Repair Priority in Distribution Systems Considering Resilience Enhancement

Abstract

When a meteorological disaster occurs and equipment becomes damaged, a significant amount of time is required to repair the damaged components as it is impossible to repair several components simultaneously. Therefore, the determination of repair priority is a significant aspect of a distribution system’s resilience. This study proposes a technique to identify the unserved areas of a radial distribution system based on the bus injection to the branch current (BIBC) matrix, as opposed to a complex optimization technique, for evaluating the repair priority determination strategy for all the possible disaster scenarios. Generally, most resilience metrics include the concept of duration; therefore, the strategy for resilience enhancement must optimize the recovery priority using an objective function that consists of the recovered capacity increment, rather than the recovered capacity. To verify the proposed method, in this paper, the resilience is evaluated under all the disaster scenarios that can occur in contingencies from N-2 to N-5. Since complex restoration or repair strategies could be simplified using the proposed method, it is expected that this study will make a significant contribution to the resilience enhancement in distribution systems.

1. Introduction

Most definitions of power-system resilience focus on the capability for anticipation, absorption, and rapid recovery from an external high-impact, low-probability (HILP) event. After a natural disaster, the primary objective of restoration is to restore the power system as fast as possible for restoring the critical loads and minimizing the unserved loads [1]. There are two approaches for accomplishing this objective: One involves the provision of transitory support during a power outage using an emergency power source or reconfiguration of the power system, whereas the other involves the repair and renovation of the damaged components [2]. M. Panteli [3] expanded the concept of resilience to include infrastructural resilience in the existing operational resilience because a critical situation might arise if a new event occurs relatively soon. The transmission system can return to its pre-event state by utilizing the available energy resources or reconfiguring the system even without repairing (or replacement) the damaged components; however, in most disasters, this is not possible for a distribution system [4,5], where repair is a factor that has the maximum effect on both operational and infrastructural resilience [6].

When evaluating power-system reliability, HILP events are ignored because the probability of the occurrence of a HILP event converges to almost zero [7], whereas resilience deals with serial failures having spatiotemporal correlations [8]. The collapse of other infrastructure (such as transportation and communication) is more time-consuming among all the processes such as damage assessment, dispatch of crews for repairing the damaged components, and component repair [9]. The resources for repairs are the personnel, trucks, tools, spares, etc., available for this work. These resources are limited and have to be carefully matched to follow the repair priority of damaged components in order to obtain acceptable outage times [10]. In HILP events, the assumption that the repair of all the damaged components commences simultaneously is not realistic.

The optimization problem of repair priority is similar to that of a power-system restoration because it rapidly restores the system to normal conditions and minimizes the unserved loads and restoration time [11]. However, with the increase in the size and complexity of the distribution system, several aspects such as cold load pickup, minimization of the number of switching operations, and overcurrent should be considered during the restoration process [12]. Repair priority optimization is simpler because the decision variables are limited to the damaged components and there are fewer constraints that must be satisfied because the repaired components do not degrade the system performance. The optimization of the repair priority is more like the restoration-path optimization, which is a part of power-system restoration, rather than the entire process. However, although the time taken to reconfigure the system structure and to start or interconnect the available energy resources is less than that required to move repair crews for restoration-path optimization, in the repair of a distribution system, the time taken to repair the damaged components is the maximum compared to any other process.

Among the various resilience metrics that quantify resilience, M. Mahzarnia [7] suggested the difference between the real and ideal performance levels of a system integrated from the start of the event until the system performance reaches an acceptable level as the most representative. Hence, this study evaluates the resilience using this metric, which applies the active power that can be supplied by the distribution system to customers (‘available capacity’) as the system performance level. To enhance this resilience metric that includes a time unit, the repair priority must be determined by comparing the increased available capacity through repair during the unit repair duration (‘recovered capacity increment’) instead of the increased available capacity through repair (‘recovered capacity’).

Many studies on restoration-path optimization and repair strategies propose techniques to iteratively select the best choice for minimizing the travel time of repair crews, while maximizing the recovered capacity for (important or all) the customers [13,14,15,16,17]. One of the problems that can occur when the optimal result is obtained using the greedy algorithm, which selects the best choice considering only the subsequent step, is that the available capacity of the entire system increases only slightly or does not increase at all for certain damaged components because the system is divided into several parts [18,19,20]. In the radial distribution system, there are customers to whom power can be supplied only if all the damaged components that are serially connected have been repaired. M. Tan [21] and F. Chaoqi [22] suggested restoration-path optimization to expand the structural connections to the maximum; however, they applied the greedy algorithm. Even if each one of the damaged components is selected sequentially as per the repair priority, it must be optimized by comparing not only the recovered capacity through component repair but also the recovered capacity through the repair of multiple components which include the component. T. C. Matisziw’s technique [23], which optimizes the repair priority based on the branch power flow, does not include the problem of convergence to the local optimum. However, the assumption that the recovered capacity through the repair of any one component is the same as the power flow of the component of a normal system may not be true for HILP events.

