HAO, Zhiguo (Xi'an Jiaotong University, Xianning west road 28Xi'an, 9, 71004, CN)
WANG, Chengen (Xi'an Jiaotong University, Xianning west road 28Xi'an, 9, 71004, CN)
BO, Zhiqian (2 Park Avenue, Bath BA2 4QD, GB)
KLIMEK, Andrzej (5918 - 149a St, Surrey, BC VS3 7W8, CA)
ZHANG, Baohui (Xi'an Jiaotong University, Xianning west road 28Xi'an, 9, 71004, CN)
HAO, Zhiguo (Xi'an Jiaotong University, Xianning west road 28Xi'an, 9, 71004, CN)
WANG, Chengen (Xi'an Jiaotong University, Xianning west road 28Xi'an, 9, 71004, CN)
BO, Zhiqian (2 Park Avenue, Bath BA2 4QD, GB)
KLIMEK, Andrzej (5918 - 149a St, Surrey, BC VS3 7W8, CA)
| CLAIMS 1. A system for controlling a real-time islanding of an electrical power network (203) for mitigating the impact of disturbances which the power network may experience, said power network comprising generator buses (209), load buses (211) and transmission lines (213) interconnecting said buses, characterised in that said system comprises: - control means (205) for identifying different classes of coherent generator buses, - control means (205) for subdividing the power network into disjoint domains (231a-231d) where each domain is composed of a single generator bus and a corresponding set of load buses, - control means (205) for separating said disjoint domains (231a-231d) into two different groups, a first group consisting of domains (231a, 231b) having coherent generator buses and a second group (231c, 231d) consisting of domains having all other generator buses, - control means (205) for determining an initial partition boundary (217a) composed of a set of partition lines which correspond to transmission lines connecting first group buses to adjacent second group buses, - control means (205) for transferring one by one load buses which are linked by said partition lines from one group of said first and second groups to the other one of said two groups while updating at each time the elements of both groups and the corresponding set of partition lines until a minimum load-generation imbalance within both updated groups has been reached, - control means (205) for determining a final partition boundary (217) composed of the updated set of partition lines, and - splitting means (207) for splitting the power network into two islands (303a, 303b) according to the final partition boundary. 2. The system according to claim 1, characterised in that it comprises measuring means (207) for measuring electrical information conveyed by the transmission lines of the power network, said measured electrical information are used by the control means (205) to identify the different classes of coherent generator buses . 3 . The method according to claim 1 or 2, characterised in that said control means (207) are configured to form each domain by associating the set of load buses to its corresponding generating bus on the basis of a power distribution representing the electrical power contribution of each generator bus to each load bus in the power network . 4. The system according to claim 3, characterised in that said control means (205) are configured to determine said power distribution by applying a flow tracing algorithm on electrical information measured upon the transmission lines. 5. The system according to claim 3 or 4, characterised in that said control means (205) are configured to associate each load bus to the generator bus from which it receives most of its electrical power. 6. The system according to claim 5, characterised in that if a load bus receives the same electrical power from a set of two or more generator buses, then said control means (205) are configured to associate said load bus to one of said set of generator buses on the basis of other criteria comprising either one of the following: the smallest numbered generator bus, the nearest generator bus, the most powerful generator bus, the generator bus belonging to the domain presenting the highest value of generation- load imbalance, or a randomly selected generator bus out of said set of generator buses. 7. The system according to anyone of the preceding claims, characterised in that said control means (205) are configured to realise the one by one transferring order by first moving the load bus which can fastest reduce the load-generation imbalance. 8. The system according to anyone of the preceding claims, characterised in that prior to each transfer of a load bus, said control means (205) are configured to determine the load-generation imbalance by calculating the difference between the total generation power and the total load power within each one of said first and second groups . 9. The system according to anyone of the preceding claims, characterised in that said control means (205) are configured to split said second group into two further islands according to the same steps used to island the initial power network if said second group is found to comprise incoherent generator buses. 10. Power distribution network equipped with an islanding system according to any of the preceding claims. 