Constructing core backbone network based on survivability of power grid

Electrical Power and Energy Systems 67 (2015) 161–167

Contents lists available at ScienceDirect

Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Constructing core backbone network based on survivability of power grid Feifei Dong, Dichen Liu, Jun Wu ⇑, Lina Ke, Chunli Song, Haolei Wang, Zhenshan Zhu School of Electric Engineering, Wuhan University, Wuhan 430072, China

a r t i c l e

i n f o

Article history: Received 31 October 2013 Received in revised form 26 October 2014 Accepted 31 October 2014

Keywords: Core backbone network Survivability of power grid Improved BBO algorithm Premature judgment mechanism Chaos optimization Cauchy optimization

a b s t r a c t Constructing core backbone network is beneficial to strengthen the construction of grid structure, raise the ability of withstanding natural disasters, as well as realize power grid’s differentiation planning reasonably and scientifically. Based on the index system of survivability, a method of constructing core backbone network with the target of the smallest line total length and the largest integrated survivability index is put forward with constraint conditions of network connectivity and power grid safe operation. The cosine migration model, the premature judgment mechanism, and the mutative scale of mutation strategy by Chaos and Cauchy optimization are introduced into the improved biogeography-based optimization algorithm (BBO) to search for the optimal solution of the core backbone network. Comparison with the traditional BBO algorithm, particle swarm optimization (PSO), binary ant colony algorithm (BACA), genetic algorithm (GA) shows that the proposed method is accurate and effective, and it has advantages in fast convergence speed and high convergence precision. Ó 2014 Elsevier Ltd. All rights reserved.

Introduction For the past few years, power grid is constantly damaged by extreme natural disasters, which is due to that the past standard of power facilities could not resist the increasingly frequent natural disasters [1,2]. Therefore, it is necessary that the resistant standards of disasters should be designed differentially, with different lines’ geographical location and climate conditions taken into consideration. The goal of differential planning design is to confirm the core backbone network, which is made up of important lines that can guarantee the continuous power supply of the important load when the major natural disasters attack [3,4]. Take the differential planning carried out for the large area blackout accident because of the ice disaster by electric power company in Quebec, Canada in the late 1990s for an example, the strategic guarantee route designed for 735 kV substation played an important role in early 2009 north American ice storm, which managed to avoid a large area blackout accident [5,6]. Therefore, constructing core backbone network is meaningful to improve the stability of the power grid’s structure, reduce the secondary investment of repairing and rebuild the harm of power grid caused by natural disasters, as well as guarantee power grid’s safe and reliable operation under severe natural disasters.

⇑ Corresponding author. Tel.: +86 13628692588. E-mail address: fl[email protected] (J. Wu). http://dx.doi.org/10.1016/j.ijepes.2014.10.056 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.

The concept of survivability is firstly put forward by Barnes and others in 1993 [7]. Survivability of system is refers to the ability that the system can complete its critical services in a timely manner, and restore its basic services as soon as possible when it is subjected to the attack, failure or a sudden accident [8]. Survivability has been widely concerned in the complex network and information system as a new research direction [9,10], but its application in the related fields of power system is relatively small. Considering survivability is helpful to keep the system alive with supportive network structure, in that case, the constructed core backbone network has strong resistance, restorative and connectivity. A key line identification method based on network survivability evaluation was proposed in the literature [11], a search model and a search method of backbone grid are also present, which has certain enlightening significance. But the survivability index of this method is relatively single, and the search algorithm is easy to fall into local optimal. The survivability index system that can systematically reflect information system is proposed in the literature [12], the formalized description and mathematical model are also given. It has a certain guiding role in establishing power system’s survivability index system. Search of core backbone network belongs to nonlinear and discontinuous optimization problem, which relies on artificial intelligence algorithm [13,14]. As a kind of new artificial intelligence algorithm, BBO algorithm based on species migration patterns