A. Arif [17] proposed a restoration-path optimization technique that includes distributed generation, the repair duration, and the travel time for dispatching multiple repair crews. This study proposes new criteria for determining the damaged component that should be repaired first by operators who must determine the repair priority urgently when a HILP disaster occurs, rather than proposing a practical but complex repair priority optimization technique. In power-system reliability, the results of all events are reflected in the reliability indices because the number of damaged components is small [24]. However, most studies on resilience evaluate it by assuming a scenario that occurred in the past or a specific scenario rather than considering as many disaster scenarios as possible. For a disconnected grid split into several unconnected parts by a disaster, repair priority optimization can commence only after determining the structural interconnection between multiple power sources and loads. The recovered capacity cannot be determined by the configuration of the normal system before disaster but can be determined only with the changed configuration by the disaster. In a radial distribution system, which has a simpler structure than the transmission system of a complex grid, the system configuration after a disaster and the changes in the system configuration with the progress of repair can be distinguished more conveniently. This study suggests a technique for identifying the interrupted area of a radial distribution system that only has a main bus as the source. Dispersed energy resources (DERs) other than the main bus can be included and the configuration may change due to power-system restoration. The improvement of the proposed technique to render it like a real system is intended for the future.

To verify the proposed repair priority optimization and the unserved area identification method, the resilience is evaluated under all the disaster scenarios that can occur in contingencies from N-2 to N-5. To prevent the resilience result from being determined by the size of a randomly assumed parameter, the suggested value of an extensively used test distribution system parameter is applied, and the recovery process is as follows:

• A repair team comprising many repair crews repairs all the damaged components in order.
• The travel time of the repair team is included in the repair duration under the assumption that the elapsed time required for accessing the damaged components is the same for all the components.
• When a disaster occurs, the system is reconfigured optimally after a certain time.
• The elapsed time required to reconfigure the system including the repaired components is included in the repair duration under the assumption that it is the same for all the components.
• All the loads have the same level of importance.

The remaining paper is organized as follows: Section 2 proposes an interrupted area identification using the BIBC (bus injection to branch current) matrix. Section 3 proposes mathematical formulations for five different comparison criteria. Section 4 provides the five solution algorithms. The numerical results are presented in Section 5. Section 6 concludes the study.

2. Interrupted Area Identification using BIBC Matrix

The proposed technique for identifying the unserved areas is explained considering the failure scenarios in three branches of a six-bus distribution system (Figure 1).

In this section, the concept is explained through an example with a simple structure, and the clearly defined equations are described in the next section.

J. Teng [25] defined the BIBC matrix to express the relationship between the bus current injections and branch currents for use in the power flow of a radial distribution system. The constant BIBC matrix is an upper triangular matrix and contains values of 0 and +1 only. In the repair priority optimization, the size of the BIBC matrix was reduced to the buses to which an active load is connected and the branches where failure can occur, instead of all buses and branches in the system.

The column vector for bus can be created by multiplying the transposed BIBC matrix by a column vector that has a value of +1 if a failure occurs and 0 otherwise for all branches (that can fail) comprising the system as follows:

$$BIB{C}^{\mathrm{Tr}}\times \left[\begin{array}{c}0\\ 0\\ 1\\ 1\\ 1\end{array}\right]={\left[\begin{array}{cccc}1& 1& 1& 1\\ 0& 1& 1& 1\\ 0& 1& 1& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}^{\mathrm{Tr}}\times \left[\begin{array}{c}0\\ 0\\ 1\\ 1\\ 1\end{array}\right]=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 1& 1& 1& 0& 0\\ 1& 1& 1& 1& 0\\ 1& 1& 0& 0& 1\end{array}\right]\times \left[\begin{array}{c}0\\ 0\\ 1\\ 1\\ 1\end{array}\right]=\left[\begin{array}{c}0\\ 1\\ 2\\ 1\end{array}\right]$$

(1)

where the superscript ^Tr denotes transposed matrix.

