Research ArticleModel and Topological Characteristics of Power DistributionSystem Security Region
Jun Xiao Guo-qiang Zu Xiao-xu Gong and Cheng-shan Wang
Key Laboratory of Smart Grid of Ministry of Education Tianjin University Nankai District Tianjin 300072 China
Correspondence should be addressed to Jun Xiao xiaojuntjueducn and Guo-qiang Zu zuguoqiang tju163com
Received 16 January 2014 Accepted 28 June 2014 Published 24 July 2014
Academic Editor H D Chiang
Copyright copy 2014 Jun Xiao et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
As an important tool of transmission system dispatching the region-based method has just been introduced into distribution areawith the ongoing smart distribution grid initiatives First a more accurate distribution system security region (DSSR) model isproposed The proposed model is based on detailed feeder-interconnected topology and both substation transformer and feederN-1 contingencies are considered Second generic characteristics ofDSSR are discussed andmathematically provedThat is DSSR isa dense set of which boundary has no suspension and can be expressed by several union subsurfaces Finally the results from both atest case and a practical case demonstrate the effectiveness of the proposedmodeling approach the shape of DSSR is also illustratedby means of 2- and 3-dimensional visualization Moreover the DSSR-based assessment and control are preliminary illustrated toshow the application of DSSRThe researches in this paper are fundamental work to develop new security region theory for futuredistribution systems
1 Introduction
This paper presents a model of power distribution systemsecurity region (DSSR) which takes both substation trans-former and feeder N-1 contingencies into account Mean-while main topological characteristics of DSSR are alsodiscussed and mathematically proved
DSSR is a newly proposed concept originated from N-1 security guideline [1 2] DSSR is defined as the set ofall operating points that make the distribution system N-1secure taking into account the capacities of substation trans-formers network topology network capacity and operationalconstraints Traditional method of security assessment andcontrol focused on service restoration in which load transferscheme is made on case by case N-1 test [3 4] Howeverthe ldquopoint-wiserdquo simulation method only provides the binaryinformation of security and insecurity which causes globalinformation and description of the security boundary to failto be obtained [5] Moreover with the number of intercon-nected feeders growing in distribution systems especiallyin urban areas the simulation time of ldquopoint-wiserdquo methodthat has to be calculated online is highly prohibitive In
contrast the ldquoregionrdquo method has advantages in dealing withthe problems above
In transmission system the research on security regionhas made substantial achievements Author of [6ndash8]described the topological characteristics for security regionand defined the practical model for engineering applicationHowever the ldquoregionrdquo method has scarcely been applied indistribution area Because early systems were little more thanan extension of SCADA beyond the substation fence mostoperations of 10 or 20 kV feeders required a high degreeof human intervention [9] Plus without high speed peer-to-peer communication system a wealth of key operationinformation cannot be timely received by dispatcherswhich limited the actual contribution of ldquoregionrdquo methodDistribution automation (DA) is an important conceptincluded in smart distribution grid and will upgrade thedistribution systems to fully information-based systems[10 11] That is a motivation for researchers to perform theresearch on ldquoregionrdquo method aiming at distribution system
References [12 13] proposed the concept of loadabilityIt is defined as the maximum loading level that can besupplied Loadability is similar to the proposed ldquoregionrdquo
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 327078 13 pageshttpdxdoiorg1011552014327078
2 Journal of Applied Mathematics
Substation 1
Traditional load
transfermode
Future load transfer mode
with FA
Substation 2
Load Bus tie
FaultTie switchFeeder
switchSectionalswitch
Load
Bus tie
T1
T2
T3
T4
2 times 40MVA 2 times 40MVA
Figure 1 Load transfer mode
concept However because of not considering N-1 cases theconcept is not suitable for urban system where N-1 security isa high concern
Total supply capability (TSC) is a newly proposed conceptwhich is defined as the capability that a system can supplythe load when the N-1 security for distribution systems isconsidered [14 15] At the TSC point the facilities in thesystem are usually fully utilized Although some TSC resultscan be used as the points of DSSR boundary [15] TSC failsto describe the integrated boundary Moreover TSC cannotobtain the relative location of an operating point in securityregion which is very useful for dispatchers tomake decisionsIn a word TSC ismore suitable to be applied in planning thanoperating process
Author of [1] made attempt to present a simplified modeland approach of security assessment with the concept ofDSSR But the papers exposed 2 main defects (1) only theload of substation transformers and their rated capacities areformulated which provide relatively limited information onthe system status (2) feeder contingency and maintenanceare ignored although feeder contingency occurs much morefrequently than transformer contingency
The DSSR theory has potential applications Howeversome wrong conclusions will be reached if the topologicalcharacteristics of DSSR are not studied Paper [16] designedan N-1 approximating approach to study the characteristicsof DSSR But those characteristics have not been provedby rigorous mathematical method and some of them areinaccurately described
This paper proposes a more accurate DSSR model Themathematical proof for the characteristics of DSSR is alsoperformed Finally a test case and a practical case are bothselected to testify the proposed model
2 Basic Concept and Method
21 N-1 Security for Distribution System The DSSR conceptis based on contingency scenarios Feeder contingency andsubstation transformer contingency are two main scenarios
considered in security analysis Here if a distribution systemis feeder N-1 secure it means that when a fault occursat outlet of any main feeder the corresponding load canbe transferred to other feeders with a series of tie-switchoperations Meanwhile all components in the system operatenormally without overload or overvoltage Similarly if adistribution is substation transformer N-1 secure it meansthat when a fault occurs at any substation transformer thecorresponding load can be transferred to other transformerswith all the components satisfying the operation constraintsThe load of a transformer can be transferred in two waysone way is to be transferred to the adjacent transformers inthe same substation with bus-tie switches the other way is tobe transferred to other substations by tie-switch operationsamong feeders
22 Load Transfer and Concept of DSSR Feeders in tra-ditional medium-voltage network are not completely auto-mated When a fault occurs at a substation transformerload will be only transferred to adjacent transformers inthe same substation even though feeders form connectionbetween substationsManual performing for complex circuit-restoration switching takes time and is not performed inpractical operation Thus it causes a lower load ratio ofsubstation transformers which also means a conservativeutilization of assets
However development of smart distribution grid hasgreatly promoted the level of feeder automation especially insome core urban areaDistributed intelligence and automatedswitching accomplish the task that is to transfer load amongdifferent substations quickly accurately and flexibly In otherwords switches that the planner formerly could not tie cannow be tied resulting in higher load ratio within securityconstraints This draws the attention to a more accuratecalculation of security boundary Figure 1 shows the loadtransfer mode after feeder automation (FA)
As is shown in Figure 1 when a fault occurs at T1 intraditional way all of the load of T1 will be only transferredto T2 because the time of load transfer through feeders is
Journal of Applied Mathematics 3
Feeder 1
Normally closed switchNormally open switchBack-feed (transfer) path
FaultS1 S2 S3
S4Feeder 2
Feeder 3
S1f
S1FS2f
S3f
S4f
Figure 2 Feeder and transfer unit load
too long In contrast with feeder automation the load of T2can be transferred not only to T1 but also to T3 in anothersubstation
The DSSR model in this paper takes both substationtransformer N-1 and feeder N-1 contingency into accountThe DSSR is defined as the set of operating points whichassure the N-1 security of distribution system An operatingpoint in transmission system is the current power injectionsof nodes while in distribution system it represents a set oftransfer unit load Consider
W = (1198781
119891 1198782
119891 119878
119899
119891)119879
(1)
whereW is the operating point which is an n-dimensionvector 119878119894
119891is transfer unit load ldquoTransfer unitrdquo is defined as
the set of feeder sections of which load has the same back-feed (transfer) path If two feeders form single loop networktransfer unit load is equivalent to feeder load If a feederis connected to two other feeders which is called two-tieconnection the feeder load can be divided into two parts ofload each of which is a transfer unit load In this paper feederload i is denoted by 119878119894
119865and transfer unitload 119894 is denoted by
119878119894119891 Take the brief case in Figure 2 as an exampleIn Figure 2 feeder 1 forms two-tie connection and feeder
2 and feeder 3 form single loop network respectively withfeeder 1 In normal state 1198781
119891and 1198782119891are both supplied by feeder
1 that is 1198781119865= 1198781119891+ 1198782119891 When a fault occurs at outlet of
feeder 1 S1 and S2 disconnect and S4 closes and then 1198781119891
changes to be supplied by feeder 3 while 1198782119891is supplied by
feeder 2 via closing S3 Transfer scheme is usually not uniqueAnother scheme is that transferring both 1198781
119891and 1198782119891to feeder
2 by disconnecting S1 and closing S3 However multischememakes the DSSR model overcomplicated Thus scheme isfixed in this paper which stipulates that each of the backupfeeders restores only one section of the whole faulted feederas the first scheme above More important this fixed schemecan usually balance the branch load in the postfault networkwhich is an important index concerned by dispatchers
The DSSR model in this paper is accurate to the safetymonitoring of transfer unit load We can certainly get feederload by summing up the transfer unit load and further get
transformer load by summing up the feeder load This is abasis for DSSR model considering both transformer N-1 andfeeder N-1 security
3 Mathematical Model for DSSR
After the load transfer incurred by tie switches against N-1 contingency all the transformers and feeders cannot beoverloading Since the tie-line is designed after the radialnetwork planning the capacity is considered as enough totransfer loadTherefore the transformer load and feeder loadshould be under a series of constraints The DSSR model canbe mathematically formulated as
ΩDSSR = W | ℎ (119909) le 0 119892 (119909) = 0 (2)
where W = (1198781119891 1198782119891 119878119899
119891)119879 is the operating point cor-
responding to 1198781119891 1198782
119891 119878
119899
119891 The inequality and equality
constraints are such that
119878119898
119865= sum119899=1
119878119898119899
119891119905119903 (3)
119878119898119899
119891119905119903+ 119878119899
119865le 119878119899
119865max (forall119898 119899) (4)
119878119894119895
119879119905119903= sum
119898isinΦ(119894)119899isinΦ(119895)
119878119898119899
119891119905119903 (5)
119878119894
119879= sum
119898isinΦ(119894)
119878119898
119865(forall119894) (6)
119878119894119895
119879119905119903+ 119878119895
119879le 119878119895
119879max (7)
where 119878119898119865
= all the load supplied by feeder m 119878119898119899119891119905119903
=the load transferred from feeder m to feeder n when anN-1 fault occurs at outlet of feeder m 119878119899
119865max = maximalthermal capacity of feeder 119899 119878119894119895
119879119905119903= the load transferred from
transformer 119894 to transformer 119895 when an N-1 fault occursat transformer 119894 Φ(119894) = the set of feeders that derive fromtransformer 119894 119878119895
119879= all of the load supplied by transformer 119894
119898 isin Φ(119894)means that feeder119898 derives from the correspondingbus of transformer 119894 119878119894
119879max = the rated capacity of transformer119894
It is notable that the correspondence relation of ldquo119878119891119905119903
rdquoand ldquo119878
119891rdquo depends on the load transfer scheme For example
in Figure 2 in the former scheme 11987812119891119905119903
Equation (3) shows that the sum of each load to betransferred away from feeder m should be equal to thetotal load supplied by feeder m in normal state inequalityconstraint (4) means that feeders cannot be overloading inthe postfault network (5) shows that path of load transferredfrom transformer 119894 to 119895 is the tie-lines between transformers 119894and 119895 Equation (6) means that load of transformer is equal tothe sum of corresponding feeder loads inequality constraint(7) describes that the load of transformer 119895 cannot exceed itsrated capacity after a substation transformerN-1 contingency
It should be pointed out that power flow and voltage dropare not taken into consideration in this DSSR model which
4 Journal of Applied Mathematics
determines that the DSSR model is completely linear Sinceoverloading under contingencies is the most critical prob-lem in urban power the simple linear model is acceptablefrom the standpoint of security-based operation [1] Moreimportantly research of [16] has demonstrated that the DSSRmodel is approximately linear even when considering factorsof power flow and voltage drop which further proves that thelinear model is highly approximate to the real DSSR model
4 Formulation for DSSR Boundary
Research on the boundary of security region has vital signif-icance because the characteristics determine the applicationway of DSSR theory [7] The proposed DSSR model whichis formulated as (2)ndash(7) cannot distinctly express the DSSRboundary Therefore derivation should be performed totransform the originalmodel to a new expression form Select119878119906V119891119905119903
as research object 119906 isin Φ(119894) and V isin Φ(119895) First formula(4) can be transformed to
119878119906V119891119905119903le 119878
V119865max minus 119878
V119865 (8)
Second substitute formulas (5) and (6) into (7) and thenperform identical deformation We obtain
Finally according to (8) and (10) the expression of DSSRboundary in direction of 119878119906V
119891119905119903can be neatly formulated as
119878119906V119891119905119903
le min
119878V119865max minus 119878
V119865 119878119895
119879max minus sum
119905isinΦ(119895)
119878119905
119865minus sum
119898isinΦ(119894)119899isinΦ(119895)
119898 =119906119899 =V
119878119898119899
119891119905119903
(11)
There are 119899 boundaries if the system consists of 119899load transfer units To express the dimension of boundaryclearly we denote all ldquo119878119906V
the transfer scheme in this paper is fixed as is illustrated inSection 22 an important equivalence holds as follows
119878119894
119891119905119903= 119878119894
119891(119894 = 1 2 119899) (12)
Through (12) 119878119891119905119903
and 119878119891have one-to-one corresponding
relation Above all the complete and succinct boundaryformulation for DSSR is expressed as
ΩDSSR
=
1198611
1198612
119861119894
119861119899
=
1198781119891le min
1198781198871119865max minus 119878
1198871
119865 1198781198871119879max minus sum
119895isinΦ(1198871)
119878119895
119865minus sum
119896 =1119896isinΘ(1)
119878119896119891
1198782
119891le min
1198781198872
119865max minus 1198781198872
119865 1198781198872
119879max minus sum
119895isinΦ(1198872)
119878119895
119865minus sum
119896 =2119896isinΘ(2)
119878119896
119891
119878119894
119891le min
119878119887119894
119865max minus 119878119887119894
119865 119878119887119894
119879max minus sum
119895isinΦ(119887119894)
119878119895
119865minus sum
119896 =119894119896isinΘ(119894)
119878119896
119891
119878119899119891le min
119878119887119899119865max minus 119878
119887119899
119865 119878119887119899119879max minus sum
119895isinΦ(119887119899)
119878119895
119865minus sum
119896 =119899119896isinΘ(119899)
119878119896119891
(13)
With the form transformation ofDSSRmodel correspondingsymbols and their meaning should be adjusted too Thus 119887119894and Θ(119894) are used for new expression 119887119894 = the number ofback-feed feeders of 119878119894
119891 119878119887119894119865= all of the load supplied by the
back-feed feeder 119887119894 Θ(119894) = the set of transfer unit loads ofwhich both normal-feed and back-feed transformer are thesame as those of 119878119894
119891 It should be pointed out that feeder 1198871
feeder 119887119899 are not 119899 different actual feeders Take the case inFigure 2 as an example again when fault occurs the back-feed feeders of 1198783
119891and 1198784
119891are both feeder 1 which means
feeder b3 = feeder b4 = feeder 1 When any equality in (13)holds the operating point is just upon the DSSR boundaryWe denote the n boundaries by 119861
1 119861
119899
The DSSR boundary formulation can also reflect bothsubstation transformer and feeder N-1 secure constraintsTake 119894 subformula of (13) to concretely explain 119878119894
119891le
119878119887119894119865max minus 119878
119887119894
119865shows that transfer of 119878119894
119891cannot cause the feeder
overloading which ensures that the feeder N-1 is secure 119878119894119891le
119878119887119894119879max minus sum119895isinΦ(119887119894) 119878
119895
119865minus sum119896 =1119896isinΘ
(119894) 119878119896
119891shows that transfer of 119878119894
119891
cannot lead to the substation transformer overloading whichensures that the transformer N-1 is secure
5 Topological Characteristics of DSSR andMathematical Proof
The research on differential topological characteristics of theDSSR focuses on several aspects as follows
Journal of Applied Mathematics 5
(1) Are there any holes inside the DSSR that is thedenseness in terms of the topology
(2) Does the boundary of the DSSR have suspension(3) Could the boundary of the DSSR be expressed with
the union of several subsurfaces
The identical deformation from formulas (2)ndash(7) to (13) isa breakthrough for the research on topological characteristicsof the DSSR However to research on these characteristicsformula (13) should be further simplified to obtain themathematical essence Take 119894 subformula of (13) as researchobject the equivalent form is
119878119894
119891+ 119878119887119894
119865le 119878119887119894
119865max
119878119894
119891+ sum
119895isinΦ(119887119894)
119878119895
119865+ sum
119896 =1119896isinΘ(119894)
119878119896
119891le 119878119887119894
119879max(14)
According to formulas (3) and (12) 119878119887119894119865
and 119878119895119865
can beexpressed as linear combination of a series of differentelement ldquo119878
119891rdquo Then formula (14) will be simplified as
119899
sum119898=1
120572119898119878119898
119891le 119878119887119894
119865max
119899
sum119898=1
120573119898119878119898
119891le 119878119887119894
119879max
(15)
where 119878119887119894119865max and 119878119887119894
119879max are positive constants Throughformula (14) coefficients 120572 and 120573 have the following features
120572119898= 1 120573
119898= 1 (119898 = 119894)
120572119898= 0 or 1 120573
119898= 0 or 1 (119898 = 119894)
(16)
LetΩDSSR be a set which consists of operating pointW DSSRboundary formulation can be abstracted as
AW le C (17)
whereW is a vector defined as formula (1)Through formulas(15)-(16) matrices A and C should satisfy
A = [119886119894119895]2119899times119899
119894 ge 1 119895 ge 1
119886119894119895= 1 (119894 = 2119895 minus 1 or 119894 = 2119895)
119886119894119895= 0 or 1 (119894 = 2119895 minus 1 119894 = 2119895)
C = [1198881 1198882 1198882119899]119879
C isin R+
(18)
Because one integrated subformula is divided into equivalenttwo parts as is shown in formula (15) the dimension ofA is 2119899 times 119899 instead of 119899 times 119899 Here ΩDSSR is both linearspace and Euclidean space which has been proved in [17]Besides this important premise we should review somepreparation definitions and theorems about topology andEuclidean space including cluster point closure dense setand hyperplane
Definition 1 (cluster point) Let x be a point in Euclideanspace 119865 sub 119877 If there is a point sequence in set 119865 convergingto x then x is the cluster point of set 119865 [17]
Definition 2 (closure) 119860 is a subset of topological space 119860represents all the points inside 119860 and cluster points of 119860 andthen 119860 is called closure of 119860 [18]
Theorem 3 If the 119861 is a subset of 119860 119861 = 119860 then 119861 is dense in119860 [18]
Theorem 4 Let 119871 be a subspace of linear space 119877 x0 is a fixedvector which does not belong to 119871 generally Considering set119867which consists of the vector x x is obtained by
x = x0 + y (19)
where vector y varies in the whole subspace 119871 Thus119867 is calledhyperplane The dimension of119867 equals that of subspace 119871 [17]
The topological characteristics of DSSR are mathemati-cally proved in the following section
Characteristic 1 TheΩDSSR is dense inside
Proof First a new setΩ1015840DSSR is defined as the set of all pointsofΩDSSR except those on boundariesΩ1015840DSSR meets
AW lt C (20)
Select randomly a vector W = (1198781119891 1198782119891 119878119899
119891)119879 from Ω1015840DSSR
and then construct a sequenceY = Y1Y2 Ym as the
following formula
Y =
Y1= (1198781119891minus1198781119891
1 1198782119891minus1198782119891
1 119878119899
119891minus119878119899119891
1)
119879
= (0 0 0)119879
Y2= (1198781
119891minus1198781119891
2 1198782
119891minus1198782119891
2 119878
119899
119891minus119878119899119891
2)
119879
Y119898= (1198781119891minus1198781119891
119898 1198782119891minus1198782119891
119898 119878119899
119891minus119878119899119891
119898)
119879
= (1 minus1
119898)W
(21)
According to (20)-(21) we obtain
AYm = A(1 minus 1119898)W lt AW lt C (22)
Therefore sequence Y = Y1Y2 Ym is inside Ω1015840
DSSRLet 120588(WYm) be the distance betweenWandYm in Euclideanspace and then
Figure 3 Shapes of 2-dimension cross-section of DSSR
Thismeans that for any vectorW inΩ1015840DSSR sequenceY existsand converges toW andY is also insideΩ1015840DSSRTherefore anypoint insideΩ1015840DSSR is cluster point of Ω
1015840
DSSRSimilarly select vector Z = (1198781
119891 1198782119891 119878119899
119891)119879 which is just
upon boundary ofΩ119863119878119878119877
AZ = C (24)
Construct a sequence U = U1U2 Um
Um = (1 minus1
119898)Z (25)
Then we obtain
AUm = A(1 minus 1119898)Z lt AZ = C (26)
According to formula (26) sequence U is also inside Ω1015840DSSRThrough the following
We obtain that any point just upon the ΩDSSR boundary isalso cluster point ofΩ1015840DSSR Above all cluster points ofΩ
1015840
DSSR
include not only all the points inside Ω1015840DSSR but also all thepoints just upon the boundaries ofΩDSSRThus the followingformula holds
Ω1015840DSSR = ΩDSSR (28)
Through Theorem 3 Ω1015840DSSR is dense in ΩDSSR This proof iscompleted
Characteristic 2 The boundary ofΩDSSR has no suspension
Proof This characteristic can be approximately simplified toprove that any 2-dimension cross-section of ΩDSSR is convexpolygon For W = (1198781
119891 1198782119891 119878119899
119891)119879 select a 2-dimension
cross-section by taking 119878119898119891
and 119878119899119891as variables and 119878
119891in
another dimension as constants According to formula (16)the coefficient of 119878
119891(including variable 119878
119891and constant ones)
can be only 1 or 0 thus in process of dimension reduction 119878119898119891
and 119878119899119891are simply constrained by
119878119898
119891+ 119878119899
119891le 1198881
0 le 119878119898
119891le 1198882
0 le 119878119899
119891le 1198883
forall1198881 1198882 1198883isin 119877+
(29)
Journal of Applied Mathematics 7
Table 1 Profile of test case
Substation Transformer Voltages (kVkV) Transformer capacity (MVA) Number of feeders Total feeder capacity (MVA)
S1 T1 3510 400 9 8028T2 3510 400 9 8028
S2 T3 3510 400 9 8028T4 3510 400 8 7136
S3 T5 11010 630 10 892T6 11010 630 10 892
S4 T7 11010 630 10 892T8 11010 630 10 892
The shape of 2-dimension cross-section varies with the valuesof 1198881 1198882 and 119888
3 as is shown in Figure 3 The dashed oblique
line represents 119878119898119891+ 119878119899119891le 1198881
As is shown in Figure 3 the shapes of 2-dimension sectionare confined to rectangle triangle ladder and convex pen-tagon This means that any 2-dimension section of DSSR isconvex polygon which proves that the boundary ofΩDSSR hasno suspension in an indirect wayThe proof is completed
Characteristic 3 The ΩDSSR boundary can be expressed withseveral subsurfaces
Proof 119867 is a set of vector W = (1198781119891 1198782119891 119878119899
119891)119879 each of
which satisfies nonhomogeneous linear equation set
AW = C (30)
where matrices A and C meet formula (18) and W is theoperating point just upon the boundary of ΩDSSR Let W0 =
(120579(0)
1 120579(0)
2 120579(0)
119899)119879 be a fixed solution of equation set (31)
Meanwhile 119871 is a subspace of 119877 which consists of y =
(1205821 1205822 120582
119899)119879 that is y varies in the whole subspace 119871
The coordinates of ymeet the following homogeneous linearequation set
Ay = 0 (31)
If y = (1205821 1205822 120582
119899)119879 is a solution of (31) W = W0
+y = (120579(0)
1+ 1205821 120579(0)
2+ 1205822 120579(0)
119899+ 120582119899)119879 is obviously
a solution of equation set (30) that is W is included inset 119867 Meanwhile x0 is not in 119871 Through Theorem 3 119867is hyperplane The boundary of ΩDSSR can be expressedwith hyperplane that is several subsurfaces Because thedimension of 119871 is 119899 the dimension of hyperplane is also 119899This proof is completed
In transmission area topological characteristics of secu-rity region (SR) have been deeply studied [7] It has shownthat the boundary of SR has no suspension and is compactand approximately linear there is no hole inside the SR Theproof above demonstrates that characteristics of DSSR aresimilar but it has its own features
(a) DSSR is a convex set of denseness in terms ofthe topology This means that DSSR has no holesinside and its boundary has no suspension Basedon this feature dispatchers can analyze the securityof operating point by judging whether the point isinside theDSSRboundarywithout being afraid of anyinsecure point inside or losing security points
(b) DSSR boundary can be described by hyperplane (sev-eral Euclidean subsurfaces) meanwhile the bound-ary ismore linear than that of SR In practical applica-tion calculation of high dimensional security regionwill incur great computation burdens and consumelarge amount of time Therefore this characteristicwill improve the computational efficiency
6 Case Study
In order to verify the effectiveness of the proposedmodel twocases are illustrated in this section The first one is a test caseand the second is a practical case
61 Test Case
611 Overview of Test Case As is shown in Figure 4 the gridis comprised of 4 substations 8 substation transformers and75 feeders 68 feeders form 34 single loop networks and 7form two-tie connections resulting in 82 load transfer unitsThe total capacity of substations is 412MVA To illustrate themodeling process clearly each load transfer unit is numberedCase profile is shown in Table 1
The conductor type of all feeders is LGJ-185 (892MVA)This type is selected based on the national electrical code inChina
612 Boundary Formulation The test case grid has 75 feed-ers 14 feeders form 7 two-tie connections Each load of these7 feeders can be divided into 2 parts to transfer in differentback-feed path So the number of load transfer units is 82which means that the dimension of operation point W is82 and the number of subformulas of DSSR boundary is82 Based on the topology of the test case we substitutethe rated capacity of substation transformers and feedersinto 1198781
119865max 119878119899
119865max and 1198781119879max 119878
119899
119879max of formula (13)
8 Journal of Applied Mathematics
Then the boundary formulation for DSSR of test case can beexpressed as
Considering the similarity of calculation approach ofeach boundary and thesis length we present 5 subformulasof 1198781119891 1198785
119891here Complete formulation is given in the
appendix (in the Supplementary Material available online athttpdxdoiorg1011552014327078) Take the first subfor-mula in (32) as an example to explain the modeling processFigure 5 shows the local topology related with 1198781
119891 including
all relative transfer units in the first subformulaFirst analyze the topology and determine the parameters
of formula (13)We know that back-feed feeder of 1198781119891is feeder
36 and the back-feed substation transformer of 1198781119891is T5 In
normal state 11987836119891 11987845
119891and 11987876
119891 11987878
119891are all load of T5
which means sum119895isinΦ(1198871) 119878119895
119865= sum45
119895=36119878119895
119891+ sum78
119895=76119878119895