162

F. Dong et al. / Electrical Power and Energy Systems 67 (2015) 161–167

has achieved good results in parameter identification [15], fault diagnosis [16], and image classification [17], as well as transmission network planning [18], and other fields. It is reflected that the algorithm has advantages of less set parameters, simple calculation, fast convergence speed and good stability. But there are problems in traditional BBO algorithm that the linear migration model cannot accurately simulate the migration process, search ability is not strong, as well as it is easy to fall into local optimum by premature [19,20]. Therefore, the cosine migration model, the premature judgment mechanism, and the mutative scale of mutation strategy by Chaos and Cauchy optimization were introduced into the improved traditional BBO algorithm, to strengthen its search ability, and make it search the optimal solution of core backbone network rapidly and accurately. A new method of constructing core backbone network considering survivability is put forward in this paper. The indexes of survivability are built from aspects of resistibility, recoverability, and connectivity. The largest integrated survivability index and the minimum total length of the backbone grid’s lines are regarded as the objective function, with network connectivity and power grid’s safe operation as constraint conditions. The improved BBO algorithm provided with strong search ability is used to search backbone grid. The simulative results show that the method can quickly and accurately search the optimal solution of core backbone network considering survivability, and its convergence speed and convergence precision are higher than that of the traditional BBO algorithm, PSO algorithm, BACA algorithm, and GA algorithm. The index system of survivability Survivability of power grid is defined as the ability of guaranteeing the electricity supply of important load relying on the high design standards of the core backbone network, as well as restoring power supply for other load gradually through the network frame when major natural disasters attacks. The index system of survivability is built from these three aspects as follows: resistibility, recoverability and connectivity. The index of resistibility Resistibility of power grid reflects the resistance of the basic service that the system provides power supply for the important load with all kinds of natural disasters. The two indicators, preserving rate of line and node are introduced to evaluate resistibility. The number of original rack’s lines is set as dim (L), and that of the node is set as dim (B). The number of failure line and failure node of the remain network frame after natural disasters compared to the original rack are respectively L_failure, B_failure. Preserving rate of line is as follows:

al ¼

dimðLÞ  L failure dimðLÞ

ð1Þ

Preserving rate of node is as follows:

ab ¼

dimðBÞ  B failure dimðBÞ

ð2Þ

ber of generator is mg. The active load and the actual active load of node j in the backbone network frame scheme that meets the safe operation conditions and ensure the rack’s load to be biggest are respectively Sjm and Sj. The generator standby indicator is as follows:

Recoverability of power grid reflects whether the power grid can recover after suffering from natural disasters, as well as how much it can recover. The generator standby indicator and load recovery degree index are introduced to evaluate recoverability. The generator is actual output and the largest capacity of the backbone grid are respectively set as Gi, and Gimax. The total num-

ð3Þ

Gi max

i¼1

The load recovery degree index is as follows:

PdimðBÞB

failure Sjm  j¼1 PdimðBÞB failure Sjm j¼1

bc ¼

Sj

ð4Þ

The index of connectivity The relative tightness degree and the relative condensation degree of the grid are introduced to evaluate connectivity. (1) The relative tightness degree of the grid Assuming that node j has Kj neighbor nodes, there are at least Kj(Kj  1)/2 lines between these nodes, but there are actually only tj lines, in that case, the clustering coefficient of the node j is as follows:

Cj ¼

2tj K j ðK j  1Þ

ð5Þ

The relative tightness degree of the grid is the weighted average of backbone grid’s all nodes’ Cj. It is the characteristic parameter that shows the connected degree of neighboring nodes, and its expression is as follows:

PdimðBÞB C¼

j¼1

failure

Cj

dimðBÞ  B failure PdimðBÞB failure 2tj

¼

j¼1

ð6Þ

K j ðK j 1Þ

dimðBÞ  B failure

If the tightness degree of the original grid is C0, the relative tightness degree of the backbone grid is as follows:

u¼

C C0

ð7Þ

(2) The relative condensation degree of the grid The relative condensation degree of the grid is the product’s reciprocal of the number of nodes and the weighted average of the shortest path, namely:

@¼

1 nl

ð8Þ

The traditional shortest path is the minimum number of edges between two nodes. Considering the actual situation of power system, the weighted average of the shortest path of backbone grid with the length of the transmission line as the line’s weight is as follows:

l¼ The index of recoverability

 mg  X Gi max  Gi

bg ¼

X 0 2 dij n  ðn  1Þ

ð9Þ

0

where dij is the weighted number of edges that the shortest path passes through; n is the total number of nodes in the grid. The formula (9) is taken into the formula (8), and then the condensation degree of the backbone grid is got as follows:

n  1 dimðBÞ  B failure  1 @¼P 0 ¼ P 0 dij dij

ð10Þ

F. Dong et al. / Electrical Power and Energy Systems 67 (2015) 161–167

If the condensation degree of the original grid is o0, the relative condensation degree of the backbone grid is as follows:

