Total supply capability and its extended indices for distribution systems: definition, model calculation and applications

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1 wwwietdlorg Published in IET Generation, Transmission & Distribution Received on 23rd November 2010 Revised on 23rd February 2011 doi: /iet-gtd ISSN Total supply capability and its extended indices for distribution systems: definition, model calculation and applications J Xiao 1 F Li 2 WZ Gu 1 CS Wang 1 P Zhang 3 1 Key Laboratory of Smart Grid of Ministry of Education, Tianjin University, People s Republic of China 2 Dept of EECS, University of Tennessee, Knoxville, TN, USA 3 Resource Operating Group, Accenture, Beijing, People s Republic of China xiaojun@tjuedncn Abstract: This study systematically proposes and defines a set of new quantitative indices named total supply capability (TSC) and its associated indices, such as available supply capability, substation supply capability, network transfer capability, maximum supply capability and expandable supply capability The concepts in the TSC family provide new approaches to analyse and optimise distribution networks TSC for distribution systems is comparable to the total transfer capability for transmission systems, but there are significant differences, such as the radial configuration and service restoration in TSC consideration TSC is defined as the maximum load to serve under the expanded N 2 1 security guideline for the distribution system, taking into account the capacities of the substation transformers, network topology, link capacity and other constraints Here, the expanded N 2 1 security means a successful, near-immediate service restoration to all loads except for the faulted section A mathematical model for TSC is then set up and an accurate method is presented to calculate the TSC with lingo Sample applications are provided to illustrate a new TSC-based approach to evaluate the link capacity and obtain the reconductoring proposal The results from a real distribution system demonstrate the effectiveness of the proposed TSC modelling and TSC-based approach 1 Introduction Distribution planning and expansion are challenging tasks for power utilities In developed countries like the US and many European countries, it is difficult and expensive, if not impossible, to expand the existing distribution systems to obtain new substation sites and underground paths for new feeders Thus, much attention has been given to the power supply capabilities (PSC) of the existing network and its potential after optimisation Meanwhile, in emerging and fast-developing markets like China, the rapid expansion of distribution systems makes it necessary to evaluate the PSC for not only the existing distribution systems, but also planned future distribution systems [1] In the past, the capacity of a distribution substation, or its transformers, was empirically analysed based on the N 2 1 guidelines, which assume that transformers in the same substation can support each other when one of them is unavailable [2] The capacity load ratio is a critical index in distribution system planning [3] This approach is practical and widely used by engineers, but is too simple and lacks the mathematical rigors Also, the supply capability of the distribution system may not be fully utilised Under the global trend towards a smart or smarter grid, this motivates research work for a more rigorous approach to evaluate the supply capabilities of distribution systems in which multiple substations are interconnected with normally open switches A practical, approximate method for PSC calculation based on N 2 1 contingency analysis of interconnected maintransformers was proposed in the literature [4, 5] However, the research in [4, 5] is confined to solve an actual engineering problem and not rigorously modelled So, it is necessary to perform research work based on a more rigorous and generalised approach This paper gives a systematic, mathematical model of total supply capability (TSC) and associated indices for distribution systems More important, the proposed optimisation-based algorithm for TSC is shown to be more accurate than the algorithm in [4, 5] Additionally, two important practical conditions, the conductor capacity constraints of feeders and the transformer overload factor, are included in the proposed TSC model, which were not considered in [4, 5] Since the original name, power supply capability, is not very precise in terms of defining whether it means the entire capability or the available capability, we re-define it as TSC to distinguish it from available supply capability (ASC), which is the TSC minus the existing capacity of load supply Hence, TSC is similar to total transfer capability (TTC) in transmission systems, whereas ASC is similar to ATC Besides the introduction of TSC and ASC, this paper also introduces other concepts in the TSC family, such as substation supply capability (SSC), network transfer capability (NTC), maximum supply capability (MSC) and expandable supply capability (ESC) With these concepts, the strength and the bottleneck of the supply capability in a distribution system can be easily identified and presented for decision makers IET Gener Transm Distrib, 2011, Vol 5, Iss 8, pp doi: /iet-gtd & The Institution of Engineering and Technology 2011