119891 11987836119891
and 11987876119891
are load of feeder 36 which means 1198781198871119865= 11987836119891+ 11987876119891 1198782119891 1198785
119891
are the load transferred fromT1 to T5when fault occurs at T1whichmeanssum
119896 =1119896isinΘ(1) 119878119896
119891= sum5
119896=2119878119896119891Therefore the formula
of B1 is preliminary expressed as
1198781
119891le min 11987836
119865max minus (11987876
119891+ 11987836
119891)
1198785
119879max minus (11987836
119891+ 11987837
119891+ 11987838
119891+ 11987839
119891+ 11987840
119891+ 11987841
11989111987842
119891
+11987843
119891+ 11987844
119891+ 11987845
119891+ 11987876
119891+ 11987877
119891+ 11987878
119891)
minus (1198782
119891+ 1198783
119891+ 1198784
119891+ 1198785
119891)
(33)
Second according to case profile substitute 11987836119865max =
892MVA and 1198785119879max = 63MVA into formula (33) and then
we obtain
1198781
119891le min 892 minus (11987876
119891+ 11987836
119891)
63 minus (11987836
119891+ 11987837
119891+ 11987838
119891+ 11987839
119891+ 11987840
119891+ 11987841
11989111987842
119891
+11987843
119891+ 11987844
119891+ 11987845
119891+ 11987876
119891+ 11987877
119891+ 11987878
119891)
minus (1198782
119891+ 1198783
119891+ 1198784
119891+ 1198785
119891)
(34)
613 Visualization and Topological Characteristics To visu-alize the shape of DSSR the dimension is reduced to 2 and 3which means that the coordinates of W should be constantsother than those to be visualized Thus an operating pointhas to be predetermined to provide the constants TSC is aspecial operating point just upon the security boundary [1]on which the facilities are usually fully utilizedThe operationof distribution system is closing to the boundaries and loadlevel is reaching TSCThus in this test case we select the TSCpoint as the predetermined operating point WTSC which isshown in Table 2The calculation approach for TSC has beenpresented in [15]
Choose randomly transfer unit loads 11987815119891
and 11987852119891
toobserve Substitute the rated capacity of transformers T1simT6capacity of all feeders and all transfer unit loads other than11987815119891
and 11987852119891 into formula (32) Then the 2D DSSR shown in
Figure 6 is obtainedAlthough the dimension is 2 the number of subformulas
which constrain the figure shape is not just 2 In fact eachof the subformulas which contains 11987815
119891or 11987852119891
should beconsidered But the subformulas withmore severe constraintswill cover the others and finally form theDSSR boundaryThefinal formulation of 2D figure is
0 le 11987815
119891le 52
0 le 11987852
119891le 3
(35)
Journal of Applied Mathematics 9
T1S1
T2
T3
T4
S 2
S4
T8
T6S3
T5
17-1
8
20-19
12ndash1
415
-16
21ndash23
41-4
3
36ndash3
8
39-4
044
-45
47-4
650
-51
52-5
3
54-5
548
-49
10-1
1
9-8
1 ndash3
4-5
6-7
76ndash7
8
79-80
58-5960-61
57-5665-62
66ndash6970-71
72ndash7524-25
29-28
26-27
30-31
32ndash35
81-8
2
Tie switch
T7
Distribution transformer
(i + 1) j are the same in the topologyin terms of calculation for DSSR
Note means that transfer units iindashj
Figure 4 Test case with eight substation transformers
Feeder 1
T1
T5
Feeder 2Feeder 36Feeder 37
S1f
S2f
S3f
S76f
S77f
S78f
S36f
S36f
S37f
S45f
S4f
S5f
middot middot middot
middot middot middot middot middot middot
Figure 5 Local topology related with f1
Similarly we choose randomly transfer unit loads 11987819119891 11987824119891
and 11987832119891
to observe and then the 3D DSSR is obtained as isshown in Figure 7
In Figure 6 it can be seen that the 2D DSSR is a denserectangle of which boundaries are linear without suspension
DSSR
0
1
2
3
1 2 3 4 52 S52f (MVA)
S15f (MVA)
Figure 6 Shape of 2D DSSR
In Figure 7 the 3D DSSR is surrounded by several subplanesforming a convex pentagon that is a special 3-dimensionhyperplane
Figure 8 The medium-voltage urban power grid of one city of southern China
62 Practical Case In this section the assessment andcontrol method based on DSSR is demonstrated on a realmedium-voltage distribution network of one city in south-ern China which consists of 12 substations 26 substationtransformers and 114 main feeders as is shown in Figure 8Total capacity of substation transformers is 10945MVA It isnotable that each two of all main feeders form a single loopnetwork in this case which leads to the fact that the numberof main feeders is equal to that of load transfer units
Paper [1] has presented concept and calculation methodof relative location of operating point in DSSR based onsubstation transformer contingency Similarly in this paperrelative location 119871
119894describes the distance from the operating
pointW to the boundary 119861119894(119894 = 1 2 119899)
119871119894= min
119878119887119894
119865max minus 119878119887119894
119865 119878119887119894
119879max minus sum
119895isinΦ(119887119894)
119878119895
119865minus sum
119896 =119894119896isinΘ(119894)
119878119896
119891
minus 119878119894
119891
(36)
119871119894is an index which is applied in security assessment
Here is an example Let W1 be an insecure operating pointEach transfer unit load of W1 is 08 times of TSC load inthe practical case except 1198781
119891and 11987810119891 while 1198781
119891= 8MVA and
11987810119891= 7MVA Total load of W1 is 5087MVA while TSC is
6289MVAAverage load rate is 046Thatmeans that current
Table 3 Distance to boundaries of 1198611and 119861
10atW1
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
operating point can be controlled back to security regionThrough formula (36) the location of a given operating pointin DSSR can be calculated We present the 119861
119894of which 119871
119894are
negative in Table 3Table 3 shows that the system should lose 3222MVA if
fault occurs at feeder 1 or feeder 10 so the system is locallyoverloading The location of W1 can be easily observed inFigure 9 To meet operational constraints under N-1 contin-gency 1198781
119891and 11987810
119891should be adjusted For example W1 is
adjusted toW1015840
1 through the dashed arrow shown in Figure 9The location ofW
1015840
1 is shown in Table 4
7 Conclusion and Further Work
This paper proposes a mathematical model for distributionsystem security region (DSSR) generic topological character-istics of DSSR are discussed and proved
First the operating point for a distribution system isdefined as the set of transfer unit loads which is a vector in the
12 Journal of Applied Mathematics
DSSR
0
2
4
6
2 4 6 8 10 11778
8
10
11778
S1f (MVA)
S10f (MVA)
W1(8 7)
W9984001 (7 4)
Figure 9 2D DSSR cross-section and preventive control
Table 4 Distance to boundaries of 1198611and 119861
10atW10158401
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
Euclidean space Then the DSSR model for both substationtransformer and feeder contingency is proposed which is theset of all operating points each ensures that the system N-1 issecure
Second to perform research on the characteristic ofDSSR the boundary formulation for DSSR is derived fromthe DSSR model Then three generic characteristics areproposed and proved by rigor mathematical approach whichare as follows (1) DSSR is dense inside (2) DSSR boundaryhas no suspension (3)DSSR boundary can be expressed withthe union of several subsurfaces
Finally the results from a test case and a practical casedemonstrate the effectiveness of the proposed model Thevisualization for DSSR boundary is also discussed to verifythe topological characteristics of DSSR Security assessmentmethod based on DSSR is preliminary exhibited to show theapplicability of DSSR for future smart distribution system
This paper improves the accuracy and understanding ofDSSR which is fundamental theoretic work for future smartdistribution system Further works to improve the accuracyof DSSR model include fully considering power flow voltageconstraints and integration of DGs
Conflict of Interests
The authors declare that there is no conflict of inter-ests regarding the publication of this paper None ofthe authors have a commercial interest financial interestandor other relationship with manufacturers of pharma-ceuticalslaboratory supplies andor medical devices or withcommercialproviders of medically related services
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (no 51277129) and National HighTechnology Research and Development Program of China(863 Program) (no 2011AA05A117)
References
[1] J Xiao W Gu C Wang and F Li ldquoDistribution systemsecurity region definition model and security assessmentrdquo IETGeneration TransmissionampDistribution vol 6 no 10 pp 1029ndash1035 2012
[2] State Grid Corporation of China Guidelines of Urban PowerNetwork Planning State Grid Corporation of China BeijingChina 2006 (Chinese)
[3] E Lakervi and E J Holmes Electricity Distribution NetworkDesign IET Peregrinus UK 1995
[4] K N Miu and H D Chiang ldquoService restoration for unbal-anced radial distribution systems with varying loads solutionalgorithmrdquo in Proceedings of the IEEE Power Engineering SocietySummer Meeting vol 1 pp 254ndash258 1999
[5] Y X Yu ldquoSecurity region of bulk power systemrdquo in Proceedingsof the International Conference on Power System Technology(PowerCon rsquo02) vol 1 pp 13ndash17 Kunming China October2002
[6] F Wu and S Kumagai ldquoSteady-state security regions of powersystemsrdquo IEEE Transactions on Circuits and Systems vol 29 no11 pp 703ndash711 1982
[7] Y Yu Y Zeng and F Feng ldquoDifferential topological character-istics of the DSR on injection space of electrical power systemrdquoScience in China E Technological Sciences vol 45 no 6 pp 576ndash584 2002
[8] S J Chen Q X Chen Q Xia and C Q Kang ldquoSteady-state security assessment method based on distance to securityregion boundariesrdquo IET Generation Transmission amp Distribu-tion vol 7 no 3 pp 288ndash297 2013
[9] D M Staszesky D Craig and C Befus ldquoAdvanced feederautomation is hererdquo IEEE Power and Energy Magazine vol 3no 5 pp 56ndash63 2005
[10] C L Smallwood and J Wennermark ldquoBenefits of distributionautomationrdquo IEEE Industry Applications Magazine vol 16 no1 pp 65ndash73 2010
[11] X Mamo S Mallet T Coste and S Grenard ldquoDistributionautomation the cornerstone for smart grid development strat-egyrdquo in Proceedings of the IEEE Power amp Energy Society GeneralMeeting (PES rsquo09) pp 1ndash6 July 2009
[12] P W Sauer B C Lesieutre and M A Pai ldquoMaximumloadability and voltage stability in power systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 15 no 3 pp145ndash153 1993
[13] K N Miu and H Chiang ldquoElectric distribution systemload capability problem formulation solution algorithm andnumerical resultsrdquo IEEE Transactions on Power Delivery vol 15no 1 pp 436ndash442 2000
[14] F Z Luo C S Wang J Xiao and S Ge ldquoRapid evaluationmethod for power supply capability of urban distribution sys-tem based on N-1 contingency analysis of main-transformersrdquoInternational Journal of Electrical Power andEnergy Systems vol32 no 10 pp 1063ndash1068 2010
Journal of Applied Mathematics 13
[15] J Xiao F X LiWZGuC SWang andP Zhang ldquoTotal supplycapability and its extended indices for distribution systemsdefinitionmodel calculation and applicationsrdquo IETGenerationTransmission and Distribution vol 5 no 8 pp 869ndash876 2011
[16] J Xiao X X Gong and C S Wang ldquoTopology properties andalgorithm of N-1 security boundary for smart gridrdquo Proceedingsof the CSEE vol 34 no 4 pp 545ndash554 2014 (Chinese)
[17] G E Shilov An Introduction to the Theory of Linear SpacesDover New York NY USA 1975
[18] J R Munkres Topology vol 2 Prentice Hall Upper SaddleRiver NJ USA 2000
concept However because of not considering N-1 cases theconcept is not suitable for urban system where N-1 security isa high concern
Total supply capability (TSC) is a newly proposed conceptwhich is defined as the capability that a system can supplythe load when the N-1 security for distribution systems isconsidered [14 15] At the TSC point the facilities in thesystem are usually fully utilized Although some TSC resultscan be used as the points of DSSR boundary [15] TSC failsto describe the integrated boundary Moreover TSC cannotobtain the relative location of an operating point in securityregion which is very useful for dispatchers tomake decisionsIn a word TSC ismore suitable to be applied in planning thanoperating process
Author of [1] made attempt to present a simplified modeland approach of security assessment with the concept ofDSSR But the papers exposed 2 main defects (1) only theload of substation transformers and their rated capacities areformulated which provide relatively limited information onthe system status (2) feeder contingency and maintenanceare ignored although feeder contingency occurs much morefrequently than transformer contingency
The DSSR theory has potential applications Howeversome wrong conclusions will be reached if the topologicalcharacteristics of DSSR are not studied Paper [16] designedan N-1 approximating approach to study the characteristicsof DSSR But those characteristics have not been provedby rigorous mathematical method and some of them areinaccurately described
This paper proposes a more accurate DSSR model Themathematical proof for the characteristics of DSSR is alsoperformed Finally a test case and a practical case are bothselected to testify the proposed model
2 Basic Concept and Method
21 N-1 Security for Distribution System The DSSR conceptis based on contingency scenarios Feeder contingency andsubstation transformer contingency are two main scenarios
considered in security analysis Here if a distribution systemis feeder N-1 secure it means that when a fault occursat outlet of any main feeder the corresponding load canbe transferred to other feeders with a series of tie-switchoperations Meanwhile all components in the system operatenormally without overload or overvoltage Similarly if adistribution is substation transformer N-1 secure it meansthat when a fault occurs at any substation transformer thecorresponding load can be transferred to other transformerswith all the components satisfying the operation constraintsThe load of a transformer can be transferred in two waysone way is to be transferred to the adjacent transformers inthe same substation with bus-tie switches the other way is tobe transferred to other substations by tie-switch operationsamong feeders
22 Load Transfer and Concept of DSSR Feeders in tra-ditional medium-voltage network are not completely auto-mated When a fault occurs at a substation transformerload will be only transferred to adjacent transformers inthe same substation even though feeders form connectionbetween substationsManual performing for complex circuit-restoration switching takes time and is not performed inpractical operation Thus it causes a lower load ratio ofsubstation transformers which also means a conservativeutilization of assets
However development of smart distribution grid hasgreatly promoted the level of feeder automation especially insome core urban areaDistributed intelligence and automatedswitching accomplish the task that is to transfer load amongdifferent substations quickly accurately and flexibly In otherwords switches that the planner formerly could not tie cannow be tied resulting in higher load ratio within securityconstraints This draws the attention to a more accuratecalculation of security boundary Figure 1 shows the loadtransfer mode after feeder automation (FA)
As is shown in Figure 1 when a fault occurs at T1 intraditional way all of the load of T1 will be only transferredto T2 because the time of load transfer through feeders is
Journal of Applied Mathematics 3
Feeder 1
Normally closed switchNormally open switchBack-feed (transfer) path
FaultS1 S2 S3
S4Feeder 2
Feeder 3
S1f
S1FS2f
S3f
S4f
Figure 2 Feeder and transfer unit load
too long In contrast with feeder automation the load of T2can be transferred not only to T1 but also to T3 in anothersubstation
The DSSR model in this paper takes both substationtransformer N-1 and feeder N-1 contingency into accountThe DSSR is defined as the set of operating points whichassure the N-1 security of distribution system An operatingpoint in transmission system is the current power injectionsof nodes while in distribution system it represents a set oftransfer unit load Consider
W = (1198781
119891 1198782
119891 119878
119899
119891)119879
(1)
whereW is the operating point which is an n-dimensionvector 119878119894
119891is transfer unit load ldquoTransfer unitrdquo is defined as
the set of feeder sections of which load has the same back-feed (transfer) path If two feeders form single loop networktransfer unit load is equivalent to feeder load If a feederis connected to two other feeders which is called two-tieconnection the feeder load can be divided into two parts ofload each of which is a transfer unit load In this paper feederload i is denoted by 119878119894
119865and transfer unitload 119894 is denoted by
119878119894119891 Take the brief case in Figure 2 as an exampleIn Figure 2 feeder 1 forms two-tie connection and feeder
2 and feeder 3 form single loop network respectively withfeeder 1 In normal state 1198781
119891and 1198782119891are both supplied by feeder
1 that is 1198781119865= 1198781119891+ 1198782119891 When a fault occurs at outlet of
feeder 1 S1 and S2 disconnect and S4 closes and then 1198781119891
changes to be supplied by feeder 3 while 1198782119891is supplied by
feeder 2 via closing S3 Transfer scheme is usually not uniqueAnother scheme is that transferring both 1198781
119891and 1198782119891to feeder
2 by disconnecting S1 and closing S3 However multischememakes the DSSR model overcomplicated Thus scheme isfixed in this paper which stipulates that each of the backupfeeders restores only one section of the whole faulted feederas the first scheme above More important this fixed schemecan usually balance the branch load in the postfault networkwhich is an important index concerned by dispatchers
The DSSR model in this paper is accurate to the safetymonitoring of transfer unit load We can certainly get feederload by summing up the transfer unit load and further get
transformer load by summing up the feeder load This is abasis for DSSR model considering both transformer N-1 andfeeder N-1 security
3 Mathematical Model for DSSR
After the load transfer incurred by tie switches against N-1 contingency all the transformers and feeders cannot beoverloading Since the tie-line is designed after the radialnetwork planning the capacity is considered as enough totransfer loadTherefore the transformer load and feeder loadshould be under a series of constraints The DSSR model canbe mathematically formulated as
ΩDSSR = W | ℎ (119909) le 0 119892 (119909) = 0 (2)
where W = (1198781119891 1198782119891 119878119899
119891)119879 is the operating point cor-
responding to 1198781119891 1198782
119891 119878
119899
119891 The inequality and equality
constraints are such that
119878119898
119865= sum119899=1
119878119898119899
119891119905119903 (3)
119878119898119899
119891119905119903+ 119878119899
119865le 119878119899
119865max (forall119898 119899) (4)
119878119894119895
119879119905119903= sum
119898isinΦ(119894)119899isinΦ(119895)
119878119898119899
119891119905119903 (5)
119878119894
119879= sum
119898isinΦ(119894)
119878119898
119865(forall119894) (6)
119878119894119895
119879119905119903+ 119878119895
119879le 119878119895
119879max (7)
where 119878119898119865
= all the load supplied by feeder m 119878119898119899119891119905119903
=the load transferred from feeder m to feeder n when anN-1 fault occurs at outlet of feeder m 119878119899
119865max = maximalthermal capacity of feeder 119899 119878119894119895
119879119905119903= the load transferred from
transformer 119894 to transformer 119895 when an N-1 fault occursat transformer 119894 Φ(119894) = the set of feeders that derive fromtransformer 119894 119878119895
119879= all of the load supplied by transformer 119894
119898 isin Φ(119894)means that feeder119898 derives from the correspondingbus of transformer 119894 119878119894
119879max = the rated capacity of transformer119894
It is notable that the correspondence relation of ldquo119878119891119905119903
rdquoand ldquo119878
119891rdquo depends on the load transfer scheme For example
in Figure 2 in the former scheme 11987812119891119905119903
Equation (3) shows that the sum of each load to betransferred away from feeder m should be equal to thetotal load supplied by feeder m in normal state inequalityconstraint (4) means that feeders cannot be overloading inthe postfault network (5) shows that path of load transferredfrom transformer 119894 to 119895 is the tie-lines between transformers 119894and 119895 Equation (6) means that load of transformer is equal tothe sum of corresponding feeder loads inequality constraint(7) describes that the load of transformer 119895 cannot exceed itsrated capacity after a substation transformerN-1 contingency
It should be pointed out that power flow and voltage dropare not taken into consideration in this DSSR model which
4 Journal of Applied Mathematics
determines that the DSSR model is completely linear Sinceoverloading under contingencies is the most critical prob-lem in urban power the simple linear model is acceptablefrom the standpoint of security-based operation [1] Moreimportantly research of [16] has demonstrated that the DSSRmodel is approximately linear even when considering factorsof power flow and voltage drop which further proves that thelinear model is highly approximate to the real DSSR model
4 Formulation for DSSR Boundary
Research on the boundary of security region has vital signif-icance because the characteristics determine the applicationway of DSSR theory [7] The proposed DSSR model whichis formulated as (2)ndash(7) cannot distinctly express the DSSRboundary Therefore derivation should be performed totransform the originalmodel to a new expression form Select119878119906V119891119905119903
as research object 119906 isin Φ(119894) and V isin Φ(119895) First formula(4) can be transformed to
119878119906V119891119905119903le 119878
V119865max minus 119878
V119865 (8)
Second substitute formulas (5) and (6) into (7) and thenperform identical deformation We obtain
Finally according to (8) and (10) the expression of DSSRboundary in direction of 119878119906V
119891119905119903can be neatly formulated as
119878119906V119891119905119903
le min
119878V119865max minus 119878
V119865 119878119895
119879max minus sum
119905isinΦ(119895)
119878119905
119865minus sum
119898isinΦ(119894)119899isinΦ(119895)
119898 =119906119899 =V
119878119898119899
119891119905119903
(11)
There are 119899 boundaries if the system consists of 119899load transfer units To express the dimension of boundaryclearly we denote all ldquo119878119906V
the transfer scheme in this paper is fixed as is illustrated inSection 22 an important equivalence holds as follows
119878119894
119891119905119903= 119878119894
119891(119894 = 1 2 119899) (12)
Through (12) 119878119891119905119903
and 119878119891have one-to-one corresponding
relation Above all the complete and succinct boundaryformulation for DSSR is expressed as
ΩDSSR
=
1198611
1198612
119861119894
119861119899
=
1198781119891le min
1198781198871119865max minus 119878
1198871
119865 1198781198871119879max minus sum
119895isinΦ(1198871)
119878119895
119865minus sum
119896 =1119896isinΘ(1)
119878119896119891
1198782
119891le min
1198781198872
119865max minus 1198781198872
119865 1198781198872
119879max minus sum
119895isinΦ(1198872)
119878119895
119865minus sum
119896 =2119896isinΘ(2)
119878119896
119891
119878119894
119891le min
119878119887119894
119865max minus 119878119887119894
119865 119878119887119894
119879max minus sum
119895isinΦ(119887119894)
119878119895
119865minus sum
119896 =119894119896isinΘ(119894)
119878119896
119891
119878119899119891le min
119878119887119899119865max minus 119878
119887119899
119865 119878119887119899119879max minus sum
119895isinΦ(119887119899)
119878119895
119865minus sum
119896 =119899119896isinΘ(119899)
119878119896119891
(13)
With the form transformation ofDSSRmodel correspondingsymbols and their meaning should be adjusted too Thus 119887119894and Θ(119894) are used for new expression 119887119894 = the number ofback-feed feeders of 119878119894
119891 119878119887119894119865= all of the load supplied by the
back-feed feeder 119887119894 Θ(119894) = the set of transfer unit loads ofwhich both normal-feed and back-feed transformer are thesame as those of 119878119894
119891 It should be pointed out that feeder 1198871
feeder 119887119899 are not 119899 different actual feeders Take the case inFigure 2 as an example again when fault occurs the back-feed feeders of 1198783
119891and 1198784
119891are both feeder 1 which means
feeder b3 = feeder b4 = feeder 1 When any equality in (13)holds the operating point is just upon the DSSR boundaryWe denote the n boundaries by 119861
1 119861
119899
The DSSR boundary formulation can also reflect bothsubstation transformer and feeder N-1 secure constraintsTake 119894 subformula of (13) to concretely explain 119878119894
119891le
119878119887119894119865max minus 119878
119887119894
119865shows that transfer of 119878119894
119891cannot cause the feeder
overloading which ensures that the feeder N-1 is secure 119878119894119891le
119878119887119894119879max minus sum119895isinΦ(119887119894) 119878
119895
119865minus sum119896 =1119896isinΘ
(119894) 119878119896
119891shows that transfer of 119878119894
119891
cannot lead to the substation transformer overloading whichensures that the transformer N-1 is secure
5 Topological Characteristics of DSSR andMathematical Proof
The research on differential topological characteristics of theDSSR focuses on several aspects as follows
Journal of Applied Mathematics 5
(1) Are there any holes inside the DSSR that is thedenseness in terms of the topology
(2) Does the boundary of the DSSR have suspension(3) Could the boundary of the DSSR be expressed with
the union of several subsurfaces
The identical deformation from formulas (2)ndash(7) to (13) isa breakthrough for the research on topological characteristicsof the DSSR However to research on these characteristicsformula (13) should be further simplified to obtain themathematical essence Take 119894 subformula of (13) as researchobject the equivalent form is
119878119894
119891+ 119878119887119894
119865le 119878119887119894
119865max
119878119894
119891+ sum
119895isinΦ(119887119894)
119878119895
119865+ sum
119896 =1119896isinΘ(119894)
119878119896
119891le 119878119887119894
119879max(14)
According to formulas (3) and (12) 119878119887119894119865
and 119878119895119865
can beexpressed as linear combination of a series of differentelement ldquo119878
119891rdquo Then formula (14) will be simplified as
119899
sum119898=1
120572119898119878119898
119891le 119878119887119894
119865max
119899
sum119898=1
120573119898119878119898
119891le 119878119887119894
119879max
(15)
where 119878119887119894119865max and 119878119887119894
119879max are positive constants Throughformula (14) coefficients 120572 and 120573 have the following features
120572119898= 1 120573
119898= 1 (119898 = 119894)
120572119898= 0 or 1 120573
119898= 0 or 1 (119898 = 119894)
(16)
LetΩDSSR be a set which consists of operating pointW DSSRboundary formulation can be abstracted as
AW le C (17)
whereW is a vector defined as formula (1)Through formulas(15)-(16) matrices A and C should satisfy
A = [119886119894119895]2119899times119899
119894 ge 1 119895 ge 1
119886119894119895= 1 (119894 = 2119895 minus 1 or 119894 = 2119895)
119886119894119895= 0 or 1 (119894 = 2119895 minus 1 119894 = 2119895)
C = [1198881 1198882 1198882119899]119879
C isin R+
(18)
Because one integrated subformula is divided into equivalenttwo parts as is shown in formula (15) the dimension ofA is 2119899 times 119899 instead of 119899 times 119899 Here ΩDSSR is both linearspace and Euclidean space which has been proved in [17]Besides this important premise we should review somepreparation definitions and theorems about topology andEuclidean space including cluster point closure dense setand hyperplane
Definition 1 (cluster point) Let x be a point in Euclideanspace 119865 sub 119877 If there is a point sequence in set 119865 convergingto x then x is the cluster point of set 119865 [17]
Definition 2 (closure) 119860 is a subset of topological space 119860represents all the points inside 119860 and cluster points of 119860 andthen 119860 is called closure of 119860 [18]
Theorem 3 If the 119861 is a subset of 119860 119861 = 119860 then 119861 is dense in119860 [18]
Theorem 4 Let 119871 be a subspace of linear space 119877 x0 is a fixedvector which does not belong to 119871 generally Considering set119867which consists of the vector x x is obtained by
x = x0 + y (19)
where vector y varies in the whole subspace 119871 Thus119867 is calledhyperplane The dimension of119867 equals that of subspace 119871 [17]
The topological characteristics of DSSR are mathemati-cally proved in the following section
Characteristic 1 TheΩDSSR is dense inside
Proof First a new setΩ1015840DSSR is defined as the set of all pointsofΩDSSR except those on boundariesΩ1015840DSSR meets
AW lt C (20)
Select randomly a vector W = (1198781119891 1198782119891 119878119899
119891)119879 from Ω1015840DSSR
and then construct a sequenceY = Y1Y2 Ym as the
following formula
Y =
Y1= (1198781119891minus1198781119891
1 1198782119891minus1198782119891
1 119878119899
119891minus119878119899119891
1)
119879
= (0 0 0)119879
Y2= (1198781
119891minus1198781119891
2 1198782
119891minus1198782119891
2 119878
119899
119891minus119878119899119891
2)
119879
Y119898= (1198781119891minus1198781119891
119898 1198782119891minus1198782119891
119898 119878119899
119891minus119878119899119891
