rce

ergzhuee

03,

a pseding

neural network (ANN)-based technique for power systemrestoration is proposed, and some workable restorationstrategies can be used as samples to train the ANN.Aapep

www.ietdl.org

IEdrule-based system is presented to generate and implementdynamic restoration plan in [6]. The mathematical

rogramming and intelligent optimisation algorithms can bembedded into the decision-making support system forower system restoration, which has shown potential in

to

cbT Gener. Transm. Distrib., 2014, Vol. 8, Iss. 1, pp. 91103oi: 10.1049/iet-gtd.2013.0065weighted power network is presented to evaluate theimportance of various nodes, but the relationship of powersupply among the nodes is not considered in power systemrestoration procedure.The regret theory was proposed in [13, 14], which is usedmake decisions in the environment with risks in the

nancial sector. The key concept of regret is to compare theurrent situation with another one which could be obtainedy choosing another strategy, and if the decision-makerand to establish a stable network, and then to prepare forcomprehensive load pick-up [3, 4]. In [5], an articial

better manner. In [12], the network cohesion degree of

system restoration process are to restore generating unitspresented to optimise the network reconguration strategy. Specically, the restoration sequence of generating units is rstoptimised so as to maximise the restored generation capacity, then the optimal restoration path is selected to restore thegenerating nodes concerned and the issues of selecting a serial or parallel restoration mode and the reconnecting failure of atransmission line are next considered. Both the restoration path selection and skeleton-network determination are implementedtogether in the proposed method, which overcomes the shortcoming of separate decision-making in the existing methods.Finally, the New England 10-unit 39-bus power system and the Guangzhou power system in South China are employed todemonstrate the basic features of the proposed method.

1 Introduction

With continuous socio-economic development, reliablepower supply is becoming more and more demanding. Bystrengthening the structure of a power system andoptimising its management, the risks of power systemfailures can be reduced. However, the growing complexityof power systems and the inevitable uncertainties in powersystem operation increase the risks of power systemfailures. The 30 July 2012 blackout in India and the 14August 2003 blackout in the USA and Canada remind usthat it is important to enhance secure and stable operationof large-scale actual power systems. Thus, it is still ofsignicant importance to study power system restorationafter a blackout [1, 2]. The main issues of networkreconguration representing the second phase in the power

solving the multi-objective optimisation problems. In [7],the optimal target system for power system restoration isformulated as a mathematical programming problem andsolved by the decompositioncoordination method, andthen a restoration method is developed by combination ofthe expert system and the mathematical programmingapproach. With the development of the complex networktheory, its applications in power systems have been paidmore and more attention [8]. In [9], betweenness, closeness,eigenvector and sub-graph indices which are employed tomeasure the importance of buses in the complex networkare introduced, and then a comprehensive index for rankingthe importance of network buses is proposed. In [10],network cohesion after node contraction is proposed toevaluate the importance of the node. In [11], it is proposedthat the weighted power network can reect the nodeimportance and the operation of actual power systems in aPublished in IET Generation, Transmission & DistributionReceived on 29th January 2013Revised on 21st May 2013Accepted on 30th May 2013doi: 10.1049/iet-gtd.2013.0065

Two-stage power networkconsidering node importancapacityCan Zhang1, Zhenzhi Lin1, Fushuan Wen1,2, G1School of Electrical Engineering, Zhejiang University, Han2School of Electrical Engineering and Computer Science, QQueensland 4001, Australia3State Grid Electric Power Research Institute, Nanjing 2100E-mail: fushuan.wen@gmail.com

Abstract: Network reconguration after complete blackout ofA new node importance evaluation method is presented baimportance of a path is employed as the objective of ndISSN 1751-8687

econfiguration strategyand restored generation

ard Ledwich2, Yusheng Xue3

ou 310027, Peoples Republic of Chinansland University of Technology, Brisbane,

Jiangsu, Peoples Republic of China

ower system is an essential step for power system restoration.on the concept of regret, and maximisation of the averagethe optimal restoration path. Then, a two-stage method is91 The Institution of Engineering and Technology 2013

path connecting two nodes. L is the maximal distance of anytwo nodes; and is the logical OR operator. Whether there is

removed; J 14 = T (0) T (0) T (w12) T (0) T (0) = 0,and this means that there exists no path connecting the nodepair (1,4) after node 3 is removed. For the node pair (i, j),J (i,j) = J ij = J ji .For the network topological characteristics in the network

reconguration procedure, two regret assessment indices aredened here to evaluate the objective node m: the regretvalue of losing topological connectivity Pm, and the regretvalue of increased restoration cost Cm. The regret value of

www.ietdl.orgrisk when reconnecting it to the power network. Thus, thecharging capacitance can be employed as the weight of theline. Larger the line weight is, larger the risk will be, andthen less possible that the line will be selected as therestoration path.Before developing a new node importance evaluation

method, the weighted network connection matrix [15] willrst be briey introduced. If there is a line lij connectingnode i and node j, and the weight of the line is wij, thenMij =Mji =wij in the connection matrix; otherwise, Mij =Mji= 0.A four-node weighted network is shown in Fig. 1, in which

wij is the line weight. The connection matrix M is

M =0 w12 w13 0w21 0 w23 0w31 w32 0 w340 0 w43 0

(1)

where w12 =w21, w13 =w31, w23 = w32 and w34 =w43.The logical connection matrix is dened as

J =0 T w12

( )T w13( )

0

T w21( )

