Abstract:
The method of determining back-feed paths is capable of efficiently solving multi-layer restoration problems by minimizing a fitness function using an iterative genetic algorithm. The method optimizes back-feeding of out-of-service areas by minimizing power loss, switching, unserved loads and voltage/current violations. The efficiency of the algorithm is further increased through the use of a reactive Tabu search to prevent duplicate candidate systems.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application is a 35 U.S.C. §371 national stage filing of International Application No. PCT/US2010/058759, filed Dec. 2, 2010, which claimed the benefit of and priority to U.S. Provisional Patent Application Ser. No. 61/266,642, filed Dec. 4, 2009. The complete disclosures of the above-identified patent applications are hereby incorporated by reference for all purposes. 
    
    
     BACKGROUND 
     An electric power distribution system generally consists of a set of distribution substations, feeders, switches (circuit breakers, reclosers, etc.) electrical loads, and monitoring and control devices. A distribution system delivers electricity from distribution substations via feeders and switches to electrical loads (customers) that connect to feeders. Feeders in a distribution system are usually configured in a radial type to ensure that each load is supplied by only one distribution substation (source) via one feeder at any instant. To maintain the radial configuration, each feeder is linked by normally open (NO) tie-switches to neighboring feeders. The feeder section that is relatively near/far to its source is referred to as upstream/downstream section, while comparing to the feeder section that is relatively far/near to its source. One or more switches in the distribution system may have an associated intelligent electronic device (IED) that has the following monitoring and control functions: (1) measuring and recording electrical and other types of switch related quantities, such as voltage, current, reclosing times (2) monitoring the switch status, (3) operating the switch between open or close, and (4) communicating information to one or more master devices. 
     Distribution system reliability can be greatly improved by automating feeder operations such as fault detection, isolation, and load restoration. In such systems, IEDs associated with switches monitor the distribution system and communicate the corresponding information to the feeder automation master controllers located in substations. If a fault occurs in the distribution system, the master controller identifies the fault location, generates fault isolation and service restoration solutions in terms of a sequence of switch operations, and sends switching commands to IEDs to control corresponding switches. 
     An example distribution network is shown in  FIG. 1 , in a normal operation mode wherein loads are omitted for simplicity, sources (S 1  to S 7 ) are oval shaped, NO switches ( 5 ,  10 ,  13 ,  16 ,  19 ,  24 ,  29 ) are square shaped, with a diagonal hatch pattern, and normally closed (NC) switches (other numbers) are square shaped, with a vertical hatch pattern. If a fault occurs between switch  1  and  2 , the protection function of switch  1  causes switch  1  to open, thereby causing the dashed line circled area to lose power. The boundary switches of the faulted portion of the distribution network include switch  1  and  2 . Switch  2  is immediately downstream of the faulted section and thus, is the isolation switch. With reference to  FIG. 2 , when switches  1  and  2  are open, the faulted portion is isolated and a remaining unserved area is bound by isolation switch  2  and NO switches  5  and  16 . 
     Switches  5  and  16  are referred to as first layer (Layer  1 ) restoration switches. If the sources, in this case S 4  and S 7 , respectively, can provide power to the area that is left unserved due to fault isolation, a first layer restoration solution is possible. If the first layer sources are not capable of providing power to the unserved area, second or even third layer solutions must be performed to provide power to the unserved area. For example, the power sources for layer  2  are S 2  and S 5 . The power sources for a third layer solution are S 3  and S 6 . As should be evident, the second and third layer restoration switches are topologically more “distant” than the first layer restoration switches. 
     The process to obtain a restoration solution beyond layer  1  is called multi-layer (or multiple-layer) restoration service analysis (RSA). This is sometimes also referred to as a multi-tier service restoration problem. Due to the potentially large number of switches involved in a multi-layer restoration solution, the process to obtain such solution is generally more challenging than a single-layer solution. With any reconfiguration problem it is desirable to achieve: computational efficiency, maximum number of restored loads, avoiding network violations, minimized switching operations and radiality of the restored network topology. 
     Network reconfiguration problems may also seek to reduce the overall system loss and relieve overloading conditions in the network. Therefore, a network reconfiguration problem may either be formulated as a loss reduction optimization problem or a load balancing optimization problem. Under normal system operation conditions, network configuration allows the periodical transfer of load from heavily loaded portions of the distribution network to relatively lightly loaded ones, and thus takes advantage of the large degree of load diversity that exists in many distribution systems. Under abnormal system operation conditions, such as planned or forced system outages, the network reconfiguration problem becomes a service restoration problem, which is a special load balancing problem, where the main objective is to restore as many out-of-service loads as possible, without violating system operating and engineering constraints. 
     As with any other type of network reconfiguration problem, service restoration is a highly complex combinatorial, non-differential, and constrained optimization problem, due to the high number of switching elements in a distribution network, and the non-linear characteristics of the constraints used to model the electrical behavior of the system. 