119898)
119879
= (1 minus1
119898)W
(21)
According to (20)-(21) we obtain
AYm = A(1 minus 1119898)W lt AW lt C (22)
Therefore sequence Y = Y1Y2 Ym is inside Ω1015840
DSSRLet 120588(WYm) be the distance betweenWandYm in Euclideanspace and then
Figure 3 Shapes of 2-dimension cross-section of DSSR
Thismeans that for any vectorW inΩ1015840DSSR sequenceY existsand converges toW andY is also insideΩ1015840DSSRTherefore anypoint insideΩ1015840DSSR is cluster point of Ω
1015840
DSSRSimilarly select vector Z = (1198781
119891 1198782119891 119878119899
119891)119879 which is just
upon boundary ofΩ119863119878119878119877
AZ = C (24)
Construct a sequence U = U1U2 Um
Um = (1 minus1
119898)Z (25)
Then we obtain
AUm = A(1 minus 1119898)Z lt AZ = C (26)
According to formula (26) sequence U is also inside Ω1015840DSSRThrough the following
We obtain that any point just upon the ΩDSSR boundary isalso cluster point ofΩ1015840DSSR Above all cluster points ofΩ
1015840
DSSR
include not only all the points inside Ω1015840DSSR but also all thepoints just upon the boundaries ofΩDSSRThus the followingformula holds
Ω1015840DSSR = ΩDSSR (28)
Through Theorem 3 Ω1015840DSSR is dense in ΩDSSR This proof iscompleted
Characteristic 2 The boundary ofΩDSSR has no suspension
Proof This characteristic can be approximately simplified toprove that any 2-dimension cross-section of ΩDSSR is convexpolygon For W = (1198781
119891 1198782119891 119878119899
119891)119879 select a 2-dimension
cross-section by taking 119878119898119891
and 119878119899119891as variables and 119878
119891in
another dimension as constants According to formula (16)the coefficient of 119878
119891(including variable 119878
119891and constant ones)
can be only 1 or 0 thus in process of dimension reduction 119878119898119891
and 119878119899119891are simply constrained by
119878119898
119891+ 119878119899
119891le 1198881
0 le 119878119898
119891le 1198882
0 le 119878119899
119891le 1198883
forall1198881 1198882 1198883isin 119877+
(29)
Journal of Applied Mathematics 7
Table 1 Profile of test case
Substation Transformer Voltages (kVkV) Transformer capacity (MVA) Number of feeders Total feeder capacity (MVA)
S1 T1 3510 400 9 8028T2 3510 400 9 8028
S2 T3 3510 400 9 8028T4 3510 400 8 7136
S3 T5 11010 630 10 892T6 11010 630 10 892
S4 T7 11010 630 10 892T8 11010 630 10 892
The shape of 2-dimension cross-section varies with the valuesof 1198881 1198882 and 119888
3 as is shown in Figure 3 The dashed oblique
line represents 119878119898119891+ 119878119899119891le 1198881
As is shown in Figure 3 the shapes of 2-dimension sectionare confined to rectangle triangle ladder and convex pen-tagon This means that any 2-dimension section of DSSR isconvex polygon which proves that the boundary ofΩDSSR hasno suspension in an indirect wayThe proof is completed
Characteristic 3 The ΩDSSR boundary can be expressed withseveral subsurfaces
Proof 119867 is a set of vector W = (1198781119891 1198782119891 119878119899
119891)119879 each of
which satisfies nonhomogeneous linear equation set
AW = C (30)
where matrices A and C meet formula (18) and W is theoperating point just upon the boundary of ΩDSSR Let W0 =
(120579(0)
1 120579(0)
2 120579(0)
119899)119879 be a fixed solution of equation set (31)
Meanwhile 119871 is a subspace of 119877 which consists of y =
(1205821 1205822 120582
119899)119879 that is y varies in the whole subspace 119871
The coordinates of ymeet the following homogeneous linearequation set
Ay = 0 (31)
If y = (1205821 1205822 120582
119899)119879 is a solution of (31) W = W0
+y = (120579(0)
1+ 1205821 120579(0)
2+ 1205822 120579(0)
119899+ 120582119899)119879 is obviously
a solution of equation set (30) that is W is included inset 119867 Meanwhile x0 is not in 119871 Through Theorem 3 119867is hyperplane The boundary of ΩDSSR can be expressedwith hyperplane that is several subsurfaces Because thedimension of 119871 is 119899 the dimension of hyperplane is also 119899This proof is completed
In transmission area topological characteristics of secu-rity region (SR) have been deeply studied [7] It has shownthat the boundary of SR has no suspension and is compactand approximately linear there is no hole inside the SR Theproof above demonstrates that characteristics of DSSR aresimilar but it has its own features
(a) DSSR is a convex set of denseness in terms ofthe topology This means that DSSR has no holesinside and its boundary has no suspension Basedon this feature dispatchers can analyze the securityof operating point by judging whether the point isinside theDSSRboundarywithout being afraid of anyinsecure point inside or losing security points
(b) DSSR boundary can be described by hyperplane (sev-eral Euclidean subsurfaces) meanwhile the bound-ary ismore linear than that of SR In practical applica-tion calculation of high dimensional security regionwill incur great computation burdens and consumelarge amount of time Therefore this characteristicwill improve the computational efficiency
6 Case Study
In order to verify the effectiveness of the proposedmodel twocases are illustrated in this section The first one is a test caseand the second is a practical case
61 Test Case
611 Overview of Test Case As is shown in Figure 4 the gridis comprised of 4 substations 8 substation transformers and75 feeders 68 feeders form 34 single loop networks and 7form two-tie connections resulting in 82 load transfer unitsThe total capacity of substations is 412MVA To illustrate themodeling process clearly each load transfer unit is numberedCase profile is shown in Table 1
The conductor type of all feeders is LGJ-185 (892MVA)This type is selected based on the national electrical code inChina
612 Boundary Formulation The test case grid has 75 feed-ers 14 feeders form 7 two-tie connections Each load of these7 feeders can be divided into 2 parts to transfer in differentback-feed path So the number of load transfer units is 82which means that the dimension of operation point W is82 and the number of subformulas of DSSR boundary is82 Based on the topology of the test case we substitutethe rated capacity of substation transformers and feedersinto 1198781
119865max 119878119899
119865max and 1198781119879max 119878
119899
119879max of formula (13)
8 Journal of Applied Mathematics
Then the boundary formulation for DSSR of test case can beexpressed as
Considering the similarity of calculation approach ofeach boundary and thesis length we present 5 subformulasof 1198781119891 1198785
119891here Complete formulation is given in the
appendix (in the Supplementary Material available online athttpdxdoiorg1011552014327078) Take the first subfor-mula in (32) as an example to explain the modeling processFigure 5 shows the local topology related with 1198781
119891 including
all relative transfer units in the first subformulaFirst analyze the topology and determine the parameters
of formula (13)We know that back-feed feeder of 1198781119891is feeder
36 and the back-feed substation transformer of 1198781119891is T5 In
normal state 11987836119891 11987845
119891and 11987876
119891 11987878
119891are all load of T5
which means sum119895isinΦ(1198871) 119878119895
119865= sum45
119895=36119878119895
119891+ sum78
119895=76119878119895
119891 11987836119891
and 11987876119891
are load of feeder 36 which means 1198781198871119865= 11987836119891+ 11987876119891 1198782119891 1198785
119891
are the load transferred fromT1 to T5when fault occurs at T1whichmeanssum
119896 =1119896isinΘ(1) 119878119896
119891= sum5
119896=2119878119896119891Therefore the formula
of B1 is preliminary expressed as
1198781
119891le min 11987836
119865max minus (11987876
119891+ 11987836
119891)
1198785
119879max minus (11987836
119891+ 11987837
119891+ 11987838
119891+ 11987839
119891+ 11987840
119891+ 11987841
11989111987842
119891
+11987843
119891+ 11987844
119891+ 11987845
119891+ 11987876
119891+ 11987877
119891+ 11987878
119891)
minus (1198782
119891+ 1198783
119891+ 1198784
119891+ 1198785
119891)
(33)
Second according to case profile substitute 11987836119865max =
892MVA and 1198785119879max = 63MVA into formula (33) and then
we obtain
1198781
119891le min 892 minus (11987876
119891+ 11987836
119891)
63 minus (11987836
119891+ 11987837
119891+ 11987838
119891+ 11987839
119891+ 11987840
119891+ 11987841
11989111987842
119891
+11987843
119891+ 11987844
119891+ 11987845
119891+ 11987876
119891+ 11987877
119891+ 11987878
119891)
minus (1198782
119891+ 1198783
119891+ 1198784
119891+ 1198785
119891)
(34)
613 Visualization and Topological Characteristics To visu-alize the shape of DSSR the dimension is reduced to 2 and 3which means that the coordinates of W should be constantsother than those to be visualized Thus an operating pointhas to be predetermined to provide the constants TSC is aspecial operating point just upon the security boundary [1]on which the facilities are usually fully utilizedThe operationof distribution system is closing to the boundaries and loadlevel is reaching TSCThus in this test case we select the TSCpoint as the predetermined operating point WTSC which isshown in Table 2The calculation approach for TSC has beenpresented in [15]
Choose randomly transfer unit loads 11987815119891
and 11987852119891
toobserve Substitute the rated capacity of transformers T1simT6capacity of all feeders and all transfer unit loads other than11987815119891
and 11987852119891 into formula (32) Then the 2D DSSR shown in
Figure 6 is obtainedAlthough the dimension is 2 the number of subformulas
which constrain the figure shape is not just 2 In fact eachof the subformulas which contains 11987815
119891or 11987852119891
should beconsidered But the subformulas withmore severe constraintswill cover the others and finally form theDSSR boundaryThefinal formulation of 2D figure is
0 le 11987815
119891le 52
0 le 11987852
119891le 3
(35)
Journal of Applied Mathematics 9
T1S1
T2
T3
T4
S 2
S4
T8
T6S3
T5
17-1
8
20-19
12ndash1
415
-16
21ndash23
41-4
3
36ndash3
8
39-4
044
-45
47-4
650
-51
52-5
3
54-5
548
-49
10-1
1
9-8
1 ndash3
4-5
6-7
76ndash7
8
79-80
58-5960-61
57-5665-62
66ndash6970-71
72ndash7524-25
29-28
26-27
30-31
32ndash35
81-8
2
Tie switch
T7
Distribution transformer
(i + 1) j are the same in the topologyin terms of calculation for DSSR
Note means that transfer units iindashj
Figure 4 Test case with eight substation transformers
Feeder 1
T1
T5
Feeder 2Feeder 36Feeder 37
S1f
S2f
S3f
S76f
S77f
S78f
S36f
S36f
S37f
S45f
S4f
S5f
middot middot middot
middot middot middot middot middot middot
Figure 5 Local topology related with f1
Similarly we choose randomly transfer unit loads 11987819119891 11987824119891
and 11987832119891
to observe and then the 3D DSSR is obtained as isshown in Figure 7
In Figure 6 it can be seen that the 2D DSSR is a denserectangle of which boundaries are linear without suspension
DSSR
0
1
2
3
1 2 3 4 52 S52f (MVA)
S15f (MVA)
Figure 6 Shape of 2D DSSR
In Figure 7 the 3D DSSR is surrounded by several subplanesforming a convex pentagon that is a special 3-dimensionhyperplane
Figure 8 The medium-voltage urban power grid of one city of southern China
62 Practical Case In this section the assessment andcontrol method based on DSSR is demonstrated on a realmedium-voltage distribution network of one city in south-ern China which consists of 12 substations 26 substationtransformers and 114 main feeders as is shown in Figure 8Total capacity of substation transformers is 10945MVA It isnotable that each two of all main feeders form a single loopnetwork in this case which leads to the fact that the numberof main feeders is equal to that of load transfer units
Paper [1] has presented concept and calculation methodof relative location of operating point in DSSR based onsubstation transformer contingency Similarly in this paperrelative location 119871
119894describes the distance from the operating
pointW to the boundary 119861119894(119894 = 1 2 119899)
119871119894= min
119878119887119894
119865max minus 119878119887119894
119865 119878119887119894
119879max minus sum
119895isinΦ(119887119894)
119878119895
119865minus sum
119896 =119894119896isinΘ(119894)
119878119896
119891
minus 119878119894
119891
(36)
119871119894is an index which is applied in security assessment
Here is an example Let W1 be an insecure operating pointEach transfer unit load of W1 is 08 times of TSC load inthe practical case except 1198781
119891and 11987810119891 while 1198781
119891= 8MVA and
11987810119891= 7MVA Total load of W1 is 5087MVA while TSC is
6289MVAAverage load rate is 046Thatmeans that current
Table 3 Distance to boundaries of 1198611and 119861
10atW1
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
operating point can be controlled back to security regionThrough formula (36) the location of a given operating pointin DSSR can be calculated We present the 119861
119894of which 119871
119894are
negative in Table 3Table 3 shows that the system should lose 3222MVA if
fault occurs at feeder 1 or feeder 10 so the system is locallyoverloading The location of W1 can be easily observed inFigure 9 To meet operational constraints under N-1 contin-gency 1198781
119891and 11987810
119891should be adjusted For example W1 is
adjusted toW1015840
1 through the dashed arrow shown in Figure 9The location ofW
1015840
1 is shown in Table 4
7 Conclusion and Further Work
This paper proposes a mathematical model for distributionsystem security region (DSSR) generic topological character-istics of DSSR are discussed and proved
First the operating point for a distribution system isdefined as the set of transfer unit loads which is a vector in the
12 Journal of Applied Mathematics
DSSR
0
2
4
6
2 4 6 8 10 11778
8
10
11778
S1f (MVA)
S10f (MVA)
W1(8 7)
W9984001 (7 4)
Figure 9 2D DSSR cross-section and preventive control
Table 4 Distance to boundaries of 1198611and 119861
10atW10158401
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
Euclidean space Then the DSSR model for both substationtransformer and feeder contingency is proposed which is theset of all operating points each ensures that the system N-1 issecure
Second to perform research on the characteristic ofDSSR the boundary formulation for DSSR is derived fromthe DSSR model Then three generic characteristics areproposed and proved by rigor mathematical approach whichare as follows (1) DSSR is dense inside (2) DSSR boundaryhas no suspension (3)DSSR boundary can be expressed withthe union of several subsurfaces
Finally the results from a test case and a practical casedemonstrate the effectiveness of the proposed model Thevisualization for DSSR boundary is also discussed to verifythe topological characteristics of DSSR Security assessmentmethod based on DSSR is preliminary exhibited to show theapplicability of DSSR for future smart distribution system
This paper improves the accuracy and understanding ofDSSR which is fundamental theoretic work for future smartdistribution system Further works to improve the accuracyof DSSR model include fully considering power flow voltageconstraints and integration of DGs
Conflict of Interests
The authors declare that there is no conflict of inter-ests regarding the publication of this paper None ofthe authors have a commercial interest financial interestandor other relationship with manufacturers of pharma-ceuticalslaboratory supplies andor medical devices or withcommercialproviders of medically related services
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (no 51277129) and National HighTechnology Research and Development Program of China(863 Program) (no 2011AA05A117)
References
[1] J Xiao W Gu C Wang and F Li ldquoDistribution systemsecurity region definition model and security assessmentrdquo IETGeneration TransmissionampDistribution vol 6 no 10 pp 1029ndash1035 2012
[2] State Grid Corporation of China Guidelines of Urban PowerNetwork Planning State Grid Corporation of China BeijingChina 2006 (Chinese)
[3] E Lakervi and E J Holmes Electricity Distribution NetworkDesign IET Peregrinus UK 1995
[4] K N Miu and H D Chiang ldquoService restoration for unbal-anced radial distribution systems with varying loads solutionalgorithmrdquo in Proceedings of the IEEE Power Engineering SocietySummer Meeting vol 1 pp 254ndash258 1999
[5] Y X Yu ldquoSecurity region of bulk power systemrdquo in Proceedingsof the International Conference on Power System Technology(PowerCon rsquo02) vol 1 pp 13ndash17 Kunming China October2002
[6] F Wu and S Kumagai ldquoSteady-state security regions of powersystemsrdquo IEEE Transactions on Circuits and Systems vol 29 no11 pp 703ndash711 1982
[7] Y Yu Y Zeng and F Feng ldquoDifferential topological character-istics of the DSR on injection space of electrical power systemrdquoScience in China E Technological Sciences vol 45 no 6 pp 576ndash584 2002
[8] S J Chen Q X Chen Q Xia and C Q Kang ldquoSteady-state security assessment method based on distance to securityregion boundariesrdquo IET Generation Transmission amp Distribu-tion vol 7 no 3 pp 288ndash297 2013
[9] D M Staszesky D Craig and C Befus ldquoAdvanced feederautomation is hererdquo IEEE Power and Energy Magazine vol 3no 5 pp 56ndash63 2005
[10] C L Smallwood and J Wennermark ldquoBenefits of distributionautomationrdquo IEEE Industry Applications Magazine vol 16 no1 pp 65ndash73 2010
[11] X Mamo S Mallet T Coste and S Grenard ldquoDistributionautomation the cornerstone for smart grid development strat-egyrdquo in Proceedings of the IEEE Power amp Energy Society GeneralMeeting (PES rsquo09) pp 1ndash6 July 2009
[12] P W Sauer B C Lesieutre and M A Pai ldquoMaximumloadability and voltage stability in power systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 15 no 3 pp145ndash153 1993
[13] K N Miu and H Chiang ldquoElectric distribution systemload capability problem formulation solution algorithm andnumerical resultsrdquo IEEE Transactions on Power Delivery vol 15no 1 pp 436ndash442 2000
[14] F Z Luo C S Wang J Xiao and S Ge ldquoRapid evaluationmethod for power supply capability of urban distribution sys-tem based on N-1 contingency analysis of main-transformersrdquoInternational Journal of Electrical Power andEnergy Systems vol32 no 10 pp 1063ndash1068 2010
Journal of Applied Mathematics 13
[15] J Xiao F X LiWZGuC SWang andP Zhang ldquoTotal supplycapability and its extended indices for distribution systemsdefinitionmodel calculation and applicationsrdquo IETGenerationTransmission and Distribution vol 5 no 8 pp 869ndash876 2011
[16] J Xiao X X Gong and C S Wang ldquoTopology properties andalgorithm of N-1 security boundary for smart gridrdquo Proceedingsof the CSEE vol 34 no 4 pp 545ndash554 2014 (Chinese)
[17] G E Shilov An Introduction to the Theory of Linear SpacesDover New York NY USA 1975
[18] J R Munkres Topology vol 2 Prentice Hall Upper SaddleRiver NJ USA 2000
Normally closed switchNormally open switchBack-feed (transfer) path
FaultS1 S2 S3
S4Feeder 2
Feeder 3
S1f
S1FS2f
S3f
S4f
Figure 2 Feeder and transfer unit load
too long In contrast with feeder automation the load of T2can be transferred not only to T1 but also to T3 in anothersubstation
The DSSR model in this paper takes both substationtransformer N-1 and feeder N-1 contingency into accountThe DSSR is defined as the set of operating points whichassure the N-1 security of distribution system An operatingpoint in transmission system is the current power injectionsof nodes while in distribution system it represents a set oftransfer unit load Consider
W = (1198781
119891 1198782
119891 119878
119899
119891)119879
(1)
whereW is the operating point which is an n-dimensionvector 119878119894
119891is transfer unit load ldquoTransfer unitrdquo is defined as
the set of feeder sections of which load has the same back-feed (transfer) path If two feeders form single loop networktransfer unit load is equivalent to feeder load If a feederis connected to two other feeders which is called two-tieconnection the feeder load can be divided into two parts ofload each of which is a transfer unit load In this paper feederload i is denoted by 119878119894
119865and transfer unitload 119894 is denoted by
119878119894119891 Take the brief case in Figure 2 as an exampleIn Figure 2 feeder 1 forms two-tie connection and feeder
2 and feeder 3 form single loop network respectively withfeeder 1 In normal state 1198781
119891and 1198782119891are both supplied by feeder
1 that is 1198781119865= 1198781119891+ 1198782119891 When a fault occurs at outlet of
feeder 1 S1 and S2 disconnect and S4 closes and then 1198781119891
changes to be supplied by feeder 3 while 1198782119891is supplied by
feeder 2 via closing S3 Transfer scheme is usually not uniqueAnother scheme is that transferring both 1198781
119891and 1198782119891to feeder
2 by disconnecting S1 and closing S3 However multischememakes the DSSR model overcomplicated Thus scheme isfixed in this paper which stipulates that each of the backupfeeders restores only one section of the whole faulted feederas the first scheme above More important this fixed schemecan usually balance the branch load in the postfault networkwhich is an important index concerned by dispatchers
The DSSR model in this paper is accurate to the safetymonitoring of transfer unit load We can certainly get feederload by summing up the transfer unit load and further get
transformer load by summing up the feeder load This is abasis for DSSR model considering both transformer N-1 andfeeder N-1 security
3 Mathematical Model for DSSR
After the load transfer incurred by tie switches against N-1 contingency all the transformers and feeders cannot beoverloading Since the tie-line is designed after the radialnetwork planning the capacity is considered as enough totransfer loadTherefore the transformer load and feeder loadshould be under a series of constraints The DSSR model canbe mathematically formulated as
ΩDSSR = W | ℎ (119909) le 0 119892 (119909) = 0 (2)
where W = (1198781119891 1198782119891 119878119899
119891)119879 is the operating point cor-
responding to 1198781119891 1198782
119891 119878
119899
119891 The inequality and equality
constraints are such that
119878119898
119865= sum119899=1
119878119898119899
119891119905119903 (3)
119878119898119899
119891119905119903+ 119878119899
119865le 119878119899
119865max (forall119898 119899) (4)
119878119894119895
119879119905119903= sum
119898isinΦ(119894)119899isinΦ(119895)
119878119898119899
119891119905119903 (5)
119878119894
119879= sum
119898isinΦ(119894)
119878119898
119865(forall119894) (6)
119878119894119895
119879119905119903+ 119878119895
119879le 119878119895
119879max (7)
where 119878119898119865
= all the load supplied by feeder m 119878119898119899119891119905119903
=the load transferred from feeder m to feeder n when anN-1 fault occurs at outlet of feeder m 119878119899
119865max = maximalthermal capacity of feeder 119899 119878119894119895
119879119905119903= the load transferred from
transformer 119894 to transformer 119895 when an N-1 fault occursat transformer 119894 Φ(119894) = the set of feeders that derive fromtransformer 119894 119878119895
119879= all of the load supplied by transformer 119894
119898 isin Φ(119894)means that feeder119898 derives from the correspondingbus of transformer 119894 119878119894
119879max = the rated capacity of transformer119894
It is notable that the correspondence relation of ldquo119878119891119905119903
rdquoand ldquo119878
119891rdquo depends on the load transfer scheme For example
in Figure 2 in the former scheme 11987812119891119905119903
Equation (3) shows that the sum of each load to betransferred away from feeder m should be equal to thetotal load supplied by feeder m in normal state inequalityconstraint (4) means that feeders cannot be overloading inthe postfault network (5) shows that path of load transferredfrom transformer 119894 to 119895 is the tie-lines between transformers 119894and 119895 Equation (6) means that load of transformer is equal tothe sum of corresponding feeder loads inequality constraint(7) describes that the load of transformer 119895 cannot exceed itsrated capacity after a substation transformerN-1 contingency
It should be pointed out that power flow and voltage dropare not taken into consideration in this DSSR model which
4 Journal of Applied Mathematics
determines that the DSSR model is completely linear Sinceoverloading under contingencies is the most critical prob-lem in urban power the simple linear model is acceptablefrom the standpoint of security-based operation [1] Moreimportantly research of [16] has demonstrated that the DSSRmodel is approximately linear even when considering factorsof power flow and voltage drop which further proves that thelinear model is highly approximate to the real DSSR model
4 Formulation for DSSR Boundary
Research on the boundary of security region has vital signif-icance because the characteristics determine the applicationway of DSSR theory [7] The proposed DSSR model whichis formulated as (2)ndash(7) cannot distinctly express the DSSRboundary Therefore derivation should be performed totransform the originalmodel to a new expression form Select119878119906V119891119905119903
as research object 119906 isin Φ(119894) and V isin Φ(119895) First formula(4) can be transformed to
119878119906V119891119905119903le 119878
V119865max minus 119878
V119865 (8)
Second substitute formulas (5) and (6) into (7) and thenperform identical deformation We obtain
Finally according to (8) and (10) the expression of DSSRboundary in direction of 119878119906V
119891119905119903can be neatly formulated as
119878119906V119891119905119903
le min
119878V119865max minus 119878
V119865 119878119895
119879max minus sum
119905isinΦ(119895)
119878119905
119865minus sum
119898isinΦ(119894)119899isinΦ(119895)
119898 =119906119899 =V
119878119898119899
119891119905119903
(11)
There are 119899 boundaries if the system consists of 119899load transfer units To express the dimension of boundaryclearly we denote all ldquo119878119906V
the transfer scheme in this paper is fixed as is illustrated inSection 22 an important equivalence holds as follows
119878119894
119891119905119903= 119878119894
119891(119894 = 1 2 119899) (12)
Through (12) 119878119891119905119903
and 119878119891have one-to-one corresponding
relation Above all the complete and succinct boundaryformulation for DSSR is expressed as
ΩDSSR
=
1198611
1198612
119861119894
119861119899
=
1198781119891le min
1198781198871119865max minus 119878
1198871
119865 1198781198871119879max minus sum
119895isinΦ(1198871)
119878119895
119865minus sum
119896 =1119896isinΘ(1)
119878119896119891
1198782
119891le min
1198781198872
119865max minus 1198781198872
119865 1198781198872
119879max minus sum
119895isinΦ(1198872)
119878119895
119865minus sum
119896 =2119896isinΘ(2)
119878119896
119891
119878119894
119891le min
119878119887119894
119865max minus 119878119887119894
119865 119878119887119894
119879max minus sum
119895isinΦ(119887119894)
119878119895
119865minus sum
119896 =119894119896isinΘ(119894)
119878119896
119891
119878119899119891le min
119878119887119899119865max minus 119878
119887119899
119865 119878119887119899119879max minus sum
119895isinΦ(119887119899)
119878119895
119865minus sum
119896 =119899119896isinΘ(119899)
119878119896119891
(13)
With the form transformation ofDSSRmodel correspondingsymbols and their meaning should be adjusted too Thus 119887119894and Θ(119894) are used for new expression 119887119894 = the number ofback-feed feeders of 119878119894
119891 119878119887119894119865= all of the load supplied by the
back-feed feeder 119887119894 Θ(119894) = the set of transfer unit loads ofwhich both normal-feed and back-feed transformer are thesame as those of 119878119894
119891 It should be pointed out that feeder 1198871
feeder 119887119899 are not 119899 different actual feeders Take the case inFigure 2 as an example again when fault occurs the back-feed feeders of 1198783
119891and 1198784
119891are both feeder 1 which means
feeder b3 = feeder b4 = feeder 1 When any equality in (13)holds the operating point is just upon the DSSR boundaryWe denote the n boundaries by 119861
1 119861
119899
The DSSR boundary formulation can also reflect bothsubstation transformer and feeder N-1 secure constraintsTake 119894 subformula of (13) to concretely explain 119878119894
119891le
119878119887119894119865max minus 119878
119887119894
119865shows that transfer of 119878119894
119891cannot cause the feeder
overloading which ensures that the feeder N-1 is secure 119878119894119891le
119878119887119894119879max minus sum119895isinΦ(119887119894) 119878
119895
119865minus sum119896 =1119896isinΘ
(119894) 119878119896
119891shows that transfer of 119878119894
119891
cannot lead to the substation transformer overloading whichensures that the transformer N-1 is secure
5 Topological Characteristics of DSSR andMathematical Proof
The research on differential topological characteristics of theDSSR focuses on several aspects as follows
Journal of Applied Mathematics 5
(1) Are there any holes inside the DSSR that is thedenseness in terms of the topology
(2) Does the boundary of the DSSR have suspension(3) Could the boundary of the DSSR be expressed with
the union of several subsurfaces
The identical deformation from formulas (2)ndash(7) to (13) isa breakthrough for the research on topological characteristicsof the DSSR However to research on these characteristicsformula (13) should be further simplified to obtain themathematical essence Take 119894 subformula of (13) as researchobject the equivalent form is
119878119894
119891+ 119878119887119894
119865le 119878119887119894
119865max
119878119894
119891+ sum
119895isinΦ(119887119894)
119878119895
119865+ sum
119896 =1119896isinΘ(119894)
119878119896
119891le 119878119887119894
119879max(14)