11. A method for controlling a real-time islanding of an electrical power network (203) for mitigating the impact of disturbances which the power network may experience, said power network comprising generator buses (209) , load buses (211) and transmission lines (213) interconnecting said buses, characterised in that said method comprises the steps of: - identifying different classes of coherent generator buses, - subdividing the power network into disjoint domains (231a-231d) where each domain is composed of a single generator bus and a corresponding set of load buses, - separating said disjoint domains into two different groups, a first group (231a, 231b) consisting of domains having coherent generator buses and a second group (231c, 231d) consisting of domains having all other generator buses, - determining an initial partition boundary (217a) composed of a set of partition lines which correspond to transmission lines connecting first group buses to adjacent second group buses, - transferring one by one load buses which are linked by said partition lines from one group of said first and second groups to the other one of said two groups while updating at each time the elements of both groups and the corresponding set of partition lines until a minimum load- generation imbalance within both updated groups has been reached, - determining a final partition boundary (217) composed of the updated set of partition lines, and - splitting the power network into two islands according to the final partition boundary. 12. Computer-assisted method for controlling a real-time islanding of an electrical power network according to the method of claims 10, characterised in that said power network is represented by a directed graph model established according to flow and topology data of the power network and in which vertices correspond to buses and arcs correspond to transmission lines, each vertex being associated with a weight representing an electric power of its corresponding generator/load bus. 13. A computer program designed to implement the method according to claim 11 or 12 when executed on a computer . |
DISCLOSURE
TECHNICAL FIELD
The present invention concerns the general field of electrical power networks, and more particularly to methods for controlling the islanding of an electrical power network .
STATE OF THE PRIOR ART
In general, electrical power networks are planned and operated to withstand contingencies especially when they are operated very close to their stability limits. However, unexpected events, weak connections or other failures may lead to severe disturbances that may cause a loss of coherence or synchronism between different regions within the power network. If such loss of synchronism occurs, it is essential to separate the incoherent regions before any propagation of disturbances that may cause the power network to lose stability leading to a widespread outage or a catastrophic failure.
One way to deal with large disturbances is to determine within a very short time which generators are coherent or synchronous and then split the power network into controlled self-sustaining islands of coherent generators. However, searching a proper islanding boundary for a large scale power network is a computationally complex problem due to a combinatorial expansion of its sample space. For example, for an IEEE 118-bus power system with 186 branches, there are 2 186 « 9.8 χ 10 55 possible outcomes of islanding boundaries.
Protective equipments such as out-of-step relays have been developed to detect loss of synchronism and to split the power network into controlled different islands to preserve the power network.
Fig. 1 schematically shows an example representing an IEEE 118 -bus electrical power network 203a. The power network comprises generator buses, load buses and transmission lines connecting said buses. Each black dot denotes a generator bus and each white dot denotes a load bus. Each transmission line is denoted by a directed segment whose direction indicates the active flow of electrical power through the transmission line.
If for example a fault occurs on the segment 23-
25 (i.e., the segment connecting dots 23 and 25) , the out- of-step relays (not shown) installed on transmission lines will detect the instability and would normally trip some transmission lines (represented by the broken lines L10- L100) in order to separate the system into two islands. However, an excessive number of transmission lines (the ten broken lines L10-L100) may be tripped causing loss of power supply to some load buses such as the load bus represented by the white dot 33, and most importantly, it may eventually create a load-generation imbalance within each island.
There are several other methods proposed in the literature for determining the unstable mode of a power network which is experiencing power disturbances. An example is given by the prior art document WO03090328 which describes a method for on-line transient stability analysis and in particular, for discriminating stable and unstable contingencies . Some other documents propose different strategies for determining the unstable mode of a power network and the islanding of this network. Such known strategies include a slow coherency method, a real-time monitoring based on WAMS (Wide Area Measurement System) , and an on-line calculation real-time matching.