     There is therefore a need in the art for a restoration switching analysis method that properly accounts for a greater number of variables and effectively processes multiple-layer RSA solutions. 
     SUMMARY OF THE INVENTION 
     According to one aspect of the invention, a method is provided for determining back-feed paths to out-of-service load areas in a network after fault isolation. The method includes determining a base network state and determining baseline values of a fitness function for the base network state. A chromosome list is initialized. Initial chromosomes are created, and for each created initial chromosome, if valid and not in list, the initial chromosome are added to an initial chromosome population and to the chromosome list. A fitness function value is generated for each chromosome in the initial chromosome population. The chromosomes are sorted by fitness function value, the chromosome having the lowest fitness function value being the best candidate chromosome. If the fitness function value of the best candidate chromosome is below a threshold fitness value, a network configuration corresponding to the best candidate chromosome is output. If the fitness function value of the best candidate chromosome is not below the threshold fitness value, genetic manipulation is used to create new chromosomes for a new chromosome population. During the creation of the new chromosome population, any new chromosomes already in the chromosome list are rejected. New chromosomes in the new population are added to the chromosome list. New generations are created until the fitness function value of the best candidate chromosome is below the threshold fitness value or until a predetermined number of new populations are created, whereupon a network configuration corresponding to the best candidate chromosome is output. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic drawing of an exemplary electrical distribution network. 
         FIG. 2  is a schematic drawing of an exemplary electrical distribution network wherein a fault has occurred and displaying the various restoration layers. 
         FIG. 3  is a schematic drawing of a second exemplary distribution network. 
         FIG. 4  is a schematic drawing of the distribution network of  FIG. 3  wherein a fault is isolated. 
         FIG. 5  is a schematic drawing of the distribution network of  FIG. 3  wherein the out-of-service areas are back-fed. 
         FIG. 6  is a schematic drawing of the distribution network of  FIG. 3  wherein swapping creates an energized loop. 
         FIG. 7  is a schematic drawing of the distribution network of  FIG. 3  wherein swapping creates an energized loop. 
         FIG. 8  is a schematic drawing of the distribution network of  FIG. 3  wherein swapping creates a deenergized loop. 
         FIG. 9  is a flowchart showing an exemplary embodiment of the algorithm of the present invention. 
         FIG. 10  is a schematic drawings of a third exemplary distribution network wherein a fault is isolated. 
         FIG. 11  is a table showing fitness function values for the base network state. 
         FIG. 12  is a schematic drawing of the distribution network of  FIG. 10  wherein switches are swapped. 
         FIG. 13  is a schematic drawing of the distribution network of  FIG. 10  wherein switches are swapped. 
         FIG. 14  is a schematic drawing of the distribution network of  FIG. 10  wherein switches are swapped. 
         FIG. 15  is a schematic drawing of the distribution network of  FIG. 10  wherein switches are swapped to create an energized loop. 
         FIG. 16  is a schematic drawing of the distribution network of  FIG. 10  wherein switches are swapped to create a deenergized loop. 
         FIG. 17  is a schematic drawing of the distribution network of  FIG. 10  wherein switches are swapped. 
         FIG. 18  is a table showing the fitness function values for the first generation. 
         FIG. 19  is a table showing the fitness function values for the second generation. 
         FIG. 20  is a schematic drawing of the distribution network of  FIG. 10  wherein the network configuration corresponds to the best chromosome of the second generation. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The methods of the present invention are principally applicable to multi-layer RSA problems, however, it should be appreciated that the methods may also be applicable to single layer RSA problems. For example of a single-layer RSA problem, reference is made to  FIG. 3 , which shows an exemplary network in a pre-fault configuration, including three sources S 1 , S 2 , and S 3  (transformer shaped); 6 feeder breakers SW 1 , SW 2 , SW 4 , SW 13 , SW 14  and SW 21  (square shaped, with a vertical hatch pattern); 12 normally closed (NC) switches (sectionalizing switching devices) SW 6 , SW 8 , SW 5 , SW 9 , SW 10 , SW 11 , SW 15 , SW 16 , SW 18 , SW 22 , SW 23  and SW 24  (circle shaped, with a vertical hatch pattern); and 7 normally open (NO) tie switches SW 3 , SW 7 , SW 17 , SW 12 , SW 19 , SW 20  and SW 25  (green circle shaped, with a vertical hatch pattern). 
     With reference to  FIG. 4 , a post-fault network configuration is shown, wherein a fault occurred at node L 16  of the distribution network which was isolated by feeder breaker SW 21  and switch (sectionalizer) SW 22 /SW 23 . In this post-fault configuration, two load areas are un-served or out-of-service. A first area includes a single load L 18 . The second area includes load nodes L 17  and L 19 . Power may be restored to the first area via NO tie switch SW 25 , and to the second area via NO tie switch SW 12  and/or SW 19  (in case there is not enough available capacity from either SW 12  and SW 19 , the out-of-service network L 17  and L 19  may be split into two to be restored by both SW 12  and SW 19 ). This restoration scheme is called the back-feed restoration because the power is restored from alternative sources. 
     Analysis of the problem begins by defining the number of out-of-service load areas as M, and the areas are identified by DE i  (1≦i≦M); the number of back-feed NO tie switches to DE i  are defined as M i   tsw , and the tie switches are identified by TSW i,k  (1≦k≦M i   tsw ). In the exemplary post-fault network of  FIG. 4 , DE 1  corresponds to load node L 18 , DE 2  corresponds to load nodes L 17  and L 19 ; M 1   tsw =1 and M 2   tsw =2; TSW 1,1 =SW 25 , TSW 2,1 =SW 12  and TSW 2,2 =SW 19 . 
     The unique fitness function is also defined, wherein the objective is to minimize the value of this function:
 