The column vector, which is the result of (1) expresses whether interruption occurs in every bus (to which a load is connected). If it is 0, the consumers at that load point are supplied with energy, and a natural number means the consumers are not receiving energy. If every natural number is converted to 1, capacity not supplied or the number of unserved customers can be calculated conveniently, as follows:

$$\left(\left[\begin{array}{c}0\\ 1\\ 2\\ 1\end{array}\right]\underset{\mathbb{N}\to 1}{\to }\left[\begin{array}{c}0\\ 1\\ 1\\ 1\end{array}\right]\right)·\left[\begin{array}{c} 50 \mathrm{kW}\\ 100 \mathrm{kW}\\ 200 \mathrm{kW}\\ 350 \mathrm{kW}\end{array}\right]=\left[\begin{array}{c}0 \\ 100 \mathrm{kW}\\ 200 \mathrm{kW}\\ 350 \mathrm{kW}\end{array}\right]$$

(2)

The recovered capacity due to the repair of branch 3 can be obtained by subtracting the capacity not supplied of a system where two branches failed from the capacity not supplied of a system where three branches failed, as follows:

$${\left[BIB{C}^{\mathrm{Tr}}\times \left[\begin{array}{c}0\\ 0\\ 1\\ 1\\ 1\end{array}\right]\right]}_{\mathbb{N}\to 1}-{\left[BIB{C}^{\mathrm{Tr}}\times \left[\begin{array}{c}0\\ 0\\ 0\\ 1\\ 1\end{array}\right]\right]}_{\mathbb{N}\to 1}=\left[\begin{array}{c}0\\ 1\\ 1\\ 1\end{array}\right]-\left[\begin{array}{c}0\\ 0\\ 1\\ 1\end{array}\right]=\left[\begin{array}{c}0\\ 1\\ 0\\ 0\end{array}\right]$$

(3)

$$\left[\begin{array}{c}0\\ 1\\ 0\\ 0\end{array}\right]·\left[\begin{array}{c}50 \mathrm{kW}\\ 100 \mathrm{kW}\\ 200 \mathrm{kW}\\ 350 \mathrm{kW}\end{array}\right]=\left[\begin{array}{c}0\\ 100 \mathrm{kW}\\ 0\\ 0\end{array}\right]$$

(4)

If the subtraction between matrices that express the failure of branches is calculated first or the column vector is composed of repaired branches only, the result becomes different from (3), as follows:

$${\left[BIB{C}^{Tr}\times \left(\left[\begin{array}{c}0\\ 0\\ 1\\ 1\\ 1\end{array}\right]-\left[\begin{array}{c}0\\ 0\\ 0\\ 1\\ 1\end{array}\right]\right)\right]}_{\mathbb{N}\to 1}={\left[BIB{C}^{\mathrm{Tr}}\times \left[\begin{array}{c}0\\ 0\\ 1\\ 0\\ 0\end{array}\right]\right]}_{\mathbb{N}\to 1}=\left[\begin{array}{c}0\\ 1\\ 1\\ 0\end{array}\right]\ne \left[\begin{array}{c}0\\ 1\\ 0\\ 0\end{array}\right]$$

(5)

The recovered capacity due to the repair of one branch or two or more branches vary by the failure and repair of other branches. Hence, they must be calculated stepwise.

3. Mathematical Formulation

3.1. Repair Strategy by Power Flow

The first repair strategy (RS1) to determine the repair priority of damaged branches is to first repair the branches with the largest power flow obtained from the steady state system. In South Korea, the repair priority is determined based on the criterion that the operator of distribution system first repairs the high-voltage lines (in case of important loads of the same level) and the main feeder. The repair priority based on power flow suggested by T. C. Matisziw [23] is an excellent method that first repairs the high-voltage main feeders. It has the advantages that the branches do not need to be classified by dichotomy, the values obtained before the disaster can be applied to disaster scenarios, and the problem of converging to the local optimum by the greedy algorithm does not occur.

Various power flow solution can be used, but in this paper, the power flow column vector $B$ was calculated by product of the BIBC matrix and the matrix of column vector $L$ composed of the active powers of buses, as follows:

$$B=BIBC\times L$$

(6)

The repair priority can be obtained instantly if the damaged branches are listed in the descending order of power flow. However, to calculate the resilience metric as well, the repair priority row vector R is obtained sequentially from the branch to be repaired first ($r=1$) to the branch to be repaired last ($r=N_r$), as follows:

$$R_r=\underset{y\in {\mathbb{E}}_r}{\mathrm{argmax}}\left(B_y+\frac{\alpha }{T_y}\right), r=1, 2, \cdots , N_r$$

(7)

where $\alpha$ is a very small positive constant and allows the branch with a shorter repair duration to be repaired first if the power flows are the same.

