1 Introduction

Power system planning analysis is an important task in the design stage of power system network and required in the extension process of the system. It also prepares utilities to deal with various un-expected events such as fault and components outages that could lead to system black-out. Different techniques had been used in formulating the solution for specific planning purposes. For example, in power system restoration planning, mathematical approach [1,2,3,4,5,6], heuristic technique [7, 8] and expert system [9,10,11] had been applied. Optimization techniques also had been widely applied in power system planning due to its ability to handle high complexity of the system and non-linearity problems. For example in restoration planning, particle swarm optimization, genetic algorithm and firefly algorithm had been used [12,13,14]. Meanwhile, evolutionary programming had been explored in load restoration [15] and network reconfiguration [16, 17].

Power system restoration planning after the occurrence of a blackout event is one of the important areas in the context of power system operator/planner. The primary restoration objective is to minimize the time required to restore the system back to normal operational state. Moreover, efficient and orderly corrective steps need to be accounted in order to reduce the possibility of another system blackout, which helps prevent the occurrence of other serious problems in the system. The system restoration planning could significantly be improved via a combined effort of actual system physical and operation knowledge, operating personnel experience, and facilities of computer simulation [18, 19].

Sectionalizing a system to sub-systems or islands for parallel restoration is commonly implemented by utilities in restoration planning [20]. In the case of widespread blackouts, the sectionalization of the affected area and the implementation of parallel restoration are beneficial in speeding up the total restoration time and avoid recurrent system collapse. Some common parallel restoration sequences are sectionalizing system to islands, restoration of each island and the resynchronization of islands. The sectionalizing method will determine which transmission lines will not be restored in order to create desired number of islands. Several restoration constraints are required to be fulfilled in sectionalizing a system in order to maintain system security and parallel restoration reliability, while also ensuring the steady state stability of future islands. The following common constraints are included [21,22,23]:

  1. 1)

    Must have at least one black start generator (BSG) as the main cranking power to non-black start generators (NBSGs) in each island. BSG unit has the capability of energizing the network from an on-site auxiliary supply (diesel).

  2. 2)

    Have sufficient matching capacity of generation and load to maintain system frequency within prescribed limits in each island.

  3. 3)

    Have sufficient voltage control capabilities to maintain a suitable voltage profile in each island.

  4. 4)

    Tie-lines between the islands should have synchronization equipment for islands interconnection.

Generally, restoration operation process begins with BSG energizing various generation units and loads through transmission lines after the identification of restoration paths. Therefore, to design a strategic sectionalizing planning after a complete blackout, the actual system topology and availability of each electrical element (e.g. BSG, NBSGs, transmission lines, loads and etc.) in the system must be considered [1,2,3,4]. A graph theory method is implemented in [2] to represent the physical properties of the system in the proposed restoration strategy. In restoration strategy, the load is commonly categorized into critical loads (CLs) and non-critical loads (NCLs). CL is the load that needs to be restored immediately and NCL must be pre-defined in order to arrange effective restoration path in optimal time. Thus, resynchronization time of the islands can be sped up.

It is important to include operating personnel experience in sectionalizing planning. The interactive mode of system restoration operating conditions, prioritized list of restoration events, and choice of executable event is investigated in [18]. Other than that, system operators can guide and participate in planning using their experience of actual systems and develop a planning process that is safe and flexible as the system grows. Thus, a minimal amount of analysis is required during online restoration. The operators require multiple sectionalizing solutions for system restoration in order to make a decision. However, methods [1,2,3,4] only provide a single solution for operators to choose from. The lack of flexibility makes it hard for the operator to work with since the single solution presented could be unfeasible based on their experience of actual systems and blackout conditions. Meanwhile, ordered binary decision diagram (OBDD)-based method [5] can produce 108 possible solutions. However, too many possibilities will fail to provide useful guidance to an operator. Thus, a short list of distinct sectionalizing solutions is required to help the operator make decisions.

The computer simulation method, which allows system operators to simulate and plan strategic system restoration in optimal time, is gaining more importance as the system grows. Various analyses can be conducted during offline simulation, which prepares the operators for real time events. Various factors or objectives are included and considered when solving parallel restoration problems. Restoration constraints, restoration operations, and optimum restoration time are some of the main factors included in the sectionalizing method. Heuristic techniques have a compact theoretical support that can include many factors and operator knowledge in their searching algorithm. The technique can provide good approximations for exact solutions. However, there are no guarantees that a successful identification of the optimal solution for a complex system topology [24]. Meanwhile, the optimization method is a method that is specially designed to find the “best” solution by minimizing one or more objective functions based on one or more dependable factors. For a complex problem with huge searching space, this method may encounter difficulty in finding the optimal solution and risk of divergence.

