Download Topological Optimization Design of a Multilevel Star Network

Topological Optimization Design of a Multilevel Star Network
JU Zhi-zhong LIU Yang
Daqing Petroleum Institute, Daqing, heilongjiang, P.R.China, 163318
[email protected]
Abstract: A topological optimization mathematic model of a multilevel star network is established, in
which the minimal total length between nodes is taken as objective function, the restrictions to the
unique relations between nodes, serving capacity, geometrical position etc. are taken as constraint
conditions. The model is a hybrid optimization design problem with disperse and continuous variables.
According to the model characteristics, the problem is divided into location layer and allocation layer,
and solved with hybrid genetic simulated annealing algorithm. In order to improve the optimization
performance, chromosome is coded in real number, while free intercross and advantage intercross are
alternatively used in the heredity operation, so are subtle mutation and violent mutation. The
reproduction strategy based on Metropolis criteria and optimal preservation strategy are also applied.
Practical example is given to verify the practicability of the method.
Key words: network, mathematic model, topological optimization, genetic algorithm, simulated
annealing
1 Introduction
Through analysis of the objective existent network system, such a network can be found, in which the
vertexes can be divided into a number of subsets at different levels, the vertexes at the same level set
have the same physical significance, low level vertexes are administered by high level vertexes, and this
affiliation is unique[1]. The network that meets those requirements is called multilevel star (MS) network.
Fig. 1 shows three-levels star crude oil gathering and transferring network which is frequently used in
the oil field, in which all dispersed wells are administered by metering stations, all metering stations are
administered by stock stations, all stock stations are administered by central treatment stations. Some
network models of military command, communications and transportation, communication systems also
have similar characteristics.
Fig. 1 Three-levels star crude oil gathering and transferring network
Usually, the cost of connecting medium between vertexes in MS network is very expensive,
generally about 50 ~ 80% of the entire system. The length of connecting medium and the positions of
vertexes are relevant to connecting relations between vertexes, therefore a topology optimization design
of the system has significant economic benefits. Based on the structural characteristics of MS network,
the topology optimization mathematical model of this network is established, and the solving strategy of
hybrid genetic simulated annealing algorithm is given.
2 Topology optimization mathematical model
The purpose of MS network topology optimization is to determine the location of the vertexes at all
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levels and connection relationship between the vertexes, as well as to obtain the shortest total length of
connecting medium between vertexes under the premise of the normal operation and management
facility. The optimization mathematical model can be expressed as:
min
F (δ, U) =
N mi mi −1
∑ ∑ ∑ lijk δ ijk
(1)
i =1 j =1 k =1
s.t.
mi
∑ δ ijk = 1
i = 1,2, L , N
j =1
；
k = 1, 2,L , mi −1
δ ijk ∈ {0,1}
(3)
mi −1
j
k
∑ qi −1δ ijk ≤ H i
i = 1,2,L, N − 1
k =1
U ∈ UD
{
{
Vi ; δ
；
mi
Where Vi is the i th set of vertexes in network, Vi = v1i , vi2 , L , vi
dimensionality of
(2)
j = 1,2, L, mi
}( i = 0,1,2,L, N ),
(4)
(5)
mi is the
is the design vector of connection relationship between nodes,
}
δ = δ ijk δ ijk ∈ (0,1), i = 1,2, L, N ; j = 1,2, L, mi ; k = 1,2, L, mi −1 , δ ijk = 1 shows that vertex vik−1 is
subordinate to vij , which means that the two vertices connect; if not, δ ijk = 0 ; U is the geometric
{
N
m
m
}
position vector of vertex V , U = U U i , U i = ( x1i , y 1i ), ( xi2 , y i2 ) L ( xi i , y i i ) , in which ( xij , yij ) is
i =1
geometric position of vertex vij ; lijk is the arc-valued function, in the condition of no obstacle
constraints, lijk can be expressed by the linear distance between vertex vij and vik−1 ; U D is the feasible
zone of U ; qik−1 is service demand of vertex vik−1 ; H i j is service capacity constraint of vertex vij .
Constraints (2)-(3) are constraints to the unique subjection relations between nodes, that is to
ensure that one vertex is only underneath to some vertex at higher level; Constraint (4) is the vertex
service capacity constraint; Constraint (5) is the geometric position constraint for vertexes.
3 Solving mathematical model
MS network topology optimization is a hybrid optimization design problem with disperse variables and
continuous variables. The issue is an extension to location-allocation problem in essence, its allocation
sub-problem is a ordinary assignment problem, which is a known NP-hard problem. In order to
effectively solve the problem, a solving strategy of hybrid genetic simulated annealing algorithm is
constructed in this paper[2-4]. The problem is divided into location layer and allocation layer. In the
location layer, the purpose is to confirm the geometric position of nodes of all levels, using genetic
simulated annealing algorithm to search the entire layout region. In the allocation layer, the purpose is to
confirm the connecting relations between nodes and connecting paths. Through this method, the entire
layout region can be effectively searched to find the global optimal solution or near global optimal
solution.
