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Optimal island partitioning of smart distribution systems to improve system restoration under emergency conditions
Electrical Power and Energy Systems 97 (2018) 155–164
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Optimal island partitioning of smart distribution systems to improve system restoration under emergency conditions Vahid Hosseinnezhada, Mansour Rafieea, Mohammad Ahmadiana, Pierluigi Sianob, a b
MARK
⁎
Faculty of Electrical and Computer Engineering, Shahid Beheshti University, A.C., Tehran, Iran Department of Industrial Engineering, University of Salerno, Fisciano, Italy
A R T I C L E I N F O
A B S T R A C T
Keywords: DG Emergency operation Restoration Island partitioning scheme Optimal load control Optimal power flow
In the case of emergency operation, intentional islanding of distributed generations is an interesting solution to maintain the reliable power supply in the smart distribution grid. This paper deals with an optimal systematic strategy to restore the post-contingency distribution grid after the occurrence of severe disturbances causing an outage. Hence, a two-stage solution procedure is developed in this paper. In the first step, the optimal partitioning scheme is addressed through a number of graph-theory related algorithms. By applying these algorithms, the loads are optimally assigned to the distributed generations and optimal load shedding programs are determined and performed if there is a shortage of generation. The second step deals with the adjustments issues. In fact, the preliminarily partitions achieved from the first stage should be checked for feasibility. Also, these partitions should be adjusted if it is necessary in order to satisfy the system related constraints. To this purpose OPF calculation is incorporated and optimal load management measures are taken to mitigate violations considering the voltage constraints, line capacities and controllable loads. Since the proposed method considers the key factors of load priorities, load controllability, power balance, voltage and line capacity constraints, it covers the requirements of practical applications. Numerical results from the PG&E-69 test system are used to verify the effectiveness of the proposed method. The comparison with previous methods demonstrates that the proposed method outperforms other techniques.
1. Introduction Maintaining system sustainability has become one of the main necessities in modern societies with the impending energy crisis and environmental deteriorations [1]. In order to comply these requirements, the application of distributed generation (DG) is considerably increased and it is expected to be higher in the near future. This tendency has led to a demand for a new electricity distribution paradigm in the framework of smart grids [2]. One of the most promising concepts, which is gaining importance in parallel to the rapid evolution of smart-gridbased solutions, is the microgrid. A microgrid is a cluster of both DGs and loads which act to cooperate with the main grid or autonomously from it [3]. The main feature of a microgrid is its ability to seamlessly separate itself from the main grid during the occurrence of an upstream network disturbance and operate as a self-controlled entity with high efficiency [1]. Inspired by this feature, a new strategy, so-called island partitioning, has been introduced as an interesting solution to help the affected customers to survive in the case of severe disturbance occurring. In this paper, the island partitioning means making, inside the
⁎
distribution system, self-sufficient areas with minimized generationload imbalance. When an outage occurs in a distribution grid, the islanding operation of DGs could rapidly restore the energy supply of important loads, and reduce the outage time. Accordingly, implementing this concept brings out reliability improvement, reduces the outage cost and mitigates the outage frequency [4–7]. In a feasible partitioning solution, key requirements such as power balance, bus voltage, line capacity restriction, load priority and load controllability should be considered [8,9]. The intentional partitioning procedure in power systems has been studied by several researchers as a restoration procedure able to defend against the catastrophe of system-wide blackout [10–13]. Recently, considering the advantages of the mentioned procedure, this subject also gained growing interest from distribution system researchers. These advantages include easier control strategy, distributed control among partitions, load routing and transfer among partitions, enhanced reliability, promoting the level of DG utilization efficiency and sustainability in response to cascading disturbances [9–13]. In [7], an adaptive intentional islanding operation system
Corresponding author. E-mail address: [email protected] (P. Siano).
https://doi.org/10.1016/j.ijepes.2017.11.003 Received 7 November 2016; Received in revised form 15 April 2017; Accepted 4 November 2017 0142-0615/ © 2017 Elsevier Ltd. All rights reserved.
Electrical Power and Energy Systems 97 (2018) 155–164
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Nomenclature
w br
A. Indices
D. Variables
b i j k ,e max,min ul,cl
α μ
cij
index for branches of system index for loads index for distributed generations (DGs) index for buses superscript for maximum and minimum values of variables, respectively superscript to represent uncontrollable and controllable loads, respectively index for controllable loads index for uncontrollable loads
d dq H p Plossb q Sl u V x kp
B. Sets:
B Br D G Y Δcl Δul
x ij δ θ
set of buses set of branches set of loads controllable DGs admittance matrix set of controllable loads set of uncontrollable loads
cost per unit of transferring the power from source j to load i active power demand reactive power demand knapsack capacity DG active power output power loss of branch b DG reactive power output capacity of lines load commitment state magnitude of bus voltage binary vector which is define the commitment of uncontrollable loads power amount transferred from source j to load i angle of bus voltage angle of admittance
E. Functions
C fcrd fad fkp fT ε
C. Parameters:
R v value
weight of each branch
resistance of each branch per unit value of each load total value of each load
cost function of DGs DGs redispatching objective function adjustment objective function knapsack problem objective function transportation problem objective function demand-generation balance objective function
procedures require applying multiple complementary rules to check and obtain the partitions boundaries. The review of the previous studies confirmed the intuition that the power supply reliability and DG power utilization in the post-contingency situation of a smart distribution grid can be significantly promoted by appropriately island partitioning strategies. However, previous results also reveal the need for a general, systematic and optimized approach to address the islanding strategy. This algorithmic approach should be able to tackle the above-mentioned deficiencies of previous studies, as well as to provide the maximum equivalent restored load. Besides, the presented solution must be reliable from the point of optimality. This paper proposes a systematic procedure for optimal island partitioning of smart distribution grids in the emergency operation. The procedure is a two-stage algorithmic solution. The first step utilizes the graph-theory related algorithms which consider the network as a weighted network matrix based on the power losses. Considering this matrix, the optimal partitioning scheme is determined through the modeling of the problem by using the transportation theory. Also, in order to manage the shortage of generation a knapsack problem is solved by the recently introduced species-based quantum particle swarm optimization (SQPSO) [19]. Then, in the second stage, an OPF calculation is carried out to check the feasibility of the formed partitions. In order to keep the practical perspective on the proposed method, it is developed such that the key factors of load priorities, load controllability, power balance, voltage, and line capacity constraints can be applied. Simulation results from the PG&E-69 test system and provided comparisons with previous works are used to verify the effectiveness of the proposed method. The key innovative contributions proposed in this paper can be summarized as follows:
configuration is provided. In [14], a procedure based on the strategy of extended sequential sampling is presented to provide the islanding operation mode of the system. However, in both [7] and [14], the optimality of islanded network partitions has not been discussed. An island partitioning plan based on exploring the power circles with DG in the center is utilized by [15] to determine the maximum restored load; however, it is difficult to find the optimal scheme in presence of multiple DGs. Besides, the procedure of [15] only considers uncontrollable loads. A graph-based procedure is introduced in [8] to execute island partitioning, however, since only one island can be planned, it may not restore a significant area in the case of the existence of DGs which are far away from each other. In [16,17], the authors introduced innovative smart grid design solutions providing virtual autonomous microgrids in a distribution system to improve power sustainability with the focus on self-healing capability. However, due to the computational complexity and uncertainties, these solutions are suitable only for planning tasks rather than to provide the optimal performance during the operation. In [18], a procedure is presented to provide and coordinate the autonomous partitions of a microgrid in the case of disconnection from the main grid. However, in this study an optimal island network adjustment procedure to make the feasible autonomous partitions is not discussed. In [9], a strategy based on the island partitioning is proposed to operate the distribution system with DGs in the case of emergency. This strategy is executed in the form of the two-stage method by using the branch and bound algorithm. The authors of [4] followed the similar approach of [9] but used a modified shuffled frog leap algorithm to address the strategy. However, both [4] and [9] acknowledged that their methods could not guarantee the optimality of the solution, since a simple sensitivity analysis is used to manage the loads instead of an optimal power flow (OPF) to get the final solution. Moreover, the procedures described in these studies could not handle the cases in which there is over generation, because they did not consider the DG adjustment possibility in their proposed models. Besides, those
• Proposing a systematic and flexible procedure in order to solve the island partitioning problem which provides the possibilities of
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• •
can survive during the disturbance occurrence (1). To this end, the maximum available capacity of the generation which is obtained from the on-line units is considered to define the objective function. Constraint (2) guarantees that the DG supply can satisfy the demands in the post-contingency system. The active and reactive power flow of the system is given in (3) and (4). The power output and voltage constraints of individual DGs and buses are depicted in (5) and (6), respectively. Eq. (7) guarantees that the capacity constraints of lines are respected during the optimization. The Eqs. (8)–(10) discuss a number of lateral-objective functions for the problem. These objective functions are not the highest priority of the defined optimization problem; however, as mentioned before, fulfilling them at satisfactory level certainly improves the quality of solution. The power losses need to be minimized as expressed in (8). Minimizing the power losses as one of the optimization goals is not only considered because of economic reasons, but it is also useful to restore more loads. In addition, supplying the loads with high priority should be considered prior to supplying ordinary loads as shown in Eq. (9). Besides, if there is a surplus of generation in the isolated part, the DG adjustment should be accomplished such that the new re-dispatch scheme minimizes the operating cost as much as possible (10). As observed above, the optimal restoring of the system in emergency conditions is formulated as a constrained nonlinear integer programming problem. Solving this problem with regard to the nonlinear constraints, the presence of mixed integer variables and considering the multiple goals creates certain complexities. On the other hand, according to the distribution operation in emergency situations, the solution procedure run time should be within between reasonable limits. In order to fulfill such requirements, in this paper, a systematic twostage method is proposed to crack down the problem and determine the distribution system operation in the event of emergency conditions. Fig. 1 shows the proposed two-stage optimal partitioning model outline for a smart distribution system. In the first stage, regardless of the system-related constraints and controllable loads, the partitioning scheme of the system after contingency is recognized ignoring power losses. In order to simplify and
applying the practical constraints, different non-linear criterions and load priority. Utilizing a weighted network matrix based on the loss of each branch of the system in order to obtain the optimal partitioning scheme with minimum losses. Integrating the nonlinear OPF simulation for determining the optimum final solution of the partitioning scheme.