This paper presents a sectionalizing planning that combines physical and operation analysis of the system, operating personnel knowledge, and computer simulation to improve the system restoration and further guide the system operators/planners. A set of transmission lines that should not be restored (cut set) is determined to sectionalize system to islands to perform parallel restoration in the shortest amount of time. The sectionalizing method using the graph theory approach is proposed in this work to represent the physical topology of the system and determine the restoration path. In order to search for the optimal solution, the discrete evolutionary programming (DEP) optimization method is applied.

Different from the conventional evolutionary programming, the DEP used a new mutation approach as proposed in [25], but with a slight modification. In order to reduce the search space, heuristic initialization is proposed to assist DEP method to converge. Similar restoration time for each island is selected as the objective function for the optimization method while fulfilling the restoration constraints of BSG availability, load-generation balance and maintaining acceptable voltage magnitude within each island. In this study, all transmission lines are assumed to possess synchronization equipment. Islands with similar restoration times can eventually speed up the resynchronization process. Multiple distinct sectionalizing solutions are presented to provide options to system operators when selecting based on their experience and knowledge of current system conditions.

This rest of the paper is organized in the following order. Section 2 introduces the sectionalizing planning method, which is divided into four sub-methods, i.e. modelling system physical topology using graph theory approach, heuristic initialization, DEP method and restoration time algorithm. In Section 3, the proposed method is applied to IEEE 39-bus and 118-bus system to demonstrate the effectiveness of the planning method. Finally, Section 4 will conclude the work.

2 Sectionalizing planning

Power system is an interconnected network of major electrical components such as generators, buses, loads, and transmission lines for delivering electricity from suppliers to customers. Due to rapid growth in the community, the network has become bigger, more complex and increasingly exposed to the possibility of a blackout. Total restoration time of system blackout is the total sum up of restorative action time of each major component in a system as presented in Table 1. In order to sectionalize the system with similar restoration time, the quantities of major components in each island need to be considered.

Table 1 Restorative action of major electrical components
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An example of a system which is sectionalized to three islands with similar restoration time (T 1 ≈ T 2 ≈ T 3) is shown in Fig. 1. Three lines are found to be the cut set. The common restoration time between all the islands is chosen as the main factor in this work in order to accelerate system resynchronization. If T 1 = α, T 2 = β and T 3 = γ, the system resynchronization time can be determined as max (α, β, γ). In order to create ‘n’ islands with similar restoration time and less number of the cut set, an objective function (∆T) is proposed in (1).

Fig. 1
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Islanded power system

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$$\Delta T = \, \left( {\hbox{max} \left\{ {T_{1} ,T_{2} , \ldots ,T_{n} } \right\}{-}\hbox{min} \, \{ T_{1} ,T_{2} , \ldots ,T_{n} \} } \right) + t_{tl} \left( z \right)$$
(1)

where z is the number of cut set; t tl is the time to connect cut set.

The sectionalizing planning in this study is divided into four main steps: model the physical topology of the system using graph theory approach; determine the initial cut set using heuristic technique; determine the optimum cut set using DEP method assisted by initial cut set; calculate the restoration time for each island.

2.1 Network modelling using graph theory approach

Physical network and connections of a system can be represented by a graph, G = (V, E, W), where V is the major electrical components (e.g. buses, generators and loads) node; E is the transmission lines (edges); W is the weight factor of the transmission lines in terms of connectivity or electrical distances. An example of an arbitrary network represented as a graph is shown in Fig. 2.