f( idv )= w   sw   SwOp   nr ( idv )+ w   loss SysLoss nr ( idv )+ w   Vvio   Vvio   nr ( idv )+ w   Ivio   Ivio   nr ( idv )+ w   shed ( P   nr ( idv ))  (Eq. 1)
 
     Where idv is the index of the individual network topology (hereafter the candidate system) to be evaluated (corresponds to the chromosomes generated in the genetic algorithm below). SysLoss nr  is the corresponding normalized system power loss, Vvio nr  is the number of voltage violations, Ivio nr  is the number of current violations, SwOp nr  is the number of switching operations and P nr  is the total unserved load after restoration. The corresponding weighting factors are represented by w loss , w Vvio , W Ivio , W Sw , W shed . The weighting factor definition in Eq. 1 allows the users of this algorithm to place an emphasis on different optimization variables, thereby increasing the application flexibility. It should be appreciated that, though the above fitness function is well suited for the present invention, fewer or additional factors may be considered in a fitness function depending on user priorities and required calculation speeds. 
     As the objective is to minimize the fitness value, the calculation of the normalized values is defined according to the following:
 
SysLoss nr ( idv )=SysLoss( idv )/SysLoss base   (Eq. 2)
 
 Vvio   nr ( idv )=NoVoltViolations( idv )/NoVoltViolations base   (Eq. 3)
 
 Ivio   nr ( idv )=NoCurrentviolations( idv )/NocurrentViolations base   (Eq. 4)
 
 SwOp   nr ( idv )=NoSwitchOperations( idv )/NoTieSwitches base   (Eq. 5)
 
 P   nr ( idv )=UnservedLoad( idv )/Load base   (Eq. 6)
 