In addition, in Section 2 a smart distribution grid optimal island partitioning model outline is described; in Section 3 the details of the proposed partitioning scheme are explained; the initial islands adjustment procedure scheme is presented in section 4; the numerical simulation results are presented and analyzed in Section 5 and finally the conclusion is presented in Section 6. 2. Problem formulation and optimal island partitioning model outline The occurrence of a serious fault in a distribution system can lead to the separation of the entire or some parts of the network from the main grid. In these conditions, the system may be divided into two parts known as connected and islanded. In the connected-part, the main parameters of the power quality, i.e. voltage and frequency, can be easily adjusted in their permitted range due to the presence of the main grid and if there exists a lack of active and reactive power, this shortage can be compensated by the main grid considered as an unlimited source. However, due to the imbalance between generation and demand, controlling the frequency and the voltage in the allowable rate for the parts of network remained isolated bring out challenging issues for the future distribution systems operation. This imbalance may also lead to the loss of isolated networks. Partitioning and creating independent sectors across the isolated area of smart distribution grid, thanks to the presence of distributed resources, is the main driving interest of the method proposed in this paper in order to provide a solution to the mentioned problem. For this area, the priority is to maintain a reliable power supply to the affected customers. However, the control actions are performed in order to also achieve economic benefits as much as possible. The problem in the emergency conditions can be formulated as an optimization problem, adapted from [18], with the objective function (1) and subject to a group of constraints as follows:
ε=
Min
∑
pj − ∑ di
j∈G
Fault
(1)
i∈D
Gridconnected part
Redispatch of DGs
Smart distribution system
subject to (2)
ε⩾0
∑
pj − ∑ di =
j∈G
i∈D
j∈G
i∈D
pj ⩽ pjmax
V min
Min
∑
(3)
∑ ∑
Vk Ve |Yke |sin (δk−δe + θke )
k∈B e∈B
j∈G
⩽ Vk ⩽
Slb ⩽ Slbmax
Vk Ve |Yke |cos(δk−δe + θke )
k∈B e∈B
qj− ∑ diq =
∑
∑ ∑
V max
∀k∈B
∀ b ∈ Br
∑ ∑ ∑ i ∈ B α ∈ Δcl μ ∈ Δul
fcrd =
∑ j∈G
Cj (pj )
Yes
(6)
Perform load shedding
(7)
Plossb
Isolated part
No
(4) (5)
Perform islands optimal adjustment
Form islands
(8)
b ∈ Br
Max
Proposed model Generation sufficiency?
dαcl uαcl vi + dμul uμul vi
Partitioning stage (Stage 1)
(9)
Feasibility checking and adjusting stage (Stage 2)
(10) Fig. 1. Proposed smart distribution grid optimal island partitioning model outline.
One of the objectives is that a high number of affected customers 157
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reduce the computational burden of the partitioning, the investigated system is modeled as an undirected, weighted tree graph and then a number of graph-theory related algorithms are applied. As observed from Fig. 1, at first the sufficiency of generation is checked for the affected part of the distribution system. If the requirement is satisfied, then the problem of optimal mapping of the loads is addressed. Through this problem, the partition that the loads are assigned is determined. In the proposed method, the transportation theory [20] is adapted to determine the partition of each DG. It is noteworthy that although the generation capacity is insufficient, it is necessary to reconsider the status of some loads to sufficiency satisfy the generation before the mapping process. In the proposed method, a tree knapsack problem [21,22] is adapted to the address the new commitments of loads based on their values. By following the stage one process, the Eqs. (1), (2) and the second term of (9) are considered in the solution methodology. In addition, by defining the weights of tree graph in the modeling of distribution system based on the power losses of branches, constraint (8) is automatically considered and satisfied in the proposed method. The implementation of the first stage is described in details in Section 3. In the second stage, the partitions obtained from the first stage are investigated to satisfy the nonlinear constraints related to system operation and their compliance. Consequently, if necessary the optimal adjustments of controllable loads are applied to obtain the satisfaction of constraints (3)–(7) in each partition. In fact, in the second stage, the optimal adjustments of controllable loads considering the first term of Eq. (9) as objective function are decided by executing OPF simulations for each partition. The output of this stage specifies the final feasible and optimal configuration of the smart distribution grid for emergency operation. Besides, if in the isolated part there is a surplus of power, the optimal re-dispatch of included DGs should be carried out which is considered in this stage by considering Eq. (10). The detailed description of the second stage is presented in Section 4.
∑
i=1
n
∑ i=1
(13)
j = 1,2,⋯,m
(14)
m
di <
∑
pj
(15)
j=1
Let Gsys = (B; Br; wbr) be a weighted graph of system, c be the length of a shortest path between Nr and Nu and Nu B, For every Nu B Nr Set
c( Nr )
0
and
c( Nu)
End For i=1,…, n-1 For each branch NuNv
Br
c( Nu) min{c( Nu), c( Nv) wbr ( Nu, Nv)} end End Output the parameter of c-the cost of transportation- for every node considering Nr as rooted node is calculated.
m j=1
j = 1,2,⋯,m
The total amount of power received by load i from all sources must be equal to the demand at this destination (12). Constraint (13) states that the total amount of power transported from the source j must be lower than its available capacity. The non-negativity conditions (14) are considered because the negative values for any x have no physical meaning. In the proposed model, for calculating the cost of supplying every load from any source (transferring cost), the problem of the shortest path is defined and solved. The solution of this problem is a spanning tree T which is rooted at a defined node Nr. The unique path from any node Nu to Nr is the shortest path from node Nu to Nr. The length of this path is the shortest distance between nodes Nu and Nr. Several algorithms are presented to solve shortest path problems. The method used for solving the shortest path in the proposed model is Bellman-FordMoore algorithm [23]. The mechanism of this algorithm is simple and also in dealing with large-scale systems is capable to handle their computational task [23]. The algorithm utilizes the weighted power network incidence matrix where the corresponding weight with each branch is obtained from the branch resistance. Obviously, the definition of the weight of each branch can be easily extended and can include risk-related parameters such as failure rate of lines. With the weighted incidence matrix, the shortest path of all nodes from each individual DG in the distribution system can be derived by using each DG as a root node. By acquiring this tree for every single load, the shortest distance (here the path with the lowest losses) can be calculated for any source. As mentioned, the value obtained for the losses is used as the cost of transporting of per unit power from that source to the target load in the calculation. The pseudocode of Bellman-Ford-Moore algorithm for calculation of transporting cost is presented in Fig. 2. Applying the partitioning mechanism based on the theory of transportation not only is a simpler method but also is more efficient from computation burden point of view so that it can be applied for large-scale distribution networks easily. For solving the transportation problem, it is required that the total demand is lower than total available sources capacity (15). Otherwise, a number of loads should be shed until (15) is established. To this purpose, in order to obtain a feasible condition, the knapsack problem is solved before the island partitioning algorithm.
In the proposed method, the transportation solution algorithm is used for allocating load demands to the energy sources in order to determine the island partition corresponding to each DG of the distribution system. As the name implies, a transportation problem aims at minimizing the cost of transporting certain merchandise from a number of origins to a number of destinations [20]. By adapting the problem to the proposed method, the origins are DGs and electrical power supposed to be transferred to the loads as destinations. The cost of the transferring is calculated based on the branch resistances. Suppose that there are m sources (origins) and n loads (destinations) in a distribution system. Let pj be the amount of power available at source j, di be the required demand at load i and cij be the cost of transferring the power from source j to load i. The aim is to determine the amount of power x ij transferred from source j to load i such that the total transportation costs are minimized (11). This problem can be formulated as follows:
∑∑
x ij ⩽ pj
x ij ⩾ 0 i = 1,2,⋯,n
3.1. Formulation of the transportation problem
fT =
(12)
i=1
The proposed island partitioning of a smart distribution system is addressed through the graph-theory related algorithms as show in stage one of Fig. 1. This procedure is designed based on the extension of the transport and knapsack problem solutions. This section describes the concepts and formulation of these problems and the proposed island partitioning procedure.
Min
i = 1,2,⋯,n
n
∑
3. Proposed partitioning scheme
n
x ij = di
j=1
cij x ij (11)
Fig. 2. The pseudocode of Bellman-Ford-Moore algorithm for calculation of transporting cost.
subject to 158
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DG. The assigned group of loads is assigned to each DG and then labeled as the partitions of that DG. It must be mentioned that, in order to consider all buses in the partitioning process, no-load buses are added to each partition based on their minimum transferring cost. Step 8. The partition of DGs are compared and if there is a common load between the clusters, then, those clusters are merged and the final islanding scheme is yielded.
3.2. Formulation of the knapsack problem Under the condition of violation of (15), a load shedding problem must be solved before running the initial island partitioning algorithm (transportation problem). In the proposed method the knapsack problem is adapted to address the load shedding problem. The knapsack problem is the problem faced by someone who has a fixed size knapsack and should fill it with the most valuable items [21,22]. In fact, considering a set of items, each with a weight and a value, this problem address the items to include in the collection so that the total value is as large as possible and the total weight is equal or lower than a given limit [21,22]. In the proposed model, the demand of each load is considered as the weight of that load and (16) is utilized to calculate the value of each bus. In (16), v is the given per unit value for each load which is considered as the load priority related coefficient in this paper and can be interpreted in the context of curtailment cost.
It is worth mentioning that if the isolated part consists of multiple separate areas then, in order to obtain the partitions of each area, the above-mentioned procedure is individually implemented for each area.
3.4. Species-based quantum particle swarm optimization The authors of this paper have introduced the SQPSO algorithm in 2014 [19]. A detailed description of the algorithm is available in [19]. In [19] it is also presented a comprehensive analysis of the exploration ability and solution quality of SQPSO applying the method on several standard benchmark functions and economic dispatch as a practical problem. Comparing the simulation results of the SQPSO with those of the other approaches in [25] depicts the superiority of SQPSO over other methods from stability, convergence characteristic and accuracy viewpoints. The results obtained in [24] with different operational limitations and concepts confirmed that the SQPSO is a very reliable tool for solving the complex practical problems. Therefore, in this paper, the binary knapsack problem included in the proposed partitioning method is addressed by using the SQPSO algorithm. However,
(16)
valuei = vi × di
The objective function of the load shedding problem is to maximize the load priority of the selected nodes (17). The specified limit for the virtual knapsack H is equal to the total capacity of available generation resources. The selected loads to be supplied should be lower than H (18). The solution is a binary vector x kp in which at each element can be assigned 0 or 1 for the shedding chosen state; therefore the problem can be mathematically formulated as follows: n
Max
fkp =
∑
valuei x ikp
i=1
(17)
subject to
Fault
n
∑
di x ikp
⩽H (18)
i=1
x ikp ∈ {0,1}
i = 1,2,⋯,n
Distribution system
Isolated part
Grid-connected part
(19) Partitioning stage start
3.3. Island partitioning scheme Gather system load data
The procedure for the island partitioning of the isolated part of the system, considering the concepts of the transportation and knapsack problems, is outlined in the partitioning stage in Fig. 3 and is delineated as follows:
Determine optimal load shedding scheme by solving (17)-(19)
Step 1. The investigated system load data are gathered in this step. Step 2. Demand and value of all nodes of the distribution system are set according to the priority list which is defined by the authority. Also, the weight of each branch is calculated considering the branch resistances. Step 3. The incidence matrix of the distribution system which is modeled as an undirected and weighted tree graph is established. Step 4. The condition of statement (15) is checked and if true, it means that there is sufficient available generation capacity to supply the total demand and the algorithm will go to step 6. Otherwise, it will go to step 5 to optimally shed the loads until the condition (15) is satisfied. Step 5. The knapsack problem is solved to optimally determine the loads that must be shed. To this purpose, the equations (17)–(19) which represent a non-deterministic polynomial NP-complete problem [21,22], must be resolved. In this paper, the SQPSO algorithm [19] is utilized to solve the knapsack problem. Step 6. The power transportation cost of supplying every load by each DG is calculated according to the Bellman-Ford-Moore algorithm which is indicated in Fig. 2. Step 7. The developed transportation problem (11)–(14) is solved by a linear programming algorithm to optimally map each load to a
n
m
di
No i 1
pj j 1
Set the value of loads and the weight of branches
Set the incidence matrix of microgird
Yes
Calculate the power transportation cost of each load according to Fig.2
Determine optimal mapping of DG-Load by solving (11)-(14) to form initial partitions
Merge the partitions have loads in common with each other to make new partitions
Feasibility checking and adjusting stage Perform partitions adjustments by maximizing (20) subjected to (3)-(7) to provide the optimal and feasible work point
End
Fig. 3. Flowchart of the proposed optimal island partitioning framework.