Fig. 2
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Arbitrary network

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By referring to Fig. 2, the arbitrary network contains ten buses, four generators, six loads and ten transmission lines. The identification of each component node and their availability status is important. This information is vital to be incorporated into restoration time calculation algorithm. The connectivity of the node i to node j inside a system can be represented as incidence matrix, M [26], which is defined as follows:

$$\varvec{M} = [m_{ij} ] = \left\{ {\begin{array}{*{20}l} 1 \hfill & {{\text{Edge}}\;e_{k} \;{\text{is}}\;{\text{inciden}}\;{\text{from}}\;{\text{node}}\;v_{i} \;{\text{to}}\;{\text{node}}\;v_{j} } \hfill \\ { - 1} \hfill & {{\text{Edge}}\;e_{k} \;{\text{is}}\;{\text{inciden}}\;{\text{from}}\;{\text{node}}\;v_{j} \;{\text{to}}\;{\text{node}}\;v_{i} } \hfill \\ 0 \hfill & {{\text{No}}\;{\text{connection}}} \hfill \\ \end{array} } \right.$$
(2)

where k = {1, 2, …, l}.

Other than the connectivity, electrical distance of each edge can also be assigned as the weight of the graph [8, 10]. The electrical distance which is defined as the line reactance x ij from node i to node j, excluding the line resistance, r ij . This is because the value of resistance is usually small, rendering it negligible.

This work model the network with n b buses and n l edges, l is the maximum number of edges in a system. The elements are V = {v 1, v 2,…, v a }, a = n b ; E = {e 1, e 2,…, e b }, b = n l ; W = {w 1, w 2,…, w c }, c = n l . Subset V G ⊂ V is defined as generator buses and V L ⊂ V as load buses. Other subsets in the system are BSG bus (V BSG ⊂ V G) and critical load buses (V CL ⊂ V L). In this work, shortest restoration path, P can be determined by using k-shortest path algorithm [9, 10]. Total weight from node i to node j in path P is calculated by:

$$\varvec{W}\left( \varvec{P} \right){ = }\sum\limits_{{k \in \varvec{P}}} {w_{k} }$$
(3)

Apart from that, system operating conditions (loads and generations) before system disturbance are essential to be known. This consists of information of the system state (bus magnitudes and phase angles), control (active and reactive power load and generation), line parameters (resistances and impedances) etc. All the information can be assigned and stored in the system matrices. The matrices can be incorporated with G to further assist the sectionalizing planning for optimization and heuristic method.

2.2 Heuristic initialization

Heuristic method is an approach to solve a problem based on expert’s knowledge and experience to explore the search space in a practical way [24]. In this proposed method, knowledge of restoration constraints, operation procedures and minimum restoration time for each island which is the main objective are included. By considering this knowledge, a reasonable initial cut set that close to the optimal solution can be determined. In power system restoration, the initial cut set is the specific transmission lines that should not be restored to separate the system into various islands. Through this initialization, the search space of the possible lines (not to be restored) can be reduced, which increase the chances to obtain optimal solution. There are two main steps required to search the initial cut set that divides the system into islands with similar restoration time. The steps are: ①group the generators according to a number of BS generators in a system and create the generator’s skeleton. The generators are grouped together based on total active power generation balance; ②search the initial cut set using proposed heuristic algorithm based on generator’s skeleton. Total restorative time for major components in each group is considered in this step. The steps are shown in Fig. 3. In order to fulfil the restoration constraint of generation-load balance, each progressive step ensures that the total active power generation is greater than the total load consumption.

Fig. 3
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Heuristic initialization method

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Power system network in Fig. 1 is used to explain the details of the proposed method. The graph theory of this network is presented in Fig. 4. The generator skeleton groups can be identified as follows.

Fig. 4
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Heuristic technique

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  1. 1)

    Step 1

Figure 4a shows a network with 4 generator units at node 1, 2, 3 and 4. The power generation for each generator is P 1 = 20 MW, P 2 = 30 MW, P 3 = 20 MW and P 4 = 30 MW, respectively. BSG units are located at node 2 and 3, thus the number of generator groups created is n = 2. The other nodes (load buses) are rated at 15 MW. CLs are located at node 6 and 10.

The identification of the generator’s skeleton group method is implemented and the result is presented in Table 2. There are 4 sets of generator groups which are found and ranked according to the total power generation balance. Group set with the lowest power generation balance is chosen as the initial generator group.

Table 2 Sets of generator’s group
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The initial skeleton point is determined using k-shortest path, connecting generator buses in group set 1. The result is as follows:

  • Group set (Fig. 4a), Set 1,

  • Skeleton point group 1, A1 = node {1, 2, 6, 7},

  • Skeleton point group 2, A2 = node {3, 4, 9}

  1. 2)

    Step 2

The searching method to find the initial cut set will start with the initial skeleton points. The points will be expanded by searching the nearest node while fulfilling the load-generation balance. It also has to ensure that the new points do not overlap with other groups. The group with the lowest restorative action time is selected to search for a new node to be added in their group. The restorative action time is determined according to major power units in each group. This step is repeated until all nodes are connected to either one of the created groups.