An example of a slow coherency method is illustrated by Bo Yang, Vijay Vittal, and Gerald T. Heydt in "Slow-coherency-based controlled islanding - A demonstration of the approach on the August 14, 2003 blackout Scenario"; IEEE Trans on Power System, vol. 21, pp. 1840-1846, Nov. 2006. In this document, the authors present an islanding method by constructing a small sub-network using a centre bus, which is one of the buses in the group boundary. This sub-network is referred to as the interface network. An exhaustive search is then conducted on the interface network to determine the cutsets where the islands are formed. For each island candidate, the total load and generation are calculated, and the island with minimum load-generation imbalance is picked up as the optimal cutest.
Another example is illustrated by Xiaoming Wang, Vijay Vittal in "System Islanding Using Minimal Cutsets with Minimum Net flow", Proceedings of the 2004 IEEE PES Power System Conference and Exposition, New York, October 2004. In this document slow coherency is also used to determine coherent generators and uses minimal cutsets technique in graph theory to deal with the islanding of the power network. In particular, the problem is converted into searching for the minimal cutset to construct the island with the minimal net flow.
Other strategies use the technique of an ordered binary decision diagram to simplify the power network into a small number of buses (less than 30 buses) and then a whole space searching is conducted to find a proper splitting boundary in a short time.
However, the above methods are complex and present some difficulties for islanding a large scale power network in a very short time and usually need to undergo system simplifications prior to finding the splitting boundary .
Another islanding strategy is described in Bo Yang, et al . "A novel slow coherency based graph theoretic islanding strategy", Proceedings of the 2004 IEEE PES General Meeting, Florida, June 2007. In this document, slow coherency analysis is used to effectively determine the slowly coherent generator groups among weak connections in the power network. A graphic theoretic method is used to simplify a large scale power network into a smaller network without losing optimal solutions . A multi- level recursive bisection graph partition method is em loyed to partition the reduced network into desired sub-networks with minimum generation- load imbalance.
However, the multi-level graph partitioning method does not seem to be satisfying for adequately considering the specific characteristics of a power network and for a proper islanding of the latter.
The problem to be solved by the present invention is therefore to propose a simple, effective and reliable system for appropriately islanding a large scale electrical power network in real-time. DESCRIPTION OF THE INVENTION
The present invention is defined by a system for controlling a real-time islanding of an electrical power network for mitigating the impact of disturbances which the power network may experience, said power network comprising generator buses, load buses and transmission lines connecting said buses, said system comprising:
- control means for identifying different classes of coherent generator buses,
- control means for subdividing the power network into disjoint domains where each domain is composed of a single generator bus and a corresponding set of load buses ,
- control means for separating said disjoint domains into two different groups, a first group consisting of domains having coherent generator buses and a second group consisting of domains having all other generator buses ,
- control means for determining an initial partition boundary composed of a set of partition lines which correspond to transmission lines connecting first group buses to adjacent second group buses,
- control means for transferring one by one load buses which are linked by said partition lines from one group of said first and second groups to the other one of said two groups while updating at each time the elements of both groups and the corresponding set of partition lines until a minimum load-generation imbalance within both updated groups has been reached, - control means for determining a final partition boundary composed of the updated set of partition lines, and
- splitting means for splitting the power network into two islands according to the final partition boundary.
This controlled islanding system permits to split a large scale power network in a short time based on a practical identification of coherent generators without supplying any intermediate simplified models. The system enables an automatic islanding at a high overall computational speed thus preventing the impact of any disturbance to propagate into the power network.
The system comprises measuring means for measuring electrical information conveyed by the transmission lines of the power network, said measured electrical information is used by the control means to identify the different classes of coherent generator buses. The electrical information (for example, voltage or current flows) in the transmission lines can be easily and rapidly measured to give direct and reliable indication of the coherency behaviour of generator buses.
According to an aspect of the present invention, said control means are configured to form each domain by associating the set of load buses to its corresponding generating bus on the basis of a power distribution representing the electrical power contribution of each generator bus to each load bus in the power network. Power distribution can be easily established in function of the electrical information in the transmission lines and thus can be used as a very reliable criterion for simulating the power network as a union of disjoint domains whose number is simply equal to that of the generator buses.
Said control means are configured to determine said power distribution by applying a flow tracing algorithm on electrical information measured upon the transmission lines .