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other reliable evolutionary algorithms can be used instead of SQPSO.
Table 1 Line capacity of IEEE 69 bus test system.
4. Islands adjustment scheme In this section, the solution of adjustment stage, stage 2 of Fig. 1, is described. 4.1. The feasibility checking and adjustment The partitions preliminarily achieved above should be checked individually for feasibility and adjusted if it is necessary in order to satisfy the system related constraints. To this purpose, considering the constraints of voltage, lines capacity and maximum output of generation units, an OPF calculation is carried out and optimal load management measures are taken to mitigate violations. In fact, in this stage the manageable parts of controllable loads are treated as control variables of the optimization problem. These variables should be adjusted to yield the maximum value of the restored loads. Besides, if there is a surplus of power, the outputs of the DGs should be adjusted considering the cost function. The objective function for the optimization process is (20) which represents the combination of (10) and the first term of (9). This objective function must be minimized subjected to the constraints (3)–(7).
Min
fad =
∑ j∈G
Cj (pj )− ∑
∑
i ∈ B α ∈ Δcl
Bus number for each line
Line capacity (kVA)
1–8 52–62 9–11, 46–48 12–16, 49 17–45, 50–51, 64–68,
4667 2223 1112 556 400
Table 2 Data of installed DG. DG number
1
2
3
4
5
6
Bus number Max. capacity (kW) Max. capacity (kVA)
47 250 300
5 50 60
19 400 500
63 1300 1600
32 40 50
42 100 120
Table 3 Controllability of loads.
dαcl uαcl vi (20)
Bus number
Load type
26–27, 34, 43–46, 50–52,54–55, 59, 64–67, 69 11, 13, 21, 49
100% controllable
Other Buses
40% controllable and 60% uncontrollable Uncontrollable
4.2. AC-OPF Table 4 Priority of loads.
In this paper, MATPOWER’s extensible architecture is used to implement and run OPF simulations [25]. The extended OPF formulation of MATPOWER is clarified in (21)–(25), which can completely cover the requirements of objective functions (20) and the constraints (3)–(7). In the below-mentioned equations, y is a vector in the form of [θ, V, p, q] that is determined according to the constraints of (22)–(25). The MINOPF package is applied to solve OPF problem, which is suitable for large-scale optimizations [26].
min f (y )
(21)
g (y ) = 0
(22)
h (y ) ⩽ 0
(23)
Priority
Priority factor
Bus number
1 2 3
100 10 1
6, 9, 12, 18, 35, 39, 48, 53, 62, 68 Other buses 7, 10, 11, 13, 16, 22, 28, 36, 37, 40, 54, 56, 57, 58, 59
y min ⩽ y ⩽ y max
(24)
lb ⩽ A (y ) ⩽ ub
(25)
Fig. 4. IEEE 69-bus test system with 6 DGs.
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5. Numerical simulations
12.66 kV distribution system with the total load of 3.8 MW and 2.69 Mvar [27]. To provide fair comparisons with previous works the following data are used [9]; Vmax = 1.05 p.u. and Vmin = 0.95 p.u., the line capacities are adjusted as shown in Table 1, the capacity and the
The IEEE 69-bus test system, depicted in Fig. 4, is selected to evaluate the performance of proposed model. The test system is a 8.7992 7.7415 7.2361 27 26 25 24 23 22 8.9724 7.2221 8.4903 7.3952 0.0093 0.4711 0.8923 3.4393 28 29 30 31 32 33 34 35 6.8805 0.0733 0.5431 1.7313 4.9133 6.6696
8.7707 7.7130 7.2076 27 26 25 24 23 22 8.9439 7.1936 8.4618 7.3667 0.0310 0.4928 0.9140 3.4610 28 29 30 31 32 33 34 35 6.8520 0.0950 0.5630 1.7530 4.9350 6.6414
0.0034 0.3945 0.8677 0.0733 0.2090 0.9391 36 37 38 39 40 41 0.0093 0.1786 0.2108
1.7360
2
15 2.6346 4.7086 5.7666
0.0251 0.3660 0.8392 0.0950 0.2307 0.9608 36 37 38 39 40 41 0.0310 0.2003 0.2325
4.7747 68 69
4.7794 7.1217 6.2269 6.6414 5.8944 5.7529 4.7467 3.0053 4 5 6 7 8 9 10 11 12 13 14 6.7550
2
6.6680
1 3
16 17 18 19 20 21
6.6699 7.0448 5.8950 47 48 49 50 51 52
6.6665 6.2754 5.8022 6.7364 6.8721 7.6022 36 37 38 39 40 41 6.6724 6.8417 6.8739
4.9339
0.3323 0.7067
2
15 4.0353 1.9613 0.9033
7.2181 6.1305 5.2304 5.6449 4.8979 4.7564 5.7626 7.5040 4 5 6 7 8 9 10 11 12 13 14 5.7585
1 3
5.6700 5.2789 4.8057 5.7399 5.8756 6.6057 36 37 38 39 40 41 5.6759 5.8452 5.8774
4.2642
1.3494 2.1779 0.9140 1.6610 1.8025 2.8087 4.5501 4 5 6 7 8 9 10 11 12 13 14
0.8889 1.2800 1.7532 0.9558 1.0915 1.8216 36 37 38 39 40 41 0.8918 1.0611 1.0933
2.6215
10.7017 9.8026
15 6.4740 8.5480 9.6060
10.0415 8.9838 8.4784 27 26 25 24 23 22 10.2147 8.4644 9.7326 8.6375 1.2486 1.7104 2.1316 4.6786 28 29 30 31 32 33 34 35 8.1228 1.3126 1.7806 2.9706 6.1526 7.9122
DG6 @bus#42
1.2491 1.6240 2.2028 47 48 49 50 51 52
7.2278 7.2231 6.8487
15 3.5201 5.5941 6.6521
66 67 3.0146 3.0099 2.1795 2.7450 5.1187 2.2907 2.1726 53 54 55 56 57 58 59 46 45 44 43 5.4229 4.3350 42 1.9765 2.4637 2.1818 2.2916 7.2694 6.4139 2.1316 65 64 63 62 61 5.8090 8.3104 6.5589 6.3165 60
4.6163 68 69 4.6210
1.7062 2.5347 1.2708 2.0178 2.1593 3.1655 4.9069 4 5 6 7 8 9 10 11 12 13 14 1.3342
1 2
1.2442
4.2595 68 69
16 17 18 19 20 21
0.8874
3
10.1817
(d)
9.6847 8.6270 8.1216 27 26 25 24 23 22 9.8579 8.1076 9.3570 8.2807 0.8830 0.4212 0.0000 2.5470 28 29 30 31 32 33 34 35 7.7660 0.8190 0.3510 0.8390 4.0210 7.5554
2
5.5754
10.5093
66 67 5.9685 5.9638 4.3794 3.8139 1.4402 7.0748 6.9567 53 54 55 56 57 58 59 46 45 44 43 1.1360 4.5824 4.0952 2.2239 42 7.0757 6.9659 6.9157 0.7105 0.1450 65 64 63 62 61 0.7499 1.7515 0.0000 0.2424 60
(c)
0.9774
15 2.6061 4.6801 5.7381
5.7399 6.2079 7.3979 10.5799 5.6734 6.0483 4.8985 7.2134 47 48 49 50 51 52 68 69
0.3276
DG5 @bus#32
1
5.9347
DG4 @bus#63
66 67 4.9526 4.9979 6.1299 6.6954 9.0691 8.0713 7.9532 53 54 55 56 57 58 59 46 45 44 43 9.3733 5.9269 6.4141 8.2854 42 8.0722 7.9624 7.9122 11.2198 10.3643 65 64 63 62 61 9.7594 12.2608 10.5093 10.266960
0.8923 1.2672 1.8460 47 48 49 50 51 52
6.3091
(b) 12.6386 11.5809 11.0755 27 26 25 24 23 22 12.8118 11.0615 12.3297 11.2346 5.6759 6.1377 6.5589 9.1059 28 29 30 31 32 33 34 35 10.7199
(a) 2.1293 1.0716 0.5662 27 26 25 24 23 22 2.3025 0.5522 1.8204 0.7253 6.6724 7.1342 7.5554 10.1024 28 29 30 31 32 33 34 35 0.2106 6.7364 7.2044 8.3944 11.5764 0.0000
1.7075
6.3138
66 67 2.1006 2.0959 1.2655 1.8310 4.2047 1.4299 1.3118 53 54 55 56 57 58 59 46 45 44 43 4.5089 1.0625 1.5497 3.4210 42 1.4308 1.3210 1.2708 6.3554 5.4999 65 64 63 62 61 4.8950 7.3964 5.6449 5.4025 60
66 67 2.1291 2.1244 1.2940 1.8595 4.2332 1.4082 1.2901 53 54 55 56 57 58 59 46 45 44 43 4.5374 1.0910 1.5782 3.4495 42 1.4091 1.2993 1.2491 6.3839 5.5284 65 64 63 62 61 4.9235 7.4249 5.6734 5.4310 60
DG3 @bus#19
3
16 17 18 19 20 21
3
3.3502 0.4856 1.2639 0.0000 0.7470 0.8885 1.8947 3.6361 4 5 6 7 8 9 10 11 12 13 14 0.1136
1
5.9632
3
16 17 18 19 20 21
0.0049
2
6.3376
3.3455 68 69
16 17 18 19 20 21
3.3787 0.4571 1.2924 0.0285 0.7755 0.9170 1.9232 3.6646 4 5 6 7 8 9 10 11 12 13 14 0.0851
1
0.0285 0.4034 0.9320 47 48 49 50 51 52
6.3423
0.0266
3.3740 68 69
DG2 @bus#5
5.6715
0.0000 0.3749 0.9605 47 48 49 50 51 52
16 17 18 19 20 21
DG1 @bus#47
1.2457 1.6368 2.1100 1.1758 1.0401 0.3100 36 37 38 39 40 41 1.2398 1.0705 1.0383
2.9783
7.5846 7.5799 7.2055
15 3.8769 5.9509 7.0089
66 67 3.3714 3.3667 2.5363 3.1018 5.4755 0.1591 0.0410 53 54 55 56 57 58 59 46 45 44 43 5.7797 4.6918 42 2.3333 2.8205 0.0502 0.1600 7.6262 6.7707 0.0000 65 64 63 62 61 6.1658 8.6672 6.9157 6.6733 60 (f)
(e)
Fig. 5. The obtained 6 short paths (a–f) with respect to individual DGs (DG1–DG6) as the root nodes and transferring cost of per unit power corresponds to each load (numbers in red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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32
33
34
33
34
35
DG5 DG1
50
49
51
3
4
3
4
5
5
6
7
6
7
8
9
8
9
37
38
11
38
39
39
40
40
41
53
42
41
42
54 43
54
43
55 44
68
27
69
12 13
11
65
35
37
10
66
53
36
10
52
DG2
36
68
52
51
67
2
26
23
49
50
1
48
46
1
48
22
27
47
47
26
32
31
25
31
25
30
24
30
24
29
23
29
22
28
66
55
44
12
13
14
14
15
15
16
16
17
17
18
18
45
19
20
20
21
DG3
67 56
19
21
28
Load priority 1 Load priority 2 Load priority 3 100% controllable load priority 2 100% controllable load priority 3 40% controllable load priority 2 40% controllable load priority 3 Load shedding
56
45
57
57
58
58
59
59
60
60
61
61
62
62
63
63
64
64
65
DG4
46
DG6 Fig. 6. Island partitioning scheme obtained by the proposed method after line 2 outage.
grounding fault on the branch 2–3 (line 2), the downstream area of the outage section temporarily loses power supply. The proposed method for optimal island partitioning is applied and its solution procedures and calculation results are presented as follows:
Table 5 Partitioning scheme after line 2 outage. Restored loads
Bus num. P (kW) Bus num. P (kW)
6 2.6 68 28
9 30 8 75
12 145 21 106.2
18 60 24 28
35 6 29 26
39 24 33 14
48 79 49 230.82
53 4.35 61 1244
62 32
• By considering the total demand and generation of the isolated part, 69
Voltage (p.u.)
since there is an insufficiency in generation, ∑i = 3 di= 2.48037 and
7, 10, 11, 13, 14, 16, 17, 20, 22, 28, 36, 37, 40
Load shedding (bus num.) Max. Min.