For group set 1, A2 has lower restorative time (2 generators and 1 load) compared to A1 (2 generators and 2 loads). A2 will search and connect nodes that are located adjacent edge/1st degree, (degree in graph theory is the number of edges incident to the node) from it skeleton points which are node 8 and 10. Now A2 has higher restorative time (2 generators and 3 loads). Finally, A1 will add node 5 to its group.

The lines that connect the nodes from different groups are chosen as an initial cut set. The created groups are called as initial islands. The initial cut set will be used in the next strategy which is the optimization method. The answers for the arbitrary network are shown as follows:

Group set 1 (Fig. 4b), cut set edges = (6–8), (4–7)

2.3 DEP optimization method

DEP is applied in this work to suit the discrete nature of the line that to be cut in order to form the island. Similar restoration time among the islands with a minimum number of the cut set is the objective function of the optimization. General DEP algorithm flowchart is shown in Fig. 5.

Fig. 5
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DEP optimization method

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In this paper, a slight modification of conventional DEP has been done. Referring to Fig. 5, initial cut set from heuristic initialization (as explain in Section 2.2) is used to create the initialization population, x. The proposed algorithm starts by generating initialization population, x based on the possibility that the optimum cut set might be less or more lines compared to the initial cut set. The initialization population stage is shown in Table 3.

Table 3 Initialization population
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Next, the first objective function is calculated based on (1) for each initial population. New population is created by the mutation stage as presented in Table 4. The objective function for each mutated population is calculated. Both populations will be combined and the new population is selected based on minimum objective function values. The mutation stage will be repeated based on a predefined number of iterations. Multiple distinct solutions can be chosen and ranked according to the objective function values. The best objective function which has the lowest value will be selected as the optimum solution.

Table 4 Mutation stage
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The replacement ‘edge’ during initialization and mutation stage plays an important role in finding the optimum solution. Random replacement with any available edges in the system or random replacement with edges that are located near the initial cut set shows a different result. Thus, both approach will be analysed and presented in this paper.

2.4 Restoration time algorithm

To achieve the main objective of restoration planning, restoration time factor needs to be considered during the sectionalizing process and at the same time fulfilling the restoration constraints. The proposed method searches for the optimum cut set which sectionalizes the system into similar restoration time. Each major component in the system plays an important role in determining the restoration time. As each type of component has its own restorative time, the idea of sectionalizing the system based on the quantities of major components can help creating balanced islands. In this work, restoration time of the backbone of each island is calculated. Restoring island’s backbone is restoring only the main equipments which are the generators and CLs in each island before resynchronization takes place. Graph theory is used to represent the system components availability and connectivity to guide the restoration time algorithm. The shortest path to restore from BSG to NBS generators, CLs and crossover lines is determined using k-shortest path algorithm. The full restoration time algorithm is shown as in Fig. 6.

Fig. 6
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Restoration time algorithm

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3 Simulation result

In order to validate the proposed method, IEEE 39-bus and 118-bus are used. Test results for 118-bus is discussed in details for each step until final solution. Meanwhile, for test system 39-bus, the focus discussion is on the influence of the heuristic initialization towards final solution.

3.1 IEEE 118-bus system

The IEEE 118 bus system as shown in Fig. 7 is used to validate the proposed sectionalizing planning. It is considered that the BSG units are located at bus 25 and 69 as per published work in [4] and the remaining generator units are defined as NBSG. Two islands are created based on the two cranking source from BSG units. The assumed CLs are assigned and identified at buses 15, 18, 23, 27, 49, 54, 59, 80, 90, 92 in [4]. Transformer feeder is excluded from the possible cut set solution. All feeders except transformer feeder are assumed to have synchro check relay. Other than that, the test network is assumed to have local reactive power compensators to compensate the reactive power imbalance. In this work, only real power balance is considered.