The set of damaged branches that remain without being selected for the repair priority, ${\mathbb{E}}_r$, is obtained by the difference of sets, as follows:

$${\mathbb{E}}_r=\left\{\begin{array}{c}\left\{x|x\in \mathrm{set} \mathrm{of} \mathrm{all} \mathrm{damaged} \mathrm{branches}\right\} \mathrm{if} r=1 \\ {\mathbb{E}}_{r-1}\setminus \left\{R_{r-1}\right\} otherwise\end{array}\right., r=1, 2, \cdots , N_r$$

(8)

3.2. Repair Strategy by Power Flow Increment

The second repair strategy (RS2) is to first repair the branch with the largest “power flow increment,” which is the power flow of a branch divided by the repair duration of the branch. In a situation where resilience is evaluated by a resilience metric that includes a time unit, a strategy to ignore the repair duration cannot expect a good result. However, because including the repair duration can interfere with selecting high-voltage main feeders, it is difficult to predict the better strategy between RS1 and RS2.

The other steps are the same as those of RS1 except for the modified (7), and if the power flow increment is the same, the branch with a larger power flow is repaired first, as follows:

$$R_r=\underset{y\in {\mathbb{E}}_r}{\mathrm{argmax}}\left(\frac{B_y}{T_y}+\alpha B_y\right), r=1, 2, \cdots , N_r$$

(9)

3.3. Repair Strategy by Recovered Capacity Increment

The third repair strategy (RS3) is to first repair the branch with the largest “recovered capacity increment (RCI)”, which is the increased amount of the available capacity of the entire system when repair is completed divided by the repair duration. Since the RCI can be different each time due to the branches repaired earlier, the repair priority must be determined sequentially from the branch to be repaired first until the branch to be repaired last. According to the greedy algorithm, the repair priority is determined by comparing the RCI due to the repair of ‘only one’ branch, as follows:

$$R_r=\underset{y\in {\mathbb{E}}_r}{\mathrm{argmax}}\left[\frac{L^{\mathrm{Tr}}\times \left(I_{-,r}^{\varnothing }-I_{-,r}^{\left\{y\right\}}\right)}{T_y}+\alpha B_y\right], r=1, 2, \cdots , N_r$$

(10)

If the RCI is the same, the branch with a larger power flow is repaired first. The matrix $I^{\mathbb{X}}$ expresses whether each bus is unserved in a system where the branches included in the set $\mathbb{X}$ have been repaired. It also includes the process of converting the element of the matrix to 1 if the element of the matrix obtained by the product of the transposed BIBC matrix and the matrix expressing the damaged branches is a natural number, as follows:

$$I_{l,r}^{\mathbb{X}}=\left\{\begin{array}{c}1 \mathrm{if} {\left[BIB{C}^{\mathrm{Tr}}\times C_{-,r}^{\mathbb{X}}\right]}_l\in N\\ 0 otherwise \end{array}\right., l=1, 2, \cdots , N_l, r=1, 2, \cdots , N_r$$

(11)

The matrix $C^{\mathbb{X}}$ has the value 1 if the branch fails, and 0 otherwise, as follows:

$$C_{b,r}^{\mathbb{X}}=\left\{\begin{array}{c}1 \mathrm{if} b\in \left({\mathbb{E}}_r\setminus \mathbb{X}\right)\\ 0 otherwise \end{array}\right., b=1, 2, \cdots , N_b, r=1, 2, \cdots , N_r$$

(12)