/¼

@ @0

ð11Þ

163

topology structure of the rack against natural disaster. Therefore, establish an index system of survivability effectively and then determine the comprehensive indicators of network survivability are the key step to construct core backbone network. Mathematical model of constructing core backbone network

Integrated index of survivability There are three indicators included in the index system of power grid survivability. Primary index is the integrated index of survivability, secondary indexes are resistibility, recoverability and connectivity index, and the tertiary indicators are made up by refining indicators of each secondary index. Among them, the indexes of the level m can be calculated by the indexes of the level m + 1 according to the following principles: Vector Rm = (Rm1, Rm2,   , Rmg)T is set as the individual risk index vector of survivability’s level m indexes. g is the total number of level m + 1 indicators. Rmi is the single index i of level m indexes. The level m index of survivability is defined as follows:

1 Rms ¼ a1  kRm k1 þ b1  kRm k1

g

ð12Þ

where a1 and b1 are weight coefficients that meet the need of a1 + b1 = 1; kRmk1 and kRmk1 are respectively 1 norm and 1 norm of vector Rm

kRm k1 ¼

g X jRmi j

ð13Þ

i¼1

kRm k1 ¼ max jRmi j

ð14Þ

Therefore, the integrated index of survivability is as follows:

1 Sur v ¼ a  kRk1 þ b  kRk1 3

ð15Þ

where a and b are weight coefficients that meet a + b = 1, R = (Resis, Recov, Connec)T. Constructing of the core backbone network Basic thoughts of constructing core backbone network The key of differential planning design is to determine the selecting principle and construction standard of affordable power supply, load and network frame, adopt the quantitative evaluation method to identify the key components and build core backbone network, increase the resistant standards of disasters of element differentially, in order to make system guarantee the delivery capacity of important load and power supply as well as the ability of the interval exchange capacity, and further makes the system recovered and reconstructed on the basis of the core backbone network. The emergency control and scheduling during disaster, the repair and reconstruction after disaster are taken place by the viability and self-healing ability of the rack itself when disaster occurs, for the sake of reducing the damage of major natural disasters to the power grid. Therefore, the core backbone network needs to satisfy the following conditions: (a) to guarantee the continuous power supply of the important load and the reliable transmission of the important power; (b) to make the rack satisfy the constraints of power grid’s safe operation as well as network’s topology connectivity; (c) to meet the total length of the son rack’s line to be smallest from the perspective of economy; (d) to meet the survivability of the son rack to be stronger from the perspective of the network frame topology’s optimization configuration. Survivability is one of the most important constraints of the core backbone network, and it has great influence in resistance ability, resilience, and the stability of the overall network frame’s

The core backbone network includes the important power source nodes, load nodes, key circuit, nodes connected to the key circuit that should be guaranteed according to the actual situation, as well as lines and nodes should be preserved considering network connectivity, tide of constraints, the total length of lines to be the smallest, the integrated index of survivability to be largest. Therefore, mathematical model of constructing core backbone network is as follows:

8 1 =L0 min f ¼ LSur > v > > > > dimðLÞL > Xfailure > > > s:t: L1 ¼ l i xi > > > > i¼1 > > < dimðLÞ X L0 ¼ l i xi > > > > i¼1 > > > /ðxÞ ¼ /ðx ; x ;    x Þ ¼ 1 > 1 2 l > > > > gðxÞ ¼ 0 > > > : hðxÞ 6 0