6 ∑ j=1
•
1.05 @ bus 19 1.0435 @ bus 49
DG locations are selected from Table 2. In Table 3 the controllable and uncontrollable parts of the loads are shown and a three level load priority factor of 100, 10 and 1 are assigned to represent the importance of the loads as presented in Table 4. It must be mentioned that to consider the reactive power in the OPF calculation of the proposed model are the extracted data of [9] modified by using a constant power factor of 0.9 and the apparent power of generators taken from [28] as shown Table 2. The numerical assessments are implemented through the proposed method based on a software module in Matlab 8.3.0 in a 2.5 GHz-Intel Core2 Duo Processor laptop.
• • •
5.1. Case 1: Outage of line 2 This case study aims to provide a descriptive and comparative case for the proposed model. To evaluate the performance of the proposed model, it is assumed that after successful isolation of three-phase
pj = 2.14, the knapsack problem must be solved during the solution procedure. Solving this problem leads to the shedding of loads of buses 7, 10, 11, 13, 14, 16, 17, 20, 22, 28, 36, 37, 40 and 41. Based on the algorithm presented in Section 3.1, the costs per unit of transferring the power from every source to each load considering the post-contingency configuration of the distribution system are obtained by using 6 DGs (affected units) as the root nodes, as presented in Fig. 5. Solving the transportation problem, considering the results of the calculated transferring cost of power and the loads decided to shed from the knapsack solution, optimally assigns the loads to suitable generation. The results of this step are depicted in Fig. 5 for each generator with highlighted areas. The zones having common demands are merged with each other to make a unit partition. The final partitioning scheme is obtained by using the solution procedure depicted in Fig. 6. As observed from this figure, the final solution just includes one partition. Considering the controllable part of the loads, final adjustments of the partition are executed by solving the OPF problem to decide about the measures to satisfy the system related constraints. According to these measures, just the loads of buses 21 and 49 are controlled to 106.2 kW and 230.82 kW, respectively, and the other controllable loads are adjusted to zeros. The results of the OPF
Table 6 Comparison of test results with different methods for line 2 outage. Methods
The method of [15]a
The method of [8]a
The method of [9]
The method of [4]
The proposed method
Total restored loads (kW) Total restored loads to the total available generation (%) Network loss (kW) Network loss to the total restored loads (%) Total load value
1077 50.33 4.96 0.4605 28831.3
2070.55 96.75 9.58 0.4627 55060.7
2111.82 98.68 5.36 0.2538 –
2062.87 96.39 6.93 0.3359 56197.4
2134.97 99.76 5.0640 0.2371 58335.2
a
The reported results for this method obtained from [4].
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33
33
34
34
35
DG5 DG1
49
49
50 51
3
3
4
4
5
5
6
6
7
7
8
9
37
37
38
9
10
10
11
66
38
39
39
40
40
41
41
42
53
42
54 43
54
43
66
55 44
68
12
44
13
14
14
15
15
16
16
17
17
18
18
45
19
19
20
20
21
DG3
67 56
26
3
69
12 13
11
65
35
53
1
36
68
52
DG2
36
52
67
2
51
50
1
48
46
1
48
22
47
47
26
32
25
32
31
25
31
24
30
24
30
23
29
23
29
22
28
21
28
27
Load priority 1 Load priority 2 Load priority 3 100% controllable load priority 2 100% controllable load priority 3 40% controllable load priority 2 40% controllable load priority 3 Load shedding
2
56
45
57
57
58
58
59
59
60
60
61
61
62
62
63
63
64
64
65
DG4
46
4
DG6 Fig. 7. Island partitioning scheme obtained by the proposed method after the outage of lines 2 and 8.
Table 7 Partitioning scheme after the outage of lines 2 and 8. Load management
Controlled
Island partitions Num. P (kW) Q (kVAr)
1 43 0 0
45 0 0
46 0 0
2
49 218 155
Island partitions Num. P (kW) Q (kVAr)
50 0 0
51 0 0
52 0 0
34 0 0
3 21 94 67
26 0 0
27 0 0
54 0 0
4 55 0 0
66 0 0
67 0 0
Shed
7, 10, 11, 13, 16, 17, 22, 28, 33, 36, 37, 40
Survived
Other loads
Total restored loads (kW) Total restored loads to the total available generation (%) Total restored load value
69 0 0
59 0 0
64 21 15
65 0 0
2127.15 99.40 58257
methods which is highlighted in the table. Moreover the simple sensitivity method used in the previous works to adjust the initial partitioning scheme to get the feasible plan is substituted by the OPF solution which increases the validity of proposed method from a systematic perspective.
calculation show that there is no constraint violation in the presented scheme shown in Fig. 6. Table 5 shows the output of the proposed island partitioning scheme. As observed this table includes restored loads and load shedding plans. Besides, this table shows the voltage variation range of the affected system which confirms the feasibility of this constraint. The time duration of proposed model simulation has been 0.59 s which is an acceptable value in order to apply it in the practical applications.
5.2. Case 2: Simultaneous outage of lines 2 and 8 This case evaluates the performance of the proposed model with multiple simultaneous faults. It is assumed that two faults happen in the system, one is in the line 2 and the other is in the line 8. The optimum system partitioning results are shown in Fig. 7. The isolated area has been sectionalized into 4 partitions which are numbered from 1 to 4. Table 7 shows the outputs of the proposed partitioning procedure. This table includes the load controlling and shedding results for partitions 1–4. As observed from Table 7, the proposed method shows a satisfactory performance by restoring 2127.15 kW with a total restored load value of 58257. As observed from the table, similar to pervious case, the proposed procedure promotes the level of DG utilization efficiency and utilizes 99.4% of the available capacity to restore the system. Fig. 8 shows the voltage profile of each partition. It can be seen that the voltage levels of each partition are within the allowable range.
In order to compare the proposed method with other methods, the reported results of the island partitioning procedures of [4,8,9] and [15] for a similar case study are presented in Table 6 alongside the results of this section. As observed from this table, the proposed method outperforms all the previous procedures by providing better results. The proposed method restores 2134.97 kW of the loads in the affected part by utilizing 99.76% of available generation capacity. In addition to a higher amount of restoration, it can be seen from Table 6 that the total load value of the proposed method is increased by 2137.8 if compared to that of [4], 3274.5 if compared to [8] and 29503.9 if compared to [15] for the same case study. Besides, since the proposed procedure decides about the partitions and mapping the loads to the generation based on the loss of branches, it provides the least loss among the other 163
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1.05 1.045 1.04 9
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Bus (c)
23
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V (p.u.)
0.954 0.952 0.95 0.948 3
4
5
6
7
8
36
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38
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40
41
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43
46
47
48
49
50
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Bus (d) Fig. 8. Voltage profile of obtained partitions after the outage of lines 2 and 8; a) partition #4 b) partition #2 c) partition #3 d) partition #1.
6. Conclusion
generations. IET Gener Transm Distrib 2012;6(3):218–25. [10] Joglekar JJ, Nerkar YP. A different approach in system restoration with special consideration of islanding schemes. Int J Electr Power Energy Syst 2008;30(9):519–24. [11] Barsali S, Poli D, Pratico A, Salvati R, Sforna M, Zaottini R. Restoration islands supplied by gas turbines. Electr Power Syst Res 2008;78(12):2004–10. [12] Ding L, Gonzalez-Longatt FM, Wall P, Terzija V. Two-step spectral clustering controlled islanding algorithm. IEEE Trans Power Syst 2013;28(1):75–84. [13] Wang CG, Zhang BH, Hao ZG, Shu J, Li P, Bo ZQ. A novel real-time searching method for power system splitting boundary. IEEE Trans Power Syst 2010;25(4):1902–9. [14] Jayaweera D, Galloway S, Burt G, McDonald JR. A sampling approach for intentional islanding of distributed generation. IEEE Trans Power Syst 2007;22(2):514–21. [15] Lu Y, Yi X, Wu JA, Lin X. An intelligent islanding technique considering load balance for distribution system with DGs. IEEE Power Engineering Society General Meeting 2006. [16] Arefifar SA, Yasser AR, El-Fouly TH. Optimum microgrid design for enhancing reliability and supply-security. IEEE Trans Smart Grid 2013;4(3):1567–75. [17] Arefifar SA, Mohamed YA, EL-Fouly TH. Comprehensive operational planning framework for self-healing control actions in smart distribution grids. IEEE Trans Power Syst 2013;28(4):4192–200. [18] Fang X, Yang Q, Wang J, Yan W. Coordinated dispatch in multiple cooperative autonomous islanded microgrids. Appl Energy 2016;162:40–8. [19] Hosseinnezhad V, Rafiee M, Ahmadian M, Ameli MT. Species-based Quantum Particle Swarm Optimization for economic load dispatch. Int J Electr Power Energy Syst 2014;63:311–22. [20] Rao SS, Rao SS. Engineering optimization: theory and practice-chapter 4. John Wiley & Sons; 2009. [21] Johnson DS, Niemi KA. On knapsacks, partitions, and a new dynamic programming technique for trees. Math Operations Res 1983;8(1):1–4. [22] Cho G, Shaw DX, Kim SL. An efficient algorithm for a capacitated subtree of a tree problem in local access telecommunication networks. Comput Oper Res 1997 Aug 31;24(8):737–48. [23] Bang-Jensen J, Gutin GZ. Digraphs: theory, algorithms and applications-chapter 2. Springer Science & Business Media; 2008. [24] Hosseinnezhad V, Rafiee M, Ahmadian M, Siano P. Optimal day-ahead operational planning of microgrids. Energy Convers Manage 2016;126:142–57. [25] Zimmerman RD, Murillo-Sánchez CE, Thomas RJ. MATPOWER: steady-state operations, planning, and analysis tools for power systems research and education. IEEE Trans Power Syst 2011;26(1):12–9. [26] MINOPF. [Online]. Available: http://www.pserc.cornell.edu/minopf/. [accessed 07.07.16]. [27] Baran ME, Wu FF. Optimal capacitor placement on radial distribution systems. IEEE Trans Power Deliv 1989 Jan;4(1):725–34. [28] Zhang M, Chen J. Islanding and scheduling of power distribution systems with distributed generation. IEEE Trans Power Syst 2015;30(6):3120–9.