2 wwwietdlorg 2 Definition for TSC and extended concepts 21 Expanded N 2 1 and TSC definition TSC defines the capability that a system can supply the load when considering an expanded concept of the N 2 1 security for distribution systems Here, if a distribution system is N 2 1 secure, it means that any fault in the primary feeders or substation transformers will lead to a service interruption of the faulted section only Hence, the service to the upstream of a faulted feeder section will be restored after opening the first switchable component upstream to the faulted section, and the service to the downstream will be restored through back feed via closing a normally open switch Thus, all sections other than the faulted one will experience a service interruption of a very short duration This distribution system N 2 1securitymeansasuccessful supply of all loads, except the ones in the faulted section, after restoration via automated switchable components Note that the national electric code in China has enforced the above-expanded N 2 1 security for all new and upgraded urban distribution systems [3] Similar rules are also observed in some European cities [6] Thus, the TSC of a distribution system is the maximum load that it can serve under the expanded N 2 1 guideline for distribution systems, taking into account the capacities of substation transformers, network topology, network capacity and operational constraints With this new TSC concept, utility engineers can evaluate the maximum load that an existing system can supply to identify its potential or hidden capacity, as well as the true maximum load it can supply, with N 2 1 security considered 22 Comparison of TSC and similar concepts There is a concept similar to TSC in transmission systems, which is called the TTC The TTC indicates the overall capability of an interconnected system to transfer power from one area to another under a set of security constraints [7, 8] The proposed TSC concept differs from the TTC because it addresses distribution systems which must remain radial before and after the restoration process, whereas transmission systems are naturally interconnected (looped) and do not require radial configuration In addition, there is no topology-oriented service restoration considered in a typical TTC analysis Also, in distribution systems, loadability is a widely used index to define the maximum loading level that can be supplied, whereas the voltages at all nodes are kept within the prescribed limits Although loadability is similar to the proposed concept of TSC, there is a significant difference Loadability is analysed or optimised based on the possible reconfiguration under the radial distribution systems [9, 10], but without the consideration of the N 2 1 cases More importantly, feeders in urban areas are usually short in length, and so the most critical problem is not voltage drop, but overloading in case of contingencies Hence, the loadability concept is more suitable for rural distribution systems with long feeders and considerable voltage drops, whereas the proposed TSC concept is more suitable for urban distribution systems where reliability or N 2 1 security is a high concern TSC also differs from service restorations Numerous algorithms have been presented in the area of service restoration when an N 2 1 contingency occurs However, although these papers produced acceptable results for the research of service restoration at a certain load level [11, 12] or varying loads [13], none of them have been applied to calculate the maximum load that the network can supply under the conditions of N 2 1 and restoration 23 Extended concepts from TSC In this paper, several concepts extended from TSC are presented as follows: ASC: The amount of that a distribution system can supply in addition to its existing load Hence, it is equal to TSC minus the existing load SSC: The TSC of a distribution system when each feeder is not linked with any other That is, any substation transformer has no connection (through normally open switches) with other substation transformers This is the TSC of the system at its lowest level NTC: The amount of TSC increased by interconnections among feeders Hence, it is equal to TSC minus SSC MSC: The TSC of a system when it is fully linked Here, a fully linked distribution system means that every substation transformer has interconnection with all other transformers in the system So, MSC means the maximum TSC that a system can reach through network linkage ESC: The maximum potential