3.1 Coding scheme of chromosome
In MS network topology optimization, location variables are continuous, so the real number coding
scheme can be adopted[5,6]. If vertexes in the network are ranged according to vertex level from low to
high in the order, the chromosome can be expressed as
(
c k = x1k , y1k , x 2k , y 2k , L , x mk , y mk
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)
Where m is the total number of vertexes from the first level to the Nth level, m = m1 + m2 + L + m N ;
(x
k k
i , yi
) is
geometric position of the i th vertex in the kth chromosome, i = 1,2, L, m .
3.2 Population initialization
Population initialization usually generates randomly. The optimal layout should be layed in the
rectangular area which includes the lowest level nodes, so the initial layout of vertexes can be arranged
anywhere in this rectangular region, repeating the process until all the population members generate. To
speed up the search procedure, a method which is more effective than random population initialization is
adopted in this paper, as: (1) generating l (l>1) chromosomes randomly, selecting the best one and
adding it to the initial population, repeating it until filling population; (2) By use of the inherent
information, putting one or more of the layout schemes of artificial design into initial population, and
generating remaining chromosome using a random method.
3.3The design of adaptive function
From the coding scheme of issues, we can see that each chromosome contains vertical layout
information at all levels. In order to evaluate the chromosome, the length of connecting medium
( )
D c k corresponding to optimum network connection in every layout scheme is taken as the judgment
criteria. So the sub allocation problem under known conditions of layout scheme needs to be solved. In
order to solve as soon as possible, Lagrange relaxation method can be adopted.
In the heredity operation process, chromosomes with better adaptability are more probable to
survive, and the objective function of MS network topology optimization is to minimize the length of
connecting medium between vertexes, so the objective function needs appropriate transformation before
application. For the chromosome c k , the following form of adaptive value function can be adopted
( )
( )
 D c k − Dmin 

f c k = exp −


t


Where Dmin is the minimum of objective function in present evolutional population; t is the
annealing temperature.
This is a preferable accelerated adaptive function, it can prevent too fast dominance genetic process
of the some ultra chromosome in heredity iterative process, and meet the need of the early competition
restriction and late competition encouraging.
3.4 Determination of initial temperature and annealing operation
According to analogy relationship with physical annealing process, the choice of initial temperature
should be sufficiently large so that almost all of the candidates can be accepted in order to ensure the
ultimate good convergence of algorithm. Experiments show that initial temperature is higher, the
probability to obtain high-quality solutions is greater, but computing time will increase. Therefore, the
initial temperature should be determined by considering both optimization quality and optimization
efficiency. To facilitate the operation, the selection of initial temperature may adopt following form
t 0 = Kδ
δ = f max − f min
Where f max is the max of objective function in present evolutional population; K is a sufficiently large
number, in the actual computation, experimental data such as K = 10,20,100, L can be chosen.
Annealing operation can adopt t k +1 = λt k method, in which 0 < λ < 1 .
3.5 Intercross operation
Intercross-operator in genetic operation is the core of the genetic algorithm. Intercross depends on
how to select parents and how to reproduce offspring. Through intercross, searching capabilities of
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genetic algorithm improve rapidly. The free intercross combined with advantage intercross strategy is
adopted here[7]. The free intercross is that the two parents are optional; advantages intercross is that the
most suitable individual is used as a fixed parent, other optional individual in the population is chosen as
another parent. The free intercross is adopted in odd number generation in genetic process, the
advantage intercross is adopted in even number generation. Assume that two chosen parents are:
(
= (x
c k1 = x1k1 , y1k1 , x 2k1 , y 2k1 , L , x mk1 , y mk1
c k2
k2
1
)
, y1k2 , x 2k 2 , y 2k 2 , L , x mk2 , y mk2
)
Generated offspring is:
c = (x1 , y1 , x 2 , y 2 , L , x m , y m )
The gene of next generation’ chromosome is identified by the following formulas:
xi = ri ⋅ xik1 + (1.0 − ri ) ⋅ xik2
y i = ri ⋅ y ik1 + (1.0 − ri ) ⋅ y ik2
Where r1 , r2 , L, rm is independent random numbers in (0,1).
3.6 Mutation operation
Two methods are adopted in mutation operation: subtle mutation and violent variation. Two
methods are used alternately[7]. Assuming that candidate for mutated chromosomes
(
)
k
k
is c k = x1k , y1k , x 2k , y 2k , L , x m
, ym
, offspring from mutation is c = (x1 , y1 , x 2 , y 2 , L , x m , y m ) , in
subtle mutation, the gene of next generation’ chromosome can be identified by the following formula:
xi = xik + random[−ε , ε ]
yi = y ik + random[−ε , ε ]
Where ε is a little positive real number.