0 T w23( )

0

T w31( )

T w32( )

0 T w34( )

0 0 T w43( )

0

(2)nds that he/she could get better results by choosing anotherstrategy, he/she would be regretful. Different results could beobtained by employing different restoration strategies in thenetwork reconguration procedure, hence, the regretconcept can be employed to evaluate node importance andoptimise the restoration path in the network recongurationprocedure in this work.Given this background, the topological characteristics of

the power system and the relationship of power supplyamong the nodes are rst addressed, and a new method forevaluating the node importance degree and optimising therestoration paths based on the concept of regret ispresented. A two-stage method is proposed to optimise thenetwork reconguration strategy. The start-up power,starting time, ramping rate and critical maximum interval ofgenerating units are considered in optimising the restorationsequence of the generating units for the purpose ofmaximising the restored generation capacity. The serial orparallel restoration mode can be selected based on therestoration state of the power system. Then, a method forevaluating the average importance degree of eachrestoration path is proposed to optimise the restoration pathsof the generating units concerned. Furthermore, the possiblefailure of reconnecting a transmission line to the powernetwork is also considered. Finally, the New England10-unit 39-bus power system and the Guangzhou actualpower system in South China are employed to demonstratethe essential features of the proposed method.

2 A method to evaluate node importancebased on the concept of regret

In the traditional shortest path optimisation method, theweights of all the lines are dened as one. Although thismethod can show the network topology, the relationship inelectricity of the nodes cannot be reasonably reected.The charging capacitance of a line can lead to overvoltage92 The Institution of Engineering and Technology 2013a path connecting two nodes can be judged from the pathjudgement matrix. If J ij = 1, this means that there will be atleast one path connecting node i with node j; otherwise,there will be no path. Taking the network shown in Fig. 1as an example, there are

J 12 = T w12( ) T w13

( ) T w23( )( )

(4)

J 14 = T w13( ) T w34

( )( )

T w12( ) T w23

( ) T w34( )( )

(5)

where is the logical AND operator; (i, j) is dened as thenode pair containing nodes i and j. Equation (4) means thatthere are two paths connecting the node pair (1, 2): l12 andl13l32. Equation (5) means that there are also two pathsconnecting the node pair (1, 4): l13l34 and l12l23l34. Theresults of the two operations are both equal to 1, and thismeans that there is at least one path connecting the relativetwo nodes.

2.1 Regret value of losing topological connectivity

In a power system, if electricity is not available in a node, thenit can be deemed that the node is removed from the powernetwork. If node 3 is removed from the network in Fig. 1,the lines directly connecting node 3 are also removed. Inthe connection matrix M, all the elements related to node 3should be set to zero. Assume that the modied pathjudgement matrix is J*, then the element J 12 =T (w12) T (0) T (0) = 1, and this means that there existpaths connecting the node pair (1, 2) after node 3 iswhere T() is a logical truthfalse operation. If the value in thebracket of T() is not equal to zero, then the result of theoperation is 1; otherwise, it is 0.The path judgement matrix is dened as

J = J1 J2 J3 JL (3)

where Jd is the path matrix, in which the distance between therelative two nodes is equal to d; d is the number of lines on the

Fig. 1 Four-node weighted networkIET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 1, pp. 91103doi: 10.1049/iet-gtd.2013.0065

m, and Hmin (i,j) is the minimum restoration cost of the nodepair (i, j) after applying the same strategy. Hmin(i,j) is theminimum restoration cost of the node pair (i, j) in case thatthe restoration of the evaluated node m is allowed. If therestoration of node m is not allowed, fewer lines can beselected to optimise the restoration path. In thiscircumstance, the shortest path obtained will not be shorterthan the path optimised with the strategy of restoring node

min min

www.ietdl.orglosing topological connectivity is dened as:

Pm =

(i,j)[Vm

J (i,j) J (i,j)( )

(6)

where Vm is the set of node pairs directly connecting node m;J (i,j) is the path judgement value of node pair (i, j) in theoriginal network; and J (i,j) is the path judgement value ofnode pair (i, j) after node m is removed. In the restorationprocedure of a power system, if there is a path connectingtwo nodes (or a node pair), the power can be delivered toeach other, hence, this kind of node pair is dened asdeliverable one; otherwise, the cranking power cannot bedelivered, that is, one node cannot be restored by the otherone, hence this kind of node pair is dened as non-deliverable one. In the phase of network reconguration,sometimes non-generating nodes should be restored withpriority, and the node is also represented by m, which is tosupply power from node m to its connecting nodes. If nodem is not restored, the whole reconguration process may beaffected, because some nodes will only be restored by nodem. In this circumstance, the number of non-deliverable nodepairs is dened as the regret value of losing topologicalconnectivity. If the regret value is large, much damage willbe caused if not restoring node m. Then, the decision-makers will be more regretful for not employing thisstrategy. As a result, node m should more likely be restored,and this means that the importance of node m is high.To evaluate the importance of node 3 in the network of

Fig. 1, the strategies of restoring node 3 and not restoringnode 3 are denoted as strategies 1 and 2, respectively. Thus,the regret values of losing topological connectivity for node3 are shown in Table 1.