     Where SysLoss is the aggregated total power loss of the candidate system, NoVoltViolations is the counted number of voltage violations in the candidate system, NoCurrentViolations is the number of current (capacity) violations in the candidate system, NoSwitchOperations is the number of switch operations needed for the transition from the fault isolated system state to the candidate system state, UnservedLoad is the aggregated out-of-service load in the candidate system. SysLoss base , NoVoltViolations base , NoCurrentViolations base , NoTieSwitches base , and Load base  and are the baseline values for the normalization, and are of a base candidate system that will be hereinafter described below. 
     The determination of a voltage violation is influenced by the settings of upper and lower voltage limits. Each element in the system (source/switch/load) has voltage limits associated with them. These voltage limits are usually known (rated values), and can be assumed to be fixed for different candidate systems. When the per unit (p.u.) voltage values are used for these fixed limits, the calculated Vvio nr  for each candidate system is comparable with each other. 
     In the calculation of UnservedLoad in Eq.6, individual load may have different priority indices (or penalty factors) associated with them. More critical load can have higher penalty factors in the calculation. For example, most critical loads may have a penalty factor of 3, loads with medium criticality may have a penalty factor of 2, and normal or low criticality loads may have a penalty factor of 1. When these penalty factors are multiplied with their associated load in the calculation, the more critical the load, the greater impact it will have on the final P nr  value. This ensures that the more critical load is less likely left unserved in the selection decision of the candidate systems. 
     A pure genetic algorithm may generate duplicate candidate networks which increase process time and reduce computation efficiency. Thus, a Reactive Tabu Search (RTS) is incorporated, by defining a search list LST RTS  that contains the indices of the candidate networks and invalid networks. The indices primarily contain information of the network configuration, that is, the statuses of each switch/breaker in terms of whether they are open or closed. 
     The method of the present invention is performed after fault isolation and proceeds according to the following steps: 
     At a first step, for each DE i , spare capacity I i,k   tsw  is calculated for each corresponding tie switch TSW i,k  (1≦k≦M i   tsw ). The spare capacity is the corresponding back-feed path&#39;s maximum spare loading capability, which is determined by tracing from the tie switch to its source, finding the lowest spare loading (current) limit for each element in that network path, and setting the lowest limit as the spare capacity. For a feeder main or lateral, the spare capacity limit is the thermal current-carrying limit minus the present loading (current); for a switch, the capacity limit is the rated AC loading capability (in amperes) minus the present loading (current); for a source, the capacity limit is the maximum available loading capacity (this value could be the thermal or stability constrained loading capacity minus the present loading). 
     In a next step, for each out-of-service load area DE i , close the NO tie switch TSW i,k     Imax    which has the maximum I i,k   tsw  among its peers (for 1≦k≦M i   tsw ). The resulting network configuration is radial and includes no out-of-service loads (current and voltage violations may still exist). This configuration is the base candidate system and its representative index is stored in the LST RTS  . The representative index may be defined as a string that consists of indices of the selected tie switches TSW i,k     Imax   . The base candidate system index thus becomes a string of tie switch indices. 
     A load flow (network) analysis is then performed on the base candidate system to calculate the total system loss SysLoss base , number of voltage violations NoVoltViolations base , number of current violations NoCurrentViolations base , number of tie switches NoTieSwitches base , and the total system load Load base . The fitness value of the base candidate system is calculated according to Eq. 1, and equals to W Sw +w loss +w Vvio +w Ivio +w shed , 
     By way of example, with reference to  FIG. 5 , the base candidate system is shown for the faulted network of  FIG. 4 . As can be seen, for DE 1 , SW 25  is the NO tie switch TSW 1,1 , which has the maximum I 1,1   tsw  (which is 200 A, i.e. the maximum loading capacity of SW 1  500 A minus the present SW 1  loading of 300 A). Likewise, for DE 2 , SW 19  is the NO tie switch TSW 2,2  which has the maximum I 2,2   tsw  (which is also 200 A, i.e. the maximum loading of SW 14  500 A minus the present SW 14  loading of 300 A). Other approaches to defining the base candidate system may be employed without impacting the operation of the present invention. 
     In a next step, for the base candidate system, swap TSW i,k  (k≠k max ) with upstream closed switches using a genetic algorithm (GA) technique to generate new candidate systems. In the present disclosure, swapping refers to the action of closing the TSW i,k  (k≠k max ) and opening an upstream NC switch. This approach will be described in greater detail below. 
     Generally, feeder circuit breakers ({ 1 ,  2 ,  4 ,  13 ,  14 } for the example network in  FIG. 5 ) may not be swapped, as it is unusual to use them as new tie switches and this also conveniently limits the size of the searching space. Multiple swaps may take place in a given candidate system, but no valid solutions can include an energized loop. In other words, candidate systems must be radial. De-energizing portions of the network are allowable, as the fitness function definition will naturally tend to exclude the corresponding candidate network, i.e. the GA will minimize load shedding if w shed  is greater than zero in Eq. 1. Each newly generated candidate system is checked against the LST RTS , and if it is already in the list, the candidate system is discarded. If the candidate system is not in the list, its index is stored in the list. For each new candidate system, a fitness value is calculated based on Eq. 1 to Eq. 6, and used in the GA evolution process and termination assessment, as will be described below in greater detail. 
     The swapping function will now be described in greater detail. In order to define the chromosomes for the genetic algorithm, a swap matrix structure is defined, wherein rows correspond to NO tie switches (index only), and columns correspond to upstream NC switches (index only) that can be swapped. In the matrix, the value 1 indicates a swap is possible and a value 0 indicates a swap is not possible. An example swap matrix for the base candidate system of  FIG. 5  is shown below: 
     
       
         
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 Chromosome Matrix 
               
               
                   
               
             
             
               
                 6 8 5 9 15 16 10 11 18 25 19 24 ← indices of NC switches 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           
                             
                               3 
                             
                           
                           
                             
                               7 
                             
                           
                           
                             
                               17 
                             
                           
                           
                             
                               20 
                             
                           
                           
                             
                               12 
                             
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 1 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 1 
                               
                               
                                 0 
                               
                               
                                 1 
                               
                               
                                 0 
                               
                               
                                 1 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 1 
                               
                               
                                 0 
                               
                               
                                 1 
                               
                               
                                 0 
                               
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 0 
                               
                               
                                 1 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                   
               
               
                 ↑ Indices of NO switches 
               
               
                   
               
             
          
         
       
     
     Based on the swap matrix, the GA chromosome structure may be defined. Each upstream NC switch can take a value (index of its downstream TSW i,k ) out of its corresponding value set, where 0 means no swap, and values other than 0 means a swap is possible with the tie switch. Table 2 shows a plurality of upstream NC switches (Line  1 ) and the structure and value set definition (Line  2 ) corresponding thereto. In this manner, Table 2 shows the chromosome definition of the genetic algorithm for the base candidate system in  FIG. 5 . 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Chromosome Definition of the Genetic Algorithm 
               
             
          
           
               
                 SW6 
                 SW8 
                 SW5 
                 SW9 
                 SW15 
                 SW16 
                 SW10 
                 SW11 
                 SW18 
                 SW25 
                 SW19 
                 SW24 
               
               
                   
               
               
                 0, 7 
                 0, 7 
                 0, 3 
                 0, 17, 20, 12 
                 0, 17 
                 0, 20, 12 
                 0 
                 0, 20, 12 
                 0, 12 
                 0 
                 0, 12 
                 0, 12 
               