ð16Þ

where xi shows the state of line’s input or excision, 1 means input, 0 means excision. li is the length of line i. L1 is the total length of the backbone grid and L0 is that of the original grid. L1/L0 is the normalization of the backbone grid’s total length compared to the original grid, in order to make its value between the (0, 1) and with the same dimension of Surv. /(x) is the judging function of connectivity. / (x) = 1 means that the network is connected, /(x) = 0 proves that the network is disconnected. g(x) = 0 is the equality constraint equation for the trend. hðxÞ 6 0 is the inequality constraint equation for the trend. The search of the core backbone network is a multi-objective optimization problem. The minimization of the objective function is equivalent to seeking the core backbone network scheme that ensures the smaller lines’ total length and the larger integrated index of survivability. Search of the backbone grid based on IBBO algorithm The basic biogeography-based optimization algorithm Biogeography-based optimization algorithm is a new type of evolutionary algorithm that uses the biogeography theory to solve optimization problems. Its basic idea is as follows: in view of the need to optimize problem, a number of relatively independent habitats are constructed as candidate solutions. The information sharing is realized depending on the species’ migration between habitats, and the information update is realized by the species’ mutation. In that case, the fitness of the habitat is improved, and the optimal solution of the problem of is obtained [21,22]. For the multi-objective optimization problems of the core backbone network, the appropriate index HIS is adopted to express the objective function of the mathematical model of constructing core back1 =L0 bone network, that is LSur v . The appropriate variable SIV is used to show variables contained in each habitat, namely, the state of lines. There are mainly two operations included in the algorithm, migration and mutation operation. (1) Migration operation The migration operation is used for information exchange with other habitats, in order to, search the solution space in the wide

164

F. Dong et al. / Electrical Power and Energy Systems 67 (2015) 161–167

Migration rate Immigration rate I λ(S) E

Migration rate I

Immigration rate λ(S)

E

Emigration rate μ(S)

Emigration rate μ(S) 0

S0

Number of speciesS

Sm Number of speciesS

Sm

0

Fig. 1. Species migration model.

Fig. 2. Cosine migration model.

area. The common species’ migration model of single habitat is shown in Fig. 1. In Fig. 1, I means the largest immigration rate, E means the biggest emigration rate, S0 and Sm respectively expresses balance and saturation point. The calculation method of kðSÞ and l(S) are shown as formula (17) and formula (18).

kðSÞ ¼ Ið1  S=Sm Þ

lðSÞ ¼ E 

S Sm

ð17Þ ð18Þ

(2) The mechanism of premature judgment is introduced The search process of BBO algorithm is easy to fall into local optimum because of premature. The mechanism of premature judgment is introduced to solve this problem. The individual number of populations is set as N, fi is the suitability of the habitat i. r2 is the suitability variance of the population, and its expression is as follows:

r2 ¼

2 N  f i  f Av g 1X f N i¼1

ð22Þ

(2) Mutation operation Mutation operation is used to increase the diversity of population. The probability of the habitat that has number of species S is written for Ps. The variation rate is calculated through the follow formula.

where fAvg is the current average suitability for species. f is the normalized scaling factor, which plays the role of limiting the size of r2. The expression of fAvg and f are relatively as follows:

MðSÞ ¼ Mmax ð1  P s =Pm Þ

ð19Þ

f Av g ¼

where Mmax is the parameter set in accordance with the requirements of different users, which is called as the biggest mutation rate. Pm is the maximum of Ps. Mutation operation makes the solution of lower HIS improved by variation, and makes the higher HIS obtain the opportunity of improving their solutions.

(

The improvement of BBO algorithm The BBO algorithm was applied to solve multi-objective optimization problem of core backbone network, and the improvements are made as follows:

f ¼

N 1X f N i¼1 i

ð23Þ

    max jf i  f Av g j ; max jf i  f Av g j > 1 1;

others

ð24Þ

The formula (22) showed that the suitability variance of the populations reflects the degree of all individuals’ ‘‘gather’’ in the population. If the value of r2 is smaller, the degree of the population’s ‘‘gather’’ is better. If the algorithm does not meet the ending conditions, this kind of ‘‘gather’’ will lead to the loss of the population’s diversity and the premature phenomenon. In that case, when it meets r2 < c01 (c0 is set as user-defined constants), the treatment for premature is carried out. (3) The mutative scale of Chaos and Cauchy mutation strategy is introduced

(1) The improvement of migration model The linear species migration model shown in Fig. 1 cannot accurately simulate material migration’s complex process of the actual biological geographical environment. Therefore, the cosine migration model more conform to the nature shown in Fig. 2 is used to calculate migration rate. The immigration rate kðSÞ and emigration rate l(S) of cosine migration model are respectively as follows:

kðSÞ ¼

    I Sp þ1 cos 2 Sm

ð20Þ

lðSÞ ¼

  E Sp  cosð Þ þ 1 2 Sm

ð21Þ

In this migration model, when there are less or more species in the habitat, immigration rate and emigration rate change smoothly, but when habitat has a certain number of species, immigration rate and emigration rate change relatively quickly.