Advantageously, said control means are configured to associate each load bus to the generator bus from which it receives most of its electrical power. This is a simple, quick and reliable criterion for associating the load buses to a generator bus within each domain.
Advantageously, if a load bus receives the same electrical power from a set of two or more generator buses, then said control means are configured to associate said load bus to one of said set of generator buses on the basis of other criteria comprising either one of the following: the smallest numbered generator bus, the nearest generator bus, the most powerful generator bus, or to the generator bus belonging to the domain presenting the highest value of generation- load imbalance, or a randomly selected generator bus out of said set of generator buses. These are also very simple and quick criteria for associating a particular load bus to a corresponding generator bus .
Advantageously, said control means are configured to realise the one by one transferring order by first moving the load bus which can fastest reduce the load- generation imbalance. This enables a quick determination of the final partition boundary with a minimal number of computational steps.
Advantageously, prior to each transfer of a load bus, said control means are configured to determine the load-generation imbalance by calculating the difference between the total generation power and the total load power within each one of said first and second groups.
According to yet another aspect of the present invention, said control means are configured to split said second group into two further islands according to the same steps used to island the initial power network if said second group is found to comprise incoherent generator buses. This enables the power network to be islanded into three or more islands in a simple manner on a one by one basis.
The invention also provides a method for controlling a real-time islanding of an electrical power network for mitigating the impact of disturbances which the power network may experience, said power network comprising generator buses, load buses and transmission lines connecting said buses, said method comprising the steps of:
- identifying different classes of coherent generator buses,
- subdividing the power network into disjoint domains where each domain is composed of a single generator bus and a corresponding set of load buses,
- separating said disjoint domains into two different groups, a first group consisting of domains having coherent generator buses and a second group consisting of domains having all other generator buses,
- determining an initial partition boundary composed of a set of partition lines which correspond to transmission lines connecting first group buses to adjacent second group buses,
- transferring one by one load buses which are linked by said partition lines from one group of said first and second groups to the other one of said two groups while updating at each time the elements of both groups and the corresponding set of partition lines until a minimum load- generation imbalance within both updated groups has been reached,
- determining a final partition boundary composed of the updated set of partition lines, and
- splitting the power network into two islands according to the final partition boundary.
The invention also concerns a computer-assisted method for controlling a real-time islanding of an electrical power network according to the above method in which said power network is represented by a directed graph model established according to flow and topology data of the power network and in which vertices correspond to buses and arcs correspond to transmission lines, each vertex being associated with a weight representing an electric power of its corresponding generator/load bus.
BRIEF DESCRIPTION OF THE DRAWINGS
Other characteristics and advantages of the invention will become apparent on reading a preferred embodiment of the invention given with reference to the appended figures amongst which:
Fig. 1 schematically shows an example of islanding known in the prior art;
Fig. 2 schematically illustrates a system that can be used for controlling a real-time islanding of an electrical power network according to the invention;
Fig. 3 schematically shows an example of an unstable mode in an IEEE 118 -bus electrical power network; Figs. 4A-4E schematically show an example of a 16-bus electrical power network for illustrating the searching scheme of the most suitable islanding boundary according to the invention; and
Figs. 5A and 5B schematically illustrate the islanding of the 118-bus electrical power network of Fig. 3 according to the invention.
DETAILED DESCRIPTION OF PARTICULAR EMBODIMENTS The present invention presents a real time islanding scheme, which is fit for controlling the islanding of a large scale power network. It is based on the following three major phases: on-line measurement of electrical information, a rapid search of the most suitable islanding boundary if a loss of synchronism is detected, and a rapid split of the power network according to the searched isianding boundary
Fig. 2 schematically shows a system 201 that can be used for controlling a real-time islanding of an electrical power network 203 according to an embodiment of the invention. The system 201 comprises an Islanding Control Centre (ICC) 205 and out-of-step relays 207.
The power network 203 comprises generator buses 209, load buses 211 and transmission lines 213 interconnecting these buses 209, 211. In particular, without loss of generality, the power network 203 is shown to comprise four generators 209a, eleven buses 209, 211, four transformers 215, and transmission lines 213 equipped with sixteen out-of-step relays 207.