3.4. Repair Strategy by RCI of Multi-Components

The fourth repair strategy (RS4) is to determine the repair priority by comparing the RCI due to the repair of two or more damaged branches, as well as the repair of one damaged branch. In the step for deciding the first repair priority, the RCI due to the repair of one branch to the RCI due to the repair of $N_r-1$ branches are compared. In the r-th repair priority, one branch to $N_r-r$ branches are compared. This is an iterative process where a subset is selected among the power sets of the set ${\mathbb{E}}_r$, and a smaller subset among the power sets of the subset is selected. The element of the final singleton set is selected as the branch to be repaired first, as follows:

$${\mathbb{D}}_r^n=\left\{\begin{array}{c}\underset{\mathbb{Y}\in \mathrm{F}\left({\mathbb{D}}_r^{n-1}\right)}{\mathrm{argmax}}\left[\frac{L^{\mathrm{Tr}}\times \left(I_{-,r}^{\varnothing }-I_{-,r}^{\mathbb{Y}}\right)}{{{\displaystyle \sum }}_{\forall i\in \mathbb{Y}}{T}_i}+\alpha {\displaystyle {\displaystyle \sum }_{\forall j\in \mathbb{Y}}}{B}_j\right] \mathrm{if} \left|{\mathbb{D}}_r^{n-1}\right|>1\\ {\mathbb{D}}_r^{n-1} \mathrm{otherwise} \end{array}\right., n=1, 2, \cdots , \left(N_r-r\right), r=1, 2, \cdots , N_r$$

(13)

$${\mathbb{D}}_r^0={\mathbb{E}}_r , r=1, 2, \cdots , N_r$$

(14)

$$R_r\in {\mathbb{D}}_r^{N_r-r} , r=1, 2, \cdots , N_r$$

(15)

If the subsets have the same RCI, the subset with a larger sum of power flows is selected. Equation (13) is repeated $N_r-r$ times, but the amount of computation is not directly proportional to the number of iterations because the size of the subset ${\mathbb{D}}_r^n$ can be decreased by two or more after one step or it can be one after only the first step. To exclude the empty set and the set composed of all elements among the power sets of a set, a function to obtain the power sets is newly defined, as follows:

$$\mathrm{F}\left(\mathbb{X}\right)=\wp \left(\mathbb{X}\right)\setminus \varnothing \setminus \mathbb{X}$$

(16)

3.5. Repair Strategy Comparing All Repair Priorities

The fifth repair strategy (RS5) is to choose the repair priority with the best result after calculating the resilience metric for all the possible repair priorities. RS5 was simulated to determine whether the repair priority given by the other four strategies is the optimal repair priority, and if not, how much it is different from the optimal repair priority. To calculate the resilience metric from one disaster scenario and one repair priority, the available capacity of the entire system must be calculated up to $N_r$. The more various constraints and decision variables are included, the more complex the calculation of the available capacity becomes and the longer it takes. The number of possible repair priorities increases sharply as the number of failures increases; therefore, it is impossible to apply RS5 for systems larger than a certain size and for failures above a certain number.

3.6. Resilience Evaluation

If the repair priority is determined by RS1~RS4, or all the repair priorities are listed by RS5, a curved resilience function can be obtained, as follows:

$$RC\left(t\right)=\left\{\begin{array}{c} {\displaystyle {\displaystyle \sum }_{\forall l}}{L}_l , t<0 \mathrm{or} t\ge \left(t_C+{\displaystyle {\displaystyle \sum }_{i=1}^{N_r}}{T}_{R_i}\right) \\ 0 , 0\le t<t_C \\ {\displaystyle {\displaystyle \sum }_{\forall l}}{L}_l-L^{\mathrm{Tr}}\times I_{-,r}^{\varnothing } , \left(t_C+{\displaystyle {\displaystyle \sum }_{j=1}^{r-1}}{T}_{R_j}\right)\le t<\left(t_C+{\displaystyle {\displaystyle \sum }_{k=1}^r}{T}_{R_k}\right)\end{array}\right.,r=1, 2, \cdots , N_r$$

(17)

If the RCI is the same, the branch with a larger power flow is repaired first. The matrix $I^{\mathbb{X}}$ expresses whether each bus is unserved in a system where the branches included in the set $\mathbb{X}$ have been repaired. It also includes the process of converting the element of the matrix to one if the element of the matrix obtained by the product of the transposed BIBC matrix and the matrix expressing the damaged branches is a natural number, as follows:

$$EENS={{\displaystyle \int }}_0^{t_C+{{\displaystyle \sum }}_{\forall r}{T}_{R_r}}\left\{{\displaystyle {\displaystyle \sum }_{\forall l}}{L}_l-RC\left(t\right)\right\}dt=t_C{\displaystyle \sum }_{\forall l}{L}_l+{\displaystyle \sum }_{\forall r}\left\{T_{R_r}\left(L^{\mathrm{Tr}}\times I_{-,r}^{\varnothing }\right)\right\}$$

(18)

4. Solution Algorithm

The solution algorithm for each repair strategy stated in Section 3 is summarized in

Table 1. Solution algorithm for each repair strategy (RS1–RS4).