The mutation strategy of BBO algorithm directly affects whether the algorithm will fall into local optimum and convergence precision. The mutative scale of chaos and Cauchy mutation strategy is introduced in this paper. In the early evolution of BBO algorithm, the ergodicity of Chaos is used for random search in the vast space, which is helpful for solving the problem of local optimal caused by premature, and then make it converge to global optimal solution. In the late stage of evolution, the center distribution characteristics of the Cauchy iteration is used for a careful search closely around the optimal solution, thus improve the convergence precision. The segmented logistic chaotic iterative equation is adopted in this paper.

( Z kþ1 j

¼

4  l  zkj  ð0:5  zkj Þ;

0 6 zkj 6 0:5

4  l  ðzkj  0:5Þ  ð1  zkj Þ; 0:5 6 zkj 6 1

ð25Þ

F. Dong et al. / Electrical Power and Energy Systems 67 (2015) 161–167

165

where zj is the j-th chaotic variable, K is the iterations of the Chaos iteration. When it meets 3:5699456 6 l 6 4; the system enters into a chaotic state. Usually take l = 4. The distribution function of Cauchy distribution is as follows:

pðxÞ ¼ 1=2 þ 1=p  arctan x

ð26Þ

The formula of the mutative scale of chaos and Cauchy mutation is as the following. 0ðkÞ

Xj

ðkÞ

¼ ð1  kk ÞX j þ kk Pc 0ðkÞ

ð27Þ ðkÞ

where X j is the newly generated Optimized individual, X j is the best individual obtained from current optimization sequence. If the 0ðkÞ suitability of X j is better than that of the original best individual, it will be remained. Otherwise, it will be abandoned. Pc is the Chaos or Cauchy iteration variable mapped to the parameters of the search space. kk is the variable scale factor, which can be obtained by the following formula:

kk ¼ 1  jðk  1Þ=kjr

ð28Þ

where r is used to control the convergence speed. As the high probability area of chaos iteration distribution is close to the edge, the high probability area of Cauchy distribution is near the center. At the beginning of the BBO algorithm’s evolution, the value of k is smaller, kk tends to be 1, Pc is made up of chaos iteration variable, it can make the algorithm jump out of local convergence as soon as possible. At the end of the algorithm, the value of k is bigger, kk tends to be 0, Pc takes Cauchy iteration variable, it can make the optimal solution fixed and improve the accuracy of the algorithm. Search of core backbone network based on IBBO algorithm The multi-objective optimization problem of constructing the core backbone network is solved by IBBO algorithm, in which the survivability is taken into account. The process algorithm is shown in Fig. 3. Specific steps are as follows: (1) Enter the original grid’s parameters needed by calculation, including the node loads, the output of generators and the proposed paths, et al. The control parameters of the BBO algorithm are initialized. The initial population P that meets the constraint condition is randomly generated. (2) The suitability index of the habitat is calculated and sorted. The individual optimal solution, Xbest, is saved, and then judge whether it meets the end condition, if meets, the results are output and then converted to backbone grid scheme, in that case, the program is over. Otherwise, continue to step (3). (3) The cosine migration model is established. The habitat’s species number, immigration rate and emigration rate are calculated. (4) The migration operation is performed in order to form a new population P1. The suitability index of the habitat is recalculated, and then the optimal solution Xbest1 is updated. (5) The average suitability r2 of the population is calculated according to the formula (22)–(24). Judge whether it falls into local optimum on the basis of the mechanism of premature judgment. If it falls, continue to step (6). Otherwise, go to step (7). (6) The mutative scale of chaos and cauchy mutation is carried out according to the formula (27), and then the optimal solution of the population Xbest2 is updated. (7) Determine whether it meets the maximum number of iterations, if satisfied, the results are output and then converted to core backbone network scheme, in that case, the program is over. Otherwise, go to step (2).

Fig. 3. Flow chart of searching core backbone network based on IBBO algorithm.