The out-of-step relays 207 are protective means
(comprising measuring and splitting means) configured to measure ■ electrical information (voltage, current, impedance, etc.) in order to detect any loss of synchronism and to eventually perform an out-of-step tripping for preserving the power network 203.
Once, a loss of synchronism is detected, the electrical information 218 measured by the out-of-step relays 207 is sent to the ICC (control means) 205 which is configured to determine the unstable mode of the power network 203. In particular, the ICC 205 comprises control and processing means for analysing the received electrical information in order to identify the different regions of synchronous generator buses 209.
Moreover, the ICC 205 is configured to search for the most suitable islanding boundary (represented by the broken line 217) on the basis of the determined unstable mode of the power network 203.
Finally, the ICC 205 sends commands to the corresponding out-of-step relays 207 to trip the lines across the determined islanding boundary 217 in order to separate the power network 203 into different self- sustaining islands.
Fig. 3 schematically illustrates the same electrical power network represented in Fig. 1 for illustrating an example of an unstable mode in the IEEE 118- bus network .
The power network is represented by a directed graph model 203a where vertices having serial numbers 1-118, correspond to buses and arcs correspond to transmission lines. Each vertex may also be associated with a weight (not shown) representing the electric power of its corresponding generator bus or load bus. Each black dot (or black vertex) denotes a generator bus and each white dot (or white vertex) denotes a load bus. Each transmission line is represented by a directed segment (or arc) whose direction indicates the active flow of electrical power through the transmission line .
If a fault occurs on the segment 23-25 (i.e., the arc connecting vertices 23 and 25) , the out-of-step relays 207 installed on the transmission lines (see Fig. 2) will detect the instability and would have normally tripped the ten transmission lines represented by the broken arcs L10-L100 as already described in reference to Fig. 1. However, according to the present invention, the ICC 205 will prevent the out-of-step relays 207 from performing such a hasty islanding and instead, will search for a suitable islanding satisfying an optimal load-generation balance.
More particularly, the electrical information 218 conveyed by the transmission lines that has been measured and transmitted by the out-of-step relays 207 will be analysed by the ICC 205 to identify the different classes of coherent generator buses. Indeed any loss of synchronism creates large variations in voltages, currents, or impedances throughout the power network and thus, the ICC 205 can rapidly identify the different classes of coherent generators in function of the electrical information 218.
According to the example of Fig. 3, the ICC 205 classifies the generator buses into two coherent classes. The first class consists of generator buses identified by the black vertices numbered 10, 12, 25, 26, and 31 whereas, the second class consists of generator buses identified by the black vertices numbered 46, 49, 54, 59, 61, 65, 66, 69, 80, 87, 89, 100, 103, and 111. The generators in each class are synchronous while the unstable mode results from the fact that the generators of the first class are asynchronous with those of the second class.
After determining the unstable mode of the power network, the ICC proceeds to search for the most suitable islanding boundary.
Indeed, Figs. 4A-4E schematically show an example of a 16-bus electrical power network 203b for illustrating the searching scheme of the most suitable islanding boundary. The power network 203b comprises four generator buses having serial numbers 1, 8, 12, and 16 as well as twelve load buses having serial numbers 2-7, 9-11, and 13-15. In addition a weight is attributed to each one of these buses.
The weight of a given bus i is denoted by P± and the weights of all buses are tabulated in Table 1. For example, the weight of bus 2 is P 2 =-2.0, the negative sign of the weight indicates that bus 2 is a load bus and it consumes 2.0 active power units from the network; the weight of bus 1 is Ρχ=10.0, which indicates that bus 1 is a generator bus and it injects 10.0 active power units into the network.