In this paper, an algorithmic two-stage procedure is proposed for post-contingency smart distribution systems. In the first stage, the partitions are obtained by solving the transportation problem and knapsack problems. The results of the first stage are presented to the second sequential sub-problem in order to analyze and adjust the obtained partition to achieve the final solution. All the key factors of the optimal islanding such as power balance, voltage, line capacity are taken into account. Besides, in order to weight the system, the power loss index is applied in the partitioning scheme which yields the least loss for the obtained solution. The proposed procedure is implemented on a test distribution system to evaluate the proposed method. The investigations carried out with significant details confirm the superiority of proposed method against the previous procedures. References [1] Zhao B, Shi Y, Dong X, Luan W, Bornemann J. Short-term operation scheduling in renewable-powered microgrids: a duality-based approach. IEEE Trans Sustain Energy 2014;5(1):209–17. [2] Conti S, Nicolosi R, Rizzo SA, Zeineldin HH. Optimal dispatching of distributed generators and storage systems for MV islanded microgrids. IEEE Transs Power Deliv 2012;27(3):1243–51. [3] Lasseter RH, Paigi P. Microgrid: a conceptual solution. In: Proc. Power electronics specialists conference, June 2004. [4] Oboudi MH, Hooshmand R, Karamad A. Feasible method for making controlled intentional islanding of microgrids based on the modified shuffled frog leap algorithm. Int J Electr Power Energy Syst 2016;78:745–54. [5] Laghari JA, Mokhlis H, Karimi M, Bakar AH, Mohamad H. Computational intelligence based techniques for islanding detection of distributed generation in distribution network: a review. Energy Convers Manage 2014;88:139–52. [6] Dorkhosh SS, Samet H. Restricting minimum size of DGs to confirm correct operation of fast directional protection switches in their simultaneous allocation with DGs. Energy Convers Manage 2015;94:482–92. [7] Caldon R, Stocco A, Turri R. Feasibility of adaptive intentional islanding operation of electric utility systems with distributed generation. Electric Power Syst Res 2008;78(12):2017–23. [8] Mao Y, Miu KN. Switch placement to improve system reliability for radial distribution systems with distributed generation. IEEE Trans Power Syst 2003;18(4):1346–52. [9] Jikeng L, Xudong W, Peng W, Shengwen L, Guang-hui S, Xin M, et al. Two-stage method for optimal island partition of distribution system with distributed
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Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Optimal island partitioning of smart distribution systems to improve system restoration under emergency conditions Vahid Hosseinnezhada, Mansour Rafieea, Mohammad Ahmadiana, Pierluigi Sianob, a b
MARK
⁎
Faculty of Electrical and Computer Engineering, Shahid Beheshti University, A.C., Tehran, Iran Department of Industrial Engineering, University of Salerno, Fisciano, Italy
A R T I C L E I N F O
A B S T R A C T
Keywords: DG Emergency operation Restoration Island partitioning scheme Optimal load control Optimal power flow
In the case of emergency operation, intentional islanding of distributed generations is an interesting solution to maintain the reliable power supply in the smart distribution grid. This paper deals with an optimal systematic strategy to restore the post-contingency distribution grid after the occurrence of severe disturbances causing an outage. Hence, a two-stage solution procedure is developed in this paper. In the first step, the optimal partitioning scheme is addressed through a number of graph-theory related algorithms. By applying these algorithms, the loads are optimally assigned to the distributed generations and optimal load shedding programs are determined and performed if there is a shortage of generation. The second step deals with the adjustments issues. In fact, the preliminarily partitions achieved from the first stage should be checked for feasibility. Also, these partitions should be adjusted if it is necessary in order to satisfy the system related constraints. To this purpose OPF calculation is incorporated and optimal load management measures are taken to mitigate violations considering the voltage constraints, line capacities and controllable loads. Since the proposed method considers the key factors of load priorities, load controllability, power balance, voltage and line capacity constraints, it covers the requirements of practical applications. Numerical results from the PG&E-69 test system are used to verify the effectiveness of the proposed method. The comparison with previous methods demonstrates that the proposed method outperforms other techniques.
1. Introduction Maintaining system sustainability has become one of the main necessities in modern societies with the impending energy crisis and environmental deteriorations [1]. In order to comply these requirements, the application of distributed generation (DG) is considerably increased and it is expected to be higher in the near future. This tendency has led to a demand for a new electricity distribution paradigm in the framework of smart grids [2]. One of the most promising concepts, which is gaining importance in parallel to the rapid evolution of smart-gridbased solutions, is the microgrid. A microgrid is a cluster of both DGs and loads which act to cooperate with the main grid or autonomously from it [3]. The main feature of a microgrid is its ability to seamlessly separate itself from the main grid during the occurrence of an upstream network disturbance and operate as a self-controlled entity with high efficiency [1]. Inspired by this feature, a new strategy, so-called island partitioning, has been introduced as an interesting solution to help the affected customers to survive in the case of severe disturbance occurring. In this paper, the island partitioning means making, inside the
⁎
distribution system, self-sufficient areas with minimized generationload imbalance. When an outage occurs in a distribution grid, the islanding operation of DGs could rapidly restore the energy supply of important loads, and reduce the outage time. Accordingly, implementing this concept brings out reliability improvement, reduces the outage cost and mitigates the outage frequency [4–7]. In a feasible partitioning solution, key requirements such as power balance, bus voltage, line capacity restriction, load priority and load controllability should be considered [8,9]. The intentional partitioning procedure in power systems has been studied by several researchers as a restoration procedure able to defend against the catastrophe of system-wide blackout [10–13]. Recently, considering the advantages of the mentioned procedure, this subject also gained growing interest from distribution system researchers. These advantages include easier control strategy, distributed control among partitions, load routing and transfer among partitions, enhanced reliability, promoting the level of DG utilization efficiency and sustainability in response to cascading disturbances [9–13]. In [7], an adaptive intentional islanding operation system
Corresponding author. E-mail address: [email protected] (P. Siano).
https://doi.org/10.1016/j.ijepes.2017.11.003 Received 7 November 2016; Received in revised form 15 April 2017; Accepted 4 November 2017 0142-0615/ © 2017 Elsevier Ltd. All rights reserved.
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Nomenclature
w br
A. Indices
D. Variables
b i j k ,e max,min ul,cl
α μ
cij
index for branches of system index for loads index for distributed generations (DGs) index for buses superscript for maximum and minimum values of variables, respectively superscript to represent uncontrollable and controllable loads, respectively index for controllable loads index for uncontrollable loads
d dq H p Plossb q Sl u V x kp
B. Sets:
B Br D G Y Δcl Δul
x ij δ θ
set of buses set of branches set of loads controllable DGs admittance matrix set of controllable loads set of uncontrollable loads
cost per unit of transferring the power from source j to load i active power demand reactive power demand knapsack capacity DG active power output power loss of branch b DG reactive power output capacity of lines load commitment state magnitude of bus voltage binary vector which is define the commitment of uncontrollable loads power amount transferred from source j to load i angle of bus voltage angle of admittance
E. Functions
C fcrd fad fkp fT ε
C. Parameters:
R v value
weight of each branch
resistance of each branch per unit value of each load total value of each load
cost function of DGs DGs redispatching objective function adjustment objective function knapsack problem objective function transportation problem objective function demand-generation balance objective function
procedures require applying multiple complementary rules to check and obtain the partitions boundaries. The review of the previous studies confirmed the intuition that the power supply reliability and DG power utilization in the post-contingency situation of a smart distribution grid can be significantly promoted by appropriately island partitioning strategies. However, previous results also reveal the need for a general, systematic and optimized approach to address the islanding strategy. This algorithmic approach should be able to tackle the above-mentioned deficiencies of previous studies, as well as to provide the maximum equivalent restored load. Besides, the presented solution must be reliable from the point of optimality. This paper proposes a systematic procedure for optimal island partitioning of smart distribution grids in the emergency operation. The procedure is a two-stage algorithmic solution. The first step utilizes the graph-theory related algorithms which consider the network as a weighted network matrix based on the power losses. Considering this matrix, the optimal partitioning scheme is determined through the modeling of the problem by using the transportation theory. Also, in order to manage the shortage of generation a knapsack problem is solved by the recently introduced species-based quantum particle swarm optimization (SQPSO) [19]. Then, in the second stage, an OPF calculation is carried out to check the feasibility of the formed partitions. In order to keep the practical perspective on the proposed method, it is developed such that the key factors of load priorities, load controllability, power balance, voltage, and line capacity constraints can be applied. Simulation results from the PG&E-69 test system and provided comparisons with previous works are used to verify the effectiveness of the proposed method. The key innovative contributions proposed in this paper can be summarized as follows:
configuration is provided. In [14], a procedure based on the strategy of extended sequential sampling is presented to provide the islanding operation mode of the system. However, in both [7] and [14], the optimality of islanded network partitions has not been discussed. An island partitioning plan based on exploring the power circles with DG in the center is utilized by [15] to determine the maximum restored load; however, it is difficult to find the optimal scheme in presence of multiple DGs. Besides, the procedure of [15] only considers uncontrollable loads. A graph-based procedure is introduced in [8] to execute island partitioning, however, since only one island can be planned, it may not restore a significant area in the case of the existence of DGs which are far away from each other. In [16,17], the authors introduced innovative smart grid design solutions providing virtual autonomous microgrids in a distribution system to improve power sustainability with the focus on self-healing capability. However, due to the computational complexity and uncertainties, these solutions are suitable only for planning tasks rather than to provide the optimal performance during the operation. In [18], a procedure is presented to provide and coordinate the autonomous partitions of a microgrid in the case of disconnection from the main grid. However, in this study an optimal island network adjustment procedure to make the feasible autonomous partitions is not discussed. In [9], a strategy based on the island partitioning is proposed to operate the distribution system with DGs in the case of emergency. This strategy is executed in the form of the two-stage method by using the branch and bound algorithm. The authors of [4] followed the similar approach of [9] but used a modified shuffled frog leap algorithm to address the strategy. However, both [4] and [9] acknowledged that their methods could not guarantee the optimality of the solution, since a simple sensitivity analysis is used to manage the loads instead of an optimal power flow (OPF) to get the final solution. Moreover, the procedures described in these studies could not handle the cases in which there is over generation, because they did not consider the DG adjustment possibility in their proposed models. Besides, those
• Proposing a systematic and flexible procedure in order to solve the island partitioning problem which provides the possibilities of
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• •
can survive during the disturbance occurrence (1). To this end, the maximum available capacity of the generation which is obtained from the on-line units is considered to define the objective function. Constraint (2) guarantees that the DG supply can satisfy the demands in the post-contingency system. The active and reactive power flow of the system is given in (3) and (4). The power output and voltage constraints of individual DGs and buses are depicted in (5) and (6), respectively. Eq. (7) guarantees that the capacity constraints of lines are respected during the optimization. The Eqs. (8)–(10) discuss a number of lateral-objective functions for the problem. These objective functions are not the highest priority of the defined optimization problem; however, as mentioned before, fulfilling them at satisfactory level certainly improves the quality of solution. The power losses need to be minimized as expressed in (8). Minimizing the power losses as one of the optimization goals is not only considered because of economic reasons, but it is also useful to restore more loads. In addition, supplying the loads with high priority should be considered prior to supplying ordinary loads as shown in Eq. (9). Besides, if there is a surplus of generation in the isolated part, the DG adjustment should be accomplished such that the new re-dispatch scheme minimizes the operating cost as much as possible (10). As observed above, the optimal restoring of the system in emergency conditions is formulated as a constrained nonlinear integer programming problem. Solving this problem with regard to the nonlinear constraints, the presence of mixed integer variables and considering the multiple goals creates certain complexities. On the other hand, according to the distribution operation in emergency situations, the solution procedure run time should be within between reasonable limits. In order to fulfill such requirements, in this paper, a systematic twostage method is proposed to crack down the problem and determine the distribution system operation in the event of emergency conditions. Fig. 1 shows the proposed two-stage optimal partitioning model outline for a smart distribution system. In the first stage, regardless of the system-related constraints and controllable loads, the partitioning scheme of the system after contingency is recognized ignoring power losses. In order to simplify and
applying the practical constraints, different non-linear criterions and load priority. Utilizing a weighted network matrix based on the loss of each branch of the system in order to obtain the optimal partitioning scheme with minimum losses. Integrating the nonlinear OPF simulation for determining the optimum final solution of the partitioning scheme.