According to formulas (3) and (12) 119878119887119894119865
and 119878119895119865
can beexpressed as linear combination of a series of differentelement ldquo119878
119891rdquo Then formula (14) will be simplified as
119899
sum119898=1
120572119898119878119898
119891le 119878119887119894
119865max
119899
sum119898=1
120573119898119878119898
119891le 119878119887119894
119879max
(15)
where 119878119887119894119865max and 119878119887119894
119879max are positive constants Throughformula (14) coefficients 120572 and 120573 have the following features
120572119898= 1 120573
119898= 1 (119898 = 119894)
120572119898= 0 or 1 120573
119898= 0 or 1 (119898 = 119894)
(16)
LetΩDSSR be a set which consists of operating pointW DSSRboundary formulation can be abstracted as
AW le C (17)
whereW is a vector defined as formula (1)Through formulas(15)-(16) matrices A and C should satisfy
A = [119886119894119895]2119899times119899
119894 ge 1 119895 ge 1
119886119894119895= 1 (119894 = 2119895 minus 1 or 119894 = 2119895)
119886119894119895= 0 or 1 (119894 = 2119895 minus 1 119894 = 2119895)
C = [1198881 1198882 1198882119899]119879
C isin R+
(18)
Because one integrated subformula is divided into equivalenttwo parts as is shown in formula (15) the dimension ofA is 2119899 times 119899 instead of 119899 times 119899 Here ΩDSSR is both linearspace and Euclidean space which has been proved in [17]Besides this important premise we should review somepreparation definitions and theorems about topology andEuclidean space including cluster point closure dense setand hyperplane
Definition 1 (cluster point) Let x be a point in Euclideanspace 119865 sub 119877 If there is a point sequence in set 119865 convergingto x then x is the cluster point of set 119865 [17]
Definition 2 (closure) 119860 is a subset of topological space 119860represents all the points inside 119860 and cluster points of 119860 andthen 119860 is called closure of 119860 [18]
Theorem 3 If the 119861 is a subset of 119860 119861 = 119860 then 119861 is dense in119860 [18]
Theorem 4 Let 119871 be a subspace of linear space 119877 x0 is a fixedvector which does not belong to 119871 generally Considering set119867which consists of the vector x x is obtained by
x = x0 + y (19)
where vector y varies in the whole subspace 119871 Thus119867 is calledhyperplane The dimension of119867 equals that of subspace 119871 [17]
The topological characteristics of DSSR are mathemati-cally proved in the following section
Characteristic 1 TheΩDSSR is dense inside
Proof First a new setΩ1015840DSSR is defined as the set of all pointsofΩDSSR except those on boundariesΩ1015840DSSR meets
AW lt C (20)
Select randomly a vector W = (1198781119891 1198782119891 119878119899
119891)119879 from Ω1015840DSSR
and then construct a sequenceY = Y1Y2 Ym as the
following formula
Y =
Y1= (1198781119891minus1198781119891
1 1198782119891minus1198782119891
1 119878119899
119891minus119878119899119891
1)
119879
= (0 0 0)119879
Y2= (1198781
119891minus1198781119891
2 1198782
119891minus1198782119891
2 119878
119899
119891minus119878119899119891
2)
119879
Y119898= (1198781119891minus1198781119891
119898 1198782119891minus1198782119891
119898 119878119899
119891minus119878119899119891
119898)
119879
= (1 minus1
119898)W
(21)
According to (20)-(21) we obtain
AYm = A(1 minus 1119898)W lt AW lt C (22)
Therefore sequence Y = Y1Y2 Ym is inside Ω1015840
DSSRLet 120588(WYm) be the distance betweenWandYm in Euclideanspace and then
Figure 3 Shapes of 2-dimension cross-section of DSSR
Thismeans that for any vectorW inΩ1015840DSSR sequenceY existsand converges toW andY is also insideΩ1015840DSSRTherefore anypoint insideΩ1015840DSSR is cluster point of Ω
1015840
DSSRSimilarly select vector Z = (1198781
119891 1198782119891 119878119899
119891)119879 which is just
upon boundary ofΩ119863119878119878119877
AZ = C (24)
Construct a sequence U = U1U2 Um
Um = (1 minus1
119898)Z (25)
Then we obtain
AUm = A(1 minus 1119898)Z lt AZ = C (26)
According to formula (26) sequence U is also inside Ω1015840DSSRThrough the following
We obtain that any point just upon the ΩDSSR boundary isalso cluster point ofΩ1015840DSSR Above all cluster points ofΩ
1015840
DSSR
include not only all the points inside Ω1015840DSSR but also all thepoints just upon the boundaries ofΩDSSRThus the followingformula holds
Ω1015840DSSR = ΩDSSR (28)
Through Theorem 3 Ω1015840DSSR is dense in ΩDSSR This proof iscompleted
Characteristic 2 The boundary ofΩDSSR has no suspension
Proof This characteristic can be approximately simplified toprove that any 2-dimension cross-section of ΩDSSR is convexpolygon For W = (1198781
119891 1198782119891 119878119899
119891)119879 select a 2-dimension
cross-section by taking 119878119898119891
and 119878119899119891as variables and 119878
119891in
another dimension as constants According to formula (16)the coefficient of 119878
119891(including variable 119878
119891and constant ones)
can be only 1 or 0 thus in process of dimension reduction 119878119898119891
and 119878119899119891are simply constrained by
119878119898
119891+ 119878119899
119891le 1198881
0 le 119878119898
119891le 1198882
0 le 119878119899
119891le 1198883
forall1198881 1198882 1198883isin 119877+
(29)
Journal of Applied Mathematics 7
Table 1 Profile of test case
Substation Transformer Voltages (kVkV) Transformer capacity (MVA) Number of feeders Total feeder capacity (MVA)
S1 T1 3510 400 9 8028T2 3510 400 9 8028
S2 T3 3510 400 9 8028T4 3510 400 8 7136
S3 T5 11010 630 10 892T6 11010 630 10 892
S4 T7 11010 630 10 892T8 11010 630 10 892
The shape of 2-dimension cross-section varies with the valuesof 1198881 1198882 and 119888
3 as is shown in Figure 3 The dashed oblique
line represents 119878119898119891+ 119878119899119891le 1198881
As is shown in Figure 3 the shapes of 2-dimension sectionare confined to rectangle triangle ladder and convex pen-tagon This means that any 2-dimension section of DSSR isconvex polygon which proves that the boundary ofΩDSSR hasno suspension in an indirect wayThe proof is completed
Characteristic 3 The ΩDSSR boundary can be expressed withseveral subsurfaces
Proof 119867 is a set of vector W = (1198781119891 1198782119891 119878119899
119891)119879 each of
which satisfies nonhomogeneous linear equation set
AW = C (30)
where matrices A and C meet formula (18) and W is theoperating point just upon the boundary of ΩDSSR Let W0 =
(120579(0)
1 120579(0)
2 120579(0)
119899)119879 be a fixed solution of equation set (31)
Meanwhile 119871 is a subspace of 119877 which consists of y =
(1205821 1205822 120582
119899)119879 that is y varies in the whole subspace 119871
The coordinates of ymeet the following homogeneous linearequation set
Ay = 0 (31)
If y = (1205821 1205822 120582
119899)119879 is a solution of (31) W = W0
+y = (120579(0)
1+ 1205821 120579(0)
2+ 1205822 120579(0)
119899+ 120582119899)119879 is obviously
a solution of equation set (30) that is W is included inset 119867 Meanwhile x0 is not in 119871 Through Theorem 3 119867is hyperplane The boundary of ΩDSSR can be expressedwith hyperplane that is several subsurfaces Because thedimension of 119871 is 119899 the dimension of hyperplane is also 119899This proof is completed
In transmission area topological characteristics of secu-rity region (SR) have been deeply studied [7] It has shownthat the boundary of SR has no suspension and is compactand approximately linear there is no hole inside the SR Theproof above demonstrates that characteristics of DSSR aresimilar but it has its own features
(a) DSSR is a convex set of denseness in terms ofthe topology This means that DSSR has no holesinside and its boundary has no suspension Basedon this feature dispatchers can analyze the securityof operating point by judging whether the point isinside theDSSRboundarywithout being afraid of anyinsecure point inside or losing security points
(b) DSSR boundary can be described by hyperplane (sev-eral Euclidean subsurfaces) meanwhile the bound-ary ismore linear than that of SR In practical applica-tion calculation of high dimensional security regionwill incur great computation burdens and consumelarge amount of time Therefore this characteristicwill improve the computational efficiency
6 Case Study
In order to verify the effectiveness of the proposedmodel twocases are illustrated in this section The first one is a test caseand the second is a practical case
61 Test Case
611 Overview of Test Case As is shown in Figure 4 the gridis comprised of 4 substations 8 substation transformers and75 feeders 68 feeders form 34 single loop networks and 7form two-tie connections resulting in 82 load transfer unitsThe total capacity of substations is 412MVA To illustrate themodeling process clearly each load transfer unit is numberedCase profile is shown in Table 1
The conductor type of all feeders is LGJ-185 (892MVA)This type is selected based on the national electrical code inChina
612 Boundary Formulation The test case grid has 75 feed-ers 14 feeders form 7 two-tie connections Each load of these7 feeders can be divided into 2 parts to transfer in differentback-feed path So the number of load transfer units is 82which means that the dimension of operation point W is82 and the number of subformulas of DSSR boundary is82 Based on the topology of the test case we substitutethe rated capacity of substation transformers and feedersinto 1198781
119865max 119878119899
119865max and 1198781119879max 119878
119899
119879max of formula (13)
8 Journal of Applied Mathematics
Then the boundary formulation for DSSR of test case can beexpressed as
Considering the similarity of calculation approach ofeach boundary and thesis length we present 5 subformulasof 1198781119891 1198785
119891here Complete formulation is given in the
appendix (in the Supplementary Material available online athttpdxdoiorg1011552014327078) Take the first subfor-mula in (32) as an example to explain the modeling processFigure 5 shows the local topology related with 1198781
119891 including
all relative transfer units in the first subformulaFirst analyze the topology and determine the parameters
of formula (13)We know that back-feed feeder of 1198781119891is feeder
36 and the back-feed substation transformer of 1198781119891is T5 In
normal state 11987836119891 11987845
119891and 11987876
119891 11987878
119891are all load of T5
which means sum119895isinΦ(1198871) 119878119895
119865= sum45
119895=36119878119895
119891+ sum78
119895=76119878119895
119891 11987836119891
and 11987876119891
are load of feeder 36 which means 1198781198871119865= 11987836119891+ 11987876119891 1198782119891 1198785
119891
are the load transferred fromT1 to T5when fault occurs at T1whichmeanssum
119896 =1119896isinΘ(1) 119878119896
119891= sum5
119896=2119878119896119891Therefore the formula
of B1 is preliminary expressed as
1198781
119891le min 11987836
119865max minus (11987876
119891+ 11987836
119891)
1198785
119879max minus (11987836
119891+ 11987837
119891+ 11987838
119891+ 11987839
119891+ 11987840
119891+ 11987841
11989111987842
119891
+11987843
119891+ 11987844
119891+ 11987845
119891+ 11987876
119891+ 11987877
119891+ 11987878
119891)
minus (1198782
119891+ 1198783
119891+ 1198784
119891+ 1198785
119891)
(33)
Second according to case profile substitute 11987836119865max =
892MVA and 1198785119879max = 63MVA into formula (33) and then
we obtain
1198781
119891le min 892 minus (11987876
119891+ 11987836
119891)
63 minus (11987836
119891+ 11987837
119891+ 11987838
119891+ 11987839
119891+ 11987840
119891+ 11987841
11989111987842
119891
+11987843
119891+ 11987844
119891+ 11987845
119891+ 11987876
119891+ 11987877
119891+ 11987878
119891)
minus (1198782
119891+ 1198783
119891+ 1198784
119891+ 1198785
119891)
(34)
613 Visualization and Topological Characteristics To visu-alize the shape of DSSR the dimension is reduced to 2 and 3which means that the coordinates of W should be constantsother than those to be visualized Thus an operating pointhas to be predetermined to provide the constants TSC is aspecial operating point just upon the security boundary [1]on which the facilities are usually fully utilizedThe operationof distribution system is closing to the boundaries and loadlevel is reaching TSCThus in this test case we select the TSCpoint as the predetermined operating point WTSC which isshown in Table 2The calculation approach for TSC has beenpresented in [15]
Choose randomly transfer unit loads 11987815119891
and 11987852119891
toobserve Substitute the rated capacity of transformers T1simT6capacity of all feeders and all transfer unit loads other than11987815119891
and 11987852119891 into formula (32) Then the 2D DSSR shown in
Figure 6 is obtainedAlthough the dimension is 2 the number of subformulas
which constrain the figure shape is not just 2 In fact eachof the subformulas which contains 11987815
119891or 11987852119891
should beconsidered But the subformulas withmore severe constraintswill cover the others and finally form theDSSR boundaryThefinal formulation of 2D figure is
0 le 11987815
119891le 52
0 le 11987852
119891le 3
(35)
Journal of Applied Mathematics 9
T1S1
T2
T3
T4
S 2
S4
T8
T6S3
T5
17-1
8
20-19
12ndash1
415
-16
21ndash23
41-4
3
36ndash3
8
39-4
044
-45
47-4
650
-51
52-5
3
54-5
548
-49
10-1
1
9-8
1 ndash3
4-5
6-7
76ndash7
8
79-80
58-5960-61
57-5665-62
66ndash6970-71
72ndash7524-25
29-28
26-27
30-31
32ndash35
81-8
2
Tie switch
T7
Distribution transformer
(i + 1) j are the same in the topologyin terms of calculation for DSSR
Note means that transfer units iindashj
Figure 4 Test case with eight substation transformers
Feeder 1
T1
T5
Feeder 2Feeder 36Feeder 37
S1f
S2f
S3f
S76f
S77f
S78f
S36f
S36f
S37f
S45f
S4f
S5f
middot middot middot
middot middot middot middot middot middot
Figure 5 Local topology related with f1
Similarly we choose randomly transfer unit loads 11987819119891 11987824119891
and 11987832119891
to observe and then the 3D DSSR is obtained as isshown in Figure 7
In Figure 6 it can be seen that the 2D DSSR is a denserectangle of which boundaries are linear without suspension
DSSR
0
1
2
3
1 2 3 4 52 S52f (MVA)
S15f (MVA)
Figure 6 Shape of 2D DSSR
In Figure 7 the 3D DSSR is surrounded by several subplanesforming a convex pentagon that is a special 3-dimensionhyperplane
Figure 8 The medium-voltage urban power grid of one city of southern China
62 Practical Case In this section the assessment andcontrol method based on DSSR is demonstrated on a realmedium-voltage distribution network of one city in south-ern China which consists of 12 substations 26 substationtransformers and 114 main feeders as is shown in Figure 8Total capacity of substation transformers is 10945MVA It isnotable that each two of all main feeders form a single loopnetwork in this case which leads to the fact that the numberof main feeders is equal to that of load transfer units
Paper [1] has presented concept and calculation methodof relative location of operating point in DSSR based onsubstation transformer contingency Similarly in this paperrelative location 119871
119894describes the distance from the operating
pointW to the boundary 119861119894(119894 = 1 2 119899)
119871119894= min
119878119887119894
119865max minus 119878119887119894
119865 119878119887119894
119879max minus sum
119895isinΦ(119887119894)
119878119895
119865minus sum
119896 =119894119896isinΘ(119894)
119878119896
119891
minus 119878119894
119891
(36)
119871119894is an index which is applied in security assessment
Here is an example Let W1 be an insecure operating pointEach transfer unit load of W1 is 08 times of TSC load inthe practical case except 1198781
119891and 11987810119891 while 1198781
119891= 8MVA and
11987810119891= 7MVA Total load of W1 is 5087MVA while TSC is
6289MVAAverage load rate is 046Thatmeans that current
Table 3 Distance to boundaries of 1198611and 119861
10atW1
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
operating point can be controlled back to security regionThrough formula (36) the location of a given operating pointin DSSR can be calculated We present the 119861
119894of which 119871
119894are
negative in Table 3Table 3 shows that the system should lose 3222MVA if
fault occurs at feeder 1 or feeder 10 so the system is locallyoverloading The location of W1 can be easily observed inFigure 9 To meet operational constraints under N-1 contin-gency 1198781
119891and 11987810
119891should be adjusted For example W1 is
adjusted toW1015840
1 through the dashed arrow shown in Figure 9The location ofW
1015840
1 is shown in Table 4
7 Conclusion and Further Work
This paper proposes a mathematical model for distributionsystem security region (DSSR) generic topological character-istics of DSSR are discussed and proved
First the operating point for a distribution system isdefined as the set of transfer unit loads which is a vector in the
12 Journal of Applied Mathematics
DSSR
0
2
4
6
2 4 6 8 10 11778
8
10
11778
S1f (MVA)
S10f (MVA)
W1(8 7)
W9984001 (7 4)
Figure 9 2D DSSR cross-section and preventive control
Table 4 Distance to boundaries of 1198611and 119861
10atW10158401
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
Euclidean space Then the DSSR model for both substationtransformer and feeder contingency is proposed which is theset of all operating points each ensures that the system N-1 issecure
Second to perform research on the characteristic ofDSSR the boundary formulation for DSSR is derived fromthe DSSR model Then three generic characteristics areproposed and proved by rigor mathematical approach whichare as follows (1) DSSR is dense inside (2) DSSR boundaryhas no suspension (3)DSSR boundary can be expressed withthe union of several subsurfaces
Finally the results from a test case and a practical casedemonstrate the effectiveness of the proposed model Thevisualization for DSSR boundary is also discussed to verifythe topological characteristics of DSSR Security assessmentmethod based on DSSR is preliminary exhibited to show theapplicability of DSSR for future smart distribution system
This paper improves the accuracy and understanding ofDSSR which is fundamental theoretic work for future smartdistribution system Further works to improve the accuracyof DSSR model include fully considering power flow voltageconstraints and integration of DGs
Conflict of Interests
The authors declare that there is no conflict of inter-ests regarding the publication of this paper None ofthe authors have a commercial interest financial interestandor other relationship with manufacturers of pharma-ceuticalslaboratory supplies andor medical devices or withcommercialproviders of medically related services
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (no 51277129) and National HighTechnology Research and Development Program of China(863 Program) (no 2011AA05A117)
References
[1] J Xiao W Gu C Wang and F Li ldquoDistribution systemsecurity region definition model and security assessmentrdquo IETGeneration TransmissionampDistribution vol 6 no 10 pp 1029ndash1035 2012
[2] State Grid Corporation of China Guidelines of Urban PowerNetwork Planning State Grid Corporation of China BeijingChina 2006 (Chinese)
[3] E Lakervi and E J Holmes Electricity Distribution NetworkDesign IET Peregrinus UK 1995
[4] K N Miu and H D Chiang ldquoService restoration for unbal-anced radial distribution systems with varying loads solutionalgorithmrdquo in Proceedings of the IEEE Power Engineering SocietySummer Meeting vol 1 pp 254ndash258 1999
[5] Y X Yu ldquoSecurity region of bulk power systemrdquo in Proceedingsof the International Conference on Power System Technology(PowerCon rsquo02) vol 1 pp 13ndash17 Kunming China October2002
[6] F Wu and S Kumagai ldquoSteady-state security regions of powersystemsrdquo IEEE Transactions on Circuits and Systems vol 29 no11 pp 703ndash711 1982
[7] Y Yu Y Zeng and F Feng ldquoDifferential topological character-istics of the DSR on injection space of electrical power systemrdquoScience in China E Technological Sciences vol 45 no 6 pp 576ndash584 2002
[8] S J Chen Q X Chen Q Xia and C Q Kang ldquoSteady-state security assessment method based on distance to securityregion boundariesrdquo IET Generation Transmission amp Distribu-tion vol 7 no 3 pp 288ndash297 2013
[9] D M Staszesky D Craig and C Befus ldquoAdvanced feederautomation is hererdquo IEEE Power and Energy Magazine vol 3no 5 pp 56ndash63 2005
[10] C L Smallwood and J Wennermark ldquoBenefits of distributionautomationrdquo IEEE Industry Applications Magazine vol 16 no1 pp 65ndash73 2010
[11] X Mamo S Mallet T Coste and S Grenard ldquoDistributionautomation the cornerstone for smart grid development strat-egyrdquo in Proceedings of the IEEE Power amp Energy Society GeneralMeeting (PES rsquo09) pp 1ndash6 July 2009
[12] P W Sauer B C Lesieutre and M A Pai ldquoMaximumloadability and voltage stability in power systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 15 no 3 pp145ndash153 1993
[13] K N Miu and H Chiang ldquoElectric distribution systemload capability problem formulation solution algorithm andnumerical resultsrdquo IEEE Transactions on Power Delivery vol 15no 1 pp 436ndash442 2000
[14] F Z Luo C S Wang J Xiao and S Ge ldquoRapid evaluationmethod for power supply capability of urban distribution sys-tem based on N-1 contingency analysis of main-transformersrdquoInternational Journal of Electrical Power andEnergy Systems vol32 no 10 pp 1063ndash1068 2010
Journal of Applied Mathematics 13
[15] J Xiao F X LiWZGuC SWang andP Zhang ldquoTotal supplycapability and its extended indices for distribution systemsdefinitionmodel calculation and applicationsrdquo IETGenerationTransmission and Distribution vol 5 no 8 pp 869ndash876 2011
[16] J Xiao X X Gong and C S Wang ldquoTopology properties andalgorithm of N-1 security boundary for smart gridrdquo Proceedingsof the CSEE vol 34 no 4 pp 545ndash554 2014 (Chinese)
[17] G E Shilov An Introduction to the Theory of Linear SpacesDover New York NY USA 1975
[18] J R Munkres Topology vol 2 Prentice Hall Upper SaddleRiver NJ USA 2000
determines that the DSSR model is completely linear Sinceoverloading under contingencies is the most critical prob-lem in urban power the simple linear model is acceptablefrom the standpoint of security-based operation [1] Moreimportantly research of [16] has demonstrated that the DSSRmodel is approximately linear even when considering factorsof power flow and voltage drop which further proves that thelinear model is highly approximate to the real DSSR model
4 Formulation for DSSR Boundary
Research on the boundary of security region has vital signif-icance because the characteristics determine the applicationway of DSSR theory [7] The proposed DSSR model whichis formulated as (2)ndash(7) cannot distinctly express the DSSRboundary Therefore derivation should be performed totransform the originalmodel to a new expression form Select119878119906V119891119905119903
as research object 119906 isin Φ(119894) and V isin Φ(119895) First formula(4) can be transformed to
119878119906V119891119905119903le 119878
V119865max minus 119878
V119865 (8)
Second substitute formulas (5) and (6) into (7) and thenperform identical deformation We obtain
Finally according to (8) and (10) the expression of DSSRboundary in direction of 119878119906V
119891119905119903can be neatly formulated as
119878119906V119891119905119903
le min
119878V119865max minus 119878
V119865 119878119895
119879max minus sum
119905isinΦ(119895)
119878119905
119865minus sum
119898isinΦ(119894)119899isinΦ(119895)
119898 =119906119899 =V
119878119898119899
119891119905119903
(11)
There are 119899 boundaries if the system consists of 119899load transfer units To express the dimension of boundaryclearly we denote all ldquo119878119906V
the transfer scheme in this paper is fixed as is illustrated inSection 22 an important equivalence holds as follows
119878119894
119891119905119903= 119878119894
119891(119894 = 1 2 119899) (12)
Through (12) 119878119891119905119903
and 119878119891have one-to-one corresponding
relation Above all the complete and succinct boundaryformulation for DSSR is expressed as
ΩDSSR
=
1198611
1198612
119861119894
119861119899
=
1198781119891le min
1198781198871119865max minus 119878
1198871
119865 1198781198871119879max minus sum
119895isinΦ(1198871)
119878119895
119865minus sum
119896 =1119896isinΘ(1)
119878119896119891
1198782
119891le min
1198781198872
119865max minus 1198781198872
119865 1198781198872
119879max minus sum
119895isinΦ(1198872)
119878119895
119865minus sum
119896 =2119896isinΘ(2)
119878119896
119891
119878119894
119891le min
119878119887119894
119865max minus 119878119887119894
119865 119878119887119894
119879max minus sum
119895isinΦ(119887119894)
119878119895
119865minus sum
119896 =119894119896isinΘ(119894)
119878119896
119891
119878119899119891le min
119878119887119899119865max minus 119878
119887119899
119865 119878119887119899119879max minus sum
119895isinΦ(119887119899)
119878119895
119865minus sum
119896 =119899119896isinΘ(119899)
119878119896119891
(13)
With the form transformation ofDSSRmodel correspondingsymbols and their meaning should be adjusted too Thus 119887119894and Θ(119894) are used for new expression 119887119894 = the number ofback-feed feeders of 119878119894
119891 119878119887119894119865= all of the load supplied by the
back-feed feeder 119887119894 Θ(119894) = the set of transfer unit loads ofwhich both normal-feed and back-feed transformer are thesame as those of 119878119894
119891 It should be pointed out that feeder 1198871
feeder 119887119899 are not 119899 different actual feeders Take the case inFigure 2 as an example again when fault occurs the back-feed feeders of 1198783
119891and 1198784
119891are both feeder 1 which means
feeder b3 = feeder b4 = feeder 1 When any equality in (13)holds the operating point is just upon the DSSR boundaryWe denote the n boundaries by 119861
1 119861
119899
The DSSR boundary formulation can also reflect bothsubstation transformer and feeder N-1 secure constraintsTake 119894 subformula of (13) to concretely explain 119878119894
119891le
119878119887119894119865max minus 119878
119887119894
119865shows that transfer of 119878119894
119891cannot cause the feeder
overloading which ensures that the feeder N-1 is secure 119878119894119891le
119878119887119894119879max minus sum119895isinΦ(119887119894) 119878
119895
119865minus sum119896 =1119896isinΘ
(119894) 119878119896
119891shows that transfer of 119878119894
119891
cannot lead to the substation transformer overloading whichensures that the transformer N-1 is secure
5 Topological Characteristics of DSSR andMathematical Proof
The research on differential topological characteristics of theDSSR focuses on several aspects as follows
Journal of Applied Mathematics 5
(1) Are there any holes inside the DSSR that is thedenseness in terms of the topology
(2) Does the boundary of the DSSR have suspension(3) Could the boundary of the DSSR be expressed with
the union of several subsurfaces
The identical deformation from formulas (2)ndash(7) to (13) isa breakthrough for the research on topological characteristicsof the DSSR However to research on these characteristicsformula (13) should be further simplified to obtain themathematical essence Take 119894 subformula of (13) as researchobject the equivalent form is
119878119894
119891+ 119878119887119894
119865le 119878119887119894
119865max
119878119894
119891+ sum
119895isinΦ(119887119894)
119878119895
119865+ sum
119896 =1119896isinΘ(119894)
119878119896
119891le 119878119887119894
119879max(14)
According to formulas (3) and (12) 119878119887119894119865
and 119878119895119865
can beexpressed as linear combination of a series of differentelement ldquo119878
119891rdquo Then formula (14) will be simplified as
119899
sum119898=1
120572119898119878119898
119891le 119878119887119894
119865max
119899
sum119898=1
120573119898119878119898
119891le 119878119887119894
119879max
(15)
where 119878119887119894119865max and 119878119887119894
119879max are positive constants Throughformula (14) coefficients 120572 and 120573 have the following features
120572119898= 1 120573
119898= 1 (119898 = 119894)
120572119898= 0 or 1 120573
119898= 0 or 1 (119898 = 119894)
(16)
LetΩDSSR be a set which consists of operating pointW DSSRboundary formulation can be abstracted as
AW le C (17)
whereW is a vector defined as formula (1)Through formulas(15)-(16) matrices A and C should satisfy
A = [119886119894119895]2119899times119899
119894 ge 1 119895 ge 1
119886119894119895= 1 (119894 = 2119895 minus 1 or 119894 = 2119895)
119886119894119895= 0 or 1 (119894 = 2119895 minus 1 119894 = 2119895)
C = [1198881 1198882 1198882119899]119879
C isin R+
(18)
Because one integrated subformula is divided into equivalenttwo parts as is shown in formula (15) the dimension ofA is 2119899 times 119899 instead of 119899 times 119899 Here ΩDSSR is both linearspace and Euclidean space which has been proved in [17]Besides this important premise we should review somepreparation definitions and theorems about topology andEuclidean space including cluster point closure dense setand hyperplane
Definition 1 (cluster point) Let x be a point in Euclideanspace 119865 sub 119877 If there is a point sequence in set 119865 convergingto x then x is the cluster point of set 119865 [17]
Definition 2 (closure) 119860 is a subset of topological space 119860represents all the points inside 119860 and cluster points of 119860 andthen 119860 is called closure of 119860 [18]
Theorem 3 If the 119861 is a subset of 119860 119861 = 119860 then 119861 is dense in119860 [18]
Theorem 4 Let 119871 be a subspace of linear space 119877 x0 is a fixedvector which does not belong to 119871 generally Considering set119867which consists of the vector x x is obtained by
x = x0 + y (19)