2.2 Regret value of increased restoration cost

If the evaluated node m is not restored, the nodes connected tonode m cannot be restored through the lines connecting nodem, and then these nodes may be supplied power through otherpaths. Thus, this strategy may involve more cost and morerisk, because some larger weighted lines or longer pathswill be selected to deliver power to the nodes. The totalweight of the lines in the restoration path is regarded as therestoration cost, and then the path with minimum cost isselected to restore the node concerned. The increasedrestoration cost by employing the strategy of not restoring

Table 1 Regret value of losing topological connectivity fornode 3

Evaluatednode

Strategy1

Strategy2

P3 =

(i,j)[V3J(i,j) J(i,j)( )

3 J(1, 2) = 1 J(1, 2) = 1 2J(1, 4) = 1 J(1, 4) = 0J(2, 4) = 1 J(2, 4) = 0the evaluated node m is dened as the regret value ofincreased restoration cost, and can be represented by

Cm =

(i,j)[Vm

Hmin (i,j) Hmin(i,j)( )

(7)

where V m is the set of the deliverable node pairs afteremploying the strategy of not restoring the evaluated node

IET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 1, pp. 91103doi: 10.1049/iet-gtd.2013.0065m, and this means that H(i,j) H(i,j) . In order to determinethe minimum restoration cost of the deliverable node pair,the minimum restoration cost matrix is dened as

Hmin = M1wQM2wQM3wQ QML

w (8)

whereMdw is the restoration cost matrix in which the distancebetween the relative two nodes is d; L is the longest distancebetween any two nodes, and the number of lines in the pathconnecting two nodes is dened as the distance between thetwo nodes concerned. It should be noted that the totalweight of the lines of the shortest path may not be theminimum one. Selecting the line with higher weight maycause more cost and would take more risk, so the path withthe minimum total weight can be regarded as the path withthe minimum restoration cost. is the operator forobtaining the minimum value from the elements on thesame location of all the matrixes. In fact, only when thereexist at least two nodes and one path connecting these twonodes after the evaluated node m is removed would theregret value of increased restoration cost be calculated.If there is no path between any two nodes after node m isremoved, this node will be extremely important, thus theregret value of increased restoration cost should be set as avery large amount (such as 100 000), and the index Hmin

dened in (8) will not be calculated.Take the network in Fig. 1 as an example, the minimum

restoration costs of the node pairs (1, 2) and (1, 4) are,respectively, as follows

Hmin(1,2) = min w12, w13 + w23( ) = min(0.4, 0.3) = 0.3

and

Hmin(1,4) = min w13 + w34, w12 + w23 + w34( ) = min(0.4, 0.9)

= 0.4

In evaluating the importance degree of node 3, only the nodepair (1, 2) is the deliverable one after node 3 is removed, andHmin (1,2) = min w12

( ) = 0.4. As a result, the regret value ofincreased restoration cost of node 3 is shown in Table 2.It can be seen from Table 2 that the regret value C3 of

increased restoration cost of node 3 is equal to 0.1. Thismeans that the restoration cost will increase if node 3 is notrestored. The increased restoration cost is an index forevaluating the importance degree of node 3. The larger theincreased cost is, the more important the evaluated nodewill be.

Table 2 Increased regret value of the restoration cost ofnode 3

Evaluatednode

Strategy 1 Strategy 2 C3 =

(i,j)[V3Hmin (i,j) Hmin(i,j)( )

3 Hmin(1,2) = 0.3 Hmin (1,2) = 0.4 0.193 The Institution of Engineering and Technology 2013

2.3 Comprehensive evaluation of the importancedegree of a node

By combining the regret value of losing topologicalconnectivity with that of increased restoration cost, acomprehensive index for evaluating the importance degreeof a node can be obtained as

bm = Pm + u Cm (9)

where u is the ratio coefcient for adjusting the relativeimportance of Pm and Cm. The comprehensive evaluatingindex of the evaluated node m can evaluate the performanceof supplying power from any node connecting with node m tothe other nodes connecting with node m, in comparing thestrategy that restoring the evaluated node m with that of notrestoring it. Larger the regret value, more the incurred cost. Inan extreme case, if there exists a node that all the unrestorednodes become isolated when it is removed, this node will beextremely important. Thus, its regret value of increasedrestoration cost as well as that of losing topologicalconnectivity should be given a very large amount, and thenthe importance degree of the node would also be very large.

The original network is shown in Fig. 2a, and the networkafter node contraction is shown in Fig. 2b. It can be seenfrom Fig. 2b that the contraction process of node 10 is tocontract node 10 and the nodes connecting with it,including nodes 4, 5, 6 and 9, as the new node 9.The node importance degree after node contraction can be

dened as

a = 1/Nnodelave (10)

where Nnode is the total number of nodes in the network; lave isthe average distance among the nodes, and dened as

lave =i,j[

_V

dmin,ij/ Nnode Nnode 1( )

/2( )

(11)

where dmin,ij is the shortest distance between nodes i and j; V_

is the set of nodes in the network.The weighted power network is shown in Fig. 2a, together

with the weights of the lines. Bigger the weight , larger thecost and risk will be, if the line is selected to restore theconcerned node. In Fig. 2a, the topological structure of

www.ietdl.orgCompared with the situation where the node can berestored, the situation where the node cannot be restoredwill lead to more costs in the restoration. Thus, the ratiocoefcient u should be set as a small number. Furthermore,another evaluation method can also be employed toevaluate the importance degree of the node, by sorting allthe nodes by Pm (or Cm if Pm is the same) in thedescending order, and then determining the order of theimportance degree of the nodes.