               
                   
               
             
          
         
       
     
     Based on this chromosome structure definition, chromosomes of the candidate systems can be generated and the systems evaluated. A chromosome is a concise mathematical representation of the network with a given combination of swaps and is represented in the form of a string of indices of the swaps, with details below. For instance, chromosome [0 0 0 0 0 20 0 12 0 0 0 0] represents that beginning at the base network state, SW 16  is swapped with SW 20 , and SW 11  is swapped with SW 12  to generate a new candidate network configuration. In this new network configuration, S 20  and SW 12  are the new tie switches. The 0&#39;s in a chromosome mean that no swapping is done for that corresponding switch. A chromosome with all 0s will be naturally excluded because it is the base network and already in the LST RST . Note that the swapping combinations can be randomly chosen from the set of possible swaps. That is, for instance, SW 16  can be swapped randomly with SW 20  or SW 12 , or not swapped at all, but not with any other normally closed switches. 
     When more than one tie switch swaps with the same upstream switch, an energized network loop will be generated. However, by design, there is naturally no corresponding chromosome generated. For example, with reference to  FIG. 6 , (for simplicity, the faulty load area and forward-feed restoration portions of the network is not shown) for the base candidate system in  FIG. 5 , if both SW 17  and SW 20  are swapped with SW 9 , (i.e. SW 17  and SW 20  are closed and SW 9  is open), the resulting network includes an energized network loop and a chromosome should not be generated for it. Because the chromosome structure only allows an upstream switch to take one downstream switch&#39;s index value in a chromosome, the circuit does not have a corresponding chromosome that can be generated.  FIG. 7  provides another example, where an energized loop is generated, but no corresponding chromosome can be generated. 
     Because load shedding is minimized using the fitness function definition, there is no need to check at the front end if a newly generated chromosome may result in portion of the network be de-energized. However, because the evaluation of each valid chromosome needs the load flow analysis, to improve the speed of the algorithm, according to one embodiment, this check could be carried out before designating a newly generated chromosome to represent a valid candidate system. 
     For example, with reference to  FIG. 8 , for the base candidate system in  FIG. 5 , when SW 11  is swapped with SW 12 , and SW 16  is swapped with SW 20 , the resulted network has a de-energized loop portion. The corresponding chromosome is valid and may be generated as [0 0 0 0 0 20 0 12 0 0 0 0] but it should be discarded as it results in a deenergized loop and put into the LST RTS . Network tracing is used to identify deenergized loops in the resulting network. If the deenergized loops are identified, the corresponding chromosome is discarded and not used in the fitness value evaluation analysis thereafter. 
     The dimension of the solution space, i.e. the total number of possible network configurations, can be calculated by multiplying the value set dimensions of all the NC switches. Thus, in the example network of  FIG. 5 , the value set dimensions (shown in Table 2) are 2*2*2*4*2*3*1*3*2*1*2*2=4,608 possible network configurations. During the execution of the GA analysis, the algorithm should be terminated gracefully, that is, it should always produce a solution that is better than or identical to the base candidate solution, and should not take too long for its analysis to avoid interference with real-time execution. In order to achieve this, the total number of chromosomes of valid candidate systems are preferably a small portion (for example, 10% or smaller) of its dimension. For example, 1% of the solution space dimension, or an absolute number, such as 10, whichever is bigger, may be used as the limit to terminate the GA in case the fitness function value cannot be reduced to less than a pre-defined ε. In other words, it is ensured that a limited number of chromosomes are created, based on a fixed value (e.g. at least 10) or a percentage of the total solution space dimension (e.g. 10% of the solution space). 
     Once the population of the first generation valid chromosomes is generated by random swapping as described above, a set of N individual chromosomes (each chromosome may take the form, for example, [0 0 0 0 0 20 0 12 0 0 0 0]) is formed. A winning chromosome is selected via fitness function evaluation, and the generation of GA offspring via crossover and mutation of chromosomes. More specifically, fitness function evaluations are performed once the chromosomes are created, using the ones which are not rejected due to energized/deenergized loops and non-radial network configurations. As is known in the art, a crossover is accomplished by merging any two generated chromosomes randomly and mutations are accomplished by randomly redoing one or more of the swappings of a chromosome. 
     The algorithm is terminated when an acceptable reconfigured network is obtained or the maximum number of valid chromosomes has been attempted. The analysis to determine whether a solution is acceptable is performed by monitoring the reduction in the fitness value and whether it has reduced below a threshold value, which can be obtained as a user-defined parameter setting. The GA results in a best chromosome (and corresponding network configuration) at every generation, even when the process is terminated on the limitation of tried chromosomes. 