where vector y varies in the whole subspace 119871 Thus119867 is calledhyperplane The dimension of119867 equals that of subspace 119871 [17]
The topological characteristics of DSSR are mathemati-cally proved in the following section
Characteristic 1 TheΩDSSR is dense inside
Proof First a new setΩ1015840DSSR is defined as the set of all pointsofΩDSSR except those on boundariesΩ1015840DSSR meets
AW lt C (20)
Select randomly a vector W = (1198781119891 1198782119891 119878119899
119891)119879 from Ω1015840DSSR
and then construct a sequenceY = Y1Y2 Ym as the
following formula
Y =
Y1= (1198781119891minus1198781119891
1 1198782119891minus1198782119891
1 119878119899
119891minus119878119899119891
1)
119879
= (0 0 0)119879
Y2= (1198781
119891minus1198781119891
2 1198782
119891minus1198782119891
2 119878
119899
119891minus119878119899119891
2)
119879
Y119898= (1198781119891minus1198781119891
119898 1198782119891minus1198782119891
119898 119878119899
119891minus119878119899119891
119898)
119879
= (1 minus1
119898)W
(21)
According to (20)-(21) we obtain
AYm = A(1 minus 1119898)W lt AW lt C (22)
Therefore sequence Y = Y1Y2 Ym is inside Ω1015840
DSSRLet 120588(WYm) be the distance betweenWandYm in Euclideanspace and then
Figure 3 Shapes of 2-dimension cross-section of DSSR
Thismeans that for any vectorW inΩ1015840DSSR sequenceY existsand converges toW andY is also insideΩ1015840DSSRTherefore anypoint insideΩ1015840DSSR is cluster point of Ω
1015840
DSSRSimilarly select vector Z = (1198781
119891 1198782119891 119878119899
119891)119879 which is just
upon boundary ofΩ119863119878119878119877
AZ = C (24)
Construct a sequence U = U1U2 Um
Um = (1 minus1
119898)Z (25)
Then we obtain
AUm = A(1 minus 1119898)Z lt AZ = C (26)
According to formula (26) sequence U is also inside Ω1015840DSSRThrough the following
We obtain that any point just upon the ΩDSSR boundary isalso cluster point ofΩ1015840DSSR Above all cluster points ofΩ
1015840
DSSR
include not only all the points inside Ω1015840DSSR but also all thepoints just upon the boundaries ofΩDSSRThus the followingformula holds
Ω1015840DSSR = ΩDSSR (28)
Through Theorem 3 Ω1015840DSSR is dense in ΩDSSR This proof iscompleted
Characteristic 2 The boundary ofΩDSSR has no suspension
Proof This characteristic can be approximately simplified toprove that any 2-dimension cross-section of ΩDSSR is convexpolygon For W = (1198781
119891 1198782119891 119878119899
119891)119879 select a 2-dimension
cross-section by taking 119878119898119891
and 119878119899119891as variables and 119878
119891in
another dimension as constants According to formula (16)the coefficient of 119878
119891(including variable 119878
119891and constant ones)
can be only 1 or 0 thus in process of dimension reduction 119878119898119891
and 119878119899119891are simply constrained by
119878119898
119891+ 119878119899
119891le 1198881
0 le 119878119898
119891le 1198882
0 le 119878119899
119891le 1198883
forall1198881 1198882 1198883isin 119877+
(29)
Journal of Applied Mathematics 7
Table 1 Profile of test case
Substation Transformer Voltages (kVkV) Transformer capacity (MVA) Number of feeders Total feeder capacity (MVA)
S1 T1 3510 400 9 8028T2 3510 400 9 8028
S2 T3 3510 400 9 8028T4 3510 400 8 7136
S3 T5 11010 630 10 892T6 11010 630 10 892
S4 T7 11010 630 10 892T8 11010 630 10 892
The shape of 2-dimension cross-section varies with the valuesof 1198881 1198882 and 119888
3 as is shown in Figure 3 The dashed oblique
line represents 119878119898119891+ 119878119899119891le 1198881
As is shown in Figure 3 the shapes of 2-dimension sectionare confined to rectangle triangle ladder and convex pen-tagon This means that any 2-dimension section of DSSR isconvex polygon which proves that the boundary ofΩDSSR hasno suspension in an indirect wayThe proof is completed
Characteristic 3 The ΩDSSR boundary can be expressed withseveral subsurfaces
Proof 119867 is a set of vector W = (1198781119891 1198782119891 119878119899
119891)119879 each of
which satisfies nonhomogeneous linear equation set
AW = C (30)
where matrices A and C meet formula (18) and W is theoperating point just upon the boundary of ΩDSSR Let W0 =
(120579(0)
1 120579(0)
2 120579(0)
119899)119879 be a fixed solution of equation set (31)
Meanwhile 119871 is a subspace of 119877 which consists of y =
(1205821 1205822 120582
119899)119879 that is y varies in the whole subspace 119871
The coordinates of ymeet the following homogeneous linearequation set
Ay = 0 (31)
If y = (1205821 1205822 120582
119899)119879 is a solution of (31) W = W0
+y = (120579(0)
1+ 1205821 120579(0)
2+ 1205822 120579(0)
119899+ 120582119899)119879 is obviously
a solution of equation set (30) that is W is included inset 119867 Meanwhile x0 is not in 119871 Through Theorem 3 119867is hyperplane The boundary of ΩDSSR can be expressedwith hyperplane that is several subsurfaces Because thedimension of 119871 is 119899 the dimension of hyperplane is also 119899This proof is completed
In transmission area topological characteristics of secu-rity region (SR) have been deeply studied [7] It has shownthat the boundary of SR has no suspension and is compactand approximately linear there is no hole inside the SR Theproof above demonstrates that characteristics of DSSR aresimilar but it has its own features
(a) DSSR is a convex set of denseness in terms ofthe topology This means that DSSR has no holesinside and its boundary has no suspension Basedon this feature dispatchers can analyze the securityof operating point by judging whether the point isinside theDSSRboundarywithout being afraid of anyinsecure point inside or losing security points
(b) DSSR boundary can be described by hyperplane (sev-eral Euclidean subsurfaces) meanwhile the bound-ary ismore linear than that of SR In practical applica-tion calculation of high dimensional security regionwill incur great computation burdens and consumelarge amount of time Therefore this characteristicwill improve the computational efficiency
6 Case Study
In order to verify the effectiveness of the proposedmodel twocases are illustrated in this section The first one is a test caseand the second is a practical case
61 Test Case
611 Overview of Test Case As is shown in Figure 4 the gridis comprised of 4 substations 8 substation transformers and75 feeders 68 feeders form 34 single loop networks and 7form two-tie connections resulting in 82 load transfer unitsThe total capacity of substations is 412MVA To illustrate themodeling process clearly each load transfer unit is numberedCase profile is shown in Table 1
The conductor type of all feeders is LGJ-185 (892MVA)This type is selected based on the national electrical code inChina
612 Boundary Formulation The test case grid has 75 feed-ers 14 feeders form 7 two-tie connections Each load of these7 feeders can be divided into 2 parts to transfer in differentback-feed path So the number of load transfer units is 82which means that the dimension of operation point W is82 and the number of subformulas of DSSR boundary is82 Based on the topology of the test case we substitutethe rated capacity of substation transformers and feedersinto 1198781
119865max 119878119899
119865max and 1198781119879max 119878
119899
119879max of formula (13)
8 Journal of Applied Mathematics
Then the boundary formulation for DSSR of test case can beexpressed as
Considering the similarity of calculation approach ofeach boundary and thesis length we present 5 subformulasof 1198781119891 1198785
119891here Complete formulation is given in the
appendix (in the Supplementary Material available online athttpdxdoiorg1011552014327078) Take the first subfor-mula in (32) as an example to explain the modeling processFigure 5 shows the local topology related with 1198781
119891 including
all relative transfer units in the first subformulaFirst analyze the topology and determine the parameters
of formula (13)We know that back-feed feeder of 1198781119891is feeder
36 and the back-feed substation transformer of 1198781119891is T5 In
normal state 11987836119891 11987845
119891and 11987876
119891 11987878
119891are all load of T5
which means sum119895isinΦ(1198871) 119878119895
119865= sum45
119895=36119878119895
119891+ sum78
119895=76119878119895
119891 11987836119891
and 11987876119891
are load of feeder 36 which means 1198781198871119865= 11987836119891+ 11987876119891 1198782119891 1198785
119891
are the load transferred fromT1 to T5when fault occurs at T1whichmeanssum
119896 =1119896isinΘ(1) 119878119896
119891= sum5
119896=2119878119896119891Therefore the formula
of B1 is preliminary expressed as
1198781
119891le min 11987836
119865max minus (11987876
119891+ 11987836
119891)
1198785
119879max minus (11987836
119891+ 11987837
119891+ 11987838
119891+ 11987839
119891+ 11987840
119891+ 11987841
11989111987842
119891
+11987843
119891+ 11987844
119891+ 11987845
119891+ 11987876
119891+ 11987877
119891+ 11987878
119891)
minus (1198782
119891+ 1198783
119891+ 1198784
119891+ 1198785
119891)
(33)
Second according to case profile substitute 11987836119865max =
892MVA and 1198785119879max = 63MVA into formula (33) and then
we obtain
1198781
119891le min 892 minus (11987876
119891+ 11987836
119891)
63 minus (11987836
119891+ 11987837
119891+ 11987838
119891+ 11987839
119891+ 11987840
119891+ 11987841
11989111987842
119891
+11987843
119891+ 11987844
119891+ 11987845
119891+ 11987876
119891+ 11987877
119891+ 11987878
119891)
minus (1198782
119891+ 1198783
119891+ 1198784
119891+ 1198785
119891)
(34)
613 Visualization and Topological Characteristics To visu-alize the shape of DSSR the dimension is reduced to 2 and 3which means that the coordinates of W should be constantsother than those to be visualized Thus an operating pointhas to be predetermined to provide the constants TSC is aspecial operating point just upon the security boundary [1]on which the facilities are usually fully utilizedThe operationof distribution system is closing to the boundaries and loadlevel is reaching TSCThus in this test case we select the TSCpoint as the predetermined operating point WTSC which isshown in Table 2The calculation approach for TSC has beenpresented in [15]
Choose randomly transfer unit loads 11987815119891
and 11987852119891
toobserve Substitute the rated capacity of transformers T1simT6capacity of all feeders and all transfer unit loads other than11987815119891
and 11987852119891 into formula (32) Then the 2D DSSR shown in
Figure 6 is obtainedAlthough the dimension is 2 the number of subformulas
which constrain the figure shape is not just 2 In fact eachof the subformulas which contains 11987815
119891or 11987852119891
should beconsidered But the subformulas withmore severe constraintswill cover the others and finally form theDSSR boundaryThefinal formulation of 2D figure is
0 le 11987815
119891le 52
0 le 11987852
119891le 3
(35)
Journal of Applied Mathematics 9
T1S1
T2
T3
T4
S 2
S4
T8
T6S3
T5
17-1
8
20-19
12ndash1
415
-16
21ndash23
41-4
3
36ndash3
8
39-4
044
-45
47-4
650
-51
52-5
3
54-5
548
-49
10-1
1
9-8
1 ndash3
4-5
6-7
76ndash7
8
79-80
58-5960-61
57-5665-62
66ndash6970-71
72ndash7524-25
29-28
26-27
30-31
32ndash35
81-8
2
Tie switch
T7
Distribution transformer
(i + 1) j are the same in the topologyin terms of calculation for DSSR
Note means that transfer units iindashj
Figure 4 Test case with eight substation transformers
Feeder 1
T1
T5
Feeder 2Feeder 36Feeder 37
S1f
S2f
S3f
S76f
S77f
S78f
S36f
S36f
S37f
S45f
S4f
S5f
middot middot middot
middot middot middot middot middot middot
Figure 5 Local topology related with f1
Similarly we choose randomly transfer unit loads 11987819119891 11987824119891
and 11987832119891
to observe and then the 3D DSSR is obtained as isshown in Figure 7
In Figure 6 it can be seen that the 2D DSSR is a denserectangle of which boundaries are linear without suspension
DSSR
0
1
2
3
1 2 3 4 52 S52f (MVA)
S15f (MVA)
Figure 6 Shape of 2D DSSR
In Figure 7 the 3D DSSR is surrounded by several subplanesforming a convex pentagon that is a special 3-dimensionhyperplane
Figure 8 The medium-voltage urban power grid of one city of southern China
62 Practical Case In this section the assessment andcontrol method based on DSSR is demonstrated on a realmedium-voltage distribution network of one city in south-ern China which consists of 12 substations 26 substationtransformers and 114 main feeders as is shown in Figure 8Total capacity of substation transformers is 10945MVA It isnotable that each two of all main feeders form a single loopnetwork in this case which leads to the fact that the numberof main feeders is equal to that of load transfer units
Paper [1] has presented concept and calculation methodof relative location of operating point in DSSR based onsubstation transformer contingency Similarly in this paperrelative location 119871
119894describes the distance from the operating
pointW to the boundary 119861119894(119894 = 1 2 119899)
119871119894= min
119878119887119894
119865max minus 119878119887119894
119865 119878119887119894
119879max minus sum
119895isinΦ(119887119894)
119878119895
119865minus sum
119896 =119894119896isinΘ(119894)
119878119896
119891
minus 119878119894
119891
(36)
119871119894is an index which is applied in security assessment
Here is an example Let W1 be an insecure operating pointEach transfer unit load of W1 is 08 times of TSC load inthe practical case except 1198781
119891and 11987810119891 while 1198781
119891= 8MVA and
11987810119891= 7MVA Total load of W1 is 5087MVA while TSC is
6289MVAAverage load rate is 046Thatmeans that current
Table 3 Distance to boundaries of 1198611and 119861
10atW1
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
operating point can be controlled back to security regionThrough formula (36) the location of a given operating pointin DSSR can be calculated We present the 119861
119894of which 119871
119894are
negative in Table 3Table 3 shows that the system should lose 3222MVA if
fault occurs at feeder 1 or feeder 10 so the system is locallyoverloading The location of W1 can be easily observed inFigure 9 To meet operational constraints under N-1 contin-gency 1198781
119891and 11987810
119891should be adjusted For example W1 is
adjusted toW1015840
1 through the dashed arrow shown in Figure 9The location ofW
1015840
1 is shown in Table 4
7 Conclusion and Further Work
This paper proposes a mathematical model for distributionsystem security region (DSSR) generic topological character-istics of DSSR are discussed and proved
First the operating point for a distribution system isdefined as the set of transfer unit loads which is a vector in the
12 Journal of Applied Mathematics
DSSR
0
2
4
6
2 4 6 8 10 11778
8
10
11778
S1f (MVA)
S10f (MVA)
W1(8 7)
W9984001 (7 4)
Figure 9 2D DSSR cross-section and preventive control
Table 4 Distance to boundaries of 1198611and 119861
10atW10158401
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
Euclidean space Then the DSSR model for both substationtransformer and feeder contingency is proposed which is theset of all operating points each ensures that the system N-1 issecure
Second to perform research on the characteristic ofDSSR the boundary formulation for DSSR is derived fromthe DSSR model Then three generic characteristics areproposed and proved by rigor mathematical approach whichare as follows (1) DSSR is dense inside (2) DSSR boundaryhas no suspension (3)DSSR boundary can be expressed withthe union of several subsurfaces
Finally the results from a test case and a practical casedemonstrate the effectiveness of the proposed model Thevisualization for DSSR boundary is also discussed to verifythe topological characteristics of DSSR Security assessmentmethod based on DSSR is preliminary exhibited to show theapplicability of DSSR for future smart distribution system
This paper improves the accuracy and understanding ofDSSR which is fundamental theoretic work for future smartdistribution system Further works to improve the accuracyof DSSR model include fully considering power flow voltageconstraints and integration of DGs
Conflict of Interests
The authors declare that there is no conflict of inter-ests regarding the publication of this paper None ofthe authors have a commercial interest financial interestandor other relationship with manufacturers of pharma-ceuticalslaboratory supplies andor medical devices or withcommercialproviders of medically related services
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (no 51277129) and National HighTechnology Research and Development Program of China(863 Program) (no 2011AA05A117)
References
[1] J Xiao W Gu C Wang and F Li ldquoDistribution systemsecurity region definition model and security assessmentrdquo IETGeneration TransmissionampDistribution vol 6 no 10 pp 1029ndash1035 2012
[2] State Grid Corporation of China Guidelines of Urban PowerNetwork Planning State Grid Corporation of China BeijingChina 2006 (Chinese)
[3] E Lakervi and E J Holmes Electricity Distribution NetworkDesign IET Peregrinus UK 1995
[4] K N Miu and H D Chiang ldquoService restoration for unbal-anced radial distribution systems with varying loads solutionalgorithmrdquo in Proceedings of the IEEE Power Engineering SocietySummer Meeting vol 1 pp 254ndash258 1999
[5] Y X Yu ldquoSecurity region of bulk power systemrdquo in Proceedingsof the International Conference on Power System Technology(PowerCon rsquo02) vol 1 pp 13ndash17 Kunming China October2002
[6] F Wu and S Kumagai ldquoSteady-state security regions of powersystemsrdquo IEEE Transactions on Circuits and Systems vol 29 no11 pp 703ndash711 1982
[7] Y Yu Y Zeng and F Feng ldquoDifferential topological character-istics of the DSR on injection space of electrical power systemrdquoScience in China E Technological Sciences vol 45 no 6 pp 576ndash584 2002
[8] S J Chen Q X Chen Q Xia and C Q Kang ldquoSteady-state security assessment method based on distance to securityregion boundariesrdquo IET Generation Transmission amp Distribu-tion vol 7 no 3 pp 288ndash297 2013
[9] D M Staszesky D Craig and C Befus ldquoAdvanced feederautomation is hererdquo IEEE Power and Energy Magazine vol 3no 5 pp 56ndash63 2005
[10] C L Smallwood and J Wennermark ldquoBenefits of distributionautomationrdquo IEEE Industry Applications Magazine vol 16 no1 pp 65ndash73 2010
[11] X Mamo S Mallet T Coste and S Grenard ldquoDistributionautomation the cornerstone for smart grid development strat-egyrdquo in Proceedings of the IEEE Power amp Energy Society GeneralMeeting (PES rsquo09) pp 1ndash6 July 2009
[12] P W Sauer B C Lesieutre and M A Pai ldquoMaximumloadability and voltage stability in power systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 15 no 3 pp145ndash153 1993
[13] K N Miu and H Chiang ldquoElectric distribution systemload capability problem formulation solution algorithm andnumerical resultsrdquo IEEE Transactions on Power Delivery vol 15no 1 pp 436ndash442 2000
[14] F Z Luo C S Wang J Xiao and S Ge ldquoRapid evaluationmethod for power supply capability of urban distribution sys-tem based on N-1 contingency analysis of main-transformersrdquoInternational Journal of Electrical Power andEnergy Systems vol32 no 10 pp 1063ndash1068 2010
Journal of Applied Mathematics 13
[15] J Xiao F X LiWZGuC SWang andP Zhang ldquoTotal supplycapability and its extended indices for distribution systemsdefinitionmodel calculation and applicationsrdquo IETGenerationTransmission and Distribution vol 5 no 8 pp 869ndash876 2011
[16] J Xiao X X Gong and C S Wang ldquoTopology properties andalgorithm of N-1 security boundary for smart gridrdquo Proceedingsof the CSEE vol 34 no 4 pp 545ndash554 2014 (Chinese)
[17] G E Shilov An Introduction to the Theory of Linear SpacesDover New York NY USA 1975
[18] J R Munkres Topology vol 2 Prentice Hall Upper SaddleRiver NJ USA 2000
(1) Are there any holes inside the DSSR that is thedenseness in terms of the topology
(2) Does the boundary of the DSSR have suspension(3) Could the boundary of the DSSR be expressed with
the union of several subsurfaces
The identical deformation from formulas (2)ndash(7) to (13) isa breakthrough for the research on topological characteristicsof the DSSR However to research on these characteristicsformula (13) should be further simplified to obtain themathematical essence Take 119894 subformula of (13) as researchobject the equivalent form is
119878119894
119891+ 119878119887119894
119865le 119878119887119894
119865max
119878119894
119891+ sum
119895isinΦ(119887119894)
119878119895
119865+ sum
119896 =1119896isinΘ(119894)
119878119896
119891le 119878119887119894
119879max(14)
According to formulas (3) and (12) 119878119887119894119865
and 119878119895119865
can beexpressed as linear combination of a series of differentelement ldquo119878
119891rdquo Then formula (14) will be simplified as
119899
sum119898=1
120572119898119878119898
119891le 119878119887119894
119865max
119899
sum119898=1
120573119898119878119898
119891le 119878119887119894
119879max
(15)
where 119878119887119894119865max and 119878119887119894
119879max are positive constants Throughformula (14) coefficients 120572 and 120573 have the following features
120572119898= 1 120573
119898= 1 (119898 = 119894)
120572119898= 0 or 1 120573
119898= 0 or 1 (119898 = 119894)
(16)
LetΩDSSR be a set which consists of operating pointW DSSRboundary formulation can be abstracted as
AW le C (17)
whereW is a vector defined as formula (1)Through formulas(15)-(16) matrices A and C should satisfy
A = [119886119894119895]2119899times119899
119894 ge 1 119895 ge 1
119886119894119895= 1 (119894 = 2119895 minus 1 or 119894 = 2119895)
119886119894119895= 0 or 1 (119894 = 2119895 minus 1 119894 = 2119895)
C = [1198881 1198882 1198882119899]119879
C isin R+
(18)
Because one integrated subformula is divided into equivalenttwo parts as is shown in formula (15) the dimension ofA is 2119899 times 119899 instead of 119899 times 119899 Here ΩDSSR is both linearspace and Euclidean space which has been proved in [17]Besides this important premise we should review somepreparation definitions and theorems about topology andEuclidean space including cluster point closure dense setand hyperplane
Definition 1 (cluster point) Let x be a point in Euclideanspace 119865 sub 119877 If there is a point sequence in set 119865 convergingto x then x is the cluster point of set 119865 [17]
Definition 2 (closure) 119860 is a subset of topological space 119860represents all the points inside 119860 and cluster points of 119860 andthen 119860 is called closure of 119860 [18]
Theorem 3 If the 119861 is a subset of 119860 119861 = 119860 then 119861 is dense in119860 [18]
Theorem 4 Let 119871 be a subspace of linear space 119877 x0 is a fixedvector which does not belong to 119871 generally Considering set119867which consists of the vector x x is obtained by
x = x0 + y (19)
where vector y varies in the whole subspace 119871 Thus119867 is calledhyperplane The dimension of119867 equals that of subspace 119871 [17]
The topological characteristics of DSSR are mathemati-cally proved in the following section
Characteristic 1 TheΩDSSR is dense inside
Proof First a new setΩ1015840DSSR is defined as the set of all pointsofΩDSSR except those on boundariesΩ1015840DSSR meets
AW lt C (20)
Select randomly a vector W = (1198781119891 1198782119891 119878119899
119891)119879 from Ω1015840DSSR
and then construct a sequenceY = Y1Y2 Ym as the
following formula
Y =
Y1= (1198781119891minus1198781119891
1 1198782119891minus1198782119891
1 119878119899
119891minus119878119899119891
1)
119879
= (0 0 0)119879
Y2= (1198781
119891minus1198781119891
2 1198782
119891minus1198782119891
2 119878
119899
119891minus119878119899119891
2)
119879
Y119898= (1198781119891minus1198781119891
119898 1198782119891minus1198782119891
119898 119878119899
119891minus119878119899119891
119898)
119879
= (1 minus1
119898)W
(21)
According to (20)-(21) we obtain
AYm = A(1 minus 1119898)W lt AW lt C (22)
Therefore sequence Y = Y1Y2 Ym is inside Ω1015840
DSSRLet 120588(WYm) be the distance betweenWandYm in Euclideanspace and then
Figure 3 Shapes of 2-dimension cross-section of DSSR
Thismeans that for any vectorW inΩ1015840DSSR sequenceY existsand converges toW andY is also insideΩ1015840DSSRTherefore anypoint insideΩ1015840DSSR is cluster point of Ω
1015840
DSSRSimilarly select vector Z = (1198781
119891 1198782119891 119878119899
119891)119879 which is just
upon boundary ofΩ119863119878119878119877
AZ = C (24)
Construct a sequence U = U1U2 Um
Um = (1 minus1
119898)Z (25)
Then we obtain
AUm = A(1 minus 1119898)Z lt AZ = C (26)
According to formula (26) sequence U is also inside Ω1015840DSSRThrough the following
We obtain that any point just upon the ΩDSSR boundary isalso cluster point ofΩ1015840DSSR Above all cluster points ofΩ
1015840
DSSR
include not only all the points inside Ω1015840DSSR but also all thepoints just upon the boundaries ofΩDSSRThus the followingformula holds
Ω1015840DSSR = ΩDSSR (28)
Through Theorem 3 Ω1015840DSSR is dense in ΩDSSR This proof iscompleted
Characteristic 2 The boundary ofΩDSSR has no suspension
Proof This characteristic can be approximately simplified toprove that any 2-dimension cross-section of ΩDSSR is convexpolygon For W = (1198781
119891 1198782119891 119878119899
119891)119879 select a 2-dimension
cross-section by taking 119878119898119891
and 119878119899119891as variables and 119878
119891in
another dimension as constants According to formula (16)the coefficient of 119878
119891(including variable 119878
119891and constant ones)
can be only 1 or 0 thus in process of dimension reduction 119878119898119891
and 119878119899119891are simply constrained by
119878119898
119891+ 119878119899
119891le 1198881
0 le 119878119898
119891le 1198882
0 le 119878119899
119891le 1198883
forall1198881 1198882 1198883isin 119877+
(29)
Journal of Applied Mathematics 7
Table 1 Profile of test case
Substation Transformer Voltages (kVkV) Transformer capacity (MVA) Number of feeders Total feeder capacity (MVA)
S1 T1 3510 400 9 8028T2 3510 400 9 8028
S2 T3 3510 400 9 8028T4 3510 400 8 7136
S3 T5 11010 630 10 892T6 11010 630 10 892
S4 T7 11010 630 10 892T8 11010 630 10 892
The shape of 2-dimension cross-section varies with the valuesof 1198881 1198882 and 119888
3 as is shown in Figure 3 The dashed oblique
line represents 119878119898119891+ 119878119899119891le 1198881
As is shown in Figure 3 the shapes of 2-dimension sectionare confined to rectangle triangle ladder and convex pen-tagon This means that any 2-dimension section of DSSR isconvex polygon which proves that the boundary ofΩDSSR hasno suspension in an indirect wayThe proof is completed
Characteristic 3 The ΩDSSR boundary can be expressed withseveral subsurfaces
Proof 119867 is a set of vector W = (1198781119891 1198782119891 119878119899
119891)119879 each of
which satisfies nonhomogeneous linear equation set
AW = C (30)
where matrices A and C meet formula (18) and W is theoperating point just upon the boundary of ΩDSSR Let W0 =
(120579(0)
1 120579(0)
2 120579(0)
119899)119879 be a fixed solution of equation set (31)
Meanwhile 119871 is a subspace of 119877 which consists of y =
(1205821 1205822 120582
119899)119879 that is y varies in the whole subspace 119871
The coordinates of ymeet the following homogeneous linearequation set
Ay = 0 (31)
If y = (1205821 1205822 120582
119899)119879 is a solution of (31) W = W0
+y = (120579(0)
1+ 1205821 120579(0)
2+ 1205822 120579(0)
119899+ 120582119899)119879 is obviously
a solution of equation set (30) that is W is included inset 119867 Meanwhile x0 is not in 119871 Through Theorem 3 119867is hyperplane The boundary of ΩDSSR can be expressedwith hyperplane that is several subsurfaces Because thedimension of 119871 is 119899 the dimension of hyperplane is also 119899This proof is completed
In transmission area topological characteristics of secu-rity region (SR) have been deeply studied [7] It has shownthat the boundary of SR has no suspension and is compactand approximately linear there is no hole inside the SR Theproof above demonstrates that characteristics of DSSR aresimilar but it has its own features
(a) DSSR is a convex set of denseness in terms ofthe topology This means that DSSR has no holesinside and its boundary has no suspension Basedon this feature dispatchers can analyze the securityof operating point by judging whether the point isinside theDSSRboundarywithout being afraid of anyinsecure point inside or losing security points
(b) DSSR boundary can be described by hyperplane (sev-eral Euclidean subsurfaces) meanwhile the bound-ary ismore linear than that of SR In practical applica-tion calculation of high dimensional security regionwill incur great computation burdens and consumelarge amount of time Therefore this characteristicwill improve the computational efficiency
6 Case Study
In order to verify the effectiveness of the proposedmodel twocases are illustrated in this section The first one is a test caseand the second is a practical case
61 Test Case
611 Overview of Test Case As is shown in Figure 4 the gridis comprised of 4 substations 8 substation transformers and75 feeders 68 feeders form 34 single loop networks and 7form two-tie connections resulting in 82 load transfer unitsThe total capacity of substations is 412MVA To illustrate themodeling process clearly each load transfer unit is numberedCase profile is shown in Table 1
The conductor type of all feeders is LGJ-185 (892MVA)This type is selected based on the national electrical code inChina
612 Boundary Formulation The test case grid has 75 feed-ers 14 feeders form 7 two-tie connections Each load of these7 feeders can be divided into 2 parts to transfer in differentback-feed path So the number of load transfer units is 82which means that the dimension of operation point W is82 and the number of subformulas of DSSR boundary is82 Based on the topology of the test case we substitutethe rated capacity of substation transformers and feedersinto 1198781