2.4 Comparisons between the proposed methodand the existing method

Traditionally, the concept of the node degree is employed toscale the importance of the node in studying the topologicalcharacteristics of a network. However, some key nodes inan actual network may not have larger node degrees thanothers [10]. Thus, network cohesion after node contractionis proposed to evaluate the importance degree of the nodesin [10]. The node contraction process is shown in Fig. 2.

Fig. 2 Node contraction process of a weighted power networka The original networkb The new network after node 10 is contractedc The new network after node 11 is contracted94 The Institution of Engineering and Technology 2013node 10 and the nodes connected with it are same withthose of node 11. If the weight of the line is not considered,the importance degrees of nodes 10 and 11 would be same.If the power is supplied from nodes 9 to 4, then the totalweight of path 9 11 3 4 would be 0.5, and that ofpath 9 10 4 would be 0.9. Thus, the path 9 113 4 should be selected to deliver power. Furthermore, ifthe cranking power is supplied from node 9 to region II,node 11 would be restored rst, and then power would besupplied from regions I to II. Thus, it can be concluded thatnode 11 is more important than node 10.The new networks after nodes 10 and 11 in Fig. 2a are

contracted as shown in Fig. 2b and c, respectively. Theimportance degrees of nodes 11 and 10 can be calculated,and are 0.2941 and 0.3271, respectively. It can be seen thatnode 10 is more important than node 11, and the result isopposite to the conclusion obtained by using the traditionalmethod. Since the lines directly connected to the node arealso contracted when the node is contracted, electricalconnection of the nodes cannot be reasonably reected inthe contracted network.IET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 1, pp. 91103doi: 10.1049/iet-gtd.2013.0065

work, a two-stage method is proposed to optimise thenetwork reconguration strategy so as to reduce the scale ofthe optimisation problem as well as the difculty degree.In the rst stage, the start-up sequence of generating units isdetermined by maximising the restored generation capacity.In the second stage, the path with maximum averageimportance degree is selected to restore the generating unitsconcerned. The optimal network reconguration strategycan then be obtained.

3.1 Optimisation model and strategy fordetermining the start-up sequence of generatingunits

3.1.1 Optimisation model for determining the

www.ietdl.orgThus, the regret values of nodes 10 and 11 can becalculated by using the proposed method, and the resultsare shown in Table 3.It can be seen from Table 3 that the Pms of nodes 10 and

11 are equal, whereas the Cm of node 11 is larger than that ofnode 10, and this means that node 11 is more important thannode 10. In summary, it can be concluded that the proposedmethod can overcome the shortcomings of the methodpresented in [10] in evaluating the importance degree of thenode, and lead to a more reasonable conclusion.

2.5 Optimisation of the skeleton network

Actually, the node importance evaluation method based onthe concept of regret can be employed to optimise theskeleton network as well. The ideal objective of optimisingthe skeleton network is to restore all generating units andthe most important nodes with the least number oftransmission lines. In this work, the skeleton network isoptimally determined by maximising the averageimportance of the skeleton network formulated as

maxss =m[Vs

bm/Nns (12)

where s is the average importance degree of the skeletonnetwork represented by s; Vs and Nns are the set andnumber of the unrestored nodes in the skeleton networkrepresented by s, respectively.The average regret value of losing topological connectivity

of the skeleton network is dened as

Pss =m[Vs

Pm/Nns (13)

The average regret value of increased restoration cost of theskeleton network is dened as

Css =m[Vs

Cm/Nns (14)

Table 3 Regret values of nodes 10 and 11

Node pairsunable tosupplypower toeach other

Pm Node pairs able to supply power toeach other

Cm

(i, j) Hmin (i ,j) Hmin(i ,j) H

min (i ,j) Hmin(i ,j)

10 (4, 5) (4, 6)(5, 6) (5, 9)

(6, 9)

5 (4, 9) 0.5 0.5 0 0

11 (1, 2) (1, 3)(1, 9) (2, 3)

(2, 9)

5 (3, 9) 1.1 0.3 0.8 0.8Thus, the average regret value of losing topologicalconnectivity as represented by Ps can be employed todetermine the best scheme from different schemes ofskeleton networks. In case that there are more than onescheme of skeleton networks having the same value of Ps,the scheme with the larger Cs will be selected.

3 Two-stage method for networkreconfiguration

The main objective of network reconguration is to restorethe generating units and establish a stable network. In this

IET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 1, pp. 91103doi: 10.1049/iet-gtd.2013.0065start-up sequence of generating units: In thenetwork reconguration phase, the start-up sequence ofgenerating units is determined by maximising the restoredgeneration capacity. The restored generation capacity Ptotalis equal to the total sum of the rated capacities minus thestart-up power required of all the generating unitsconcerned, is formulated as

Ptotal =Nsbn=1

Psbn +Nsnbk=1

Psnbk Nsnbk=1

Psnbjk (15)

where Nsb and Nsnb are the numbers of the black-startgenerating units and non-black-start generating unitsrestored in the network reconguration phase, respectively;Psbn and P

snbk are the restored outputs of the black-start

generating unit n and the non-black-start generating unit k,respectively; Psnbjk is the required start-up power of thenon-black-start generating unit k. The complex start-upprocess of a thermal generating unit is shown in Fig. 3 [16,17]. It can be seen from Fig. 3 that the generating unit isrestarted at T1 when it is supplied with the cranking power;the unit starts to output power at T1 + Tgp, and thenincreases the output gradually with the ramping rate untilthe maximum output Pmax at T2.Thus, the problem of optimising the restoration sequence of

the generating units can be described as

maxPtotal =Nsbn=1

Psbn +Nsnbk=1

Psnbk Nsnbk=1

Psnbjk (16)

The following constraints should be respected in the networkreconguration phase:

Fig. 3 Start-up characteristic of a thermal generating unit95 The Institution of Engineering and Technology 2013

(1) The generation output constraints

PminGn PGn PmaxGn , n [ CG (17)QminGn QGn QmaxGn , n [ CG (18)

importance for maximising the restored generation capacity[17]. Thus, the following restoration strategy of thegenerating units can be employed:

(1) The black-start generating units should restart rst to

units.