Table 1 Bus weights of a 16-bus power network
Bus serial
1 2 3 4 5 6 7 8
number
i 10.0 -2.0 -4.0 -2.0 -5.0 -5.0 -3.0 1 5.0
Bus serial
9 10 1 1 12 13 14 15 16
number
Pi -8.0 -8.0 -3.0 8.0 -3.0 -1.0 -4.0 15.0 The searching scheme is mainly made up of three main steps: determining domains of the generator buses; searching initial partition boundary; and constructing a final partition boundary. At first, a Wide Area Measure System (WAMS) is used at different intervals to collect on line power flow data and topology data of the power network 203b. Then, based on these data, a graph model of the power network is established in the same manner as the one shown in the example of Fig. 3.
Then, based on the graph model and a flow tracing algorithm, the ICC 205 subdivides the power network 203b into disjoint domains 231a-231d (see closed contours in Fig. 4B) where each domain is composed of a single generator bus and a corresponding set of load buses.
Thus, the number of disjoint domains of the power network 203b (or its graph model) is equal to the number of generator buses and each bus necessarily belongs to only one corresponding domain.
More particularly, the ICC 205 is configured to form each domain by associating the set of load buses to its corresponding generating bus on the basis of a power distribution representing the electrical power contribution of each generator bus to each load bus in the power network 203b. The power distribution is determined by applying the flow tracing algorithm on the electrical information 218 (see Fig. 2) measured on the transmission lines 213. The flow tracing algorithm is a known technique that uses power flows in the transmission lines for analysing the power distribution of generator and load buses throughout the power network (see for example, the document published by Bialek, "Tracing the flow of electricity", IEEE Proceedings Generation Transmission Distribution, vol. 143, pp. 313-320, July, 1996) . According to a first simple and reliable criterion, each load bus is associated to the generator bus from which it receives most of its electrical power.
However, if a given load bus receives the same electrical power from a given set of two or more generator buses then the ICC 205 associates the load bus to one of these generator buses on the basis of other criteria. For example, the given load bus may be associated to the smallest numbered generator bus, or to the nearest generator bus, or to the most powerful generator bus, or to the generator bus belonging to the domain presenting the highest value of generation- load imbalance, or may eventually be associated to a randomly selected generator bus out of the given set of generator buses. Advantageously, the choice may be determined according to the computer program used to implement the method of the present invention.
Based on the application of the flow tracing algorithm to the 16-bus network, the contribution of each generator bus to each load bus is shown in Table 2.
Table 2 : Power distribution determination based on power flow tracing
^ ~~~~--- ^ _ Load bus
2 3 4 5 6 7 9 10 11 13 14 15 Generator bus~~~-— ____
1 2.0 4.0 1.0 0 1.0 2.0 0 0 0 0 0 0
8 0 0 1.0 5.0 3.67 0 5.33 0 0 0 0 0
12 0 0 0 0 0.02 1.0 0.17 0.48 1.0 3.0 1.0 1.33
16 0 0 0 0 0.31 0 2.5 7.52 2.0 0 0 2.67
Table 2 shows for example that the active powers that load bus 6 receives from generator buses 1, 8, 12, and 16 are 1, 3.67, 0.02, and 0.31 respectively. Load bus 6 receives most of its power (3.67 power units) from generator bus 8 and thus, according to the criterion of generator bus domains determination, bus 6 will be in the domain 231b of the generator bus 8. The domains 231a-231d of all generator buses are shown in Table 3 in which the domain of a given generator bus i is denoted by (i) . For example, the domain 231b of generator bus 8 is denoted by (8) .
Table 3 Domains of various generator buses
Domain' s name Buses in each domain
(1) 1, 2, 3, 4, 7
(8) 8, 5, 6, 9
(12) 12, 13, 14
(16) 16, 10, 11, 15
In practice, the power flow data and topology data of the power network can be collected on line at different intervals, knowing that these data do not change significantly in a short time. However, at each interval and after collecting new data, the domains 231a-231d of the various generator buses are updated.
Fig- 4C shows that once the different domains
231a-231d are determined, the ICC 205 separates these disjoint domains into two different groups: a first group consisting of domains 231a and 231b having coherent (synchronous) generator buses (serial numbers 8 and 10) , and a second group consisting of domains 231c and 231d having all other generator buses (serial numbers 12 and 16) .