In addition, in Section 2 a smart distribution grid optimal island partitioning model outline is described; in Section 3 the details of the proposed partitioning scheme are explained; the initial islands adjustment procedure scheme is presented in section 4; the numerical simulation results are presented and analyzed in Section 5 and finally the conclusion is presented in Section 6. 2. Problem formulation and optimal island partitioning model outline The occurrence of a serious fault in a distribution system can lead to the separation of the entire or some parts of the network from the main grid. In these conditions, the system may be divided into two parts known as connected and islanded. In the connected-part, the main parameters of the power quality, i.e. voltage and frequency, can be easily adjusted in their permitted range due to the presence of the main grid and if there exists a lack of active and reactive power, this shortage can be compensated by the main grid considered as an unlimited source. However, due to the imbalance between generation and demand, controlling the frequency and the voltage in the allowable rate for the parts of network remained isolated bring out challenging issues for the future distribution systems operation. This imbalance may also lead to the loss of isolated networks. Partitioning and creating independent sectors across the isolated area of smart distribution grid, thanks to the presence of distributed resources, is the main driving interest of the method proposed in this paper in order to provide a solution to the mentioned problem. For this area, the priority is to maintain a reliable power supply to the affected customers. However, the control actions are performed in order to also achieve economic benefits as much as possible. The problem in the emergency conditions can be formulated as an optimization problem, adapted from [18], with the objective function (1) and subject to a group of constraints as follows:
ε=
Min
∑
pj − ∑ di
j∈G
Fault
(1)
i∈D
Gridconnected part
Redispatch of DGs
Smart distribution system
subject to (2)
ε⩾0
∑
pj − ∑ di =
j∈G
i∈D
j∈G
i∈D
pj ⩽ pjmax
V min
Min
∑
(3)
∑ ∑
Vk Ve |Yke |sin (δk−δe + θke )
k∈B e∈B
j∈G
⩽ Vk ⩽
Slb ⩽ Slbmax
Vk Ve |Yke |cos(δk−δe + θke )
k∈B e∈B
qj− ∑ diq =
∑
∑ ∑
V max
∀k∈B
∀ b ∈ Br
∑ ∑ ∑ i ∈ B α ∈ Δcl μ ∈ Δul
fcrd =
∑ j∈G
Cj (pj )
Yes
(6)
Perform load shedding
(7)
Plossb
Isolated part
No
(4) (5)
Perform islands optimal adjustment
Form islands
(8)
b ∈ Br
Max
Proposed model Generation sufficiency?
dαcl uαcl vi + dμul uμul vi
Partitioning stage (Stage 1)
(9)
Feasibility checking and adjusting stage (Stage 2)
(10) Fig. 1. Proposed smart distribution grid optimal island partitioning model outline.
One of the objectives is that a high number of affected customers 157
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reduce the computational burden of the partitioning, the investigated system is modeled as an undirected, weighted tree graph and then a number of graph-theory related algorithms are applied. As observed from Fig. 1, at first the sufficiency of generation is checked for the affected part of the distribution system. If the requirement is satisfied, then the problem of optimal mapping of the loads is addressed. Through this problem, the partition that the loads are assigned is determined. In the proposed method, the transportation theory [20] is adapted to determine the partition of each DG. It is noteworthy that although the generation capacity is insufficient, it is necessary to reconsider the status of some loads to sufficiency satisfy the generation before the mapping process. In the proposed method, a tree knapsack problem [21,22] is adapted to the address the new commitments of loads based on their values. By following the stage one process, the Eqs. (1), (2) and the second term of (9) are considered in the solution methodology. In addition, by defining the weights of tree graph in the modeling of distribution system based on the power losses of branches, constraint (8) is automatically considered and satisfied in the proposed method. The implementation of the first stage is described in details in Section 3. In the second stage, the partitions obtained from the first stage are investigated to satisfy the nonlinear constraints related to system operation and their compliance. Consequently, if necessary the optimal adjustments of controllable loads are applied to obtain the satisfaction of constraints (3)–(7) in each partition. In fact, in the second stage, the optimal adjustments of controllable loads considering the first term of Eq. (9) as objective function are decided by executing OPF simulations for each partition. The output of this stage specifies the final feasible and optimal configuration of the smart distribution grid for emergency operation. Besides, if in the isolated part there is a surplus of power, the optimal re-dispatch of included DGs should be carried out which is considered in this stage by considering Eq. (10). The detailed description of the second stage is presented in Section 4.
∑
i=1
n
∑ i=1
(13)
j = 1,2,⋯,m
(14)
m
di <
∑
pj
(15)
j=1
Let Gsys = (B; Br; wbr) be a weighted graph of system, c be the length of a shortest path between Nr and Nu and Nu B, For every Nu B Nr Set
c( Nr )
0
and
c( Nu)
End For i=1,…, n-1 For each branch NuNv
Br
c( Nu) min{c( Nu), c( Nv) wbr ( Nu, Nv)} end End Output the parameter of c-the cost of transportation- for every node considering Nr as rooted node is calculated.
m j=1
j = 1,2,⋯,m
The total amount of power received by load i from all sources must be equal to the demand at this destination (12). Constraint (13) states that the total amount of power transported from the source j must be lower than its available capacity. The non-negativity conditions (14) are considered because the negative values for any x have no physical meaning. In the proposed model, for calculating the cost of supplying every load from any source (transferring cost), the problem of the shortest path is defined and solved. The solution of this problem is a spanning tree T which is rooted at a defined node Nr. The unique path from any node Nu to Nr is the shortest path from node Nu to Nr. The length of this path is the shortest distance between nodes Nu and Nr. Several algorithms are presented to solve shortest path problems. The method used for solving the shortest path in the proposed model is Bellman-FordMoore algorithm [23]. The mechanism of this algorithm is simple and also in dealing with large-scale systems is capable to handle their computational task [23]. The algorithm utilizes the weighted power network incidence matrix where the corresponding weight with each branch is obtained from the branch resistance. Obviously, the definition of the weight of each branch can be easily extended and can include risk-related parameters such as failure rate of lines. With the weighted incidence matrix, the shortest path of all nodes from each individual DG in the distribution system can be derived by using each DG as a root node. By acquiring this tree for every single load, the shortest distance (here the path with the lowest losses) can be calculated for any source. As mentioned, the value obtained for the losses is used as the cost of transporting of per unit power from that source to the target load in the calculation. The pseudocode of Bellman-Ford-Moore algorithm for calculation of transporting cost is presented in Fig. 2. Applying the partitioning mechanism based on the theory of transportation not only is a simpler method but also is more efficient from computation burden point of view so that it can be applied for large-scale distribution networks easily. For solving the transportation problem, it is required that the total demand is lower than total available sources capacity (15). Otherwise, a number of loads should be shed until (15) is established. To this purpose, in order to obtain a feasible condition, the knapsack problem is solved before the island partitioning algorithm.
In the proposed method, the transportation solution algorithm is used for allocating load demands to the energy sources in order to determine the island partition corresponding to each DG of the distribution system. As the name implies, a transportation problem aims at minimizing the cost of transporting certain merchandise from a number of origins to a number of destinations [20]. By adapting the problem to the proposed method, the origins are DGs and electrical power supposed to be transferred to the loads as destinations. The cost of the transferring is calculated based on the branch resistances. Suppose that there are m sources (origins) and n loads (destinations) in a distribution system. Let pj be the amount of power available at source j, di be the required demand at load i and cij be the cost of transferring the power from source j to load i. The aim is to determine the amount of power x ij transferred from source j to load i such that the total transportation costs are minimized (11). This problem can be formulated as follows:
∑∑
x ij ⩽ pj
x ij ⩾ 0 i = 1,2,⋯,n
3.1. Formulation of the transportation problem
fT =
(12)
i=1
The proposed island partitioning of a smart distribution system is addressed through the graph-theory related algorithms as show in stage one of Fig. 1. This procedure is designed based on the extension of the transport and knapsack problem solutions. This section describes the concepts and formulation of these problems and the proposed island partitioning procedure.
Min
i = 1,2,⋯,n
n
∑
3. Proposed partitioning scheme
n
x ij = di
j=1
cij x ij (11)
Fig. 2. The pseudocode of Bellman-Ford-Moore algorithm for calculation of transporting cost.
subject to 158
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DG. The assigned group of loads is assigned to each DG and then labeled as the partitions of that DG. It must be mentioned that, in order to consider all buses in the partitioning process, no-load buses are added to each partition based on their minimum transferring cost. Step 8. The partition of DGs are compared and if there is a common load between the clusters, then, those clusters are merged and the final islanding scheme is yielded.
3.2. Formulation of the knapsack problem Under the condition of violation of (15), a load shedding problem must be solved before running the initial island partitioning algorithm (transportation problem). In the proposed method the knapsack problem is adapted to address the load shedding problem. The knapsack problem is the problem faced by someone who has a fixed size knapsack and should fill it with the most valuable items [21,22]. In fact, considering a set of items, each with a weight and a value, this problem address the items to include in the collection so that the total value is as large as possible and the total weight is equal or lower than a given limit [21,22]. In the proposed model, the demand of each load is considered as the weight of that load and (16) is utilized to calculate the value of each bus. In (16), v is the given per unit value for each load which is considered as the load priority related coefficient in this paper and can be interpreted in the context of curtailment cost.
It is worth mentioning that if the isolated part consists of multiple separate areas then, in order to obtain the partitions of each area, the above-mentioned procedure is individually implemented for each area.