Fig. 7
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Restoration strategy for IEEE 118-bus system

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First, the 118-bus system is converted to a graph and system operating matric is created. Availability status and system operating information (load and power generation, voltage magnitude, phase angle, line parameter etc.) of each major equipment in the bus system are determined. This step is required to assist the proposed method to be conducted in actual test conditions after total system blackout. All major equipment in the bus system is assumed to be disconnected and the breakers are open.

After that, heuristic initialization is implemented. In the first step, generator units are divided into different groups according to the number of BSG. The availability of BSG in each created group must be ensured. The smallest total active power generation difference between the created groups is chosen as the initial group set as shown in Table 5.

Table 5 Heuristic initialization for IEEE 118-bus system
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This is the first step to creating a balanced island size in terms of major equipment quantities and similar restoration time for each island. The two generator groups are located at bus {46, 69, 80, 87, 89, 100, 103, 111} and bus {10, 12, 25, 26, 31, 49, 54, 59, 61, 65, 66}. Generators in each group will be connected using k-shortest path, thus two initial generator skeletons are created. Generator skeleton, An (n = 1, number of islands) for each group are as follows:

  • A1: bus{46, 47, 69, 77, 80, 82, 83, 85, 86, 87, 89, 98, 100, 103, 110, 111}

  • A2: bus{10, 9, 8, 5, 3, 12, 30, 26, 25, 17, 31, 38, 65, 66, 49, 54, 55, 59, 61, 62, 65, 66}

Both skeletons are used in the second step of heuristic initialization to search the initial cut set which will sectionalize the bus system into two islands. Total restorative time for all major equipment (generators, loads, and lines) contained in each group as per Table 1 is calculated and compared. Restorative time for skeleton group A1 and A2 are 315 and 375 min, respectively. The group with the lowest restorative time will carry out the second step. The second step is carried out by searching the buses that are adjacent to buses in group A1 with 1st degree of connection. The buses identified will be added up to group A1 one by one while checking if the generation is greater than the load. If the load is greater than the generation, the bus identified will not be added to the group and total restorative time will be calculated again. The procedure will be repeated until all buses have been selected by their respective groups. Lines that connect buses from different groups will be selected as the cut set. The initial cut set for the bus system is shown in Table 5.

The initial cut set is used by the DEP optimization method to determine the optimum cut set. In this work, only restoration of system backbone (BSG, NBSG and CLs units) is conducted before resynchronization of the islands. Other equipment will be restored in parallel after the system is reconstructed. The initial cut set will be used as the initial population in the DEP optimization to reduce the searching space and assist the convergence. To show the contribution of the initial solution, there are three cases carried out in this work. In the first case, set of edges which are connected to the adjacent bus (1st degree) from initial cut set buses are chosen randomly as a replacement in the mutation stage of DEP algorithm. For example, set of edges of 1st degree connections of the first initial cut set {22–23} taken from Table 5, is {21–22, 23–24, 23–25, 23–32}. The full list set of edges is continued to search for the rest of initial cut set. The second case is conducted by expanding the set of edges which are connected to 1st and 2nd degree connections from the initial cut set (one and two edge incident from the cut set buses) as a replacement in mutation stage. Meanwhile, in the last case, random edges are chosen to replace in the mutation stage. 10 iterations are conducted for each case to search for the optimum cut set. 50 trials are repeated to show the consistency of all cases. Objective function of each population is calculated and the lowest value for each trial is compared. The result for both cases is presented in a graph shown in Fig. 8.

Fig. 8
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Comparison result of objective function for three different cases for IEEE 118-bus system

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Based on the results, it is shown that the optimum cut set, {22–23, 23–25, 23–32, 45–46, 47–49, 48–49, 65–68, 49–69} (lowest objective function) was found near the initial solution. The consistency of the DEP method in finding the optimum solution (275 min) for the first case (98%) is higher compared to the second and third case. The second case shows that the algorithm starts to diverge. Its consistency to find the optimum solution (275 min) is only 38%. Meanwhile, in the last case, the optimum answer is not found. The final solution found (300 min) is equivalent to the initial solution value with no further improvement. It can be concluded that the heuristic initialization is able to reduce the search space and guide the DEP optimization method to converge to the optimum solution. Other than that, by comparing the initial solution and the final solution, only two lines are different between them. It is shown the optimum solution is close to initial solution.