RS1 Algorithm	RS2 Algorithm	Step	RS3 Algorithm	RS4 Algorithm
Input: BIBC, L, T, Nl, Nb		1.	Input: BIBC, L, T, Nl, Nb
B by (6)		2.	B by (6)
for all disaster scenarios		3.	for all disaster scenarios
Nr		4.	Nr
for r = 1 to Nr		5.	for r = 1 to Nr
${\mathbb{E}}_r$ by (8)		6.	${\mathbb{E}}_r$ by (8)
$R_r$ by (7)	$R_r$ by (9)	7.	${\mathbf{C}}_{-,r}^{\varnothing }$$\mathrm{by}\left(12\right),{\mathbf{I}}_{-,r}^{\varnothing }$ by (11)
:		8.	:	for n = 1 to Nr-r
:		9.	:	$\mathrm{if}n=1,{\mathbb{D}}_r^0$ by (14), end if n
:		10.	:	$\mathrm{if}{\mathbb{D}}_r^{n-1}$>1
:		11.	:	$\mathrm{F}\left({\mathbb{D}}_r^{n-1}\right)$ by (16)
:		12.	$\mathrm{for}\mathrm{all}y\in {\mathbb{E}}_r$	$\mathrm{for}\mathrm{all}\mathbb{Y}\in \mathrm{F}\left({\mathbb{D}}_r^{n-1}\right)$
:		13.	${\mathbf{C}}_{-,r}^{\left\{y\right\}}$$\mathrm{by}\left(12\right),{\mathbf{I}}_{-,r}^{\left\{y\right\}}$ by (11)	${\mathbf{C}}_{-,r}^{\mathbb{Y}}$$\mathrm{by}\left(12\right),{\mathbf{I}}_{-,r}^{\mathbb{Y}}$ by (11)
:		14.	end for y	end for $\mathbb{Y}$
:		15.	:	$\mathrm{end}\mathrm{if}{\mathbb{D}}_r^{n-1}$
:		16.	:	${\mathbb{D}}_r^n$ by (13)
:		17.	:	end for n
${\mathbf{C}}_{-,r}^{\varnothing }$$\mathrm{by}\left(12\right),{\mathbf{I}}_{-,r}^{\varnothing }$ by (11)		18.	$R_r$ by (10)	$R_r$ by (15)
end for r		19.	end for r
EENS by (18)		20.	EENS by (18)
end for scenario		21.	end for scenario

and

Table 2. Solution algorithm for the fifth repair strategy (RS5).

Step	RS5 Algorithm
1.	Input: BIBC, L, T, Nl, Nb
2.	for all disaster scenarios
3.	Nr$,{\mathbb{E}}_1$$\mathrm{by}\left(8\right),{\mathbf{C}}_{-,1}^{\varnothing }$$\mathrm{by}\left(12\right),{\mathbf{I}}_{-,1}^{\varnothing }$ by (11)
4.	for all R
5.	for r = 2 to Nr
6.	${\mathbb{E}}_r$ by (8)
7.	${\mathbf{C}}_{-,r}^{\varnothing }$$\mathrm{by}\left(12\right),{\mathbf{I}}_{-,r}^{\varnothing }$ by (11)
8.	end for r
9.	EENS by (18)
10.	end for R
11.	Choosing the best R
12	end for scenario

.

The power flow of the branch in the disaster scenario in the six-bus system in Figure 1 is [700, 650, 300, 200, 350] kW; therefore, the repair priority column vector by RS1 is [5, 3, 4]^Tr. The operator can see that the repair priority [3, 4, 5] is the optimal repair priority empirically or by comparing the resilience metric for all repair priorities (RS5). RS1 is disadvantageous in a resilience metric that includes time unit because it repairs a branch first if its power flow is even minutely larger than that of the others, despite the infinitely long repair duration taken by the branch. If branch 5 is repaired first, 350 kW is supplied to the customers after 4 h, and if branches 3 and 4 are repaired first, 300 kW is supplied after 3 h. Since RS1 ignores the repair duration and considers only the power flow, the repair priority is decided to repair branch 5 first of all.