It is to be noted that usually when a fault occurs, loss of synchronism takes place between only two classes of coherent generator buses. In that case, the generators of the two different groups simply correspond to the two identified classes of coherent generator buses.
However, if more than two coherent classes are identified then the first group is chosen to correspond to one of these classes (for example, the class the largest class of synchronous generators) and all other classes are grouped in the second group. The second group will then be split into further other groups, two groups at a time, according to the same procedures until there are no asynchronous generators in any group.
After grouping the different domains 231a-231d into two different groups (231a, 231b) and (231c, 231d) , the ICC 205 determines a coarse or initial partition boundary (represented by the broken line 217a in Fig. 4C) for isolating the asynchronous generators.
The initial partition boundary 217a is composed of a set of partition lines E c which correspond to transmission lines connecting first group buses to adjacent second group buses such that if these lines are removed, then the power network 203b will be islanded into two islands corresponding to the two groups (231a, 231b) and (231c, 231d) .
It is to be noted that searching for the initial partition boundary can be easily executed after having subdivided the graph model of the power network into disjoint domains. It is only required to find all transmission lines 213 whose incident buses belong to two different domains having asynchronous generators.
For example, if the unstable mode is identified on the one hand, by generator buses 1 and 8 being coherent, and on the other hand, by generator buses 12 and 16 being coherent, then the initial partition boundary 217a will be chosen to separate the domains comprising generator buses 1 and 8 from those comprising generator buses 12 and 16.
According to this initial boundary 217a, the power network 203b would be partitioned into two islands: a first island denoted by [1] in which generator buses 1 and 8 are located and a second island denoted by [2] in which generator buses 12 and 16 are located. Thus, buses in domains (1) and (8) belong to island [1] , and buses in domain (12) and (16) belong to island [2] . This can be represented by a partition vector U in which each coordinate denotes to which island the corresponding bus belongs :
U=[[l] [1] [1] [1] [1] [1] [1] [1] [1] [2] [2] [2] [2] [2] [2] [2]]
The dimension of the partition vector U corresponds to the total number of buses in the power network 203b, and for a given bus referenced by the number i, the coordinate U[i] indicates to which island this bus belongs. For example, 17 [10 ]=[2] means that the bus having the serial number 10 belongs to island [2] .
When the partition vector U is determined, then all the arcs (representing the transmission lines) of the graph model are taken into account. Thus, if the incident buses of an arc belong to different islands, this arc will be stored into the set of partition lines E c .
According to the above example, the initial partition boundary 217a isolating the asynchronous generators of the power network 203b is defined by the following set of partition lines : E c = {I12-7, I10-9} , where li- j indicates the arc joining vertices i and j or in other words, the transmission line whose incident buses have serial numbers i and j .
Although the initial partition boundary 217a can isolate asynchronous generators, still the generation-load imbalance within each isolated group may not be satisfying.
Thus, the ICC 205 is configured to search for the most suitable partition boundary in the vicinity of the initial partition boundary 217a separating the asynchronous generators which is a reasonable starting point that largely simplifies the computational steps.
Indeed, the ICC 205 is configured to transfer individually (i.e., one by one) the load buses which are linked by lines belonging to the set of partition lines E c from one group of the first (231a, 231b) and second (231c, 231d) groups to the other one of the two groups while updating at each time the elements of both groups and the corresponding set of partition lines E c until a minimum load-generation imbalance within both updated groups (231a, 231b) and (231c, 231d) has been reached. The load-generation imbalance is determined by calculating the difference between the total generation power and the total load power within each one of the first and second groups .
In order to simplify the computational steps, the following definition of the neighbourhood N{i) of a given load bus ϊ is introduced:
This means that the neighbourhood N{i) of a given load bus i corresponds to the set of islands in which at least one other load bus j is adjacent to the given load bus i. The neighbourhood of each one of the load buses of Fig. 4C according to the initial partition boundary 217a is tabulated in Table 4.
Table 4: Neighborhood N(i) of i according to the initial partition boundary.