Example analysis In order to verify the effectiveness of the proposed method, the core backbone network of IEEE118 node system is constructed based on IBBO algorithm with power grid’s survivability taken into account. The system contains 118 nodes and 179 branches, including 53 generator nodes, in addition to the nodes of 17, 22, 23, 57, 58, 84, 102, 108, 109, 114, other load nodes are all important loads. The parameters of IBBO algorithm are set as follows: the population size N is set as 50, the largest number of iterations kmax is 200, the biggest mutation rate Mmax is 0.01, the biggest immigration rate I and emigration rate E are both set as 1, the elite reserves Z is 2, the global mobility Pmod is 1, and the coefficient of controlling the convergence rate r is set as 5. The optimal scheme of core backbone network considering survivability searched by IBBO algorithm is shown in Fig. 4, the solid point ‘‘’’ represents generator node, the hollow point ‘‘s’’ means load node, the reserved lines

166

F. Dong et al. / Electrical Power and Energy Systems 67 (2015) 161–167

Fig. 4. Core backbone network scheme for IEEE 118-bus power system based on improved BBO optimization.

BBO PSO IBBO BACA GA

14

Suitability index

of the core backbone network is expressed by the solid lines. This core backbone network scheme is composed of 71 nodes and 70 lines, including 19 generator nodes and 44 load nodes. In order to verify IBBO algorithm’s performance, the traditional BBO algorithm, PSO algorithm, BACA algorithm and GA algorithm are used to search the core backbone network, their population size and their maximum number of iterations are set as the same as IBBO algorithm. Based on the randomness of the algorithm itself, the statistical results of the five kinds of algorithms’ independent running 50 times are listed in Table 1. The contrast curves of five kinds of algorithms’ optimal scheme’s suitability index are shown in Fig. 5. Table 1 and Fig. 5 show as follows: the best value and the worst value of IBBO search’s objective function is relatively smaller, and its average value is less than the best value of most methods, which means the precision of objective function searched by IBBO algorithm is higher. Meanwhile, the IBBO algorithm searches 34 times of the optimal solution in the 50 times independent operation, which shows that this algorithm has stronger convergence compared with other four kinds of algorithms. Especially compared to the traditional BBO algorithm’s 12 times of optimal solution, it shows that the search ability of IBBO algorithm is enhanced by adding the mechanism of premature judgment and the mutative scale of chaos and Cauchy mutation strategy. In addition, it can be seen from Fig. 5 that the IBBO algorithm has obvious advantages in convergence rate compared with other four kinds of algorithms.

12

10

8

6 0

50

100

150

200

The number of iterations Fig. 5. Contrast of suitability index curve of 5 kinds of algorithm’ optimal solution.

The collection of lines of three objective functions’ core backbone network optimal scheme searched by IBBO algorithm are presented in Table 2. Three kinds of objective functions are as follows: min f1 = a1 means that the number of lines is least, in which a1 1 =L0 expresses the total number of lines. min f 2 ¼ LSur v means the smallest lines’ total length and the largest integrated survivability index mentioned earlier in this paper. min f3 = L1 means the minimum of lines’ length.

Table 1 Comparison of objective function solution by different algorithms. Algorithm

BBO PSO IBBO BACA GA

Objective function Best value

Worst value

Average value

Times of searching optimal solution

5.7331 5.6774 5.6206 5.8591 6.3829

6.4462 6.1864 5.9595 6.2013 6.7064

6.0402 5.965 5.6933 5.9831 6.5054

12 20 34 18 25

167

F. Dong et al. / Electrical Power and Energy Systems 67 (2015) 161–167 Table 2 Comparison of core backbone network’s optimal scheme under different objective functions based on IBBO. Objective function

The collection of the optimal scheme’s lines

Lines

min f1 = a1

5–8, 25–26, 37–38, 3–5, 8–9, 9–10, 5–11, 2–12, 3–12, 7–12, 11–13, 12–14, 12–16, 19–20, 20–21, 25–27, 27–28, 28–29, 8–30, 26–30, 29–31, 19–34, 35–37, 33–37, 34–37, 37–39, 37–40, 30–38, 40–41, 41–42, 34–43, 44–45, 47–49, 42–49, 45–49, 48–49, 49–50, 49–51, 51–52, 52–53, 60–62, 49–66, 62–67, 66–67, 47–69, 69–75, 75–77, 77–78, 78–79, 77–82, 82–83, 83–85, 85–86, 85–88, 88–89, 93–94, 94–95, 82–96, 94–96, 94–100, 96–97, 98–100, 100–101, 100–106, 27–115, 12–117, 75–118