     With reference now to  FIG. 9 , a flow-chart is shown which discloses the process algorithm according to the present invention. As discussed above, the methods of the present invention are performed after a fault is isolated on a network. Thereafter, the process according to the present invention is evoked at  100 . At a first step  102 , it is determined whether a single layer solution is obtainable. An exemplary solution strategy for obtaining a single-layer solution is disclosed in U.S. Provisional Application 61/086,175, owned by the assignee of the present invention. 
     If a single layer solution is not available, the process proceeds to step  104 , wherein the Searched Chromosome List LST RTS  is initiated. At step  106  the base network is identified in the manner described above. Accordingly, for each out-of-service load area DE i  in the post-fault-isolation network, the NO tie switch TSW i,k     Imax    which has the maximum capacity I i,k   tsw  among its peers (for 1≦k≦M i   tsw ) is closed. In this base network configuration the base values and weightings for the fitness function are identified. The chromosome corresponding to the base network configuration is also saved in LST RTS . 
     At step  108  a generation number GN is initialized to 1. Further, a population number PN is calculated. The population number according to one embodiment is determined by the following equation, PN=6n+1, wherein n is an integer value set by the user. At step  110 , a candidate network chromosome is created by the switch swap method described above. The candidate network chromosome is checked against the chromosomes in LST RTS  and if a duplicate chromosome already resides in LST RTS  the chromosome is not added to the generation population. If the chromosome does not already exist in the LST RTS  it is added to the LST RTS  and if it is valid (ie. contains no energized loops), it is added to the generation population. This process is repeated until the full population number of chromosomes is reached. 
     At step  112 , the fitness function is evaluated for the networks corresponding to each chromosome in the population. The chromosomes are then sorted in ascending order from lowest (best) fitness value to highest (worst) fitness value. Next at step  114  it is determined if the generation number is greater than a pre-set maximum generation number. If so, the algorithm proceeds to step  122  wherein the chromosome with the lowest fitness value is selected as the multi-layer restoration solution. In this manner, a limit is placed on the number of iterations that process will cycle through. 
     If the generation number is less than the pre-set maximum generation number, the process proceeds to step  116 , wherein it is determined if the lowest fitness value in the generation population is below a fitness threshold value. If it is below the fitness threshold value, the network configuration corresponding to that chromosome is determined to be a good solution and the algorithm proceeds to step  122 . If at step  116  it is determined that the lowest fitness function value of the generation population is not below the fitness threshold, the algorithm proceeds to step  118 , wherein the generation number is incremented by 1. At step  120 , a new population is created for the new generation. The new generation is created using genetic functions as described below. 
     According to one embodiment, the new generation is populated according to the following sequence. The top 2n+1 chromosomes from the previous generation (i.e. if n=1, the three chromosomes having the lowest fitness function value) are carried over to the new generation. A cross-over is then performed using the topmost (lowest fitness function value) chromosome with the next 2n chromosomes to generate 2n new chromosomes. Next, the middle 2n chromosomes (i.e. the next chromosomes after the top 2n+1 are added to the new generation) are mutated to create 2n new chromosomes in the new generation. 
     As the functions within the described genetic algorithm each create new chromosomes, they are checked against the stored chromosomes in LST RTS  to ensure no repeats. If the chromosome does not exist in LST RTS , it is added to LST RTS . Each chromosome is also checked to ensure it is a valid solution (no energized loop). If valid and not already in the LST RTs  the chromosome is added to the new generation. If, after the above genetic manipulations are completed, the population number PN of the generation is not reached, additional chromosomes are created using the random switch swapping method described above. These chromosomes are checked against LST RTS  and for validity, and then added to the generation until the PN of the new generation is met. Thereafter, the algorithm returns to step  112 . This loop continues until the maximum number of generations (iterations) is reached, or a chromosome (corresponding to a particular network configuration) satisfies the fitness function criteria. 
     With reference now to  FIG. 10 , an example network is shown wherein a permanent fault has occurred at L 16  and is isolated by SW 23  and SW 22 . The example network of  FIG. 10  is substantially identical to the network of  FIGS. 3-8 , except that the capacities of source S 1 , and switches SW 4  and SW 14  are 950 A, 550 A and 450 A respectively. As this network does not have a single-layer solution that satisfies capacity constraints while restoring all the un-faulted loads, it qualifies for the multi-layer reconfiguration analysis of the present invention. The base network is determined as described above, wherein NO tie switches having the highest spare capacity, SW 25  and SW 19 , are closed to backfeed the unserved areas. 
     As discussed above, the fitness function is determined according to (Eq. 1). In the present example the following base network values for the fitness function as well as the user definable weights are shown. As can be seen, greater emphasis is placed on current capacity violations, amount of load that remains unserved and number of switching operations, than voltage violations and network (resistive) power loss in the system. Note that all the weight factors sum up to unity.
 