amount of TSC can be increased by expanding the distribution network until it is fully linked without increasing the substation capacity Hence, ESC is equal to MSC minus the TSC of the current system ASC is designed for security assessment on distribution systems and indicates the available load to supply as the system is SSC and NTC are two parts of TSC, which are defined to analyse TSC composition and can be indication of the strength or bottleneck of the system TSC MSC represents the maximum load that a system can supply after a full network linkage is implemented ESC indicates the potential growth of TSC by enhancing or expanding distribution networks via connecting two feeders with a normally open switch 3 Mathematical model for TSC 31 Links of distribution systems In a distribution system, a transformer can be connected with a transformer in the same substation or a transformer in another substation through feeders with tie lines The interconnection between transformers is designated as a link [5] One link may stand for one or several tie lines Two types of links exist, the link within substation (LWS) and the link through feeders (LTF) The set consisting of transformer i and all transformers that have links with transformer i is designated as Link Unit i (LU i ) The security and reliability of a distribution system are enhanced by these links In a typical N 2 1 case (ie the fault occurs at transformer i), the load of transformer i is transferred to transformers within the local substation and transformers in other substations through the LWS and/or LTF After this transfer, loads served by transformers in the same substation with transformer i cannot exceed their allowed short-term capacity, whereas transformers in other substations cannot be overloaded as well The transfer is also constrained by the capacity of the LTFs 32 Modelling of TSC Therefore the objective function and constraints of the TSC for a distribution system can be mathematically expressed as max TSC = R i T i (1) 870 IET Gener Transm Distrib, 2011, Vol 5, Iss 8, pp & The Institution of Engineering and Technology 2011 doi: /iet-gtd

3 wwwietdlorg such that R i T i = tr ij + tr ij ( i) (2) j[v (i) 1 j[v (i) 2 tr ij + R j T j kr j ( i, j [ V (i) 1 ) (3) tr ij + R j T j R j ( i, j [ V (i) 2 ) (4) tr ij RL ij ( i, j [ V (i) 1 < V(i) 2 ) (5) T min T i T max ( i) (6) where R i ¼ rated capacity of transformer i; T i ¼ loading rate of transformer i; tr ij ¼ quantity of the load that is transferred from transformer i to transformer j when a N 2 1 fault occurs to transformer i; k ¼ overload factor of transformers allowed for a short time, assuming all transformers have the same k value (usually k ¼ 13); RL ij ¼ transfer capacity of the LTF between transformer i and transformer j; V (i) 1 ¼ set of transformers that have a link (LWS) to transformer i within the local substation and; V (i) 2 ¼ set of transformers that have a link (LTF) to transformer i in other substations; T min ¼ lower bound of T i ;andt max ¼ upper bound of T i The model is defined as a linear programming problem, which can be solved by software tools such as Lingo Equation (1) shows the objective function of the TSC, which is determined by the capacity and loading rate of the substation transformers in a distribution system Equations (2) (6) represent the equality and inequality constraints of the TSC Equation (2) shows that the sum of the load transferred to other transformers must be equal to the load on transformer i where the fault occurs Inequality constraints (3) and (4) describe that the load on transformers in the same substation with transformer i cannot exceed their short-term capacity rating, whereas transformers in other substations also cannot be overloaded after the load transfer in an N 2 1 case If the right side of inequality (3) or (4) increases (ie transformers capacity increase), TSC will increase The inequality constraint in (5) represents that the load transferred in an N 2 1 case cannot exceed the capacity of any LTF that assumes the transferred load If RL i,j is not large enough, it will limit the TSC, which is discussed in Section 5 The inequality constraint (6) shows that loading rate of each transformer should be in a reasonable range T min is 05 and T max is 09 in the study case Our research has shown that loading rate of some transformer will be 0 whereas some of them will be 10, if this model is solved directly by Lingo without considering (6) It is observed that TSC will increase with a more broad range of load rate (ie T min decreases or T max increases) Several operational constraints also need to be considered One constraint is that the load transfer through buses in the same substation has a higher priority than the load transfer to other substations through feeders Additionally, a load can only be transferred once so that there is no chained transfer and the total number of switch operations will not