67

1 =L0 min f 2 ¼ LSur v

5–8, 25–26, 37–38, 61–64, 65–66, 68–69, 3–5, 8–9, 9–10, 5–11, 11–12, 2–12, 7–12, 11–13, 12–14, 12–16, 19–20, 15–19, 20–21, 25–27, 27–28, 28–29, 8–30, 26–30, 15–33, 35–37, 33–37, 34–37, 37–39, 30–38, 39–40, 40–41, 43–44, 34–43, 44–45, 47–49, 48–49, 49–50, 49–51, 51–52, 52–53, 60–61, 38–65, 64–65, 49–66, 66–67, 65–68, 69–75, 69–77, 77–78, 78–79, 77–80, 77–82, 82–83, 83–85, 85–86, 85–88, 88–89, 93–94, 94–95, 82–96, 94–96, 80–97, 80–98, 94–100, 100–101, 100–106, 27–115, 12–117, 75–118

70

min f3 = L1

5–8, 25–26, 37–38, 61–64, 65–66, 68–69, 80–81, 3–5, 8–9, 9–10, 5–11, 11–12, 2–12, 7–12, 11–13, 12–14, 12–16, 19–20, 15–19, 20–21, 25–27, 27–28, 28–29, 8–30, 26–30, 29–31, 15–33, 35–37, 33–37, 34–37, 37–39, 30–38, 39–40, 40–41, 34–43, 44–45, 47–49, 45–49, 48–49, 49–50, 49–51, 51–52, 52–53, 60–61, 38–65, 64–65, 49–66, 66–67, 65–68, 69–75, 77–78, 78–79, 77–80, 68–81, 77–82, 82–83, 85–86, 85–88, 88–89, 89–92, 92–93, 93–94, 94–95, 82–96, 94–96, 80–98, 94–100, 96–97, 100–101, 100–103, 103–105, 105–106, 27–115, 12–117, 75–118

75

As it can be seen from Table 2, when the objective function is min f1 = a1, the number of lines of the core backbone network’s opti1 =L0 mal scheme is 67. When the objective function is min f 2 ¼ LSur v, there are 70 lines in the lines’ collection corresponding to the optimal scheme of the core backbone network. What’s more, there are 75 lines in the core backbone network’s optimal scheme of the objective function min f3 = L1. For the objective function f2 reflects the content of the objective function f3, and its preserving rate indicators of survivability also reflects the content of the objective function f1, so the searched number of lines between the two is reasonable, by which the correct and effectiveness of search algorithm proposed in this paper is further verified. On the other hand, the objective function of constructing the core backbone network set up in this paper also takes into account the resistibility and recoverability of the grid as well as the reasonable configuration of the network frame’s topology compared to the commonly used objective function considering economy such as f1 and f3. It considers the characteristic of network frame more comprehensively, and has more practical significance in improving the resistance of power grid against natural disasters and differentiation design. Conclusions Based on the index system of power grid survivability, the model of constructing core backbone network with the target of the smallest lines’ total length and the largest integrated survivability index is built. The results of the example show that the model can well balance the economy, resistibility, recoverability and the reasonable configuration of the network frame’s topology in constructing network frame, which provides differentiated planning with reasonable design scheme. The improved BBO algorithm is put forward, in which the cosine migration model, the premature judgment mechanism, and the mutative scale of mutation strategy by Chaos and Cauchy optimization are introduced to enhance the search ability of BBO algorithm. Compared with other four kinds of algorithms, it can be found that the IBBO algorithm has higher precision and better convergence speed. The system of IEEE118 nodes is taken as an example, in which the multiple operational results of the five different algorithms and the search results for three kinds of objective functions by IBBO algorithm are contrasted. In that case, the rationality of the proposed model and the superiority of search algorithm are verified. Acknowledgement The authors would like to acknowledge the supports from the Natural Science Foundation of China (51207114) for the work reported in this paper.