119865max 119878119899
119865max and 1198781119879max 119878
119899
119879max of formula (13)
8 Journal of Applied Mathematics
Then the boundary formulation for DSSR of test case can beexpressed as
Considering the similarity of calculation approach ofeach boundary and thesis length we present 5 subformulasof 1198781119891 1198785
119891here Complete formulation is given in the
appendix (in the Supplementary Material available online athttpdxdoiorg1011552014327078) Take the first subfor-mula in (32) as an example to explain the modeling processFigure 5 shows the local topology related with 1198781
119891 including
all relative transfer units in the first subformulaFirst analyze the topology and determine the parameters
of formula (13)We know that back-feed feeder of 1198781119891is feeder
36 and the back-feed substation transformer of 1198781119891is T5 In
normal state 11987836119891 11987845
119891and 11987876
119891 11987878
119891are all load of T5
which means sum119895isinΦ(1198871) 119878119895
119865= sum45
119895=36119878119895
119891+ sum78
119895=76119878119895
119891 11987836119891
and 11987876119891
are load of feeder 36 which means 1198781198871119865= 11987836119891+ 11987876119891 1198782119891 1198785
119891
are the load transferred fromT1 to T5when fault occurs at T1whichmeanssum
119896 =1119896isinΘ(1) 119878119896
119891= sum5
119896=2119878119896119891Therefore the formula
of B1 is preliminary expressed as
1198781
119891le min 11987836
119865max minus (11987876
119891+ 11987836
119891)
1198785
119879max minus (11987836
119891+ 11987837
119891+ 11987838
119891+ 11987839
119891+ 11987840
119891+ 11987841
11989111987842
119891
+11987843
119891+ 11987844
119891+ 11987845
119891+ 11987876
119891+ 11987877
119891+ 11987878
119891)
minus (1198782
119891+ 1198783
119891+ 1198784
119891+ 1198785
119891)
(33)
Second according to case profile substitute 11987836119865max =
892MVA and 1198785119879max = 63MVA into formula (33) and then
we obtain
1198781
119891le min 892 minus (11987876
119891+ 11987836
119891)
63 minus (11987836
119891+ 11987837
119891+ 11987838
119891+ 11987839
119891+ 11987840
119891+ 11987841
11989111987842
119891
+11987843
119891+ 11987844
119891+ 11987845
119891+ 11987876
119891+ 11987877
119891+ 11987878
119891)
minus (1198782
119891+ 1198783
119891+ 1198784
119891+ 1198785
119891)
(34)
613 Visualization and Topological Characteristics To visu-alize the shape of DSSR the dimension is reduced to 2 and 3which means that the coordinates of W should be constantsother than those to be visualized Thus an operating pointhas to be predetermined to provide the constants TSC is aspecial operating point just upon the security boundary [1]on which the facilities are usually fully utilizedThe operationof distribution system is closing to the boundaries and loadlevel is reaching TSCThus in this test case we select the TSCpoint as the predetermined operating point WTSC which isshown in Table 2The calculation approach for TSC has beenpresented in [15]
Choose randomly transfer unit loads 11987815119891
and 11987852119891
toobserve Substitute the rated capacity of transformers T1simT6capacity of all feeders and all transfer unit loads other than11987815119891
and 11987852119891 into formula (32) Then the 2D DSSR shown in
Figure 6 is obtainedAlthough the dimension is 2 the number of subformulas
which constrain the figure shape is not just 2 In fact eachof the subformulas which contains 11987815
119891or 11987852119891
should beconsidered But the subformulas withmore severe constraintswill cover the others and finally form theDSSR boundaryThefinal formulation of 2D figure is
0 le 11987815
119891le 52
0 le 11987852
119891le 3
(35)
Journal of Applied Mathematics 9
T1S1
T2
T3
T4
S 2
S4
T8
T6S3
T5
17-1
8
20-19
12ndash1
415
-16
21ndash23
41-4
3
36ndash3
8
39-4
044
-45
47-4
650
-51
52-5
3
54-5
548
-49
10-1
1
9-8
1 ndash3
4-5
6-7
76ndash7
8
79-80
58-5960-61
57-5665-62
66ndash6970-71
72ndash7524-25
29-28
26-27
30-31
32ndash35
81-8
2
Tie switch
T7
Distribution transformer
(i + 1) j are the same in the topologyin terms of calculation for DSSR
Note means that transfer units iindashj
Figure 4 Test case with eight substation transformers
Feeder 1
T1
T5
Feeder 2Feeder 36Feeder 37
S1f
S2f
S3f
S76f
S77f
S78f
S36f
S36f
S37f
S45f
S4f
S5f
middot middot middot
middot middot middot middot middot middot
Figure 5 Local topology related with f1
Similarly we choose randomly transfer unit loads 11987819119891 11987824119891
and 11987832119891
to observe and then the 3D DSSR is obtained as isshown in Figure 7
In Figure 6 it can be seen that the 2D DSSR is a denserectangle of which boundaries are linear without suspension
DSSR
0
1
2
3
1 2 3 4 52 S52f (MVA)
S15f (MVA)
Figure 6 Shape of 2D DSSR
In Figure 7 the 3D DSSR is surrounded by several subplanesforming a convex pentagon that is a special 3-dimensionhyperplane
Figure 8 The medium-voltage urban power grid of one city of southern China
62 Practical Case In this section the assessment andcontrol method based on DSSR is demonstrated on a realmedium-voltage distribution network of one city in south-ern China which consists of 12 substations 26 substationtransformers and 114 main feeders as is shown in Figure 8Total capacity of substation transformers is 10945MVA It isnotable that each two of all main feeders form a single loopnetwork in this case which leads to the fact that the numberof main feeders is equal to that of load transfer units
Paper [1] has presented concept and calculation methodof relative location of operating point in DSSR based onsubstation transformer contingency Similarly in this paperrelative location 119871
119894describes the distance from the operating
pointW to the boundary 119861119894(119894 = 1 2 119899)
119871119894= min
119878119887119894
119865max minus 119878119887119894
119865 119878119887119894
119879max minus sum
119895isinΦ(119887119894)
119878119895
119865minus sum
119896 =119894119896isinΘ(119894)
119878119896
119891
minus 119878119894
119891
(36)
119871119894is an index which is applied in security assessment
Here is an example Let W1 be an insecure operating pointEach transfer unit load of W1 is 08 times of TSC load inthe practical case except 1198781
119891and 11987810119891 while 1198781
119891= 8MVA and
11987810119891= 7MVA Total load of W1 is 5087MVA while TSC is
6289MVAAverage load rate is 046Thatmeans that current
Table 3 Distance to boundaries of 1198611and 119861
10atW1
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
operating point can be controlled back to security regionThrough formula (36) the location of a given operating pointin DSSR can be calculated We present the 119861
119894of which 119871
119894are
negative in Table 3Table 3 shows that the system should lose 3222MVA if
fault occurs at feeder 1 or feeder 10 so the system is locallyoverloading The location of W1 can be easily observed inFigure 9 To meet operational constraints under N-1 contin-gency 1198781
119891and 11987810
119891should be adjusted For example W1 is
adjusted toW1015840
1 through the dashed arrow shown in Figure 9The location ofW
1015840
1 is shown in Table 4
7 Conclusion and Further Work
This paper proposes a mathematical model for distributionsystem security region (DSSR) generic topological character-istics of DSSR are discussed and proved
First the operating point for a distribution system isdefined as the set of transfer unit loads which is a vector in the
12 Journal of Applied Mathematics
DSSR
0
2
4
6
2 4 6 8 10 11778
8
10
11778
S1f (MVA)
S10f (MVA)
W1(8 7)
W9984001 (7 4)
Figure 9 2D DSSR cross-section and preventive control
Table 4 Distance to boundaries of 1198611and 119861
10atW10158401
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
Euclidean space Then the DSSR model for both substationtransformer and feeder contingency is proposed which is theset of all operating points each ensures that the system N-1 issecure
Second to perform research on the characteristic ofDSSR the boundary formulation for DSSR is derived fromthe DSSR model Then three generic characteristics areproposed and proved by rigor mathematical approach whichare as follows (1) DSSR is dense inside (2) DSSR boundaryhas no suspension (3)DSSR boundary can be expressed withthe union of several subsurfaces
Finally the results from a test case and a practical casedemonstrate the effectiveness of the proposed model Thevisualization for DSSR boundary is also discussed to verifythe topological characteristics of DSSR Security assessmentmethod based on DSSR is preliminary exhibited to show theapplicability of DSSR for future smart distribution system
This paper improves the accuracy and understanding ofDSSR which is fundamental theoretic work for future smartdistribution system Further works to improve the accuracyof DSSR model include fully considering power flow voltageconstraints and integration of DGs
Conflict of Interests
The authors declare that there is no conflict of inter-ests regarding the publication of this paper None ofthe authors have a commercial interest financial interestandor other relationship with manufacturers of pharma-ceuticalslaboratory supplies andor medical devices or withcommercialproviders of medically related services
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (no 51277129) and National HighTechnology Research and Development Program of China(863 Program) (no 2011AA05A117)
References
[1] J Xiao W Gu C Wang and F Li ldquoDistribution systemsecurity region definition model and security assessmentrdquo IETGeneration TransmissionampDistribution vol 6 no 10 pp 1029ndash1035 2012
[2] State Grid Corporation of China Guidelines of Urban PowerNetwork Planning State Grid Corporation of China BeijingChina 2006 (Chinese)
[3] E Lakervi and E J Holmes Electricity Distribution NetworkDesign IET Peregrinus UK 1995
[4] K N Miu and H D Chiang ldquoService restoration for unbal-anced radial distribution systems with varying loads solutionalgorithmrdquo in Proceedings of the IEEE Power Engineering SocietySummer Meeting vol 1 pp 254ndash258 1999
[5] Y X Yu ldquoSecurity region of bulk power systemrdquo in Proceedingsof the International Conference on Power System Technology(PowerCon rsquo02) vol 1 pp 13ndash17 Kunming China October2002
[6] F Wu and S Kumagai ldquoSteady-state security regions of powersystemsrdquo IEEE Transactions on Circuits and Systems vol 29 no11 pp 703ndash711 1982
[7] Y Yu Y Zeng and F Feng ldquoDifferential topological character-istics of the DSR on injection space of electrical power systemrdquoScience in China E Technological Sciences vol 45 no 6 pp 576ndash584 2002
[8] S J Chen Q X Chen Q Xia and C Q Kang ldquoSteady-state security assessment method based on distance to securityregion boundariesrdquo IET Generation Transmission amp Distribu-tion vol 7 no 3 pp 288ndash297 2013
[9] D M Staszesky D Craig and C Befus ldquoAdvanced feederautomation is hererdquo IEEE Power and Energy Magazine vol 3no 5 pp 56ndash63 2005
[10] C L Smallwood and J Wennermark ldquoBenefits of distributionautomationrdquo IEEE Industry Applications Magazine vol 16 no1 pp 65ndash73 2010
[11] X Mamo S Mallet T Coste and S Grenard ldquoDistributionautomation the cornerstone for smart grid development strat-egyrdquo in Proceedings of the IEEE Power amp Energy Society GeneralMeeting (PES rsquo09) pp 1ndash6 July 2009
[12] P W Sauer B C Lesieutre and M A Pai ldquoMaximumloadability and voltage stability in power systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 15 no 3 pp145ndash153 1993
[13] K N Miu and H Chiang ldquoElectric distribution systemload capability problem formulation solution algorithm andnumerical resultsrdquo IEEE Transactions on Power Delivery vol 15no 1 pp 436ndash442 2000
[14] F Z Luo C S Wang J Xiao and S Ge ldquoRapid evaluationmethod for power supply capability of urban distribution sys-tem based on N-1 contingency analysis of main-transformersrdquoInternational Journal of Electrical Power andEnergy Systems vol32 no 10 pp 1063ndash1068 2010
Journal of Applied Mathematics 13
[15] J Xiao F X LiWZGuC SWang andP Zhang ldquoTotal supplycapability and its extended indices for distribution systemsdefinitionmodel calculation and applicationsrdquo IETGenerationTransmission and Distribution vol 5 no 8 pp 869ndash876 2011
[16] J Xiao X X Gong and C S Wang ldquoTopology properties andalgorithm of N-1 security boundary for smart gridrdquo Proceedingsof the CSEE vol 34 no 4 pp 545ndash554 2014 (Chinese)
[17] G E Shilov An Introduction to the Theory of Linear SpacesDover New York NY USA 1975
[18] J R Munkres Topology vol 2 Prentice Hall Upper SaddleRiver NJ USA 2000
Figure 3 Shapes of 2-dimension cross-section of DSSR
Thismeans that for any vectorW inΩ1015840DSSR sequenceY existsand converges toW andY is also insideΩ1015840DSSRTherefore anypoint insideΩ1015840DSSR is cluster point of Ω
1015840
DSSRSimilarly select vector Z = (1198781
119891 1198782119891 119878119899
119891)119879 which is just
upon boundary ofΩ119863119878119878119877
AZ = C (24)
Construct a sequence U = U1U2 Um
Um = (1 minus1
119898)Z (25)
Then we obtain
AUm = A(1 minus 1119898)Z lt AZ = C (26)
According to formula (26) sequence U is also inside Ω1015840DSSRThrough the following
We obtain that any point just upon the ΩDSSR boundary isalso cluster point ofΩ1015840DSSR Above all cluster points ofΩ
1015840
DSSR
include not only all the points inside Ω1015840DSSR but also all thepoints just upon the boundaries ofΩDSSRThus the followingformula holds
Ω1015840DSSR = ΩDSSR (28)
Through Theorem 3 Ω1015840DSSR is dense in ΩDSSR This proof iscompleted
Characteristic 2 The boundary ofΩDSSR has no suspension
Proof This characteristic can be approximately simplified toprove that any 2-dimension cross-section of ΩDSSR is convexpolygon For W = (1198781
119891 1198782119891 119878119899
119891)119879 select a 2-dimension
cross-section by taking 119878119898119891
and 119878119899119891as variables and 119878
119891in
another dimension as constants According to formula (16)the coefficient of 119878
119891(including variable 119878
119891and constant ones)
can be only 1 or 0 thus in process of dimension reduction 119878119898119891
and 119878119899119891are simply constrained by
119878119898
119891+ 119878119899
119891le 1198881
0 le 119878119898
119891le 1198882
0 le 119878119899
119891le 1198883
forall1198881 1198882 1198883isin 119877+
(29)
Journal of Applied Mathematics 7
Table 1 Profile of test case
Substation Transformer Voltages (kVkV) Transformer capacity (MVA) Number of feeders Total feeder capacity (MVA)
S1 T1 3510 400 9 8028T2 3510 400 9 8028
S2 T3 3510 400 9 8028T4 3510 400 8 7136
S3 T5 11010 630 10 892T6 11010 630 10 892
S4 T7 11010 630 10 892T8 11010 630 10 892
The shape of 2-dimension cross-section varies with the valuesof 1198881 1198882 and 119888
3 as is shown in Figure 3 The dashed oblique
line represents 119878119898119891+ 119878119899119891le 1198881
As is shown in Figure 3 the shapes of 2-dimension sectionare confined to rectangle triangle ladder and convex pen-tagon This means that any 2-dimension section of DSSR isconvex polygon which proves that the boundary ofΩDSSR hasno suspension in an indirect wayThe proof is completed
Characteristic 3 The ΩDSSR boundary can be expressed withseveral subsurfaces
Proof 119867 is a set of vector W = (1198781119891 1198782119891 119878119899
119891)119879 each of
which satisfies nonhomogeneous linear equation set
AW = C (30)
where matrices A and C meet formula (18) and W is theoperating point just upon the boundary of ΩDSSR Let W0 =
(120579(0)
1 120579(0)
2 120579(0)
119899)119879 be a fixed solution of equation set (31)
Meanwhile 119871 is a subspace of 119877 which consists of y =
(1205821 1205822 120582
119899)119879 that is y varies in the whole subspace 119871
The coordinates of ymeet the following homogeneous linearequation set
Ay = 0 (31)
If y = (1205821 1205822 120582
119899)119879 is a solution of (31) W = W0
+y = (120579(0)
1+ 1205821 120579(0)
2+ 1205822 120579(0)
119899+ 120582119899)119879 is obviously
a solution of equation set (30) that is W is included inset 119867 Meanwhile x0 is not in 119871 Through Theorem 3 119867is hyperplane The boundary of ΩDSSR can be expressedwith hyperplane that is several subsurfaces Because thedimension of 119871 is 119899 the dimension of hyperplane is also 119899This proof is completed
In transmission area topological characteristics of secu-rity region (SR) have been deeply studied [7] It has shownthat the boundary of SR has no suspension and is compactand approximately linear there is no hole inside the SR Theproof above demonstrates that characteristics of DSSR aresimilar but it has its own features
(a) DSSR is a convex set of denseness in terms ofthe topology This means that DSSR has no holesinside and its boundary has no suspension Basedon this feature dispatchers can analyze the securityof operating point by judging whether the point isinside theDSSRboundarywithout being afraid of anyinsecure point inside or losing security points
(b) DSSR boundary can be described by hyperplane (sev-eral Euclidean subsurfaces) meanwhile the bound-ary ismore linear than that of SR In practical applica-tion calculation of high dimensional security regionwill incur great computation burdens and consumelarge amount of time Therefore this characteristicwill improve the computational efficiency
6 Case Study
In order to verify the effectiveness of the proposedmodel twocases are illustrated in this section The first one is a test caseand the second is a practical case
61 Test Case
611 Overview of Test Case As is shown in Figure 4 the gridis comprised of 4 substations 8 substation transformers and75 feeders 68 feeders form 34 single loop networks and 7form two-tie connections resulting in 82 load transfer unitsThe total capacity of substations is 412MVA To illustrate themodeling process clearly each load transfer unit is numberedCase profile is shown in Table 1
The conductor type of all feeders is LGJ-185 (892MVA)This type is selected based on the national electrical code inChina
612 Boundary Formulation The test case grid has 75 feed-ers 14 feeders form 7 two-tie connections Each load of these7 feeders can be divided into 2 parts to transfer in differentback-feed path So the number of load transfer units is 82which means that the dimension of operation point W is82 and the number of subformulas of DSSR boundary is82 Based on the topology of the test case we substitutethe rated capacity of substation transformers and feedersinto 1198781
119865max 119878119899
119865max and 1198781119879max 119878
119899
119879max of formula (13)
8 Journal of Applied Mathematics
Then the boundary formulation for DSSR of test case can beexpressed as
Considering the similarity of calculation approach ofeach boundary and thesis length we present 5 subformulasof 1198781119891 1198785
119891here Complete formulation is given in the
appendix (in the Supplementary Material available online athttpdxdoiorg1011552014327078) Take the first subfor-mula in (32) as an example to explain the modeling processFigure 5 shows the local topology related with 1198781
119891 including
all relative transfer units in the first subformulaFirst analyze the topology and determine the parameters
of formula (13)We know that back-feed feeder of 1198781119891is feeder
36 and the back-feed substation transformer of 1198781119891is T5 In
normal state 11987836119891 11987845
119891and 11987876
119891 11987878
119891are all load of T5
which means sum119895isinΦ(1198871) 119878119895
119865= sum45
119895=36119878119895
119891+ sum78
119895=76119878119895
119891 11987836119891
and 11987876119891
are load of feeder 36 which means 1198781198871119865= 11987836119891+ 11987876119891 1198782119891 1198785
119891
are the load transferred fromT1 to T5when fault occurs at T1whichmeanssum
119896 =1119896isinΘ(1) 119878119896
119891= sum5
119896=2119878119896119891Therefore the formula
of B1 is preliminary expressed as
1198781
119891le min 11987836
119865max minus (11987876
119891+ 11987836
119891)
1198785
119879max minus (11987836
119891+ 11987837
119891+ 11987838
119891+ 11987839
119891+ 11987840
119891+ 11987841
11989111987842
119891
+11987843
119891+ 11987844
119891+ 11987845
119891+ 11987876
119891+ 11987877
119891+ 11987878
119891)
minus (1198782
119891+ 1198783
119891+ 1198784
119891+ 1198785
119891)
(33)
Second according to case profile substitute 11987836119865max =
892MVA and 1198785119879max = 63MVA into formula (33) and then
we obtain
1198781
119891le min 892 minus (11987876
119891+ 11987836
119891)
63 minus (11987836
119891+ 11987837
119891+ 11987838
119891+ 11987839
119891+ 11987840
119891+ 11987841
11989111987842
119891
+11987843
119891+ 11987844
119891+ 11987845
119891+ 11987876
119891+ 11987877
119891+ 11987878
119891)
minus (1198782
119891+ 1198783
119891+ 1198784
119891+ 1198785
119891)
(34)
613 Visualization and Topological Characteristics To visu-alize the shape of DSSR the dimension is reduced to 2 and 3which means that the coordinates of W should be constantsother than those to be visualized Thus an operating pointhas to be predetermined to provide the constants TSC is aspecial operating point just upon the security boundary [1]on which the facilities are usually fully utilizedThe operationof distribution system is closing to the boundaries and loadlevel is reaching TSCThus in this test case we select the TSCpoint as the predetermined operating point WTSC which isshown in Table 2The calculation approach for TSC has beenpresented in [15]
Choose randomly transfer unit loads 11987815119891
and 11987852119891
toobserve Substitute the rated capacity of transformers T1simT6capacity of all feeders and all transfer unit loads other than11987815119891
and 11987852119891 into formula (32) Then the 2D DSSR shown in
Figure 6 is obtainedAlthough the dimension is 2 the number of subformulas
which constrain the figure shape is not just 2 In fact eachof the subformulas which contains 11987815
119891or 11987852119891
should beconsidered But the subformulas withmore severe constraintswill cover the others and finally form theDSSR boundaryThefinal formulation of 2D figure is
0 le 11987815
119891le 52
0 le 11987852
119891le 3
(35)
Journal of Applied Mathematics 9
T1S1
T2
T3
T4
S 2
S4
T8
T6S3
T5
17-1
8
20-19
12ndash1
415
-16
21ndash23
41-4
3
36ndash3
8
39-4
044
-45
47-4
650
-51
52-5
3
54-5
548
-49
10-1
1
9-8
1 ndash3
4-5
6-7
76ndash7
8
79-80
58-5960-61
57-5665-62
66ndash6970-71
72ndash7524-25
29-28
26-27
30-31
32ndash35
81-8
2
Tie switch
T7
Distribution transformer
(i + 1) j are the same in the topologyin terms of calculation for DSSR
Note means that transfer units iindashj
Figure 4 Test case with eight substation transformers
Feeder 1
T1
T5
Feeder 2Feeder 36Feeder 37
S1f
S2f
S3f
S76f
S77f
S78f
S36f
S36f
S37f
S45f
S4f
S5f
middot middot middot
middot middot middot middot middot middot
Figure 5 Local topology related with f1
Similarly we choose randomly transfer unit loads 11987819119891 11987824119891
and 11987832119891
to observe and then the 3D DSSR is obtained as isshown in Figure 7
In Figure 6 it can be seen that the 2D DSSR is a denserectangle of which boundaries are linear without suspension
DSSR
0
1
2
3
1 2 3 4 52 S52f (MVA)
S15f (MVA)
Figure 6 Shape of 2D DSSR
In Figure 7 the 3D DSSR is surrounded by several subplanesforming a convex pentagon that is a special 3-dimensionhyperplane
Figure 8 The medium-voltage urban power grid of one city of southern China
62 Practical Case In this section the assessment andcontrol method based on DSSR is demonstrated on a realmedium-voltage distribution network of one city in south-ern China which consists of 12 substations 26 substationtransformers and 114 main feeders as is shown in Figure 8Total capacity of substation transformers is 10945MVA It isnotable that each two of all main feeders form a single loopnetwork in this case which leads to the fact that the numberof main feeders is equal to that of load transfer units
Paper [1] has presented concept and calculation methodof relative location of operating point in DSSR based onsubstation transformer contingency Similarly in this paperrelative location 119871
119894describes the distance from the operating
pointW to the boundary 119861119894(119894 = 1 2 119899)
119871119894= min
119878119887119894
119865max minus 119878119887119894
119865 119878119887119894
119879max minus sum
119895isinΦ(119887119894)
119878119895
119865minus sum
119896 =119894119896isinΘ(119894)
119878119896
119891
minus 119878119894
119891
(36)
119871119894is an index which is applied in security assessment
Here is an example Let W1 be an insecure operating pointEach transfer unit load of W1 is 08 times of TSC load inthe practical case except 1198781
119891and 11987810119891 while 1198781
119891= 8MVA and
11987810119891= 7MVA Total load of W1 is 5087MVA while TSC is
6289MVAAverage load rate is 046Thatmeans that current
Table 3 Distance to boundaries of 1198611and 119861
10atW1
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
operating point can be controlled back to security regionThrough formula (36) the location of a given operating pointin DSSR can be calculated We present the 119861
119894of which 119871
119894are
negative in Table 3Table 3 shows that the system should lose 3222MVA if
fault occurs at feeder 1 or feeder 10 so the system is locallyoverloading The location of W1 can be easily observed inFigure 9 To meet operational constraints under N-1 contin-gency 1198781
119891and 11987810
119891should be adjusted For example W1 is
adjusted toW1015840
1 through the dashed arrow shown in Figure 9The location ofW
1015840
1 is shown in Table 4
7 Conclusion and Further Work
This paper proposes a mathematical model for distributionsystem security region (DSSR) generic topological character-istics of DSSR are discussed and proved
First the operating point for a distribution system isdefined as the set of transfer unit loads which is a vector in the
12 Journal of Applied Mathematics
DSSR
0
2
4
6
2 4 6 8 10 11778
8
10
11778
S1f (MVA)
S10f (MVA)
W1(8 7)
W9984001 (7 4)
Figure 9 2D DSSR cross-section and preventive control
Table 4 Distance to boundaries of 1198611and 119861
10atW10158401
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
Euclidean space Then the DSSR model for both substationtransformer and feeder contingency is proposed which is theset of all operating points each ensures that the system N-1 issecure
Second to perform research on the characteristic ofDSSR the boundary formulation for DSSR is derived fromthe DSSR model Then three generic characteristics areproposed and proved by rigor mathematical approach whichare as follows (1) DSSR is dense inside (2) DSSR boundaryhas no suspension (3)DSSR boundary can be expressed withthe union of several subsurfaces
Finally the results from a test case and a practical casedemonstrate the effectiveness of the proposed model Thevisualization for DSSR boundary is also discussed to verifythe topological characteristics of DSSR Security assessmentmethod based on DSSR is preliminary exhibited to show theapplicability of DSSR for future smart distribution system
This paper improves the accuracy and understanding ofDSSR which is fundamental theoretic work for future smartdistribution system Further works to improve the accuracyof DSSR model include fully considering power flow voltageconstraints and integration of DGs
Conflict of Interests
The authors declare that there is no conflict of inter-ests regarding the publication of this paper None ofthe authors have a commercial interest financial interestandor other relationship with manufacturers of pharma-ceuticalslaboratory supplies andor medical devices or withcommercialproviders of medically related services
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (no 51277129) and National HighTechnology Research and Development Program of China(863 Program) (no 2011AA05A117)
References
[1] J Xiao W Gu C Wang and F Li ldquoDistribution systemsecurity region definition model and security assessmentrdquo IETGeneration TransmissionampDistribution vol 6 no 10 pp 1029ndash1035 2012
[2] State Grid Corporation of China Guidelines of Urban PowerNetwork Planning State Grid Corporation of China BeijingChina 2006 (Chinese)
[3] E Lakervi and E J Holmes Electricity Distribution NetworkDesign IET Peregrinus UK 1995
[4] K N Miu and H D Chiang ldquoService restoration for unbal-anced radial distribution systems with varying loads solutionalgorithmrdquo in Proceedings of the IEEE Power Engineering SocietySummer Meeting vol 1 pp 254ndash258 1999
[5] Y X Yu ldquoSecurity region of bulk power systemrdquo in Proceedingsof the International Conference on Power System Technology(PowerCon rsquo02) vol 1 pp 13ndash17 Kunming China October2002
[6] F Wu and S Kumagai ldquoSteady-state security regions of powersystemsrdquo IEEE Transactions on Circuits and Systems vol 29 no11 pp 703ndash711 1982
[7] Y Yu Y Zeng and F Feng ldquoDifferential topological character-istics of the DSR on injection space of electrical power systemrdquoScience in China E Technological Sciences vol 45 no 6 pp 576ndash584 2002
[8] S J Chen Q X Chen Q Xia and C Q Kang ldquoSteady-state security assessment method based on distance to securityregion boundariesrdquo IET Generation Transmission amp Distribu-tion vol 7 no 3 pp 288ndash297 2013
[9] D M Staszesky D Craig and C Befus ldquoAdvanced feederautomation is hererdquo IEEE Power and Energy Magazine vol 3no 5 pp 56ndash63 2005
[10] C L Smallwood and J Wennermark ldquoBenefits of distributionautomationrdquo IEEE Industry Applications Magazine vol 16 no1 pp 65ndash73 2010