3.4. Species-based quantum particle swarm optimization The authors of this paper have introduced the SQPSO algorithm in 2014 [19]. A detailed description of the algorithm is available in [19]. In [19] it is also presented a comprehensive analysis of the exploration ability and solution quality of SQPSO applying the method on several standard benchmark functions and economic dispatch as a practical problem. Comparing the simulation results of the SQPSO with those of the other approaches in [25] depicts the superiority of SQPSO over other methods from stability, convergence characteristic and accuracy viewpoints. The results obtained in [24] with different operational limitations and concepts confirmed that the SQPSO is a very reliable tool for solving the complex practical problems. Therefore, in this paper, the binary knapsack problem included in the proposed partitioning method is addressed by using the SQPSO algorithm. However,
(16)
valuei = vi × di
The objective function of the load shedding problem is to maximize the load priority of the selected nodes (17). The specified limit for the virtual knapsack H is equal to the total capacity of available generation resources. The selected loads to be supplied should be lower than H (18). The solution is a binary vector x kp in which at each element can be assigned 0 or 1 for the shedding chosen state; therefore the problem can be mathematically formulated as follows: n
Max
fkp =
∑
valuei x ikp
i=1
(17)
subject to
Fault
n
∑
di x ikp
⩽H (18)
i=1
x ikp ∈ {0,1}
i = 1,2,⋯,n
Distribution system
Isolated part
Grid-connected part
(19) Partitioning stage start
3.3. Island partitioning scheme Gather system load data
The procedure for the island partitioning of the isolated part of the system, considering the concepts of the transportation and knapsack problems, is outlined in the partitioning stage in Fig. 3 and is delineated as follows:
Determine optimal load shedding scheme by solving (17)-(19)
Step 1. The investigated system load data are gathered in this step. Step 2. Demand and value of all nodes of the distribution system are set according to the priority list which is defined by the authority. Also, the weight of each branch is calculated considering the branch resistances. Step 3. The incidence matrix of the distribution system which is modeled as an undirected and weighted tree graph is established. Step 4. The condition of statement (15) is checked and if true, it means that there is sufficient available generation capacity to supply the total demand and the algorithm will go to step 6. Otherwise, it will go to step 5 to optimally shed the loads until the condition (15) is satisfied. Step 5. The knapsack problem is solved to optimally determine the loads that must be shed. To this purpose, the equations (17)–(19) which represent a non-deterministic polynomial NP-complete problem [21,22], must be resolved. In this paper, the SQPSO algorithm [19] is utilized to solve the knapsack problem. Step 6. The power transportation cost of supplying every load by each DG is calculated according to the Bellman-Ford-Moore algorithm which is indicated in Fig. 2. Step 7. The developed transportation problem (11)–(14) is solved by a linear programming algorithm to optimally map each load to a
n
m
di
No i 1
pj j 1
Set the value of loads and the weight of branches
Set the incidence matrix of microgird
Yes
Calculate the power transportation cost of each load according to Fig.2
Determine optimal mapping of DG-Load by solving (11)-(14) to form initial partitions
Merge the partitions have loads in common with each other to make new partitions
Feasibility checking and adjusting stage Perform partitions adjustments by maximizing (20) subjected to (3)-(7) to provide the optimal and feasible work point
End
Fig. 3. Flowchart of the proposed optimal island partitioning framework.
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other reliable evolutionary algorithms can be used instead of SQPSO.
Table 1 Line capacity of IEEE 69 bus test system.
4. Islands adjustment scheme In this section, the solution of adjustment stage, stage 2 of Fig. 1, is described. 4.1. The feasibility checking and adjustment The partitions preliminarily achieved above should be checked individually for feasibility and adjusted if it is necessary in order to satisfy the system related constraints. To this purpose, considering the constraints of voltage, lines capacity and maximum output of generation units, an OPF calculation is carried out and optimal load management measures are taken to mitigate violations. In fact, in this stage the manageable parts of controllable loads are treated as control variables of the optimization problem. These variables should be adjusted to yield the maximum value of the restored loads. Besides, if there is a surplus of power, the outputs of the DGs should be adjusted considering the cost function. The objective function for the optimization process is (20) which represents the combination of (10) and the first term of (9). This objective function must be minimized subjected to the constraints (3)–(7).
Min
fad =
∑ j∈G
Cj (pj )− ∑
∑
i ∈ B α ∈ Δcl
Bus number for each line
Line capacity (kVA)
1–8 52–62 9–11, 46–48 12–16, 49 17–45, 50–51, 64–68,
4667 2223 1112 556 400
Table 2 Data of installed DG. DG number
1
2
3
4
5
6
Bus number Max. capacity (kW) Max. capacity (kVA)
47 250 300
5 50 60
19 400 500
63 1300 1600
32 40 50
42 100 120
Table 3 Controllability of loads.
dαcl uαcl vi (20)
Bus number
Load type
26–27, 34, 43–46, 50–52,54–55, 59, 64–67, 69 11, 13, 21, 49
100% controllable
Other Buses
40% controllable and 60% uncontrollable Uncontrollable
4.2. AC-OPF Table 4 Priority of loads.
In this paper, MATPOWER’s extensible architecture is used to implement and run OPF simulations [25]. The extended OPF formulation of MATPOWER is clarified in (21)–(25), which can completely cover the requirements of objective functions (20) and the constraints (3)–(7). In the below-mentioned equations, y is a vector in the form of [θ, V, p, q] that is determined according to the constraints of (22)–(25). The MINOPF package is applied to solve OPF problem, which is suitable for large-scale optimizations [26].
min f (y )
(21)
g (y ) = 0
(22)
h (y ) ⩽ 0
(23)
Priority
Priority factor
Bus number
1 2 3
100 10 1
6, 9, 12, 18, 35, 39, 48, 53, 62, 68 Other buses 7, 10, 11, 13, 16, 22, 28, 36, 37, 40, 54, 56, 57, 58, 59
y min ⩽ y ⩽ y max
(24)
lb ⩽ A (y ) ⩽ ub
(25)
Fig. 4. IEEE 69-bus test system with 6 DGs.
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5. Numerical simulations
12.66 kV distribution system with the total load of 3.8 MW and 2.69 Mvar [27]. To provide fair comparisons with previous works the following data are used [9]; Vmax = 1.05 p.u. and Vmin = 0.95 p.u., the line capacities are adjusted as shown in Table 1, the capacity and the
The IEEE 69-bus test system, depicted in Fig. 4, is selected to evaluate the performance of proposed model. The test system is a 8.7992 7.7415 7.2361 27 26 25 24 23 22 8.9724 7.2221 8.4903 7.3952 0.0093 0.4711 0.8923 3.4393 28 29 30 31 32 33 34 35 6.8805 0.0733 0.5431 1.7313 4.9133 6.6696
8.7707 7.7130 7.2076 27 26 25 24 23 22 8.9439 7.1936 8.4618 7.3667 0.0310 0.4928 0.9140 3.4610 28 29 30 31 32 33 34 35 6.8520 0.0950 0.5630 1.7530 4.9350 6.6414
0.0034 0.3945 0.8677 0.0733 0.2090 0.9391 36 37 38 39 40 41 0.0093 0.1786 0.2108
1.7360
2
15 2.6346 4.7086 5.7666
0.0251 0.3660 0.8392 0.0950 0.2307 0.9608 36 37 38 39 40 41 0.0310 0.2003 0.2325
4.7747 68 69
4.7794 7.1217 6.2269 6.6414 5.8944 5.7529 4.7467 3.0053 4 5 6 7 8 9 10 11 12 13 14 6.7550
2
6.6680
1 3
16 17 18 19 20 21
6.6699 7.0448 5.8950 47 48 49 50 51 52
6.6665 6.2754 5.8022 6.7364 6.8721 7.6022 36 37 38 39 40 41 6.6724 6.8417 6.8739
4.9339
0.3323 0.7067
2
15 4.0353 1.9613 0.9033
7.2181 6.1305 5.2304 5.6449 4.8979 4.7564 5.7626 7.5040 4 5 6 7 8 9 10 11 12 13 14 5.7585
1 3
5.6700 5.2789 4.8057 5.7399 5.8756 6.6057 36 37 38 39 40 41 5.6759 5.8452 5.8774
4.2642
1.3494 2.1779 0.9140 1.6610 1.8025 2.8087 4.5501 4 5 6 7 8 9 10 11 12 13 14
0.8889 1.2800 1.7532 0.9558 1.0915 1.8216 36 37 38 39 40 41 0.8918 1.0611 1.0933
2.6215
10.7017 9.8026
15 6.4740 8.5480 9.6060
10.0415 8.9838 8.4784 27 26 25 24 23 22 10.2147 8.4644 9.7326 8.6375 1.2486 1.7104 2.1316 4.6786 28 29 30 31 32 33 34 35 8.1228 1.3126 1.7806 2.9706 6.1526 7.9122
DG6 @bus#42
1.2491 1.6240 2.2028 47 48 49 50 51 52
7.2278 7.2231 6.8487
15 3.5201 5.5941 6.6521
66 67 3.0146 3.0099 2.1795 2.7450 5.1187 2.2907 2.1726 53 54 55 56 57 58 59 46 45 44 43 5.4229 4.3350 42 1.9765 2.4637 2.1818 2.2916 7.2694 6.4139 2.1316 65 64 63 62 61 5.8090 8.3104 6.5589 6.3165 60
4.6163 68 69 4.6210
1.7062 2.5347 1.2708 2.0178 2.1593 3.1655 4.9069 4 5 6 7 8 9 10 11 12 13 14 1.3342
1 2
1.2442
4.2595 68 69
16 17 18 19 20 21
0.8874
3
10.1817
(d)
9.6847 8.6270 8.1216 27 26 25 24 23 22 9.8579 8.1076 9.3570 8.2807 0.8830 0.4212 0.0000 2.5470 28 29 30 31 32 33 34 35 7.7660 0.8190 0.3510 0.8390 4.0210 7.5554
2
5.5754
10.5093
66 67 5.9685 5.9638 4.3794 3.8139 1.4402 7.0748 6.9567 53 54 55 56 57 58 59 46 45 44 43 1.1360 4.5824 4.0952 2.2239 42 7.0757 6.9659 6.9157 0.7105 0.1450 65 64 63 62 61 0.7499 1.7515 0.0000 0.2424 60
(c)
0.9774
15 2.6061 4.6801 5.7381
5.7399 6.2079 7.3979 10.5799 5.6734 6.0483 4.8985 7.2134 47 48 49 50 51 52 68 69
0.3276
DG5 @bus#32
1
5.9347
DG4 @bus#63
66 67 4.9526 4.9979 6.1299 6.6954 9.0691 8.0713 7.9532 53 54 55 56 57 58 59 46 45 44 43 9.3733 5.9269 6.4141 8.2854 42 8.0722 7.9624 7.9122 11.2198 10.3643 65 64 63 62 61 9.7594 12.2608 10.5093 10.266960
0.8923 1.2672 1.8460 47 48 49 50 51 52
6.3091
(b) 12.6386 11.5809 11.0755 27 26 25 24 23 22 12.8118 11.0615 12.3297 11.2346 5.6759 6.1377 6.5589 9.1059 28 29 30 31 32 33 34 35 10.7199
(a) 2.1293 1.0716 0.5662 27 26 25 24 23 22 2.3025 0.5522 1.8204 0.7253 6.6724 7.1342 7.5554 10.1024 28 29 30 31 32 33 34 35 0.2106 6.7364 7.2044 8.3944 11.5764 0.0000
1.7075
6.3138
66 67 2.1006 2.0959 1.2655 1.8310 4.2047 1.4299 1.3118 53 54 55 56 57 58 59 46 45 44 43 4.5089 1.0625 1.5497 3.4210 42 1.4308 1.3210 1.2708 6.3554 5.4999 65 64 63 62 61 4.8950 7.3964 5.6449 5.4025 60
66 67 2.1291 2.1244 1.2940 1.8595 4.2332 1.4082 1.2901 53 54 55 56 57 58 59 46 45 44 43 4.5374 1.0910 1.5782 3.4495 42 1.4091 1.2993 1.2491 6.3839 5.5284 65 64 63 62 61 4.9235 7.4249 5.6734 5.4310 60
DG3 @bus#19
3
16 17 18 19 20 21
3
3.3502 0.4856 1.2639 0.0000 0.7470 0.8885 1.8947 3.6361 4 5 6 7 8 9 10 11 12 13 14 0.1136
1
5.9632
3
16 17 18 19 20 21
0.0049
2
6.3376
3.3455 68 69
16 17 18 19 20 21
3.3787 0.4571 1.2924 0.0285 0.7755 0.9170 1.9232 3.6646 4 5 6 7 8 9 10 11 12 13 14 0.0851
1
0.0285 0.4034 0.9320 47 48 49 50 51 52
6.3423
0.0266
3.3740 68 69
DG2 @bus#5
5.6715
0.0000 0.3749 0.9605 47 48 49 50 51 52
16 17 18 19 20 21
DG1 @bus#47
1.2457 1.6368 2.1100 1.1758 1.0401 0.3100 36 37 38 39 40 41 1.2398 1.0705 1.0383
2.9783
7.5846 7.5799 7.2055
15 3.8769 5.9509 7.0089
66 67 3.3714 3.3667 2.5363 3.1018 5.4755 0.1591 0.0410 53 54 55 56 57 58 59 46 45 44 43 5.7797 4.6918 42 2.3333 2.8205 0.0502 0.1600 7.6262 6.7707 0.0000 65 64 63 62 61 6.1658 8.6672 6.9157 6.6733 60 (f)
(e)
Fig. 5. The obtained 6 short paths (a–f) with respect to individual DGs (DG1–DG6) as the root nodes and transferring cost of per unit power corresponds to each load (numbers in red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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32
33
34
33
34
35
DG5 DG1
50
49
51
3
4
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7
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9
37
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11
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54 43
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55 44
68
27
69
12 13
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65
35
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66
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10
52
DG2
36
68
52
51
67
2
26
23
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50
1
48
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1
48
22
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47
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22
28
66
55
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12
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45
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DG3
67 56
19
21
28
Load priority 1 Load priority 2 Load priority 3 100% controllable load priority 2 100% controllable load priority 3 40% controllable load priority 2 40% controllable load priority 3 Load shedding
56
45
57
57
58
58
59
59
60
60
61
61
62
62
63
63
64
64
65
DG4
46
DG6 Fig. 6. Island partitioning scheme obtained by the proposed method after line 2 outage.