In violent mutation, the gene of next generation’ chromosome can be identified by the following
method:
xi = random[ xmin , x max ]
y i = random[ y min , y max ]
3.7 Selection and copy operation
Selection is to select superior and eliminate the bad one from population. It is established on the
basis of individual adaptive value assessment, individual with greater adaptability has more opportunity
to be selected. This is different from standard genetic algorithm. Here the offspring is selected from
the parents and superior offspring. The relatively prohibited selection strategy is adopted in order to save
the best chromosome and to avoid degeneration[8]. The prohibited neighborhood of chromosome s k
can be defined as:
{
Ω(s k , α , γ ) = s | s − s k ≤ γ , f (s k ) − f (s ) < α , s ∈ R 2 m
}
Where α and γ are two given positive parameters. In the process of selection, if s k is elected to the
next generation, the chromosomes on neighborhood are prohibited to be selected. The value of
γ defines neighborhood of s k in the sense of layout, to avoid to select individual which has little
difference in layout. The value of α defines neighborhood of s k in the sense of adaptive value, to
avoid to select individual which has little difference in adaptive value.
3.8 The copy strategy based on the Metropolis decision criterion
Every individual, which is generated from the intercross, mutation and selection operation in the
genetic algorithm, is searched randomly in a certain neighborhood in currant temperature using
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Metropolis sampling strategy, until the sample is stabilized. The optimal preservation strategy is adopted
for the algorithm convergence in the whole region. This strategy can guarantee the best individual
entering into next generation and diversity of population. It accepts optimal solution, accepts poor
quality solutions conditionally at the same time. This way can avoid local optimum solutions.
3.9 Convergence criterion of algorithm
The hybrid genetic simulated annealing algorithm is a repeatedly iteration searching method. It
evolves many times to approximate to optimal solution rather than just is equal to the optimal solution,
the algorithm convergence criteria should be given to determine whether or not to terminate the
evolutionary process. Here two criterions are adopted: (1) the maximum iteration frequency is given; (2)
judgment is achieved by monitoring the changes of the optimum individual objective functions. When
the number of iterations achieves the specified maximum number of iterations or optimal individual
objective function value does not change obviously for steps, the algorithm can be judged to have met
the convergence conditions, then population evolutionary process ends.
3.10 Algorithm flowchart
In summary, algorithms flowchart which adopts a hybrid genetic simulated annealing algorithm to
solve MS network topology optimization is shown in Fig. 2.
Give algorithm parameters
Initialize population and SA temperature
Assess all individuals in current population
Lagrange relaxation method
Yes
Has the iteration converged
Result Output
No
Implement the GA intercross operation
Implement the GA mutation operation
Implement the selection and copy operation
Obtain the SA initial population, operate copy
strategy to all individual based on the
Metropolis rule to generate new individual
Annealing operation
Fig. 2 Hybrid genetic simulated annealing algorithm structure flowchart
4 Calculation example
In an oilfield block there are 157 wells, according to oil production rate and production
management convenience, two levels station-distribution is adopted, in which there are 13 metering
stations, 1 central treatment station, a star gathering network is to be designed. Optimization design is
made using the method that this paper mentions. The calculation parameters are: group size is 60,
mutation rate is 5%, intercross rate is 90%, annealing operation coefficient is λ = 0.8 , the cost of pipe
per mile from well to metering station is 14.16 × 104 RMB, cost of pipe per mile from metering station
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to central treatment station is 22.37 × 104RMB. Fig.3 and Fig.4 show the optimization results of artificial
design and hybrid genetic simulated annealing algorithm respectively, Tab.1 is comparison of data. From
optimization results, compared the results using this paper’s method with artificial design results, the
total length of pipeline decreases 14.72 km, pipeline investment decreases 233.8 × 10 4 RMB, decreasing
10.86% and 10.49% respectively, the optimization effect is evident.
Fig. 3 Artificial design result
Fig. 4 Optimization result of hybrid genetic simulated annealing algorithm
Tab.1 Optimization design result
Metering stationTotal
central treatment station
Total length
investment of
of
pipeline
Pipeline
pipeline
Pipeline length Pipeline investment Pipeline length
(km)
investment
( × 104RMB)
(km)
( × 104RMB)
(km)
4
( × 10 RMB)
Well-metering station
Artificial
design
Optimization
result
97.95
1386.97
37.63
841.78
135.58
2228.75
86.32
1222.29
34.54
772.66
120.86
1994.95
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5 Conclusion
(1) A class of topological optimization mathematic model of a multilevel star network is established,
in which the minimal total length between nodes is taken as objective function, the restrictions to the
unique relations between nodes, serving capacity, geometrical position etc. are taken as constraint
conditions.
(2) According to configuration characteristics of the model, the problem is divided into location
layer and allocation layer, and solved with a hybrid genetic simulated annealing algorithm which
organically combined genetic algorithm, simulated annealing with conventional optimization algorithm.
This model is more efficient, and can be used effectively to search entire layout region to find global
optimum solution or near global optimum solution.
(3) The optimization design of crude oil gathering system in an oilfield block shows that the total
length of pipeline and pipeline investment decreases 10.86% and 10.49% after optimization respectively.
This result describes that the mathematical model is correct, and the optimization algorithm is effective
and reliable.
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