)+

www.ietdl.orgwhereG is the set of the generators; PGn andQGn are the activeand reactive power outputs of generator n, respectively; PmaxGn ,PminGn , Q

maxGn and Q

minGn are the maximum active power,

minimum active power, maximum reactive power andminimum reactive power of generator n, respectively.

(2) The constraints of voltage amplitudes and phase angles atnodes

Vmini Vi Vmaxi , i [ CN (19)

uib u jb

ub, b [ CL (20)

whereN is the set of all nodes; Vi is the voltage amplitude ofnode i; L is the set of all lines; ib and jb are the voltageangles at the two ends of line b.

(3) The constraints of transmission line capacities

Plb Pmaxlb , b [ CL (21)

where L is the set of all lines; Plb and Pmaxlb are the active

power and the transmission capacity of line b, respectively.

(4) The constraint of the system frequency

f min f f max (22)

where f is the system frequency; fmax and fmin are the upperand lower limits of the frequency, respectively.

(5) The constraint of the critical maximum interval of agenerator

0 Tn1 Tnmax, n [ CG (23)

where Tn1 is the time when generator n obtains the crankingpower; Tnmax is the critical maximum interval of generator n.If the generator can obtain the cranking power within thecritical maximum interval, it will restart and supply powerto the system quickly.There exist several ways to keep the system voltage within

the accepted range, such as deactivating switched staticcapacitors, connecting shunt reactors, adjusting transformertaps to appropriate positions and picking up loads withlagging power factors [18]. Restoring loads gradually with asmall increment can maintain the system frequency withinthe accepted limit.

3.1.2 Optimisation strategy of the restorationsequence of generating units: Quickly restoring thepower system is the main objective of the networkreconguration phase. The characteristics of variousgenerators can be quite different, thus optimising therestoration sequence of the generating units is of great

lG =m[CG

(i,j)[Vm

J (i,j) J (i,j)(96 The Institution of Engineering and Technology 2013It should be noted that the restoration path for a unit in therst part of the unit restoration sequence may contain a nodewith a unit in the later part. Indeed, the later unit will not berestarted rst in this case, and the node with the later unit isonly regarded as the one to deliver the cranking power toother nodes, and the units are restarted following thedetermined restoration sequence. In addition, the crankingpower may not be large enough to restart the unit in therestoration path, and hence the cranking power is justemployed to restart the unit at the end of the restorationpath. Furthermore, if the unit in the later part is restartedrst, it may have negative impact on the restoration of otherunits. As a result, the restored generation capacity woulddecrease because the restoration sequence of the units is notproperly organised. Moreover, generating units are usuallylocated at the terminal nodes which could not be selectedinto the restoration path of other units. In summary, therestoration sequence of the units is rst dened, and therestoration paths next determined based on the restorationsequence of the generation units. In addition, the restorationof the path should not delay the restoration of generatingunits. If no path could be found without delaying therestoration of generating units, the restoration sequence ofthe remained unrestored generating units will be optimisedagain to maximise the restored generation capacity. Then,the restoration path will be determined based on the newrestoration sequence of generating units.

3.2 Optimisation model and strategy of therestoration paths

3.2.1 Optimisation model of the restoration paths:After the restoration sequence of the generating units isdetermined, the next step is to optimise the restorationpaths. In this work, the path with maximum averageimportance degree is selected to direct the optimisation ofthe path restoration. The average importance degree of therestoration path is dened as (see (24))where and N are the set and number of the unrestorednodes in path , respectively. In the restoration process, the

u

(i,j)[VmHmin (i,j) Hmin(i,j)( )/NG (24)provide the cranking power to the power system;(2) The generating units with the critical maximum intervals

should have the priority to be restarted, and the units withsmall critical maximum intervals should be rst restarted.

(3) The generating units with larger ramping rates should berestarted with priority under the strategy (2).

(4) If the generating units with the critical maximumintervals cannot be restarted because of the insufcientcranking power, the other type of units can beconsidered to be restarted.

(5) If the cranking power is sufcient, the parallel restorationstrategy can be applied; otherwise, the series restorationstrategy should be employed to restore the generatingIET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 1, pp. 91103doi: 10.1049/iet-gtd.2013.0065

longer path may restore more nodes, however it wouldrequire more operations of the lines, and lead to more risk.

The average regret value of losing topological connectivityof the path is dened as

PG =m[CG

(i,j)[Vm

J (i,j) J (i,j)( )

/NG (26)

The average regret value of increased restoration cost of thepath is dened as

CG =m[CG

(i,j)[Vm

Hmin (i,j) Hmin(i,j)( )

/NG (27)

The evaluating method presented before can also beemployed to evaluate the importance degree of the path.In this method, all the paths are sorted by P (or C if P is

www.ietdl.orgAs a result, the path with maximum average importancedegree is employed to optimise the restoration path.The objective function for optimising the restoration path is

formulated as (see (25))

3.2.2 Optimisation strategy of the restoration paths:The restoration paths to be selected are those connecting therestored region with the generating units unrestored. Thepath with maximum average importance degree is selectedto supply power to the generating unit concerned after theimportant nodes should be restored as many as possible, andthe operating time and the risk caused by reconnecting thelines with the power system should also be considered. Theideal restoration objective is to restore the most importantnodes with the least operations of the lines. Charging a

Fig. 4 New England 10-unit 39-bus power systemaverage importance degree of the candidate paths iscalculated.