In the scenario of Figure 1, the power flow increments are [700, 650, 150, 200, 87.5] kW/h, and the repair priority by RS2 is [4, 3, 5]. Repairing branch 4 earlier than branch 3 does not improve the resilience at all. If branch 4 is repaired first, no power is restored to the customers after 1 h. The repair duration was included to solve the disadvantage of RS1, but the advantage of being able to find high-voltage main feeders was lost in the process. Power flow can be a criterion for identification whether a branch corresponds to a high-voltage main feeder, but the power flow increment cannot fulfill its role properly. In RS2, the repair priorities are often obtained in which branch with very short repair duration is repaired first of all.

In Figure 1 using RS3, when deciding the first repair priority, the RCIs of the three damaged branches is [50, 0, 87.5] kW/h. Therefore, branch 5 is repaired first. For the second repair priority, the RCIs of branches 3 and 4 ([50, 0] kW/h) are compared and branch 3 is selected. The final repair priority is [5, 3, 4], and when the damaged branches are repaired in this order, the resilience becomes as shown in Figure 2a. If two damaged branches are serially connected, even if the branch close to the power source is repaired, the recovered capacity is small (due to the distal branch which is still in a damaged state), so the greedy algorithm selects an alternative from the remaining branches.

In Figure 1 using RS4, the set of damaged branches that can be selected as the first repair priority is {3,4,5}, and the power sets of this set are {∅, {3}, {4}, {5}, {3,4}, {3,5}, {4,5}, and {3,4,5}}. Among the subsets of the power sets excluding ∅ and {3,4,5}, the subset with the largest RCI, {3,4}, is selected (Figure 3a). Among the power sets of the subset {3,4}, the subset of a smaller size, {3}, is selected (Figure 3a). As a result, branch 3 is selected as the first repair priority. The set that can be selected as the second repair priority is {4,5}, and after the subset {4} is selected, branch 4 is selected as the second repair priority (Figure 3b). Since the set that can be selected as the third repair priority is {5}, branch 5 is immediately selected as the third repair priority. The final repair priority is [3, 4, 5], and when the damaged branches are repaired in this order, the resilience becomes as shown in Figure 2b.

RS1 and RS2 calculate the available capacity of the entire system to obtain the resilience metric and not to determine the repair priority. When the number of calculations for the available capacity in RS3 to RS5 to simulate one disaster scenario are compared (Figure 4), we find that as the number of damaged branches increases, the number of calculations increases exponentially in RS5. In Figure 3, the number of calculations for the available capacity in RS4 is the value in the worst case, where it is assumed that the size of subsets always decreases by 1, and the number of calculations for most disaster scenarios is significantly fewer than this.

5. Numerical Results

5.1. Input Data

We conducted case studies for the IEEE 13-bus feeder, the IEEE 34-bus feeder, and the IEEE 37-feeder [26,27]. It was assumed that the restoration process took 30 min to reconfigure the system after a disaster. To determine the repair priority and evaluate resilience, the time it takes to repair the branch is required. Since the IEEE test distribution system does not provide repair duration, the reliability data of Roy Billinton Test System (RBTS) was applied as shown in

Table 3. Reliability parameters for IEEE test systems.

Distribution System	Repair Duration
13-bus Feeder	4.16 kV overhead line: 5 h (repair) 4.16 kV underground line: 30 h (repair) 4.16/0.48 kV transformer: 10 h (replacement) Switch: 4 h (replacement)
34-bus Feeder	4.16 kV overhead line: 5 h (repair) 24.9 kV overhead line: 8 h (repair) 24.9/4.16 kV transformer: 15 h (replacement) Regulator: 15 h (replacement)
37-bus Feeder	4.8 kV underground line: 30 h (repair) 4.8/0.48 kV transformer: 10 h (replacement) Regulator: 10 h (replacement)

.

5.2. Results of Resilience Metrics

Table 4. The calculated result of average EENS per scenario of the five strategies in the contingencies from N-2 to N-5.

Type of Test System	Level of Contingencies	RS1 [kWh]	RS2 [kWh]	RS3 [kWh]	RS4 [kWh]	RS5 [kWh]
13-bus	N-2	19,920	19,603	19,552	19,552	19,552
	N-3	30,762	29,894	29,770	29,733	29,733
	N-4	42,571	41,005	40,813	40,666	40,666
	N-5	55,198	52,879	52,642	52,283	52,283
34-bus	N-2	15,268	15,314	15,260	15,260	15,260
	N-3	22,844	22,972	22,944	22,819	22,819
	N-4	30,655	30,890	31,059	30,599	30,599
	N-5	38,685	39,046	39,633	38,582	38,582
37-bus	N-2	25,954	25,954	25,954	25,954	25,954
	N-3	41,140	41,140	41,193	41,089	41,089
	N-4	58,038	58,038	58,218	57,835	57,835
	N-5	76,528	76,528	76,917	76,041	76,041

shows the average values of resilience metric per scenario of the five strategies in all disaster scenarios which can occur in the contingencies from N-2 to N-5.