[11] X Mamo S Mallet T Coste and S Grenard ldquoDistributionautomation the cornerstone for smart grid development strat-egyrdquo in Proceedings of the IEEE Power amp Energy Society GeneralMeeting (PES rsquo09) pp 1ndash6 July 2009
[12] P W Sauer B C Lesieutre and M A Pai ldquoMaximumloadability and voltage stability in power systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 15 no 3 pp145ndash153 1993
[13] K N Miu and H Chiang ldquoElectric distribution systemload capability problem formulation solution algorithm andnumerical resultsrdquo IEEE Transactions on Power Delivery vol 15no 1 pp 436ndash442 2000
[14] F Z Luo C S Wang J Xiao and S Ge ldquoRapid evaluationmethod for power supply capability of urban distribution sys-tem based on N-1 contingency analysis of main-transformersrdquoInternational Journal of Electrical Power andEnergy Systems vol32 no 10 pp 1063ndash1068 2010
Journal of Applied Mathematics 13
[15] J Xiao F X LiWZGuC SWang andP Zhang ldquoTotal supplycapability and its extended indices for distribution systemsdefinitionmodel calculation and applicationsrdquo IETGenerationTransmission and Distribution vol 5 no 8 pp 869ndash876 2011
[16] J Xiao X X Gong and C S Wang ldquoTopology properties andalgorithm of N-1 security boundary for smart gridrdquo Proceedingsof the CSEE vol 34 no 4 pp 545ndash554 2014 (Chinese)
[17] G E Shilov An Introduction to the Theory of Linear SpacesDover New York NY USA 1975
[18] J R Munkres Topology vol 2 Prentice Hall Upper SaddleRiver NJ USA 2000
Substation Transformer Voltages (kVkV) Transformer capacity (MVA) Number of feeders Total feeder capacity (MVA)
S1 T1 3510 400 9 8028T2 3510 400 9 8028
S2 T3 3510 400 9 8028T4 3510 400 8 7136
S3 T5 11010 630 10 892T6 11010 630 10 892
S4 T7 11010 630 10 892T8 11010 630 10 892
The shape of 2-dimension cross-section varies with the valuesof 1198881 1198882 and 119888
3 as is shown in Figure 3 The dashed oblique
line represents 119878119898119891+ 119878119899119891le 1198881
As is shown in Figure 3 the shapes of 2-dimension sectionare confined to rectangle triangle ladder and convex pen-tagon This means that any 2-dimension section of DSSR isconvex polygon which proves that the boundary ofΩDSSR hasno suspension in an indirect wayThe proof is completed
Characteristic 3 The ΩDSSR boundary can be expressed withseveral subsurfaces
Proof 119867 is a set of vector W = (1198781119891 1198782119891 119878119899
119891)119879 each of
which satisfies nonhomogeneous linear equation set
AW = C (30)
where matrices A and C meet formula (18) and W is theoperating point just upon the boundary of ΩDSSR Let W0 =
(120579(0)
1 120579(0)
2 120579(0)
119899)119879 be a fixed solution of equation set (31)
Meanwhile 119871 is a subspace of 119877 which consists of y =
(1205821 1205822 120582
119899)119879 that is y varies in the whole subspace 119871
The coordinates of ymeet the following homogeneous linearequation set
Ay = 0 (31)
If y = (1205821 1205822 120582
119899)119879 is a solution of (31) W = W0
+y = (120579(0)
1+ 1205821 120579(0)
2+ 1205822 120579(0)
119899+ 120582119899)119879 is obviously
a solution of equation set (30) that is W is included inset 119867 Meanwhile x0 is not in 119871 Through Theorem 3 119867is hyperplane The boundary of ΩDSSR can be expressedwith hyperplane that is several subsurfaces Because thedimension of 119871 is 119899 the dimension of hyperplane is also 119899This proof is completed
In transmission area topological characteristics of secu-rity region (SR) have been deeply studied [7] It has shownthat the boundary of SR has no suspension and is compactand approximately linear there is no hole inside the SR Theproof above demonstrates that characteristics of DSSR aresimilar but it has its own features
(a) DSSR is a convex set of denseness in terms ofthe topology This means that DSSR has no holesinside and its boundary has no suspension Basedon this feature dispatchers can analyze the securityof operating point by judging whether the point isinside theDSSRboundarywithout being afraid of anyinsecure point inside or losing security points
(b) DSSR boundary can be described by hyperplane (sev-eral Euclidean subsurfaces) meanwhile the bound-ary ismore linear than that of SR In practical applica-tion calculation of high dimensional security regionwill incur great computation burdens and consumelarge amount of time Therefore this characteristicwill improve the computational efficiency
6 Case Study
In order to verify the effectiveness of the proposedmodel twocases are illustrated in this section The first one is a test caseand the second is a practical case
61 Test Case
611 Overview of Test Case As is shown in Figure 4 the gridis comprised of 4 substations 8 substation transformers and75 feeders 68 feeders form 34 single loop networks and 7form two-tie connections resulting in 82 load transfer unitsThe total capacity of substations is 412MVA To illustrate themodeling process clearly each load transfer unit is numberedCase profile is shown in Table 1
The conductor type of all feeders is LGJ-185 (892MVA)This type is selected based on the national electrical code inChina
612 Boundary Formulation The test case grid has 75 feed-ers 14 feeders form 7 two-tie connections Each load of these7 feeders can be divided into 2 parts to transfer in differentback-feed path So the number of load transfer units is 82which means that the dimension of operation point W is82 and the number of subformulas of DSSR boundary is82 Based on the topology of the test case we substitutethe rated capacity of substation transformers and feedersinto 1198781
119865max 119878119899
119865max and 1198781119879max 119878
119899
119879max of formula (13)
8 Journal of Applied Mathematics
Then the boundary formulation for DSSR of test case can beexpressed as
Considering the similarity of calculation approach ofeach boundary and thesis length we present 5 subformulasof 1198781119891 1198785
119891here Complete formulation is given in the
appendix (in the Supplementary Material available online athttpdxdoiorg1011552014327078) Take the first subfor-mula in (32) as an example to explain the modeling processFigure 5 shows the local topology related with 1198781
119891 including
all relative transfer units in the first subformulaFirst analyze the topology and determine the parameters
of formula (13)We know that back-feed feeder of 1198781119891is feeder
36 and the back-feed substation transformer of 1198781119891is T5 In
normal state 11987836119891 11987845
119891and 11987876
119891 11987878
119891are all load of T5
which means sum119895isinΦ(1198871) 119878119895
119865= sum45
119895=36119878119895
119891+ sum78
119895=76119878119895
119891 11987836119891
and 11987876119891
are load of feeder 36 which means 1198781198871119865= 11987836119891+ 11987876119891 1198782119891 1198785
119891
are the load transferred fromT1 to T5when fault occurs at T1whichmeanssum
119896 =1119896isinΘ(1) 119878119896
119891= sum5
119896=2119878119896119891Therefore the formula
of B1 is preliminary expressed as
1198781
119891le min 11987836
119865max minus (11987876
119891+ 11987836
119891)
1198785
119879max minus (11987836
119891+ 11987837
119891+ 11987838
119891+ 11987839
119891+ 11987840
119891+ 11987841
11989111987842
119891
+11987843
119891+ 11987844
119891+ 11987845
119891+ 11987876
119891+ 11987877
119891+ 11987878
119891)
minus (1198782
119891+ 1198783
119891+ 1198784
119891+ 1198785
119891)
(33)
Second according to case profile substitute 11987836119865max =
892MVA and 1198785119879max = 63MVA into formula (33) and then
we obtain
1198781
119891le min 892 minus (11987876
119891+ 11987836
119891)
63 minus (11987836
119891+ 11987837
119891+ 11987838
119891+ 11987839
119891+ 11987840
119891+ 11987841
11989111987842
119891
+11987843
119891+ 11987844
119891+ 11987845
119891+ 11987876
119891+ 11987877
119891+ 11987878
119891)
minus (1198782
119891+ 1198783
119891+ 1198784
119891+ 1198785
119891)
(34)
613 Visualization and Topological Characteristics To visu-alize the shape of DSSR the dimension is reduced to 2 and 3which means that the coordinates of W should be constantsother than those to be visualized Thus an operating pointhas to be predetermined to provide the constants TSC is aspecial operating point just upon the security boundary [1]on which the facilities are usually fully utilizedThe operationof distribution system is closing to the boundaries and loadlevel is reaching TSCThus in this test case we select the TSCpoint as the predetermined operating point WTSC which isshown in Table 2The calculation approach for TSC has beenpresented in [15]
Choose randomly transfer unit loads 11987815119891
and 11987852119891
toobserve Substitute the rated capacity of transformers T1simT6capacity of all feeders and all transfer unit loads other than11987815119891
and 11987852119891 into formula (32) Then the 2D DSSR shown in
Figure 6 is obtainedAlthough the dimension is 2 the number of subformulas
which constrain the figure shape is not just 2 In fact eachof the subformulas which contains 11987815
119891or 11987852119891
should beconsidered But the subformulas withmore severe constraintswill cover the others and finally form theDSSR boundaryThefinal formulation of 2D figure is
0 le 11987815
119891le 52
0 le 11987852
119891le 3
(35)
Journal of Applied Mathematics 9
T1S1
T2
T3
T4
S 2
S4
T8
T6S3
T5
17-1
8
20-19
12ndash1
415
-16
21ndash23
41-4
3
36ndash3
8
39-4
044
-45
47-4
650
-51
52-5
3
54-5
548
-49
10-1
1
9-8
1 ndash3
4-5
6-7
76ndash7
8
79-80
58-5960-61
57-5665-62
66ndash6970-71
72ndash7524-25
29-28
26-27
30-31
32ndash35
81-8
2
Tie switch
T7
Distribution transformer
(i + 1) j are the same in the topologyin terms of calculation for DSSR
Note means that transfer units iindashj
Figure 4 Test case with eight substation transformers
Feeder 1
T1
T5
Feeder 2Feeder 36Feeder 37
S1f
S2f
S3f
S76f
S77f
S78f
S36f
S36f
S37f
S45f
S4f
S5f
middot middot middot
middot middot middot middot middot middot
Figure 5 Local topology related with f1
Similarly we choose randomly transfer unit loads 11987819119891 11987824119891
and 11987832119891
to observe and then the 3D DSSR is obtained as isshown in Figure 7
In Figure 6 it can be seen that the 2D DSSR is a denserectangle of which boundaries are linear without suspension
DSSR
0
1
2
3
1 2 3 4 52 S52f (MVA)
S15f (MVA)
Figure 6 Shape of 2D DSSR
In Figure 7 the 3D DSSR is surrounded by several subplanesforming a convex pentagon that is a special 3-dimensionhyperplane
Figure 8 The medium-voltage urban power grid of one city of southern China
62 Practical Case In this section the assessment andcontrol method based on DSSR is demonstrated on a realmedium-voltage distribution network of one city in south-ern China which consists of 12 substations 26 substationtransformers and 114 main feeders as is shown in Figure 8Total capacity of substation transformers is 10945MVA It isnotable that each two of all main feeders form a single loopnetwork in this case which leads to the fact that the numberof main feeders is equal to that of load transfer units
Paper [1] has presented concept and calculation methodof relative location of operating point in DSSR based onsubstation transformer contingency Similarly in this paperrelative location 119871
119894describes the distance from the operating
pointW to the boundary 119861119894(119894 = 1 2 119899)
119871119894= min
119878119887119894
119865max minus 119878119887119894
119865 119878119887119894
119879max minus sum
119895isinΦ(119887119894)
119878119895
119865minus sum
119896 =119894119896isinΘ(119894)
119878119896
119891
minus 119878119894
119891
(36)
119871119894is an index which is applied in security assessment
Here is an example Let W1 be an insecure operating pointEach transfer unit load of W1 is 08 times of TSC load inthe practical case except 1198781
119891and 11987810119891 while 1198781
119891= 8MVA and
11987810119891= 7MVA Total load of W1 is 5087MVA while TSC is
6289MVAAverage load rate is 046Thatmeans that current
Table 3 Distance to boundaries of 1198611and 119861
10atW1
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
operating point can be controlled back to security regionThrough formula (36) the location of a given operating pointin DSSR can be calculated We present the 119861
119894of which 119871
119894are
negative in Table 3Table 3 shows that the system should lose 3222MVA if
fault occurs at feeder 1 or feeder 10 so the system is locallyoverloading The location of W1 can be easily observed inFigure 9 To meet operational constraints under N-1 contin-gency 1198781
119891and 11987810
119891should be adjusted For example W1 is
adjusted toW1015840
1 through the dashed arrow shown in Figure 9The location ofW
1015840
1 is shown in Table 4
7 Conclusion and Further Work
This paper proposes a mathematical model for distributionsystem security region (DSSR) generic topological character-istics of DSSR are discussed and proved
First the operating point for a distribution system isdefined as the set of transfer unit loads which is a vector in the
12 Journal of Applied Mathematics
DSSR
0
2
4
6
2 4 6 8 10 11778
8
10
11778
S1f (MVA)
S10f (MVA)
W1(8 7)
W9984001 (7 4)
Figure 9 2D DSSR cross-section and preventive control
Table 4 Distance to boundaries of 1198611and 119861
10atW10158401
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
Euclidean space Then the DSSR model for both substationtransformer and feeder contingency is proposed which is theset of all operating points each ensures that the system N-1 issecure
Second to perform research on the characteristic ofDSSR the boundary formulation for DSSR is derived fromthe DSSR model Then three generic characteristics areproposed and proved by rigor mathematical approach whichare as follows (1) DSSR is dense inside (2) DSSR boundaryhas no suspension (3)DSSR boundary can be expressed withthe union of several subsurfaces
Finally the results from a test case and a practical casedemonstrate the effectiveness of the proposed model Thevisualization for DSSR boundary is also discussed to verifythe topological characteristics of DSSR Security assessmentmethod based on DSSR is preliminary exhibited to show theapplicability of DSSR for future smart distribution system
This paper improves the accuracy and understanding ofDSSR which is fundamental theoretic work for future smartdistribution system Further works to improve the accuracyof DSSR model include fully considering power flow voltageconstraints and integration of DGs
Conflict of Interests
The authors declare that there is no conflict of inter-ests regarding the publication of this paper None ofthe authors have a commercial interest financial interestandor other relationship with manufacturers of pharma-ceuticalslaboratory supplies andor medical devices or withcommercialproviders of medically related services
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (no 51277129) and National HighTechnology Research and Development Program of China(863 Program) (no 2011AA05A117)
References
[1] J Xiao W Gu C Wang and F Li ldquoDistribution systemsecurity region definition model and security assessmentrdquo IETGeneration TransmissionampDistribution vol 6 no 10 pp 1029ndash1035 2012
[2] State Grid Corporation of China Guidelines of Urban PowerNetwork Planning State Grid Corporation of China BeijingChina 2006 (Chinese)
[3] E Lakervi and E J Holmes Electricity Distribution NetworkDesign IET Peregrinus UK 1995
[4] K N Miu and H D Chiang ldquoService restoration for unbal-anced radial distribution systems with varying loads solutionalgorithmrdquo in Proceedings of the IEEE Power Engineering SocietySummer Meeting vol 1 pp 254ndash258 1999
[5] Y X Yu ldquoSecurity region of bulk power systemrdquo in Proceedingsof the International Conference on Power System Technology(PowerCon rsquo02) vol 1 pp 13ndash17 Kunming China October2002
[6] F Wu and S Kumagai ldquoSteady-state security regions of powersystemsrdquo IEEE Transactions on Circuits and Systems vol 29 no11 pp 703ndash711 1982
[7] Y Yu Y Zeng and F Feng ldquoDifferential topological character-istics of the DSR on injection space of electrical power systemrdquoScience in China E Technological Sciences vol 45 no 6 pp 576ndash584 2002
[8] S J Chen Q X Chen Q Xia and C Q Kang ldquoSteady-state security assessment method based on distance to securityregion boundariesrdquo IET Generation Transmission amp Distribu-tion vol 7 no 3 pp 288ndash297 2013
[9] D M Staszesky D Craig and C Befus ldquoAdvanced feederautomation is hererdquo IEEE Power and Energy Magazine vol 3no 5 pp 56ndash63 2005
[10] C L Smallwood and J Wennermark ldquoBenefits of distributionautomationrdquo IEEE Industry Applications Magazine vol 16 no1 pp 65ndash73 2010
[11] X Mamo S Mallet T Coste and S Grenard ldquoDistributionautomation the cornerstone for smart grid development strat-egyrdquo in Proceedings of the IEEE Power amp Energy Society GeneralMeeting (PES rsquo09) pp 1ndash6 July 2009
[12] P W Sauer B C Lesieutre and M A Pai ldquoMaximumloadability and voltage stability in power systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 15 no 3 pp145ndash153 1993
[13] K N Miu and H Chiang ldquoElectric distribution systemload capability problem formulation solution algorithm andnumerical resultsrdquo IEEE Transactions on Power Delivery vol 15no 1 pp 436ndash442 2000
[14] F Z Luo C S Wang J Xiao and S Ge ldquoRapid evaluationmethod for power supply capability of urban distribution sys-tem based on N-1 contingency analysis of main-transformersrdquoInternational Journal of Electrical Power andEnergy Systems vol32 no 10 pp 1063ndash1068 2010
Journal of Applied Mathematics 13
[15] J Xiao F X LiWZGuC SWang andP Zhang ldquoTotal supplycapability and its extended indices for distribution systemsdefinitionmodel calculation and applicationsrdquo IETGenerationTransmission and Distribution vol 5 no 8 pp 869ndash876 2011
[16] J Xiao X X Gong and C S Wang ldquoTopology properties andalgorithm of N-1 security boundary for smart gridrdquo Proceedingsof the CSEE vol 34 no 4 pp 545ndash554 2014 (Chinese)
[17] G E Shilov An Introduction to the Theory of Linear SpacesDover New York NY USA 1975
[18] J R Munkres Topology vol 2 Prentice Hall Upper SaddleRiver NJ USA 2000
Considering the similarity of calculation approach ofeach boundary and thesis length we present 5 subformulasof 1198781119891 1198785
119891here Complete formulation is given in the
appendix (in the Supplementary Material available online athttpdxdoiorg1011552014327078) Take the first subfor-mula in (32) as an example to explain the modeling processFigure 5 shows the local topology related with 1198781
119891 including
all relative transfer units in the first subformulaFirst analyze the topology and determine the parameters
of formula (13)We know that back-feed feeder of 1198781119891is feeder
36 and the back-feed substation transformer of 1198781119891is T5 In
normal state 11987836119891 11987845
119891and 11987876
119891 11987878
119891are all load of T5
which means sum119895isinΦ(1198871) 119878119895
119865= sum45
119895=36119878119895
119891+ sum78
119895=76119878119895
119891 11987836119891
and 11987876119891
are load of feeder 36 which means 1198781198871119865= 11987836119891+ 11987876119891 1198782119891 1198785
119891
are the load transferred fromT1 to T5when fault occurs at T1whichmeanssum
119896 =1119896isinΘ(1) 119878119896
119891= sum5
119896=2119878119896119891Therefore the formula
of B1 is preliminary expressed as
1198781
119891le min 11987836
119865max minus (11987876
119891+ 11987836
119891)
1198785
119879max minus (11987836
119891+ 11987837
119891+ 11987838
119891+ 11987839
119891+ 11987840
119891+ 11987841
11989111987842
119891
+11987843
119891+ 11987844
119891+ 11987845
119891+ 11987876
119891+ 11987877
119891+ 11987878
119891)
minus (1198782
119891+ 1198783
119891+ 1198784
119891+ 1198785
119891)
(33)
Second according to case profile substitute 11987836119865max =
892MVA and 1198785119879max = 63MVA into formula (33) and then
we obtain
1198781
119891le min 892 minus (11987876
119891+ 11987836
119891)
63 minus (11987836
119891+ 11987837
119891+ 11987838
119891+ 11987839
119891+ 11987840
119891+ 11987841
11989111987842
119891
+11987843
119891+ 11987844
119891+ 11987845
119891+ 11987876
119891+ 11987877
119891+ 11987878
119891)
minus (1198782
119891+ 1198783
119891+ 1198784
119891+ 1198785
119891)
(34)
613 Visualization and Topological Characteristics To visu-alize the shape of DSSR the dimension is reduced to 2 and 3which means that the coordinates of W should be constantsother than those to be visualized Thus an operating pointhas to be predetermined to provide the constants TSC is aspecial operating point just upon the security boundary [1]on which the facilities are usually fully utilizedThe operationof distribution system is closing to the boundaries and loadlevel is reaching TSCThus in this test case we select the TSCpoint as the predetermined operating point WTSC which isshown in Table 2The calculation approach for TSC has beenpresented in [15]
Choose randomly transfer unit loads 11987815119891
and 11987852119891
toobserve Substitute the rated capacity of transformers T1simT6capacity of all feeders and all transfer unit loads other than11987815119891
and 11987852119891 into formula (32) Then the 2D DSSR shown in
Figure 6 is obtainedAlthough the dimension is 2 the number of subformulas
which constrain the figure shape is not just 2 In fact eachof the subformulas which contains 11987815
119891or 11987852119891
should beconsidered But the subformulas withmore severe constraintswill cover the others and finally form theDSSR boundaryThefinal formulation of 2D figure is
0 le 11987815
119891le 52
0 le 11987852
119891le 3
(35)
Journal of Applied Mathematics 9
T1S1
T2
T3
T4
S 2
S4
T8
T6S3
T5
17-1
8
20-19
12ndash1
415
-16
21ndash23
41-4
3
36ndash3
8
39-4
044
-45
47-4
650
-51
52-5
3
54-5
548
-49
10-1
1
9-8
1 ndash3
4-5
6-7
76ndash7
8
79-80
58-5960-61
57-5665-62
66ndash6970-71
72ndash7524-25
29-28
26-27
30-31
32ndash35
81-8
2
Tie switch
T7
Distribution transformer
(i + 1) j are the same in the topologyin terms of calculation for DSSR
Note means that transfer units iindashj
Figure 4 Test case with eight substation transformers
Feeder 1
T1
T5
Feeder 2Feeder 36Feeder 37
S1f
S2f
S3f
S76f
S77f
S78f
S36f
S36f
S37f
S45f
S4f
S5f
middot middot middot
middot middot middot middot middot middot
Figure 5 Local topology related with f1
Similarly we choose randomly transfer unit loads 11987819119891 11987824119891
and 11987832119891
to observe and then the 3D DSSR is obtained as isshown in Figure 7
In Figure 6 it can be seen that the 2D DSSR is a denserectangle of which boundaries are linear without suspension
DSSR
0
1
2
3
1 2 3 4 52 S52f (MVA)
S15f (MVA)
Figure 6 Shape of 2D DSSR
In Figure 7 the 3D DSSR is surrounded by several subplanesforming a convex pentagon that is a special 3-dimensionhyperplane
Figure 8 The medium-voltage urban power grid of one city of southern China
62 Practical Case In this section the assessment andcontrol method based on DSSR is demonstrated on a realmedium-voltage distribution network of one city in south-ern China which consists of 12 substations 26 substationtransformers and 114 main feeders as is shown in Figure 8Total capacity of substation transformers is 10945MVA It isnotable that each two of all main feeders form a single loopnetwork in this case which leads to the fact that the numberof main feeders is equal to that of load transfer units
Paper [1] has presented concept and calculation methodof relative location of operating point in DSSR based onsubstation transformer contingency Similarly in this paperrelative location 119871
119894describes the distance from the operating
pointW to the boundary 119861119894(119894 = 1 2 119899)
119871119894= min
119878119887119894
119865max minus 119878119887119894
119865 119878119887119894
119879max minus sum
119895isinΦ(119887119894)
119878119895
119865minus sum
119896 =119894119896isinΘ(119894)
119878119896
119891
minus 119878119894
119891
(36)
119871119894is an index which is applied in security assessment
Here is an example Let W1 be an insecure operating pointEach transfer unit load of W1 is 08 times of TSC load inthe practical case except 1198781
119891and 11987810119891 while 1198781
119891= 8MVA and
11987810119891= 7MVA Total load of W1 is 5087MVA while TSC is
6289MVAAverage load rate is 046Thatmeans that current
Table 3 Distance to boundaries of 1198611and 119861
10atW1
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
operating point can be controlled back to security regionThrough formula (36) the location of a given operating pointin DSSR can be calculated We present the 119861
119894of which 119871
119894are
negative in Table 3Table 3 shows that the system should lose 3222MVA if
fault occurs at feeder 1 or feeder 10 so the system is locallyoverloading The location of W1 can be easily observed inFigure 9 To meet operational constraints under N-1 contin-gency 1198781
119891and 11987810
119891should be adjusted For example W1 is
adjusted toW1015840
1 through the dashed arrow shown in Figure 9The location ofW
1015840
1 is shown in Table 4
7 Conclusion and Further Work
This paper proposes a mathematical model for distributionsystem security region (DSSR) generic topological character-istics of DSSR are discussed and proved
First the operating point for a distribution system isdefined as the set of transfer unit loads which is a vector in the
12 Journal of Applied Mathematics
DSSR
0
2
4
6
2 4 6 8 10 11778
8
10
11778
S1f (MVA)
S10f (MVA)
W1(8 7)
W9984001 (7 4)
Figure 9 2D DSSR cross-section and preventive control
Table 4 Distance to boundaries of 1198611and 119861
10atW10158401
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
Euclidean space Then the DSSR model for both substationtransformer and feeder contingency is proposed which is theset of all operating points each ensures that the system N-1 issecure
Second to perform research on the characteristic ofDSSR the boundary formulation for DSSR is derived fromthe DSSR model Then three generic characteristics areproposed and proved by rigor mathematical approach whichare as follows (1) DSSR is dense inside (2) DSSR boundaryhas no suspension (3)DSSR boundary can be expressed withthe union of several subsurfaces
Finally the results from a test case and a practical casedemonstrate the effectiveness of the proposed model Thevisualization for DSSR boundary is also discussed to verifythe topological characteristics of DSSR Security assessmentmethod based on DSSR is preliminary exhibited to show theapplicability of DSSR for future smart distribution system
This paper improves the accuracy and understanding ofDSSR which is fundamental theoretic work for future smartdistribution system Further works to improve the accuracyof DSSR model include fully considering power flow voltageconstraints and integration of DGs
Conflict of Interests
The authors declare that there is no conflict of inter-ests regarding the publication of this paper None ofthe authors have a commercial interest financial interestandor other relationship with manufacturers of pharma-ceuticalslaboratory supplies andor medical devices or withcommercialproviders of medically related services
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (no 51277129) and National HighTechnology Research and Development Program of China(863 Program) (no 2011AA05A117)
References
[1] J Xiao W Gu C Wang and F Li ldquoDistribution systemsecurity region definition model and security assessmentrdquo IETGeneration TransmissionampDistribution vol 6 no 10 pp 1029ndash1035 2012
[2] State Grid Corporation of China Guidelines of Urban PowerNetwork Planning State Grid Corporation of China BeijingChina 2006 (Chinese)
[3] E Lakervi and E J Holmes Electricity Distribution NetworkDesign IET Peregrinus UK 1995
[4] K N Miu and H D Chiang ldquoService restoration for unbal-anced radial distribution systems with varying loads solutionalgorithmrdquo in Proceedings of the IEEE Power Engineering SocietySummer Meeting vol 1 pp 254ndash258 1999
[5] Y X Yu ldquoSecurity region of bulk power systemrdquo in Proceedingsof the International Conference on Power System Technology(PowerCon rsquo02) vol 1 pp 13ndash17 Kunming China October2002
[6] F Wu and S Kumagai ldquoSteady-state security regions of powersystemsrdquo IEEE Transactions on Circuits and Systems vol 29 no11 pp 703ndash711 1982
[7] Y Yu Y Zeng and F Feng ldquoDifferential topological character-istics of the DSR on injection space of electrical power systemrdquoScience in China E Technological Sciences vol 45 no 6 pp 576ndash584 2002
[8] S J Chen Q X Chen Q Xia and C Q Kang ldquoSteady-state security assessment method based on distance to securityregion boundariesrdquo IET Generation Transmission amp Distribu-tion vol 7 no 3 pp 288ndash297 2013
[9] D M Staszesky D Craig and C Befus ldquoAdvanced feederautomation is hererdquo IEEE Power and Energy Magazine vol 3no 5 pp 56ndash63 2005
[10] C L Smallwood and J Wennermark ldquoBenefits of distributionautomationrdquo IEEE Industry Applications Magazine vol 16 no1 pp 65ndash73 2010
[11] X Mamo S Mallet T Coste and S Grenard ldquoDistributionautomation the cornerstone for smart grid development strat-egyrdquo in Proceedings of the IEEE Power amp Energy Society GeneralMeeting (PES rsquo09) pp 1ndash6 July 2009
[12] P W Sauer B C Lesieutre and M A Pai ldquoMaximumloadability and voltage stability in power systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 15 no 3 pp145ndash153 1993
[13] K N Miu and H Chiang ldquoElectric distribution systemload capability problem formulation solution algorithm andnumerical resultsrdquo IEEE Transactions on Power Delivery vol 15no 1 pp 436ndash442 2000