Table 4 Parameters of the generating units

Generatingunit

Active powercapacity, MW

Ramping rate,MW/h

Start-requi

30 350 7231 1145.55 17532 750 10233 750 9634 660 9035 750 9636 660 9037 640 7838 930 12039 1100 140

max lG =m[CG

(i,j)[Vm

J (i,j) J (i,j)( )

IET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 1, pp. 91103doi: 10.1049/iet-gtd.2013.0065the same) in the descending order, and then the order of theimportance of the paths can be determined. Finally, the pathwith he maximum average importance degree is selected tobe restored. If some lines cannot be charged successfully,the candidate restoration paths would be optimised based onthe current network and the recalculated regret values.It should be noted that the restoration path may have impact

on the restoration of the generating units, because the requiredtime may be long to energise a path with many transmissionlines, and this may delay the restoration of the concernedgenerating units. Hence, the candidate restoration pathsshould be among the paths which would not delay therestoration of the concerned generating units. If no path canbe found without delaying the restoration of the generatingunits, it would be delayed until the cranking power isdelivered to the generating units. However, the negativeimpacts of energising the line can be avoided to someextent by employing certain strategies. For example, somelines in the restoration path of a generating unit in the laterpart of the start-up sequence may be energisedsimultaneously when restoring the path of the generatingunit in the anterior part.The restoration time of each linecan be obtained by using the method proposed in [19]. Anoptimistic time A, a most likely time M and a pessimistictime B of restoring each line can be determined by systemoperators based on their experience, and then the expected

up powerred, MW

Start-up timerequired, min

Critical maximuminterval, min

9 10 2532 65 9520 30 60 16 10 1522 30 6018 20 5516 25 5528 45 7030 55 80

+ u

(i,j)[VmHmin (i,j) Hmin(i,j)( )/NG (25)97 The Institution of Engineering and Technology 2013

3 and 4 min, then the expected restoration time of line b ist = (2 + 4 3 + 4)/6 = 3 min.

3.3 Steps of the two-stage network recongurationoptimisation

In summary, the steps of the two-stage networkreconguration optimisation are as follows:

(1) Calculate the regret value of losing topologicalconnectivity Pm, and the regret value of increasedrestoration cost Cm of the nodes based on the data ofthe power system;

Table 5 Parameters of node importance degrees

Node Regret value oflosing

topologicalconnectivity

Pm

Regret valueof increasedrestorationcost Cm

Nodedegree

Ranking ofthe

importancedegree

1 0 0.8921 2 182 3 6.3032 4 23 0 2.9382 3 144 0 2.0470 3 155 0 0.9220 3 176 3 1.0416 4 37 0 0 2 278 0 5.6930 3 129 0 0.6287 2 2210 2 0.3522 3 611 0 0.6984 3 2012 0 0 2 2713 0 0.5648 3 2314 0 1.3153 3 1615 0 0.3025 2 2416 8 1.2454 5 117 0 3.3570 3 1318 0 0.6851 2 2119 3 0 3 420 1 0 2 1121 0 0.1023 2 2622 2 0.2427 3 723 2 0.0337 3 8

Table 7 Restoration sequence of the generating units

Generatingunit

Restorationtime, min

Generatingunit

Restorationtime, min

33 0 35 5034 10 37 5030 20 38 5032 30 31 6036 30 39 60

www.ietdl.orgrestoration time of the line can be obtained as

24 0 0.2669 2 2525 2 0.5051 3 526 2 0 3 927 0 0.7007 2 1928 0 0 2 27"29 2 0 3 9t = (A+ 4M + B)/6 (28)

For example, if the optimistic time, the most likely time andthe pessimistic time of restoring line b are, respectively, 2,

Table 6 Restoration time of each line

Starting node ofthe line

End node ofthe line

Restoration operation time ofthe line/min

S

2 1 339 1 13 2 325 2 24 3 218 3 45 4 2 26 25 314 4 2 27 26 26 5 28 5 27 6 411 6 18 7 39 8 239 9 311 10 213 10 214 13 315 14 216 15 217 16 319 16 321 16 3

98 The Institution of Engineering and Technology 201328 26 329 26 329 28 312 11 3(2) Optimise the restoration sequence of the generating unitsby maximising the restored generation capacity;

(3) Divide the power network into the restored region and theunrestored region, and nd the candidate pathsconnecting the restored region with the unrestoredgenerating units. Then, calculate P and C, and selectthe optimal path to restore the generating unitconcerned. Check the constraints and determine theload to be restored so as to ensure the stability andsecurity of the power system. If some constraints areviolated, the optimal power ow program could be usedto determine the outputs of the generators and/or theload shedding amounts. In the restoration procedure, thelimits of constraints could be relaxed in order to recoverpower supply quickly, and this is one of the major

tarting node ofthe line

End node ofthe line

Restoration operation time ofthe line/min

24 16 318 17 427 17 222 21 423 22 324 23 312 13 46 31 210 32 119 33 320 34 222 35 323 36 225 37 22 30 229 38 219 20 2

IET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 1, pp. 91103doi: 10.1049/iet-gtd.2013.0065

differences between the system restoration and normaloperation scenarios.