References [1] Brostrom E. Modeling of ice storms and their impact applied to a part of the Swedish transmission network. Proceedings of IEEE lausanne power tech conference, July 1–5; 2007. [2] Billinton R, Singh G. Application of adverse and extreme adverse weather: modeling in transmission and distribution system reliability evaluation. Proc Inst Elect Eng Gen Transm Distrib 2006;153(1):115–20. [3] Hooshmand Rahmat-Allah, Hemmati Reza, Parastegari Moein, et al. Combination of AC transmission expansion planning and reactive power planning in the restructured power system. Energy Convers Manage 2012;55:26–35. [4] Kjolle Gerd, Rolfseng Lars, Dahl Eyolf. The economic aspect of reliability in distribution system planning. IEEE Trans Power Delivery 1990;5(2):1153–7. [5] Trudel Gilles, Gingras Jean-Pierre, Pierre Jean-Robert. Designing a reliable power system: Hydro-Québec’s integrated approach. Proc IEEE 2005;93(5):907–17. [6] Trudel Gilles, Bernard Serge, Scott Guy, et al. Hydro-Quebec’s defence plan against extreme contingencies. IEEE Trans Power Syst 1999;14(3):958–66. [7] Hollway BA, Neumann PG. Survivable computer-communication systems: the problem working group recommendations. Washington: US Army Research Laboratory; 1993. [8] Ellison RJ, Fisher DA, Linger RC, et al. Survivable network systems: an emerging discipline. Pittsburgh, PA: Software Engineering Institute. Carnegie Mellon University; 1997. [9] Schroeder MA, Netport KT. Tactical network survivability through connectivity optimization. New York: IEEE; 1987. [10] Hagin AA. Performability, reliability and survivability of communication networks: system of methods and models for evaluation. IEEE Trans 1994;11(7):562–73. [11] Yang Wenhui, Bi Tianshu, Huang Shaofeng, et al. An approach for critical lines identification based on the survivability of power grid. Proc CSEE 2011;31(7):29–35. [12] Wang Jian, Wang Huiqiang, Zhao Guosheng, et al. Index system of quantitative evaluation for information systems survivability. Comput Eng 2009;35(3):54–6. [13] Soroudi A, Afrasiab M. Binary PSO-based dynamic multi-objective model for distributed generation planning under uncertainty. IET Renew Power Generation 2012;6(2):67–78. [14] Maciel Renan S, Rosa Mauro, Miranda Vladimiro, et al. Multi-objective evolutionary particle swarm optimization in the assessment of the impact of distributed generation. Electric Power Syst Res 2012;89:100–8. [15] Wang Ling, Xu Ye. An effective hybrid biogeography based optimization algorithm for parameter estimation of chaotic systems. Expert Syst Appl 2011;38(12):15103–9. [16] Mirela O, Simon D. Biogeography-based optimization of neuro-fuzzy system parameters for diagnosis of cardiac disease. In: Proceedings of the 12th annual genetic and evolutionary computation conference. Portland, USA: ACM; 2010. [17] Sonakshi G, Anuja A, Panchal VK, et al. Extended biogeography based optimization for natural terrain feature classification from satellite remote sensing images. Proceedings contemporary computing-4th international conference. Noida, India. IC3; 2011. p. 262–9. [18] Simon D. Biogeography based optimization. IEEE Trans Evolutionary Comput 2008;12(6):702–13. [19] Boussaid I, Chatterjee A, Siarry P, et al. Hybridizing biogeography-based optimization with differential evolution for optimal power allocation in wireless sensor networks. IEEE Trans Vehicular Technol 2011;60(5):2347–53. [20] Mukherjee Rajarshi, Chakraborty Shankar. Selection of the optimal electrochemical machining process parameters using biogeography-based optimization algorithm. Int J Adv Manuf Technol 2013;64(5/8):781–91. [21] Müller A. On the performance of linear adaptive filters driven by the ergodic chaotic logistic map[J]. Int J Bifurcation Chaos Appl Sci Eng 2012;22(12): 1250290-1–1250290-11. [22] Roy PK, Ghoshal SP, Thakur SS, et al. Optimal reactive power dispatch considering flexible AC transmission system devices using biogeographybased optimization. Electric Power Components Syst 2011;39(5/8):733–50.