be excessive Finally, the network must remain radial both before and after the restoration process Result of this model includes not only the TSC, but also a vector {T i }, which are maximum transformer loading rates when the system TSC reaches its maximum value This vector is named as the TSC transformer loading rate, denoted as T TSC T TSC is the optimal transformer loading rates considering N 2 1 security, which can be used as the security limit for distribution system operation 33 ASC and security assessment As can be seen from the above model, the N 2 1 security is applied as a constraint This is different from several common practices in planning, which verifies the N 2 1 after a planning or expansion scheme is obtained Hence, the traditional N 2 1 verification is a necessary condition for security that tends to be conservative and does not guarantee the maximum utilisation of system capacity Therefore the load levels below the TSC should pass the N 2 1 security check In this sense, the TSC represents the true security region of the distribution system Since the TSC represents the security region of a distribution system, ASC can be defined for security assessment compared to available transfer capability (ATC) in the transmission system ASC is the amount of that a distribution system can supply in addition to its existing load ASC = TSC Ld (7) where Ld is the current total load of a distribution system The ASC transformer loading rate can be defined as T ASC = T T TSC (8) where T is the vector of current transformer loading rates; T TSC is the TSC transformer loading rate T ASC represents available loading capacity within the system N 2 1 security margin for each transformer, which is very useful for operators 34 SSC, NTC, MSC and ESC It is necessary to find efficient ways to improve or optimise the TSC for distribution systems The TSC for a distribution system can be attributed to two factors One is the capacities of transformers; the other is the links among transformers As previously explained, the security and reliability of a distribution system are enhanced by links More links means more paths for load transfer in the N 2 1 cases, which can enhance the loading rate of transformers and TSC as well Thus, there should be enough number of links in a distribution system to guarantee the TSC meeting a criterion In most cases, a transformer always has LWS within the substation, and the transfer capacity of LWS is large enough Obviously, if any transformer in a distribution system has no LTF with other transformers, the TSC of the system is at its lowest level, which is the SSC The NTC of a distribution system is defined as the amount of TSC increased by LTFs, which is expressed as NTC = TSC SSC (9) When a distribution system is fully linked, which means every transformer has LWSs or LTFs with all other transformers in the system, and the link capacity is enough for load transfer, the TSC will reach its highest level, that is, MSC The TSC of a distribution system can be increased by either increasing SSC or NTC To increase SSC, it is necessary to IET Gener Transm Distrib, 2011, Vol 5, Iss 8, pp doi: /iet-gtd & The Institution of Engineering and Technology 2011

4 wwwietdlorg build new substations or expand existing substations, which is costly In contrast, optimising the distribution network to achieve higher NTC is much more economic Hence, this paper focuses on the method of NTC exploration The expandable network TSC for a distribution system is defined as the maximum TSC that can be increased by building more LTFs or tie lines This is denoted as ESC, which is given by ESC = (MSC TSC) (10) Further, the ESC% is the ratio of ESC against SSC ESC% = 100 ESC/SSC (11) Obviously, when a distribution system has no tie line, or operators do not perform load transfer through network, the ESC will reach its highest value, which is equal to MSC SSC This is defined as maximum expandable TSC, which is denoted as ESC max and ESC% max ESC max = (MSC SSC) (12) ESC% max = 100 (MSC SSC)/SSC (13) The maximum ESC is the theoretically maximum TSC range that can be increased by means of network transfer 4 Calculation for TSC When TSC is obtained, it is easy to calculate ASC, SSC, NTC, MSC and ESC by their definitions So this section focuses on TSC calculation This paper transforms the TSC model in (1 6) into a standard linear programming problem so that common linear programming tools like lingo can be applied to calculate TSC Suppose there are n substations in a distribution system and N 1, N 2,, N n transformers in each of these n substations, respectively Hence, the total number of transformers in this system is N S ¼ N 1 + N 2 + +N n We first define a matrix named L to represent links among transformers Since there are two types of links, LWS and LTF, L can be expressed as L 1,1 L 1,i L 1,NS L = L 1,i L i,i L i,ns L NS,1 L NS,i L NS,N S = L in + L out (14) where L i,j ¼ 1 when