 w   sw 0.1, w   loss =0, w   Vvio =0, w   Ivio =0.35, w   shed =0.55
 
NoTieSwitches base =5,NoCurrentViolations base =2,Load base =1700
 
     Based on the parameter values as above and referring to (Eq. 1) to (Eq. 6) above, the fitness function for future calculations becomes:
 
 f ( idv )=0.1*NoSwitchOperations( idv )/5+0.35*NoCurrentViolations( idv )/2+0.55*UnservedLoad( idv )/1700
 
     This network corresponds to the base chromosome as shown in  FIG. 11 . For the present example the input n=1. Thus, the population number is calculated according to: PN=6(1)+1=7. The generation number is initiated to GN=1. For the present example, the user selected fitness function threshold is 0.2. In other words, the fitness function of a candidate network must be 0.2 or lower to be considered a satisfactory restoration solution. 
     As described above, a switch swap matrix is derived for the base candidate network. 
                     TABLE 3               Chromosome Matrix for Base Network                   6 8 5 9 15 16 10 11 18 25 19 24 ← indices of NC switches                             3           7           17           20           12         ⁡     [         0       0       1       0       0       0       0       0       0       0       0       0           1       1       0       0       0       0       0       0       0       0       0       0           0       0       0       1       1       0       0       0       0       0       0       0           0       0       0       1       0       1       0       1       0       0       0       0           0       0       0       1       0       1       0       1       1       0       1       1         ]                         ↑ Indices of NO switches                    
Using the above matrix, which indicates the allowed switch swapping pairs (e.g., SW 6  with SW 7 , SW 8  with SW 7 , SW 5  with SW 3 , but not SW 6  with SW 3 , for instance), a first generation of PN=7 chromosomes are randomly generated as described below.
 
     Chromosome 1.a: This is generated by swapping SW 6  with SW 7 . The corresponding network is shown in  FIG. 12  and the chromosome is displayed as [7 0 0 0 0 0 0 0 0 0 0 0]. The resulting network includes three capacity violations and no unserved load. The fitness function therefore evaluates to 0.1*4/5+0.35*3/2+0.55*0/1700=0.605 
     Chromosome 1.b: This is generated by swapping SW 12  with SW 24 . The corresponding network is shown in  FIG. 13  and the chromosome is displayed as [0 0 0 0 0 0 0 0 0 0 0 12]. The resulting network includes one capacity violation and no unserved load. The fitness function therefore evaluates to 0.1*4/5+0.35*1/2+0.55*0/1700=0.255 
     Chromosome 1.c: This is generated by swapping SW 12  with SW 18 . The corresponding network is shown in  FIG. 14  and the chromosome is displayed as [0 0 0 0 0 0 0 0 12 0 0 0]. The resulting network includes three capacity violations and no unserved load. The fitness function evaluates to 0.1*4/5+0.35*3/2+0.55*0/1700=0.605 
     Chromosome 1.d: This is generated by swapping SW 9  with SW 17 , and SW 9  with SW 20 . The corresponding network is shown in  FIG. 15 . Because the network includes an energized loop, this case is discarded and no chromosome is created. 
     Chromosome 1.e: This is generated by swapping SW 11  with SW 12 , and SW 16  with SW 20 . The corresponding network is shown in  FIG. 16  and the chromosome is displayed as [0 0 0 0 0 20 0 12 0 0 0 0]. The resulting network includes a deenergized loop, and thus a total 500 A of unserved load. The fitness function evaluates to 0.1*6/5+0.35*0/2+0.55*500/1700=0.282. 
     Chromosome 1.f: This is generated by swapping SW 9  with SW 17 , and SW 11  with SW 12 . The corresponding network is shown in  FIG. 17  and the chromosome is displayed as [0 0 0 17 0 0 0 12 0 0 0 0]. The resulting network includes two capacity violations, but no unserved load. The fitness function evaluates to 0.1*6/5+0.35*2/2+0.55*0/1700=0.47. 
     Chromosome 1.g: This is generated by swapping SW 9  with SW 17  and the chromosome is [0 0 0 17 0 0 0 0 0 0 0 0]. The resulting network includes one capacity violation, but no unserved load. The fitness function evaluates to 0.1*4/5+0.35*1/2+0.55*0/1700=0.255. 
     Chromosome 1.h: This is generated by swapping SW 5  with SW 3  and the chromosome is [0 0 3 0 0 0 0 0 0 0 0 0]. The resulting network includes one capacity violation, but no unserved load. The fitness function evaluates to 0.1*4/5+0.35*2/2+0.55*0/1700=0.43. 
     In this manner, seven (7) valid chromosomes are created. The Searched Chromosome List LST RTS  is initially empty, so all valid chromosomes are moved to LST RTS  . At this point in the process, the chromosomes are arranged in ascending order of their fitness function values, as shown in  FIG. 18 . As can be seen, chromosomes 1.b and 1.g are the best in the population (i.e. lowest fitness function values). However, because the best fitness function value 0.255 is higher than the pre-defined threshold 0.2, the analysis is continued to the next generation GN=2. 
     In order to form the chromosome population for the next generation, the top 2n+1=2(1)+1=3 chromosomes from  FIG. 18  are carried over to the GN=2 population. These are shown in the top three rows (2.a through 2.c) in  FIG. 19 . Next, a crossover operation is performed between the top chromosome and the following 2n=2(1)=2 chromosomes to obtain new chromosomes 2.d and 2.e. The crossover operation between 2.a and 2.b produces 2.d, and between 2.a and 2.c to produces 2.e. This is further described in the following. 
     Chromosome 2.d: This is generated by performing a crossover operation between 2.a and 2.b. The crossover point is randomly selected.
     2.a [0 0 0 0 |0 0 0 0 0 0 0 12] (Chromosome 2.a and a crossover place “|”)   2.b [0 0 0 17 |0 0 0 0 0 0 0 0] (Chromosome 2.b)   =&gt;2.d [0 0 0 17 |0 0 0 0 0 0 0 12] (Chromosome 2.d as a result of crossover)   