(4) Put the restored lines and nodes into the restored region.If some lines cannot be charged successfully, go back tostep (3) based on the current network, and then nd thenew optimal path;

(5) Repeat steps (3) and (4) until all the generating units arerestored.

4 Case studies

4.1 Case 1: the New England 10-unit 39-bus powersystem

The New England 10-unit 39-bus power system shown inFig. 4 is employed to demonstrate the proposed method.The characteristics of the lines can be found in [20].Suppose that there is only one generator in each generatingnode, and the characteristics of the generators are shown inTable 4. The charging capacitance of a line is regarded asits weight, and the importance degree of the node iscalculated by the proposed method and shown in Table 5.The restoration time of each line is shown in Table 6.The black-start generator is located at node 33. Suppose

that the black-start generator restarts right after the blackout,and provides cranking power to the power system. Thus,

the restoration sequence and the restoration time of thegenerating units can be optimised and are shown in Table 7.After the restoration sequence of the generating units is

optimised, the path with maximum average importancedegree is selected to restore the generating unit concerned.Comparisons between the proposed method and thetraditional shortest path method are shown in Table 8.It can be seen from Table 8 that the restoration pathsoptimised by the proposed method are different from theones optimised by the traditional method, in which only thecharging capacitance of the path is considered.Comparisons of the proposed restoration path selecting

method with the traditional charging capacitanceminimisation method are carried out in terms of thegeneration outputs. In Fig. 5, the real line and the dashedline, respectively, represent the generation outputs obtainedby the proposed method and by the traditional chargingcapacitance minimisation method. It can be seen fromFig. 5 that, in the early period, part of the dashed line isabove the real line, because some generating units aredelayed to be restored because of the long restoration timeof the paths. In the traditional method only the chargingcapacitance in the path are considered, whereas the pathwith the minimal charging capacitance may take a longertime to deliver the cranking power to the generating units.For example, 22 min are used to deliver the cranking powerto the generating unit at node 30 by employing the

Table 8 Comparisons between the two methods for optimising the restoration paths of generating units

Generating unit Proposed method Shortest path method

Restoration path P C Restoration path

34 33 19 20 34 2.0000 0 33 19 20 3430 19 16 17 27 26 25 2 30 2.5000 2.0186 19 16 17 18 3 2 3032 2 3 4 5 6 11 10 32 0.8333 1.3332 3 4 14 13 10 3236 16 21 22 23 36 1.3333 0.1262 16 24 23 3635 22 35 0 0 23 22 3537 25 37 0 0 2 25 3738 26 29 38 2.0000 0 25 26 29 38

eth

www.ietdl.org31 6 3139 5 8 9 39

Fig. 5 Generation outputs optimised by traditional and proposed mIET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 1, pp. 91103doi: 10.1049/iet-gtd.2013.00650 0 4 5 6 310 3.1609 2 1 39

ods99 The Institution of Engineering and Technology 2013

traditional method, whereas only 20 min are required torestart this unit by employing the proposed method; thusthe restoration of this unit has to be delayed if thetraditional method is applied. In the later period, the dashedline is below the real line, because the restoration of somegenerating units is delayed, and then the moments theseunits start to output power are also delayed. At 130 min, theoutputs of all generating units optimised by the proposedmethod reaches 812 MW, whereas that optimised by thetraditional method reaches 783 MW; this means that theproposed method leads to a quicker restoration ofgeneration outputs than the traditional charging capacitanceminimisation method.Take the restoration of node 39 as an example, there are

three candidate paths and the average importance degree isshown in Table 9. All the P s of the three paths are 0, sothe optimal path cannot be determined only by P. Then,the C of the paths are compared, and the path (5 89 39) with maximum C is selected to restore node 39.In Table 8, the P s and Cs of the restoration paths ofnode 35, 37 and 31 are all 0, because the nodes directlyconnecting with the generating nodes are restored. Finally,the destination restoration network can be obtained and isshown in Fig. 6.The restoration of node 32 is taken as an example to show

how to deal with the failure of transmission line charging(restoration) in the power system restoration procedure. Therestoration path of node 32 is 2 3 4 5 6 1110 32. Suppose that the charging (restoration) of lines611 is failed, then the restored lines and nodes are put intothe restored region, and the restoration path is optimisedagain based on the current restored network. The averageimportance degree of the four candidate paths is shown in

Table 10, and it can be seen that the candidate path (414 13 10 32) is optimal.The skeleton network can also be obtained by using the

proposed skeleton-network optimisation method, as shown inFig. 7; the skeleton network importance degree is listed inTable 11. The importance degree of the skeleton networkobtained by the restoration path optimisation method is alsoincluded in Table 11. It can be seen from Tables 11 that Psobtained with the skeleton-network optimisation method islarger than the other one, and this means that the skeletonnetwork shown in Fig. 7 is better than that shown in Fig. 6.This is because that the skeleton network shown in Fig. 7 is

Fig. 7 Optimised skeleton network

Table 11 Importance degrees of the skeleton networksobtained by two methods

Method Lines involved Ps Cs

Skeleton-networkoptimisationmethod

L3319, L1920, L2034,L1916, L1617, L1727,

L2726, L2625, L252, L2

1.5789 1.1340

Table 10 Average importance degree of the candidaterestoring paths for node 32 (after lines 611 fail to be restored)