there is a link between transformers i and j L in and L out represent the LWS and LTF, respectively Transformers are usually allowed to be overloaded for a short period of time when a fault occurs Hence, we may multiply the LWS by an overload factor k such that L is modified to L with the following expression L = k(l in I) + L out + I (15) Next, we may define matrices R and R to show the capacity of each transformer in the system, which can be, respectively, expressed as R = diag(r 1, R 2, R i, R NS ) (16) where R i is the rated capacity of transformer i R = L linkr (17) where R i,j is the capacity of transformer j in LU i taking the overload factor k into account A matrix RL is defined to represent the capacities of the tie lines in the system 0 RL 1,i RL 1,NS RL = RL i,1 0 RL i,ns RL NS,1 RL NS,i 0 (18) where RL i,j is the total transfer capacity of the tie line(s) between transformers i and j A matrix Tr is defined by the following expression 0 Tr 1,i Tr 1,NS Tr = Tr i,1 0 Tr i,ns Tr NS,1 Tr NS,i 0 (19) where Tr i,j represents the load to be transferred from transformer i to j when the fault occurs at transformer i Define a vector named T where T i is the loading rate of transformer i Define T avg to describe the loading rate of the entire distribution system On the basis of the mathematical model of TSC and definitions of the matrix above, the objective function and constraints of the TSC for a distribution system can be transformed into the following expression which can be input into lingo such that max TSC = R i T i (20) R i T i = N S i=1 L i, j Tr i, j (21) L i,j (Tr i,j + R j T j ) R i,j (22) Tr ij RL ij (23) T min T i T max (24) Then the objective value calculated by lingo is the TSC for the distribution system 872 IET Gener Transm Distrib, 2011, Vol 5, Iss 8, pp & The Institution of Engineering and Technology 2011 doi: /iet-gtd

5 wwwietdlorg 5 TSC-based applications to reconductoring The TSC can be used as a new techno-economical index together with reliability, loss ratio etc to evaluate and optimise distribution systems The most common and difficult problem for a distribution planner may not be the network planning from scratch, but the expansion or optimisation of currently operating networks [14] This section discusses applications based on the TSC concept to illustrate how to analyse and evaluate the link capacity and obtain the reconductoring proposal for a currently operating network Resizing the conductor in distribution systems is a common approach to supply more load in an existing network The optimal conductor determined will maximise the total savings of the cost of conductor materials and energy losses [14] However, most research on conductor sizing in distribution systems is performed on the basis of the radial network [15] It is an increasing trend to interconnect radial circuits in urban distribution systems to improve security and reliability The capacity of some links may not be sufficient for a load transfer in the case of an N 2 1 contingency It is not surprising that the load transfer under a presumable fault may be limited by various bottleneck links It is a challenge to locate these bottleneck links for a large-scale interconnected distribution system Planners have to determine the locations of bottleneck links based on their engineering knowledge and experience, but this is not optimal or accurate Another approach can be an extensive N 2 1 contingency analysis However, the information given by an N 2 1 analysis is limited and indirect for locating bottleneck links, because it is not easy to summarise many N 2 1 cases to discern the cause of the bottleneck links Furthermore, the N 2 1analysis results depend on a certain load level, which means the result can vary when the load changes Obviously, the time-consuming N 2 1 approach is not suitable for this application, though it is acceptable for verification purposes The above challenge can be addressed well by the TSC-based approach Below is a discussion on how it differs from a traditional N 2 1 security analysis The proposed TSC-based link capacity analysis method consists of two parts: location of the weak links and the determination of the necessary capacity of the links Tr i,j represents the load to be transferred from transformer i to j when the fault occurs at transformer i To calculate matrix Tr rapidly, suppose all transformers in the same LU have the same loading rate, we have Tr i, j = R i, j R SNS j=1,j=i R i, j S N S j=1 R (i = j) j L i, j 0 (i = j) (25) The existence of weak links decreases the TSC In other words, this approach can automatically find all weak links that should be given higher priority in the construction or planning stage This will result in a better TSC of the whole distribution system with less investment 52 Necessary capacity of links On the other hand, using large-capacity conductors anywhere in a distribution network is not economical A new approach