     Chromosome 2.e: This is generated by performing a crossover operation between 2.a and 2.c. The crossover point is randomly selected.
     2.a [0 0 0 0 0 0 |0 0 0 0 0 12] (Chromosome 2.a and a crossover place “|”)   2.c [0 0 0 0 0 20 |0 12 0 0 0 0] (Chromosome 2.c)   =&gt;2.e [0 0 0 0 0 20 |0 0 0 0 0 12] (Chromosome 2.e as a result of crossover)   

     Next, the following 2n=2(1)=2 chromosomes from GN=1 (that is, 1.h and 1.f) are used to perform mutations and create the remaining chromosomes (2.f and 2.g) in the population of GN=2. 
     Chromosome 2.f: This is generated by performing a mutation operation on 1.h. The mutation is randomly done with allowed switch swapping operations as defined in switch swap matrix in Table 3.
     1.h [0 0 3 0 0 0 0 0 0 0 0 0] (Chromosome 1.h)   =&gt;2.f [0 0 3 17 0 0 0 0 0 0 0 0] (mutation by switch swap of SW 9  &amp; SW 17 )   

     Chromosome 2.g: This is generated by performing a mutation operation on 1.f. The mutation is randomly done with allowed switch swapping operations as defined in switch swap matrix in Table 3.
     1.f [0 0 0 17 0 0 0 12 0 0 0 0] (Chromosome 1.h)   =&gt;2.g [0 0 0 0 0 0 0 12 0 0 0 0] (mutation by switch swap of SW 9  &amp; SW 17 )
 
The operations above obtain a full population PN=7 of valid chromosomes in second generation GN=2.  FIG. 19  shows the chromosomes and the corresponding fitness function values. When the chromosomes in the table are arranged in the ascending order of fitness function values, the topmost chromosome (having the least fitness function value) is 2.d. This is generated by swapping SW 9  with SW 17 , and SW 24  with SW 12 , with the corresponding network shown in  FIG. 20 . This results in no capacity violations, as well as no unserved load. The fitness function evaluates to (also shown in  FIG. 19 ) 0.1*6/5+0.35*0/2+0.55*0/1700=0.12. As 0.12 is less than the fitness function threshold value of 0.2, this chromosome (and its corresponding restoration switching scheme) are the restoration solution.
   

     This process results in the identification of a solution as the closing of SW 12  and SW 9 , and opening of SW 24  and SW 17 . Note that the entire load would be restored (except L 16 , where the permanent fault exists) via alternate sources, and all the alternate sources as well as the breakers/switches would carry currents within their capacity limits. Thus this qualifies as a good reconfiguration solution. 
     The method of the present invention requires network load flow calculations (in order to evaluate the fitness function) only for the valid chromosomes in question, as opposed to many more load flow calculations if approaches such as classical genetic algorithm, network tracing or deterministic optimization methods are used. This increases the speed of solution finding, thus making it appropriate for real-time restoration switching applications. It is especially practical for multi-layer RSA, when the network topology is complex (for example, many tie switches between adjacent feeders) and many different alternatives for back-feed restoration exist. The functionality of multi-layer RSA would reside at either at DMS or in a sub-station. 
     As will be appreciated by one of ordinary skill in the art, the present invention may be embodied as or take the form of the method and system previously described, as well as of a computer readable medium having computer-readable instructions stored thereon which, when executed by a processor, carry out the operations of the present inventions as previously described. The computer-readable medium may be any medium that can contain, store, communicate, propagate, or transport the user-interface program instruction for use by or in connection with the instruction execution system, apparatus, or device and may by way of example but without limitation, be an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, device, or propagation medium or other suitable medium upon which the program is printed. More specific examples (a non-exhaustive list) of the computer-readable medium would include: a portable computer diskette, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), an optical fiber, a portable compact disc read-only memory (CD-ROM), an optical storage device, a transmission media such as those supporting the Internet or an intranet, or a magnetic storage device. Computer program code or instructions for carrying out operations of the present invention may be written in any suitable programming language provided it allows achieving the previously described technical results. 
     It is to be understood that the description of the foregoing exemplary embodiment(s) is (are) intended to be only illustrative, rather than exhaustive, of the present invention. Those of ordinary skill will be able to make certain additions, deletions, and/or modifications to the embodiment(s) of the disclosed subject matter without departing from the spirit of the invention or its scope, as defined by the appended claims.