grounding fault on the branch 2–3 (line 2), the downstream area of the outage section temporarily loses power supply. The proposed method for optimal island partitioning is applied and its solution procedures and calculation results are presented as follows:
Table 5 Partitioning scheme after line 2 outage. Restored loads
Bus num. P (kW) Bus num. P (kW)
6 2.6 68 28
9 30 8 75
12 145 21 106.2
18 60 24 28
35 6 29 26
39 24 33 14
48 79 49 230.82
53 4.35 61 1244
62 32
• By considering the total demand and generation of the isolated part, 69
Voltage (p.u.)
since there is an insufficiency in generation, ∑i = 3 di= 2.48037 and
7, 10, 11, 13, 14, 16, 17, 20, 22, 28, 36, 37, 40
Load shedding (bus num.) Max. Min.
6 ∑ j=1
•
1.05 @ bus 19 1.0435 @ bus 49
DG locations are selected from Table 2. In Table 3 the controllable and uncontrollable parts of the loads are shown and a three level load priority factor of 100, 10 and 1 are assigned to represent the importance of the loads as presented in Table 4. It must be mentioned that to consider the reactive power in the OPF calculation of the proposed model are the extracted data of [9] modified by using a constant power factor of 0.9 and the apparent power of generators taken from [28] as shown Table 2. The numerical assessments are implemented through the proposed method based on a software module in Matlab 8.3.0 in a 2.5 GHz-Intel Core2 Duo Processor laptop.
• • •
5.1. Case 1: Outage of line 2 This case study aims to provide a descriptive and comparative case for the proposed model. To evaluate the performance of the proposed model, it is assumed that after successful isolation of three-phase
pj = 2.14, the knapsack problem must be solved during the solution procedure. Solving this problem leads to the shedding of loads of buses 7, 10, 11, 13, 14, 16, 17, 20, 22, 28, 36, 37, 40 and 41. Based on the algorithm presented in Section 3.1, the costs per unit of transferring the power from every source to each load considering the post-contingency configuration of the distribution system are obtained by using 6 DGs (affected units) as the root nodes, as presented in Fig. 5. Solving the transportation problem, considering the results of the calculated transferring cost of power and the loads decided to shed from the knapsack solution, optimally assigns the loads to suitable generation. The results of this step are depicted in Fig. 5 for each generator with highlighted areas. The zones having common demands are merged with each other to make a unit partition. The final partitioning scheme is obtained by using the solution procedure depicted in Fig. 6. As observed from this figure, the final solution just includes one partition. Considering the controllable part of the loads, final adjustments of the partition are executed by solving the OPF problem to decide about the measures to satisfy the system related constraints. According to these measures, just the loads of buses 21 and 49 are controlled to 106.2 kW and 230.82 kW, respectively, and the other controllable loads are adjusted to zeros. The results of the OPF
Table 6 Comparison of test results with different methods for line 2 outage. Methods
The method of [15]a
The method of [8]a
The method of [9]
The method of [4]
The proposed method
Total restored loads (kW) Total restored loads to the total available generation (%) Network loss (kW) Network loss to the total restored loads (%) Total load value
1077 50.33 4.96 0.4605 28831.3
2070.55 96.75 9.58 0.4627 55060.7
2111.82 98.68 5.36 0.2538 –
2062.87 96.39 6.93 0.3359 56197.4
2134.97 99.76 5.0640 0.2371 58335.2
a
The reported results for this method obtained from [4].
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DG5 DG1
49
49
50 51
3
3
4
4
5
5
6
6
7
7
8
9
37
37
38
9
10
10
11
66
38
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42
54 43
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55 44
68
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44
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17
18
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45
19
19
20
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21
DG3
67 56
26
3
69
12 13
11
65
35
53
1
36
68
52
DG2
36
52
67
2
51
50
1
48
46
1
48
22
47
47
26
32
25
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25
31
24
30
24
30
23
29
23
29
22
28
21
28
27
Load priority 1 Load priority 2 Load priority 3 100% controllable load priority 2 100% controllable load priority 3 40% controllable load priority 2 40% controllable load priority 3 Load shedding
2
56
45
57
57
58
58
59
59
60
60
61
61
62
62
63
63
64
64
65
DG4
46
4
DG6 Fig. 7. Island partitioning scheme obtained by the proposed method after the outage of lines 2 and 8.
Table 7 Partitioning scheme after the outage of lines 2 and 8. Load management
Controlled
Island partitions Num. P (kW) Q (kVAr)
1 43 0 0
45 0 0
46 0 0
2
49 218 155
Island partitions Num. P (kW) Q (kVAr)
50 0 0
51 0 0
52 0 0
34 0 0
3 21 94 67
26 0 0
27 0 0
54 0 0
4 55 0 0
66 0 0
67 0 0
Shed
7, 10, 11, 13, 16, 17, 22, 28, 33, 36, 37, 40
Survived
Other loads
Total restored loads (kW) Total restored loads to the total available generation (%) Total restored load value
69 0 0
59 0 0
64 21 15
65 0 0
2127.15 99.40 58257
methods which is highlighted in the table. Moreover the simple sensitivity method used in the previous works to adjust the initial partitioning scheme to get the feasible plan is substituted by the OPF solution which increases the validity of proposed method from a systematic perspective.
calculation show that there is no constraint violation in the presented scheme shown in Fig. 6. Table 5 shows the output of the proposed island partitioning scheme. As observed this table includes restored loads and load shedding plans. Besides, this table shows the voltage variation range of the affected system which confirms the feasibility of this constraint. The time duration of proposed model simulation has been 0.59 s which is an acceptable value in order to apply it in the practical applications.
5.2. Case 2: Simultaneous outage of lines 2 and 8 This case evaluates the performance of the proposed model with multiple simultaneous faults. It is assumed that two faults happen in the system, one is in the line 2 and the other is in the line 8. The optimum system partitioning results are shown in Fig. 7. The isolated area has been sectionalized into 4 partitions which are numbered from 1 to 4. Table 7 shows the outputs of the proposed partitioning procedure. This table includes the load controlling and shedding results for partitions 1–4. As observed from Table 7, the proposed method shows a satisfactory performance by restoring 2127.15 kW with a total restored load value of 58257. As observed from the table, similar to pervious case, the proposed procedure promotes the level of DG utilization efficiency and utilizes 99.4% of the available capacity to restore the system. Fig. 8 shows the voltage profile of each partition. It can be seen that the voltage levels of each partition are within the allowable range.
In order to compare the proposed method with other methods, the reported results of the island partitioning procedures of [4,8,9] and [15] for a similar case study are presented in Table 6 alongside the results of this section. As observed from this table, the proposed method outperforms all the previous procedures by providing better results. The proposed method restores 2134.97 kW of the loads in the affected part by utilizing 99.76% of available generation capacity. In addition to a higher amount of restoration, it can be seen from Table 6 that the total load value of the proposed method is increased by 2137.8 if compared to that of [4], 3274.5 if compared to [8] and 29503.9 if compared to [15] for the same case study. Besides, since the proposed procedure decides about the partitions and mapping the loads to the generation based on the loss of branches, it provides the least loss among the other 163
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Bus (d) Fig. 8. Voltage profile of obtained partitions after the outage of lines 2 and 8; a) partition #4 b) partition #2 c) partition #3 d) partition #1.
6. Conclusion
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In this paper, an algorithmic two-stage procedure is proposed for post-contingency smart distribution systems. In the first stage, the partitions are obtained by solving the transportation problem and knapsack problems. The results of the first stage are presented to the second sequential sub-problem in order to analyze and adjust the obtained partition to achieve the final solution. All the key factors of the optimal islanding such as power balance, voltage, line capacity are taken into account. Besides, in order to weight the system, the power loss index is applied in the partitioning scheme which yields the least loss for the obtained solution. The proposed procedure is implemented on a test distribution system to evaluate the proposed method. The investigations carried out with significant details confirm the superiority of proposed method against the previous procedures. References [1] Zhao B, Shi Y, Dong X, Luan W, Bornemann J. Short-term operation scheduling in renewable-powered microgrids: a duality-based approach. IEEE Trans Sustain Energy 2014;5(1):209–17. [2] Conti S, Nicolosi R, Rizzo SA, Zeineldin HH. Optimal dispatching of distributed generators and storage systems for MV islanded microgrids. IEEE Transs Power Deliv 2012;27(3):1243–51. [3] Lasseter RH, Paigi P. Microgrid: a conceptual solution. In: Proc. Power electronics specialists conference, June 2004. [4] Oboudi MH, Hooshmand R, Karamad A. Feasible method for making controlled intentional islanding of microgrids based on the modified shuffled frog leap algorithm. Int J Electr Power Energy Syst 2016;78:745–54. [5] Laghari JA, Mokhlis H, Karimi M, Bakar AH, Mohamad H. Computational intelligence based techniques for islanding detection of distributed generation in distribution network: a review. Energy Convers Manage 2014;88:139–52. [6] Dorkhosh SS, Samet H. Restricting minimum size of DGs to confirm correct operation of fast directional protection switches in their simultaneous allocation with DGs. Energy Convers Manage 2015;94:482–92. [7] Caldon R, Stocco A, Turri R. Feasibility of adaptive intentional islanding operation of electric utility systems with distributed generation. Electric Power Syst Res 2008;78(12):2017–23. [8] Mao Y, Miu KN. Switch placement to improve system reliability for radial distribution systems with distributed generation. IEEE Trans Power Syst 2003;18(4):1346–52. [9] Jikeng L, Xudong W, Peng W, Shengwen L, Guang-hui S, Xin M, et al. Two-stage method for optimal island partition of distribution system with distributed
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