Candidate paths P C

16 15 14 13 12 11 10 32 0.3333 0.538916 15 14 13 10 32 0.5000 0.63374 14 13 12 11 10 32 0.4000 0.58614 14 13 10 32 0.6667 0.7441

www.ietdl.orgFig. 6 Destination restoration network

Table 9 Average importance degree of the candidate restoringpaths for node 39

Restoration path P C

2 1 39 0 0.89216 7 8 9 39 0 2.10725 8 9 39 0 3.1609100 The Institution of Engineering and Technology 201330, L23, L34, L45, L56,L611, L1110, L1032, L1624, L2423, L2336, L2322,L2235, L2537, L2629,

L2938, L631, L21, L139Restoration pathoptimisationmethod

L3319, L1920, L2034,L1916, L1617, L1727,

L2726, L2625, L252, L230, L23, L34, L45, L56,L611, L1110, L1032, L1621, L2122, L2223, L2336,L2235, L2537, L2629,L2938, L631, L58, L89,

L939

1.5000 1.3406IET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 1, pp. 91103doi: 10.1049/iet-gtd.2013.0065

Fig. 8 Guangzhou power system in south China

Table 12 Restoration sequence of the generating units and the restoration paths of the Guangzhou power system in south China

Restorationstep

Restoration time,min

Cranking power,MW

Generatingunits

Restoration paths

1 0 0 XNAP 2 4 2 TPP XNAPBEJ JIHRH TR TPP3 14 5 BHP BEJ LYGT JCBHP

XC BEJ LYXCGZP BEJ LYGZPMZCP BEJ LYWXMRBCSYC TAXBISKY

GQMZCP4 34 18.3 ZJP TAX TC PY ZJP

HYBP KYHYBPHH GT TX FRHHBJ GT TX FRHHBJ

5 44 11 HPAP ZJPYF FS FCHPAPYCP TAXYCP

6 59 61.2 HPBP BISHPBPHYCP KYHYCPSTP CSXZSTPLHSP CSXZ LHS LHSPLJP PYSIJ LJPNS PYSIJDYHDZNSMS PYSIJDYHDZHGMSZNP FSSQSW ZNP

MZABP GQMZABPGBP KYGBPLCP TAX ZEC LC LCPGPTP TAX ZEC LC LCPDF ZG PTGPTPJLP TAX ZEC LCASJSIT JLPXTP TAX ZECXTXIHXTPDMP BEJCHWQ LIK LVTDMPLXHP BEJCHWQ LIK LXHPDZBP BEJCHWQ TLDZBP

www.ietdl.org

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101 The Institution of Engineering and Technology 2013

www.ietdl.orgoptimised with the objective of maximising the importancedegree of the skeleton network, whereas that shown in Fig. 6is obtained by selecting the restoration paths for thegenerating units. However, if the restoration of any line(s) ornode(s) in Fig. 7 fails, the skeleton network will not be theoptimal one, whereas a failure in the line or node restorationof Fig. 6 will not have any impact on the restoration ofprevious restoration paths, although the previous restorationpaths are still optimal. Hence, the restoration pathoptimisation method may be more practical in the networkreconguration procedure of actual power systems.

4.2 Case 2: the Guangzhou power system in southChina

The Guangzhou power system in south China as shown inFig. 8 is also employed to demonstrate the proposedmethod. There are 29 generating units, 163 buses (nodes)and 212 transmission lines. The black-start generator islocated at XNAP, and it restarts right after a blackout so asto provide the cranking power to this system. Therestoration sequence of the generating units and therestoration paths of this actual system are optimised byusing the proposed method, and are shown in Table 12. Therestoration procedure is divided into six steps as shown inTable 12, and the generating units are restored step by step.If there are several generating units in the same step, thesegenerating units can be restored with a parallel mode. Thedestination restoration network such obtained is shownin Fig. 9.

Fig. 9 Destination restoration network of the Guangzhou power system

102 The Institution of Engineering and Technology 20135 Concluding remarks

A new method to evaluate the node importance degree basedon the concept of regret is proposed, so as to properly takeinto account the network characteristics of the power systemconcerned. The quantitative node importance degree can beobtained by using the proposed method. Then, a two-stagenetwork reconguration strategy is presented. In theproposed strategy, the start-up sequence of generating unitsis rst determined by maximising the restored generationcapacity, and then a serial or parallel restoration mode canbe employed based on the restoration state of the powersystem; then, the path with maximum average importancedegree is selected to restore the generating units concerned.Many factors, such as the restoration of important nodesand operating risks, are considered. The problem associatedwith the existing methods in which the restoration paths andthe skeleton network are separately determined could thenbe solved at least to some extent by using the proposedstrategy. The results of the case studies demonstrate thefeasibility and efciency of the proposed method. Thedeveloped method could be employed in power systemrestoration after a complete blackout or local outages.

6 Acknowledgments

This work is jointly supported by National High TechnologyResearch and Development Program (863 Program)(2011AA05A105), National Natural Science Foundation ofChina (51007080), the Fundamental Research Funds for the

in south China

IET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 1, pp. 91103doi: 10.1049/iet-gtd.2013.0065

Central Universities (2012QNA4011) and a key project fromState Grid Corporation of China (ZDK/GW002-2012).

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www.ietdl.orgIET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 1, pp. 91103doi: 10.1049/iet-gtd.2013.0065103 The Institution of Engineering and Technology 2013