based on the TSC can determine the largest capacity that a link needs with any capacity more than this value considered as an unnecessary investment Define a matrix U ¼ max{tr, Tr T }, where U i,j describes the maximum quantity of the load that may be transferred through the tie line(s) between transformers i and j Thus, U i,j is the necessary capacity of the link A tie line with a capacity equal to U i,j is enough to guarantee the TSC of the given distribution system It is necessary to point out that if there is more than one tie line between two transformers, the necessary capacity is the sum of their capacities Define a matrix S ¼ max{rl 2 U, 0}(if L in i, j = 0, S i,j ¼ 0), where S i,j is named the spare capacity of the link between transformers i and j If S i,j is too large, it means there is a waste of material in the link from transformer i to j 6 Case study of a real system 61 Overview of the test system In this section, the evaluation method of the TSC concept is tested on a real distribution system with 44 nodes and six transformers in three substations (Fig 1) The data of the test system are shown in Tables 1 and 2 Both the overload factor (k ¼ 13) and the conductor name are based on the national electrical code in China The conductors of all the main feeders are JKLYJ-185 with a capacity of Location of weak links Define a matrix V ¼ max{tr 2 RL, 0}and let V i,j represent the shortfall of the link from transformer i to j for the load transfer needed in the LU i Thus, W i,j ¼ max{v i,j, V j,i } is the shortfall of the link (i, j) in capacity When W i,j 0, the total capacity of the tie line(s) between transformers i and j is not sufficient for load transfer and link (i, j) is named the weak link Fig 1 Test distribution system with 44 nodes IET Gener Transm Distrib, 2011, Vol 5, Iss 8, pp doi: /iet-gtd & The Institution of Engineering and Technology 2011

6 wwwietdlorg Table 1 Table 2 case Substation data of the network Substation Transformer Voltage, kv/kv 62 TSC and ASC calculation Capacity, Based on (14 19), we have matrices L, R, RL, Tr L = RL = Load, S1 1 35/ / S2 3 35/ / S / / Conductors and capacities of the tie lines in the study Tie line Conductor Capacity, JKLV JKLYJ JKLYJ JKLYJ JKLYJ JKLYJ With the aid of lingo, the TSC is calculated through (20) (24) The TSC of the test system is and the TSC transformer loading rate T TSC = (0650, 0650, 0840, 0832, 0740, 0726) ASC of the test system is 492 and the ASC transformer loading rate T ASC = (0150, 0025, 0465, 0082, 0264, 0059) The ASC and T ASC have shown the overall and individual quantitative distance to the N 2 1 security margin of a distribution system, which are very useful to operators and the planners The data have shown that the current loading of the test system is secure and load of transformer two and six are near their N 2 1 security margin 63 Comparison of TSC with previous research Although TSC and PSC share the same N 2 1 security concept, their mathematical approaches are different To compare TSC in this paper and the PSC in [4, 5], itis assumed that k (the overload factor of transformers) is 10 and the capacities of tie lines are large enough Hence, the TSC has the same assumptions as the PSC The TSC result for the case system in [5] is 915, which is the same as the result in [5] Nevertheless, the results of PSC and TSC can be different For instance, if we remove tie line (3, 5) and (4, 6) from the case system in [5], the TSC is still 915, whereas the PSC decreases to 8538 That means building link (3, 5) and (4, 6) will not increase the TSC of the system, but in PSC calculation these two links are necessary to guarantee the PSC, which will bring more investment in the construction Table 3 shows the difference between PSC and TSC when the number of links changes in the test system It can be seen from Table 3 that the PSC may give significant error (up to 108%) with respect to the TSC algorithm proposed in this paper It tends to be lower than or equal to TSC The error is because of approximation in the PSC algorithm in [4, 5] In 42 of [5], when analysing the maximum load of each LU, the method assumes all the transformers in the same LU have the same loading rate, which is too ideal and may miss the maximum objective value of supply capability In short, the model-based TSC gives more accurate results than the approximate PSC calculation The main reason is that the TSC approach is based on an optimisation model while the previous PSC employed a direct algorithm with assumptions and approximations 64 SSC, NTC, MSC and ESC When there are no LTFs in the test system, the TSC reaches the lowest, where SSC is 143 NTC is ¼ 6820, that is the SSC and NTC accounting for 678 and 323%, respectively, of the TSC Table 3 of links Difference between PSC and TSC of different number Number of links TSC, PSC, PSC error, PSC error, % 0 211% 2108% 274% 240% 0 Table 4 Maximum ESC (k ¼ 13) Scale of system SSC, MSC, ESCmax, ESC%max, % 6-transformer transformer IET Gener Transm Distrib, 2011, Vol 5, Iss 8, pp & The Institution of Engineering and Technology 2011 doi: /iet-gtd