DEP method can produce several distinct solutions according to its objective function. The solutions can be ranked as per Table 6. Operators can examine the ranked solution using their knowledge of the actual system and current blackout conditions to further decide on the appropriate corrective action. For example, if a fault occurs in a substation at bus 32, the operator cannot choose the 1st or 2nd solution since lines {23–33} is required to perform resynchronization. Instead, the operator can choose the 3rd solution as the final solution.

Table 6 Results of proposed method
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In addition, the proposed DEP also has the ability to exclude the lines that not fulfil technical operation condition such as unavailability of synchronizing equipment for islands interconnection. To show this capability, analysis on IEEE 118 is repeated with the same initial solution but considering unavailability of synchronizing equipment on lines 23–25. The final solutions are presented in Table 7. Results in Tables 6 and 7 are compared. It shown that, both results are different as lines 23–25 not appear as solution in Table 7. It can be concluded that the proposed method is flexible to the system technical conditions in determining the final solutions.

Table 7 Results of proposed method with synchronizing equipment unavailability condition
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Few published methods are chosen and compared with the proposed method’s result. Its comparison is presented in Table 8. It is shown that the proposed method produces the lowest restoration time for created islands. Island 2 requires 325 min to restore and subsequently followed by resynchronization between the two islands. Meanwhile, 375 min is required for the method in [2, 4, 5]. The proposed method managed to sectionalize the system with better restoration time.

Table 8 Method for island’s ‘backbone’ restoration time
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Power flow is conducted on the optimum solution to ensure that the voltage profile of the islanded areas is within the acceptable range of ±5%–10% of the rated voltage. The generator bus with the highest generation capacity in each island is selected as the slack bus. The results are presented in Table 9.

Table 9 Voltage profile for each bus at created islands
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3.2 IEEE 39-bus system

Three islands as per Fig. 9 are determined by considering the availability of three units of BS (located at buses 32, 33 and 37) while other generators are defined as NBS units. CLs are assumed at buses 7, 18, 21, 23 and 26. The results of the proposed method, heuristic initialization and DEP are shown in Tables 10 and 11, respectively. Power flow analysis is carried out and the voltage profile for each island after final sectionalizing is within an acceptable range as shown in Table 12.

Fig. 9
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Restoration strategy for IEEE 39-bus system

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Table 10 Heuristic initialization for IEEE 39-bus system
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Table 11 Results of DEP method
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Table 12 Voltage profile for each bus node at created island
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From Tables 10 and 11, the heuristic initialization solution and final solution are {1–39, 3–4, 14–15, 17–18, 17–27} and {1–39, 3–4, 14–15, 16–17} respectively. It can be observed that the final solution is close to the initial solution with the difference of two lines.

Meanwhile, to show the contribution of initialization solution in assisting DEP method to converge, 3 cases similar tested using the 118-bus system are carried out. 10 iterations are conducted for each case to search for the optimum cut set. 50 trials are repeated to show the consistency of all cases. Objective function of each population is calculated and the lowest value for each trial is compared. The result for both cases is presented in a graph shown in Fig. 10.

Fig. 10
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Comparison result of objective function for three different cases for IEEE-39 bus system

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The consistency of the DEP method in finding the optimum solution (130 min) for the first case is 96%. The second case shows that the algorithm starts to diverge. Its consistency to find the optimum solution (130 min) is 54%. Meanwhile, in the last case, differ from the 118-bus case, the optimum answer (130 min) is found by 42%. It can be concluded that the heuristic initialization is able to guide the DEP optimization method to converge to the optimum solution.

4 Conclusion

This paper presents a sectionalizing planning that determines cut set (transmission lines that should not be restored) to sectionalize the system into different islands for parallel power system restoration. Islands with similar restoration time were identified to speed up the resynchronization time using DEP optimization method assisted by heuristic initialization and graph theory approach. Graph theory is used to model the physical properties of the power system network in the form of a graph and to determine the restoration path. The heuristic initialization scours the search space and provides the best initial cut set. The DEP optimization method further uses this information to find the optimal cut set while fulfilling the restoration constraints. Other than that, the proposed method also able to produce multiple distinct solutions to give options to system operators to choose from based on their experience and knowledge of current system conditions. The proposed strategy was tested using IEEE 39-bus and 118-bus systems. The results for both test system show that the initial solution is close to the optimum solution. In the 118-bus test system, the robustness of the proposed method is highlighted when the method is able to find optimal cut set with lowest restoration time with 13% faster compared to published results [8, 10].