the two is identical at this time.
The present invention is reasonably allocated to each processor under the premise of finite energy, by energy, so that is individually handled gulps down
The amount of spitting is higher, while whole handling capacity is also relatively high.
Claims (5)
1. the energy distributing method based on cooperative game in a kind of cloud computing system of renewable energy supply, which is characterized in that the party
Method the following steps are included:
Step 1: the time interval of this energy distribution is determined；
Step 2: predicting the utilisable energy in this time interval, energy source of the utilisable energy as system；
Step 3: judging whether energy is sufficient, four gone to step when inadequate, turns distribution according to need energy when sufficient, and go to step six；
Step 4: modeling energy distribution with game theory, and be converted to the optimization problem of belt restraining, then converts it into pair
Even problem；
Step 5: dual problem is solved with gradient projection method, and verifies the Global Optimality for seeking solution；
Step 6: distribution terminates；Wherein:
The step 4 specifically includes:
Step D1: energy assignment problem is modeled with game theory, and is converted into the optimization problem of a belt restraining, it may be assumed that
s.t:∑I=1,2 ..., MEaloc,i=Eavl(Δt)
Wherein, Ealoc,iFor the energy distributed on i-th of server, μi 0For the handling capacity that i-th of server subsistence level meets, M
For the number of server, siFor the frequency of server, NiFor the number of task on server, c is the execution frequency of task, δiFor
Circuit efficiency factor；
Step D2: the corresponding Lagrangian of the optimization problem are as follows:
Local derviation is asked to Lagrangian, and local derviation is made to be equal to 0, obtains allocation plan are as follows:Know that allocation plan is Lagrange multiplier (α, βi, γi) function, it is right by its
Even problem seeks these variables；Wherein, Ealoc,iFor the energy distributed on i-th of server, μi 0For the minimum need of i-th of server
The handling capacity to be met, M are the number of server, siFor the frequency of server, NiFor the number of task on server, c is task
Execution frequency, δiFor circuit efficiency factor；
Step D3: willLagrangian is substituted into, its dual problem is obtained are as follows:
s.t.βi>=0, i=1,2 ..., M
Wherein, M is the number of server, siFor the frequency of server, c is the execution frequency of task, δiFor circuit efficiency factor,
NiFor the number of task on server.
2. energy distributing method as described in claim 1, which is characterized in that the step 1 specifically includes:
Step A1: minimum energy required for computing system:
Emin(Δ t)=M*min { δisi 2c*Ni|, i=1,2 ..., M }
Wherein: M is the number of server, siFor the frequency of server, NiFor the number of task on server, c is the execution of task
Frequency, δiFor circuit efficiency factor；
Step A2: most energy required for computing system:
Emax(Δ t)=M*max { δisi 2c*Ni|, i=1,2 ..., M }
Wherein: M is the number of server, siFor the frequency of server, NiFor the number of task on server, c is the execution of task
Frequency, δiFor circuit efficiency factor；
Step A3: energy distribution time interval Δ t is defined as Eharv(Δ t) is in Emin(Δ t) and Emax(time between Δ t):
Δ t=min Δ t | Eharv(t+Δt)∈YEmin(Δt),Emax(Δt)]}。
3. energy distributing method as described in claim 1, which is characterized in that the step 2 specifically includes:
Step B1: the utilisable energy in this time interval predicted are as follows:
Wherein, Δ t is the time interval of this energy distribution, and t is current time, PharvIt (t) is energy acquisition power, Pharv(Δ
It t) is time interval energy acquisition power.
4. energy distributing method as described in claim 1, which is characterized in that the step 3 specifically includes:
Step C1: the energy that computing system needs in total:
Econs,i=Ni*Econs(τ,Si)
Wherein, Edemand(Δ t) is gross energy required for M server, Econs,iFor the energy that i-th of server needs, Econs
(τ,si) it is the energy that each task consumes, when task τ is s in frequencyiServer on when running；NiFor on i-th of server
The number of task；Δ t is the time interval of this energy distribution, and M is the number of server, δiFor circuit efficiency factor, c is to appoint
The execution frequency of business；
Step C2: judge whether energy is sufficient
Work as Eharv(Δ t) > Edemand(when Δ t), system capacity is abundance, and otherwise system capacity is inadequate；
Wherein, Eharv(Δ t) is utilisable energy, Edemand(Δ t) is required gross energy.
5. energy distributing method as described in claim 1, which is characterized in that the step 5 specifically includes:
Step E1: dual problem is solved with gradient projection method；The initial value of algorithm is provided first, can be obtained after algorithmic statement pair
The solution of even problem, α*, βi *, γi *, substituted intoObtain final energy distribution side
Case；
Wherein, Ealoc,iFor the energy distributed on i-th of server, siFor the frequency of server, c is the execution frequency of task, δi
For circuit efficiency factor；
Step E2: the Global Optimality of solution is sought in verifying
Firstly, finding out the Hessian matrix of dual problem objective function, i.e. function Hessian matrix；Each single item calculates such as in Hessian matrix
Under:
As it can be seen that each single item is both greater than equal to 0 in Hessian matrix, Hessian matrix positive semidefinite, objective function is convex letter in dual problem
Number, therefore, the solution α asked by gradient projection*, βi *, γi *It is global optimum, therefore, final energy allocation planIt is also global optimum；Wherein, Ealoc,iFor the energy distributed on i-th of server,
siFor the frequency of server, c is the execution frequency of task, δiFor circuit efficiency factor, M is the number of server, NiFor server
The number of upper task.
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| CN108415763B (en) * | 2018-02-11 | 2020-12-11 | 中南大学 | Distribution method of edge computing system |
| CN109116730B (en) * | 2018-07-11 | 2021-10-12 | 南京航空航天大学 | Hybrid execution mechanism energy optimization management method based on TU cooperative game |
| CN109657861B (en) * | 2018-12-20 | 2021-06-08 | 北京航空航天大学 | Sequential game based selective maintenance method for multi-stage continuous task equipment cluster |
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