Bus number 2 3 4 5 6 7
Neighborhood Φ Φ Φ Φ Φ {[¾}
Bus number 9 10 11 13 14 15
Neighborhood {[¾} {[1]} Φ Φ Φ Φ Table 4 shows that only the neighbourhoods of load buses 7, 9, and 10 are nonempty sets. Thus, these load buses can be directly moved into either side of the initial partition boundary 217a without affecting the configuration of the graph model representing the power network 203b. For example, load bus 7 can be directly moved into island [2] .
Thus load buses in island [i] whose neighbourhoods are not empty are transferred into island [j] in a one by one process. The one by one transferring order is realised by first moving the load bus which can fastest reduce the load-generation imbalance within island [i] .
After a load bus is moved out, the partition vector U, the partition boundary E CI and the neighbourhood of each load bus are updated. The transferring steps are repeated until the imbalance degrees of island [i] cannot be reduced anymore .
For example, the generation load imbalance according to the initial partition boundary 217a is equal to -4 in island [1] , and 4 in island [2] . Generation ' power in island [1] being deficient, it is advantageous to move out from [1] either one of load buses 7 and 9. If load bus 7 is moved into island [2] , then the generation- load imbalances in [1] and [2] will be -1 and 1, respectively. However, if load bus 9 is moved into [2] , then the imbalances in [1] and [2] will be 4 and -4, respectively. Thus, in the first place, load bus 7 should be transferred into island [2] because it secures a better generation- load balance than transferring load bus 9 into island [2] . After moving load bus 7 into island [2] , the partition vector U and the corresponding partition boundary Ec are updated as follows:
U= [ [1] [1] [1] [1] [1] [1] [2] [1] [1] [2] [2] [2] [2] [2] [2] [2] ] and I10-9} · Consequently, the neighbourhood of each load bus is also updated. Now, by moving load bus 7 into island [2] , the generation- load imbalance in island [1] (as well as in island [2] ) has been reduced but still there is a generation deficiency in [1] and a surplus in [2] . Thus, load buses 3 and 9 in island [1] can be chosen to be moved into island [2] . However, moving anyone of them into [2] cannot reduce the imbalance of either one of islands [1] and [2] and thus, the process of partition boundary searching is ended.
Fig- 4D shows the final partition boundary
(denoted by a poly-line 217) which corresponds to the updated set of partition lines given by ¾={1 3 _ 7 , l 10 . g } .
Once, the final partition boundary 217 is determined, the ICC 205 sends commands to the corresponding relays (splitting means) 207 for splitting the power network 203b into two islands 303a and 303b according to this final partition boundary 217 as depicted in Fig. 4E.
The determination of the final partition boundary 217 has been described on the basis of islanding the power network 203b into two islands. In practice a power network may be islanded into three or more islands. However, when a fault occurs, loss of synchronism between coherent generator groups usually takes place one by one. It is unlikely that more than two coherent groups become asynchronous simultaneously. However, if a second group is found to comprise incoherent generator buses, then the partition boundary within this second group can be determined again on a one by one basis. Thus, the second group can be split into two further islands according to the same steps used to island the initial power network. The same procedure can be repeated until there are no asynchronous generators in any island. It should be noted that the Islanding Control Centre (ICC) 205 comprises processing means (for example, a workstation or computer) which can be used for executing a computer program designed to implement the method of the invention. The computer program may be designed according to the directed graph model representing the power network.
A computer program according to the islanding control method of the present invention has been tested on the IEEE 118 -bus power network experiencing the unstable mode described in relation to Fig. 3.
The islanding boundary resulting out of this program is illustrated by the broken lines L1-L7 in Fig. 5A, and the resulting islands are shown in Fig. 5B. The islanding is realised in real time with a minimal generation- load imbalance within each island. Furthermore, unlike the islanding illustrated in Fig. 1, the number of tripped transmission lines L1-L7 is minimal (seven lines instead of ten) and no load bus experienced any loss of power.
By way of example, the computer program adapted to the IEEE 118-bus power network, was written in a C++ language and implemented on an ordinary computer. The resulting running time of the program was not more than 0.012s, and this clearly shows that the present islanding control method can adequately split a large scale power network in a very short time.
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