7 wwwietdlorg in the test system Also, ESC is 118 and ESC% is 559% On the contrary, when the system is fully linked, TSC reaches the highest, where MSC is 223 Another system with 10 substations and 22 transformers is also employed The maximum ESC of the two systems is in Table 4 The maximum potential of TSC increased by optimising distribution network is very considerable, which is more than 30% (if k ¼ 13) or 50% (if k ¼ 10) of the SSC Also, a larger distribution system has more potential TSC to be used 65 TSC-based application: reconductoring The matrix W is calculated as follows W = Based on the W, the weak links can be located and the shortfall in capacity of these links can be obtained as well It is determined that there are two weak links: the link from transformer three to six and the link from four to six, which will limit the load transfer when a fault occurs Next, different solutions and a comparison of the weak links are discussed Solution to the weak links: Strengthen the weak link (3, 6): This is to change the conductor size of the tie line between transformers 3 to 6 from JKLYJ-70 to JKLYJ-150 Thus, the capacity of link (3, 6) increases from 443 to 883 Then, the TSC increases from to Strengthen the weak link (4, 6): This is to change the conductor of between transformers 4 to 6 from JKLYJ-95 to JKLYJ-150 Thus, the capacity of link (3, 6) increases from 602 to 883 Then, the TSC of the test system increases from to Strengthen both the weak links (3, 6) and (4, 6): This is to change the conductors of (3, 6) and (4, 6) to JKLYJ-150 After this upgrade, no weak link exists in the system, and W ¼ 0 Then, the TSC is Comparison of weak links: Through the W matrix, the weak links of the test system are identified, eg (3, 6) and (4, 6) with 4133 and 2543 shortfalls in capacity, respectively Since the shortfall of the link (3, 6) is greater than that of the link (4, 6), a conclusion can be reached that the weakness on link (3, 6) is the most critical It is advisable to strengthen the weakest link in the system first in order to efficiently enhance the TSC of a distribution system Necessary capacity of tie lines: Matrix U can be helpful for choosing the right conductor size of the tie lines between feeders or transformers A tie line with a capacity equal to U ij is sufficient to guarantee an adequate TSC of a distribution system It is inadvisable to choose a tie line with a capacity less than or greater than the necessary capacity The spare capacity of the tie lines in the current system can be obtained as shown in matrix S U = S = Links with a great spare capacity can be identified through matrix S For example, a tie line between transformers 2 and 5 is located with a spare capacity of 1661, which is unnecessary Assuming that the conductor of link (2, 5) can be changed from JKLYJ-185 to JKLV-185 to reduce costs, the capacity of link (2, 5) will decrease from 113 to 102 The TSC of the test system is still 21120, which has not decreased 7 Conclusions This paper proposes the concept and index named TSC as well as its extended indices for distribution systems The TSC is the maximum load that it can serve under the expanded N 2 1 guideline for distribution systems The ASC is the quantitative distance to the N 2 1 security margin, which is very useful to operators and the planners SSC and NTC can collectively give an overview of the strength and weakness of the distribution system supply capability such as whether the substation is strong or the network interconnection is strong The MSC is the maximum TSC that a system can reach through network linkage ESC is defined to help utility engineers to exploit the potential TSC The maximum ESC can be increased by network enforcement for a typical distribution system is considerable, which is about 30 50% of the SSC An optimisation-based model for TSC is then set up This rigorously model-based approach gives the true optimal result on supply capability for distribution systems In addition, the TSC calculation is fast Calculation for the case takes less than 1 s in a PC (Pentium4 30 GHz-CPU-1GBRAM) and 140 s for another system with 100 substations and 1389 nodes Finally, this paper regards TSC family not only as new indices but also as a new methodology for the evaluation and optimisation of distribution systems Samples on conductor selection are given to demonstrate TSC-based applications, in which both weak and unnecessary large links in a distribution network are efficiently located 8 Acknowledgments This work was supported by the National Basic Research Program of China (973 Program, No 2009CB219703) IET Gener Transm Distrib, 2011, Vol 5, Iss 8, pp doi: /iet-gtd & The Institution of Engineering and Technology 2011

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