[14] F Z Luo C S Wang J Xiao and S Ge ldquoRapid evaluationmethod for power supply capability of urban distribution sys-tem based on N-1 contingency analysis of main-transformersrdquoInternational Journal of Electrical Power andEnergy Systems vol32 no 10 pp 1063ndash1068 2010
Journal of Applied Mathematics 13
[15] J Xiao F X LiWZGuC SWang andP Zhang ldquoTotal supplycapability and its extended indices for distribution systemsdefinitionmodel calculation and applicationsrdquo IETGenerationTransmission and Distribution vol 5 no 8 pp 869ndash876 2011
[16] J Xiao X X Gong and C S Wang ldquoTopology properties andalgorithm of N-1 security boundary for smart gridrdquo Proceedingsof the CSEE vol 34 no 4 pp 545ndash554 2014 (Chinese)
[17] G E Shilov An Introduction to the Theory of Linear SpacesDover New York NY USA 1975
[18] J R Munkres Topology vol 2 Prentice Hall Upper SaddleRiver NJ USA 2000
Figure 8 The medium-voltage urban power grid of one city of southern China
62 Practical Case In this section the assessment andcontrol method based on DSSR is demonstrated on a realmedium-voltage distribution network of one city in south-ern China which consists of 12 substations 26 substationtransformers and 114 main feeders as is shown in Figure 8Total capacity of substation transformers is 10945MVA It isnotable that each two of all main feeders form a single loopnetwork in this case which leads to the fact that the numberof main feeders is equal to that of load transfer units
Paper [1] has presented concept and calculation methodof relative location of operating point in DSSR based onsubstation transformer contingency Similarly in this paperrelative location 119871
119894describes the distance from the operating
pointW to the boundary 119861119894(119894 = 1 2 119899)
119871119894= min
119878119887119894
119865max minus 119878119887119894
119865 119878119887119894
119879max minus sum
119895isinΦ(119887119894)
119878119895
119865minus sum
119896 =119894119896isinΘ(119894)
119878119896
119891
minus 119878119894
119891
(36)
119871119894is an index which is applied in security assessment
Here is an example Let W1 be an insecure operating pointEach transfer unit load of W1 is 08 times of TSC load inthe practical case except 1198781
119891and 11987810119891 while 1198781
119891= 8MVA and
11987810119891= 7MVA Total load of W1 is 5087MVA while TSC is
6289MVAAverage load rate is 046Thatmeans that current
Table 3 Distance to boundaries of 1198611and 119861
10atW1
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
operating point can be controlled back to security regionThrough formula (36) the location of a given operating pointin DSSR can be calculated We present the 119861
119894of which 119871
119894are
negative in Table 3Table 3 shows that the system should lose 3222MVA if
fault occurs at feeder 1 or feeder 10 so the system is locallyoverloading The location of W1 can be easily observed inFigure 9 To meet operational constraints under N-1 contin-gency 1198781
119891and 11987810
119891should be adjusted For example W1 is
adjusted toW1015840
1 through the dashed arrow shown in Figure 9The location ofW
1015840
1 is shown in Table 4
7 Conclusion and Further Work
This paper proposes a mathematical model for distributionsystem security region (DSSR) generic topological character-istics of DSSR are discussed and proved
First the operating point for a distribution system isdefined as the set of transfer unit loads which is a vector in the
12 Journal of Applied Mathematics
DSSR
0
2
4
6
2 4 6 8 10 11778
8
10
11778
S1f (MVA)
S10f (MVA)
W1(8 7)
W9984001 (7 4)
Figure 9 2D DSSR cross-section and preventive control
Table 4 Distance to boundaries of 1198611and 119861
10atW10158401
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
Euclidean space Then the DSSR model for both substationtransformer and feeder contingency is proposed which is theset of all operating points each ensures that the system N-1 issecure
Second to perform research on the characteristic ofDSSR the boundary formulation for DSSR is derived fromthe DSSR model Then three generic characteristics areproposed and proved by rigor mathematical approach whichare as follows (1) DSSR is dense inside (2) DSSR boundaryhas no suspension (3)DSSR boundary can be expressed withthe union of several subsurfaces
Finally the results from a test case and a practical casedemonstrate the effectiveness of the proposed model Thevisualization for DSSR boundary is also discussed to verifythe topological characteristics of DSSR Security assessmentmethod based on DSSR is preliminary exhibited to show theapplicability of DSSR for future smart distribution system
This paper improves the accuracy and understanding ofDSSR which is fundamental theoretic work for future smartdistribution system Further works to improve the accuracyof DSSR model include fully considering power flow voltageconstraints and integration of DGs
Conflict of Interests
The authors declare that there is no conflict of inter-ests regarding the publication of this paper None ofthe authors have a commercial interest financial interestandor other relationship with manufacturers of pharma-ceuticalslaboratory supplies andor medical devices or withcommercialproviders of medically related services
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (no 51277129) and National HighTechnology Research and Development Program of China(863 Program) (no 2011AA05A117)
References
[1] J Xiao W Gu C Wang and F Li ldquoDistribution systemsecurity region definition model and security assessmentrdquo IETGeneration TransmissionampDistribution vol 6 no 10 pp 1029ndash1035 2012
[2] State Grid Corporation of China Guidelines of Urban PowerNetwork Planning State Grid Corporation of China BeijingChina 2006 (Chinese)
[3] E Lakervi and E J Holmes Electricity Distribution NetworkDesign IET Peregrinus UK 1995
[4] K N Miu and H D Chiang ldquoService restoration for unbal-anced radial distribution systems with varying loads solutionalgorithmrdquo in Proceedings of the IEEE Power Engineering SocietySummer Meeting vol 1 pp 254ndash258 1999
[5] Y X Yu ldquoSecurity region of bulk power systemrdquo in Proceedingsof the International Conference on Power System Technology(PowerCon rsquo02) vol 1 pp 13ndash17 Kunming China October2002
[6] F Wu and S Kumagai ldquoSteady-state security regions of powersystemsrdquo IEEE Transactions on Circuits and Systems vol 29 no11 pp 703ndash711 1982
[7] Y Yu Y Zeng and F Feng ldquoDifferential topological character-istics of the DSR on injection space of electrical power systemrdquoScience in China E Technological Sciences vol 45 no 6 pp 576ndash584 2002
[8] S J Chen Q X Chen Q Xia and C Q Kang ldquoSteady-state security assessment method based on distance to securityregion boundariesrdquo IET Generation Transmission amp Distribu-tion vol 7 no 3 pp 288ndash297 2013
[9] D M Staszesky D Craig and C Befus ldquoAdvanced feederautomation is hererdquo IEEE Power and Energy Magazine vol 3no 5 pp 56ndash63 2005
[10] C L Smallwood and J Wennermark ldquoBenefits of distributionautomationrdquo IEEE Industry Applications Magazine vol 16 no1 pp 65ndash73 2010
[11] X Mamo S Mallet T Coste and S Grenard ldquoDistributionautomation the cornerstone for smart grid development strat-egyrdquo in Proceedings of the IEEE Power amp Energy Society GeneralMeeting (PES rsquo09) pp 1ndash6 July 2009
[12] P W Sauer B C Lesieutre and M A Pai ldquoMaximumloadability and voltage stability in power systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 15 no 3 pp145ndash153 1993
[13] K N Miu and H Chiang ldquoElectric distribution systemload capability problem formulation solution algorithm andnumerical resultsrdquo IEEE Transactions on Power Delivery vol 15no 1 pp 436ndash442 2000
[14] F Z Luo C S Wang J Xiao and S Ge ldquoRapid evaluationmethod for power supply capability of urban distribution sys-tem based on N-1 contingency analysis of main-transformersrdquoInternational Journal of Electrical Power andEnergy Systems vol32 no 10 pp 1063ndash1068 2010
Journal of Applied Mathematics 13
[15] J Xiao F X LiWZGuC SWang andP Zhang ldquoTotal supplycapability and its extended indices for distribution systemsdefinitionmodel calculation and applicationsrdquo IETGenerationTransmission and Distribution vol 5 no 8 pp 869ndash876 2011
[16] J Xiao X X Gong and C S Wang ldquoTopology properties andalgorithm of N-1 security boundary for smart gridrdquo Proceedingsof the CSEE vol 34 no 4 pp 545ndash554 2014 (Chinese)
[17] G E Shilov An Introduction to the Theory of Linear SpacesDover New York NY USA 1975
[18] J R Munkres Topology vol 2 Prentice Hall Upper SaddleRiver NJ USA 2000
Figure 8 The medium-voltage urban power grid of one city of southern China
62 Practical Case In this section the assessment andcontrol method based on DSSR is demonstrated on a realmedium-voltage distribution network of one city in south-ern China which consists of 12 substations 26 substationtransformers and 114 main feeders as is shown in Figure 8Total capacity of substation transformers is 10945MVA It isnotable that each two of all main feeders form a single loopnetwork in this case which leads to the fact that the numberof main feeders is equal to that of load transfer units
Paper [1] has presented concept and calculation methodof relative location of operating point in DSSR based onsubstation transformer contingency Similarly in this paperrelative location 119871
119894describes the distance from the operating
pointW to the boundary 119861119894(119894 = 1 2 119899)
119871119894= min
119878119887119894
119865max minus 119878119887119894
119865 119878119887119894
119879max minus sum
119895isinΦ(119887119894)
119878119895
119865minus sum
119896 =119894119896isinΘ(119894)
119878119896
119891
minus 119878119894
119891
(36)
119871119894is an index which is applied in security assessment
Here is an example Let W1 be an insecure operating pointEach transfer unit load of W1 is 08 times of TSC load inthe practical case except 1198781
119891and 11987810119891 while 1198781
119891= 8MVA and
11987810119891= 7MVA Total load of W1 is 5087MVA while TSC is
6289MVAAverage load rate is 046Thatmeans that current
Table 3 Distance to boundaries of 1198611and 119861
10atW1
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
operating point can be controlled back to security regionThrough formula (36) the location of a given operating pointin DSSR can be calculated We present the 119861
119894of which 119871
119894are
negative in Table 3Table 3 shows that the system should lose 3222MVA if
fault occurs at feeder 1 or feeder 10 so the system is locallyoverloading The location of W1 can be easily observed inFigure 9 To meet operational constraints under N-1 contin-gency 1198781
119891and 11987810
119891should be adjusted For example W1 is
adjusted toW1015840
1 through the dashed arrow shown in Figure 9The location ofW
1015840
1 is shown in Table 4
7 Conclusion and Further Work
This paper proposes a mathematical model for distributionsystem security region (DSSR) generic topological character-istics of DSSR are discussed and proved
First the operating point for a distribution system isdefined as the set of transfer unit loads which is a vector in the
12 Journal of Applied Mathematics
DSSR
0
2
4
6
2 4 6 8 10 11778
8
10
11778
S1f (MVA)
S10f (MVA)
W1(8 7)
W9984001 (7 4)
Figure 9 2D DSSR cross-section and preventive control
Table 4 Distance to boundaries of 1198611and 119861
10atW10158401
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
Euclidean space Then the DSSR model for both substationtransformer and feeder contingency is proposed which is theset of all operating points each ensures that the system N-1 issecure
Second to perform research on the characteristic ofDSSR the boundary formulation for DSSR is derived fromthe DSSR model Then three generic characteristics areproposed and proved by rigor mathematical approach whichare as follows (1) DSSR is dense inside (2) DSSR boundaryhas no suspension (3)DSSR boundary can be expressed withthe union of several subsurfaces
Finally the results from a test case and a practical casedemonstrate the effectiveness of the proposed model Thevisualization for DSSR boundary is also discussed to verifythe topological characteristics of DSSR Security assessmentmethod based on DSSR is preliminary exhibited to show theapplicability of DSSR for future smart distribution system
This paper improves the accuracy and understanding ofDSSR which is fundamental theoretic work for future smartdistribution system Further works to improve the accuracyof DSSR model include fully considering power flow voltageconstraints and integration of DGs
Conflict of Interests
The authors declare that there is no conflict of inter-ests regarding the publication of this paper None ofthe authors have a commercial interest financial interestandor other relationship with manufacturers of pharma-ceuticalslaboratory supplies andor medical devices or withcommercialproviders of medically related services
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (no 51277129) and National HighTechnology Research and Development Program of China(863 Program) (no 2011AA05A117)
References
[1] J Xiao W Gu C Wang and F Li ldquoDistribution systemsecurity region definition model and security assessmentrdquo IETGeneration TransmissionampDistribution vol 6 no 10 pp 1029ndash1035 2012
[2] State Grid Corporation of China Guidelines of Urban PowerNetwork Planning State Grid Corporation of China BeijingChina 2006 (Chinese)
[3] E Lakervi and E J Holmes Electricity Distribution NetworkDesign IET Peregrinus UK 1995
[4] K N Miu and H D Chiang ldquoService restoration for unbal-anced radial distribution systems with varying loads solutionalgorithmrdquo in Proceedings of the IEEE Power Engineering SocietySummer Meeting vol 1 pp 254ndash258 1999
[5] Y X Yu ldquoSecurity region of bulk power systemrdquo in Proceedingsof the International Conference on Power System Technology(PowerCon rsquo02) vol 1 pp 13ndash17 Kunming China October2002
[6] F Wu and S Kumagai ldquoSteady-state security regions of powersystemsrdquo IEEE Transactions on Circuits and Systems vol 29 no11 pp 703ndash711 1982
[7] Y Yu Y Zeng and F Feng ldquoDifferential topological character-istics of the DSR on injection space of electrical power systemrdquoScience in China E Technological Sciences vol 45 no 6 pp 576ndash584 2002
[8] S J Chen Q X Chen Q Xia and C Q Kang ldquoSteady-state security assessment method based on distance to securityregion boundariesrdquo IET Generation Transmission amp Distribu-tion vol 7 no 3 pp 288ndash297 2013
[9] D M Staszesky D Craig and C Befus ldquoAdvanced feederautomation is hererdquo IEEE Power and Energy Magazine vol 3no 5 pp 56ndash63 2005
[10] C L Smallwood and J Wennermark ldquoBenefits of distributionautomationrdquo IEEE Industry Applications Magazine vol 16 no1 pp 65ndash73 2010
[11] X Mamo S Mallet T Coste and S Grenard ldquoDistributionautomation the cornerstone for smart grid development strat-egyrdquo in Proceedings of the IEEE Power amp Energy Society GeneralMeeting (PES rsquo09) pp 1ndash6 July 2009
[12] P W Sauer B C Lesieutre and M A Pai ldquoMaximumloadability and voltage stability in power systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 15 no 3 pp145ndash153 1993
[13] K N Miu and H Chiang ldquoElectric distribution systemload capability problem formulation solution algorithm andnumerical resultsrdquo IEEE Transactions on Power Delivery vol 15no 1 pp 436ndash442 2000
[14] F Z Luo C S Wang J Xiao and S Ge ldquoRapid evaluationmethod for power supply capability of urban distribution sys-tem based on N-1 contingency analysis of main-transformersrdquoInternational Journal of Electrical Power andEnergy Systems vol32 no 10 pp 1063ndash1068 2010
Journal of Applied Mathematics 13
[15] J Xiao F X LiWZGuC SWang andP Zhang ldquoTotal supplycapability and its extended indices for distribution systemsdefinitionmodel calculation and applicationsrdquo IETGenerationTransmission and Distribution vol 5 no 8 pp 869ndash876 2011
[16] J Xiao X X Gong and C S Wang ldquoTopology properties andalgorithm of N-1 security boundary for smart gridrdquo Proceedingsof the CSEE vol 34 no 4 pp 545ndash554 2014 (Chinese)
[17] G E Shilov An Introduction to the Theory of Linear SpacesDover New York NY USA 1975
[18] J R Munkres Topology vol 2 Prentice Hall Upper SaddleRiver NJ USA 2000
Figure 8 The medium-voltage urban power grid of one city of southern China
62 Practical Case In this section the assessment andcontrol method based on DSSR is demonstrated on a realmedium-voltage distribution network of one city in south-ern China which consists of 12 substations 26 substationtransformers and 114 main feeders as is shown in Figure 8Total capacity of substation transformers is 10945MVA It isnotable that each two of all main feeders form a single loopnetwork in this case which leads to the fact that the numberof main feeders is equal to that of load transfer units
Paper [1] has presented concept and calculation methodof relative location of operating point in DSSR based onsubstation transformer contingency Similarly in this paperrelative location 119871
119894describes the distance from the operating
pointW to the boundary 119861119894(119894 = 1 2 119899)
119871119894= min
119878119887119894
119865max minus 119878119887119894
119865 119878119887119894
119879max minus sum
119895isinΦ(119887119894)
119878119895
119865minus sum
119896 =119894119896isinΘ(119894)
119878119896
119891
minus 119878119894
119891
(36)
119871119894is an index which is applied in security assessment
Here is an example Let W1 be an insecure operating pointEach transfer unit load of W1 is 08 times of TSC load inthe practical case except 1198781
119891and 11987810119891 while 1198781
119891= 8MVA and
11987810119891= 7MVA Total load of W1 is 5087MVA while TSC is
6289MVAAverage load rate is 046Thatmeans that current
Table 3 Distance to boundaries of 1198611and 119861
10atW1
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
operating point can be controlled back to security regionThrough formula (36) the location of a given operating pointin DSSR can be calculated We present the 119861
119894of which 119871
119894are
negative in Table 3Table 3 shows that the system should lose 3222MVA if
fault occurs at feeder 1 or feeder 10 so the system is locallyoverloading The location of W1 can be easily observed inFigure 9 To meet operational constraints under N-1 contin-gency 1198781
119891and 11987810
119891should be adjusted For example W1 is
adjusted toW1015840
1 through the dashed arrow shown in Figure 9The location ofW
1015840
1 is shown in Table 4
7 Conclusion and Further Work
This paper proposes a mathematical model for distributionsystem security region (DSSR) generic topological character-istics of DSSR are discussed and proved
First the operating point for a distribution system isdefined as the set of transfer unit loads which is a vector in the
12 Journal of Applied Mathematics
DSSR
0
2
4
6
2 4 6 8 10 11778
8
10
11778
S1f (MVA)
S10f (MVA)
W1(8 7)
W9984001 (7 4)
Figure 9 2D DSSR cross-section and preventive control
Table 4 Distance to boundaries of 1198611and 119861
10atW10158401
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
Euclidean space Then the DSSR model for both substationtransformer and feeder contingency is proposed which is theset of all operating points each ensures that the system N-1 issecure
Second to perform research on the characteristic ofDSSR the boundary formulation for DSSR is derived fromthe DSSR model Then three generic characteristics areproposed and proved by rigor mathematical approach whichare as follows (1) DSSR is dense inside (2) DSSR boundaryhas no suspension (3)DSSR boundary can be expressed withthe union of several subsurfaces
Finally the results from a test case and a practical casedemonstrate the effectiveness of the proposed model Thevisualization for DSSR boundary is also discussed to verifythe topological characteristics of DSSR Security assessmentmethod based on DSSR is preliminary exhibited to show theapplicability of DSSR for future smart distribution system
This paper improves the accuracy and understanding ofDSSR which is fundamental theoretic work for future smartdistribution system Further works to improve the accuracyof DSSR model include fully considering power flow voltageconstraints and integration of DGs
Conflict of Interests
The authors declare that there is no conflict of inter-ests regarding the publication of this paper None ofthe authors have a commercial interest financial interestandor other relationship with manufacturers of pharma-ceuticalslaboratory supplies andor medical devices or withcommercialproviders of medically related services
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (no 51277129) and National HighTechnology Research and Development Program of China(863 Program) (no 2011AA05A117)
References
[1] J Xiao W Gu C Wang and F Li ldquoDistribution systemsecurity region definition model and security assessmentrdquo IETGeneration TransmissionampDistribution vol 6 no 10 pp 1029ndash1035 2012
[2] State Grid Corporation of China Guidelines of Urban PowerNetwork Planning State Grid Corporation of China BeijingChina 2006 (Chinese)
[3] E Lakervi and E J Holmes Electricity Distribution NetworkDesign IET Peregrinus UK 1995
[4] K N Miu and H D Chiang ldquoService restoration for unbal-anced radial distribution systems with varying loads solutionalgorithmrdquo in Proceedings of the IEEE Power Engineering SocietySummer Meeting vol 1 pp 254ndash258 1999
[5] Y X Yu ldquoSecurity region of bulk power systemrdquo in Proceedingsof the International Conference on Power System Technology(PowerCon rsquo02) vol 1 pp 13ndash17 Kunming China October2002
[6] F Wu and S Kumagai ldquoSteady-state security regions of powersystemsrdquo IEEE Transactions on Circuits and Systems vol 29 no11 pp 703ndash711 1982
[7] Y Yu Y Zeng and F Feng ldquoDifferential topological character-istics of the DSR on injection space of electrical power systemrdquoScience in China E Technological Sciences vol 45 no 6 pp 576ndash584 2002
[8] S J Chen Q X Chen Q Xia and C Q Kang ldquoSteady-state security assessment method based on distance to securityregion boundariesrdquo IET Generation Transmission amp Distribu-tion vol 7 no 3 pp 288ndash297 2013
[9] D M Staszesky D Craig and C Befus ldquoAdvanced feederautomation is hererdquo IEEE Power and Energy Magazine vol 3no 5 pp 56ndash63 2005
[10] C L Smallwood and J Wennermark ldquoBenefits of distributionautomationrdquo IEEE Industry Applications Magazine vol 16 no1 pp 65ndash73 2010
[11] X Mamo S Mallet T Coste and S Grenard ldquoDistributionautomation the cornerstone for smart grid development strat-egyrdquo in Proceedings of the IEEE Power amp Energy Society GeneralMeeting (PES rsquo09) pp 1ndash6 July 2009
[12] P W Sauer B C Lesieutre and M A Pai ldquoMaximumloadability and voltage stability in power systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 15 no 3 pp145ndash153 1993
[13] K N Miu and H Chiang ldquoElectric distribution systemload capability problem formulation solution algorithm andnumerical resultsrdquo IEEE Transactions on Power Delivery vol 15no 1 pp 436ndash442 2000
[14] F Z Luo C S Wang J Xiao and S Ge ldquoRapid evaluationmethod for power supply capability of urban distribution sys-tem based on N-1 contingency analysis of main-transformersrdquoInternational Journal of Electrical Power andEnergy Systems vol32 no 10 pp 1063ndash1068 2010
Journal of Applied Mathematics 13
[15] J Xiao F X LiWZGuC SWang andP Zhang ldquoTotal supplycapability and its extended indices for distribution systemsdefinitionmodel calculation and applicationsrdquo IETGenerationTransmission and Distribution vol 5 no 8 pp 869ndash876 2011
[16] J Xiao X X Gong and C S Wang ldquoTopology properties andalgorithm of N-1 security boundary for smart gridrdquo Proceedingsof the CSEE vol 34 no 4 pp 545ndash554 2014 (Chinese)
[17] G E Shilov An Introduction to the Theory of Linear SpacesDover New York NY USA 1975
[18] J R Munkres Topology vol 2 Prentice Hall Upper SaddleRiver NJ USA 2000
Figure 9 2D DSSR cross-section and preventive control
Table 4 Distance to boundaries of 1198611and 119861
10atW10158401
119861119894
1198611
11986110
119871119894(MVA) 0778 0778
Euclidean space Then the DSSR model for both substationtransformer and feeder contingency is proposed which is theset of all operating points each ensures that the system N-1 issecure
Second to perform research on the characteristic ofDSSR the boundary formulation for DSSR is derived fromthe DSSR model Then three generic characteristics areproposed and proved by rigor mathematical approach whichare as follows (1) DSSR is dense inside (2) DSSR boundaryhas no suspension (3)DSSR boundary can be expressed withthe union of several subsurfaces
Finally the results from a test case and a practical casedemonstrate the effectiveness of the proposed model Thevisualization for DSSR boundary is also discussed to verifythe topological characteristics of DSSR Security assessmentmethod based on DSSR is preliminary exhibited to show theapplicability of DSSR for future smart distribution system
This paper improves the accuracy and understanding ofDSSR which is fundamental theoretic work for future smartdistribution system Further works to improve the accuracyof DSSR model include fully considering power flow voltageconstraints and integration of DGs
Conflict of Interests
The authors declare that there is no conflict of inter-ests regarding the publication of this paper None ofthe authors have a commercial interest financial interestandor other relationship with manufacturers of pharma-ceuticalslaboratory supplies andor medical devices or withcommercialproviders of medically related services
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (no 51277129) and National HighTechnology Research and Development Program of China(863 Program) (no 2011AA05A117)
References
[1] J Xiao W Gu C Wang and F Li ldquoDistribution systemsecurity region definition model and security assessmentrdquo IETGeneration TransmissionampDistribution vol 6 no 10 pp 1029ndash1035 2012
[2] State Grid Corporation of China Guidelines of Urban PowerNetwork Planning State Grid Corporation of China BeijingChina 2006 (Chinese)
[3] E Lakervi and E J Holmes Electricity Distribution NetworkDesign IET Peregrinus UK 1995
[4] K N Miu and H D Chiang ldquoService restoration for unbal-anced radial distribution systems with varying loads solutionalgorithmrdquo in Proceedings of the IEEE Power Engineering SocietySummer Meeting vol 1 pp 254ndash258 1999
[5] Y X Yu ldquoSecurity region of bulk power systemrdquo in Proceedingsof the International Conference on Power System Technology(PowerCon rsquo02) vol 1 pp 13ndash17 Kunming China October2002
[6] F Wu and S Kumagai ldquoSteady-state security regions of powersystemsrdquo IEEE Transactions on Circuits and Systems vol 29 no11 pp 703ndash711 1982
[7] Y Yu Y Zeng and F Feng ldquoDifferential topological character-istics of the DSR on injection space of electrical power systemrdquoScience in China E Technological Sciences vol 45 no 6 pp 576ndash584 2002
[8] S J Chen Q X Chen Q Xia and C Q Kang ldquoSteady-state security assessment method based on distance to securityregion boundariesrdquo IET Generation Transmission amp Distribu-tion vol 7 no 3 pp 288ndash297 2013
[9] D M Staszesky D Craig and C Befus ldquoAdvanced feederautomation is hererdquo IEEE Power and Energy Magazine vol 3no 5 pp 56ndash63 2005
[10] C L Smallwood and J Wennermark ldquoBenefits of distributionautomationrdquo IEEE Industry Applications Magazine vol 16 no1 pp 65ndash73 2010
[11] X Mamo S Mallet T Coste and S Grenard ldquoDistributionautomation the cornerstone for smart grid development strat-egyrdquo in Proceedings of the IEEE Power amp Energy Society GeneralMeeting (PES rsquo09) pp 1ndash6 July 2009
[12] P W Sauer B C Lesieutre and M A Pai ldquoMaximumloadability and voltage stability in power systemsrdquo InternationalJournal of Electrical Power amp Energy Systems vol 15 no 3 pp145ndash153 1993
[13] K N Miu and H Chiang ldquoElectric distribution systemload capability problem formulation solution algorithm andnumerical resultsrdquo IEEE Transactions on Power Delivery vol 15no 1 pp 436ndash442 2000
[14] F Z Luo C S Wang J Xiao and S Ge ldquoRapid evaluationmethod for power supply capability of urban distribution sys-tem based on N-1 contingency analysis of main-transformersrdquoInternational Journal of Electrical Power andEnergy Systems vol32 no 10 pp 1063ndash1068 2010
Journal of Applied Mathematics 13
[15] J Xiao F X LiWZGuC SWang andP Zhang ldquoTotal supplycapability and its extended indices for distribution systemsdefinitionmodel calculation and applicationsrdquo IETGenerationTransmission and Distribution vol 5 no 8 pp 869ndash876 2011
[16] J Xiao X X Gong and C S Wang ldquoTopology properties andalgorithm of N-1 security boundary for smart gridrdquo Proceedingsof the CSEE vol 34 no 4 pp 545ndash554 2014 (Chinese)
[17] G E Shilov An Introduction to the Theory of Linear SpacesDover New York NY USA 1975
[18] J R Munkres Topology vol 2 Prentice Hall Upper SaddleRiver NJ USA 2000
[15] J Xiao F X LiWZGuC SWang andP Zhang ldquoTotal supplycapability and its extended indices for distribution systemsdefinitionmodel calculation and applicationsrdquo IETGenerationTransmission and Distribution vol 5 no 8 pp 869ndash876 2011
[16] J Xiao X X Gong and C S Wang ldquoTopology properties andalgorithm of N-1 security boundary for smart gridrdquo Proceedingsof the CSEE vol 34 no 4 pp 545ndash554 2014 (Chinese)
[17] G E Shilov An Introduction to the Theory of Linear SpacesDover New York NY USA 1975
[18] J R Munkres Topology vol 2 Prentice Hall Upper SaddleRiver NJ USA 2000