Table 5. The number of scenarios where the resilience of each strategy is the same as the resilience of the optimal repair priority.

Type of Test System	Level of Contingencies	RS1 [count]	RS2 [count]	RS3 [count]	RS4 [count]	RS5 [count]
13-bus	N-2	70	77	78	78	78
	N-3	214	261	284	286	286
	N-4	419	547	696	715	715
	N-5	554	733	1207	1287	1287
34-bus	N-2	525	519	528	528	528
	N-3	5350	5180	5350	5456	5456
	N-4	39,166	36,825	38,067	40,920	40,920
	N-5	219,118	198,383	200,724	237,336	237,336
37-bus	N-2	666	666	666	666	666
	N-3	7617	7617	7653	7770	7770
	N-4	61,482	61,482	62,652	66,045	66,045
	N-5	370,460	370,460	387,764	435,897	435,897

shows the number of scenarios (‘optimal scenarios’) where the resilience of each strategy is the same as the resilience of the optimal repair priority. Since the repair priority of RS5 corresponds to the optimal repair priority, the values for RS5 are the same as the total number of disaster scenarios.

When the resilience metric is compared, RS1 and RS2 can be considered slightly better strategies than RS3, but when the number of scenarios where the repair priority is not optimized (‘non-optimal scenarios’) are compared, the opposite is true. Figure 5 shows a graph which rearranged the non-optimal scenarios for each RS1~RS3 in the ascending order of the increased EENS among the N-5 disaster scenarios. RS3 has a low probability of not optimizing the repair priority, but it significantly increases the EENS if it does not optimize it. The number of non-optimal scenarios and the decrease in the resilience metric (increased EENS) are not proportional. Non-optimization due to greedy algorithm makes the resilience metric relatively worse than non-optimization by other strategies.

5.3. Examples of Non-Optimization

RS1 and RS2 have a non-optimal scenario even in the N-2 disaster, which is the simplest. Among the N-2 disaster scenarios, which occurred in the IEEE 13-bus feeder, the number of cumulative times where each branch was included in the scenario where RS1 or RS2 failed to optimize the repair priority is shown in Figure 6. In the scenario where a failure occurred in the components (underground line and transformer) that have longer repair durations than the overhead line which accounts for the majority of the remaining components, RS1 failed to optimize the repair priority. RS2 repairs the important (but less important than main bus) overhead line at the center of the system earlier than the regulator, which is directly connected to the main bus.

The proposed strategy RS4 optimizes the repair priority perfectly in all disaster scenarios which can occur in the contingencies from N-2 to N-5. RS3 optimizes the repair priority in all scenarios only in the N-2 disaster, and in N-3 or higher disasters, the proportion of non-optimal scenarios increases with the number of failures. Among the N-3 disaster scenarios, the cumulative times where each branch was included in the scenarios where RS3 failed to optimize the repair priority are shown in Figure 7 and Figure 8. If failure occurs in two serially connected branches which are connected to a load of large capacity, RS3 may not optimize the repair priority.

5.4. Repair Duration Proportional to the Length of Branch

In the IEEE 37-bus feeder, the regulator that is directly connected to the main bus is always repaired first, and the transformer that is not connected to a load is always repaired last. In this distribution system, because all the branches affecting the repair priority have the same repair duration (as shown in Table 4 and Table 5), RS1 and RS2 show the same results in all levels of disaster scenarios and derive the optimal repair priority in all N-2 disaster scenarios. The relationship between the length of an overhead line or an underground line and the repair duration does not have a widely used correlation function yet. To avoid the reversal of the superiority and inferiority of strategies by intentional assumptions, we assumed that the branch length and repair duration are unrelated. To examine what results are derived by each strategy when the repair durations of branches are different, in the IEEE 37-bus feeder, only the repair duration of the underground line was changed from 30 h to 30 h/1000 ft, and the values in Table 4 and Table 5 were calculated again. The results are outlined in

Table 6. The calculated values of average EENS and the number of optimal scenarios by repair duration proportional to the length.

Type of Test System	Level of Contingencies	RS1	RS2	RS3	RS4	RS5
		[kWh]	[kWh]	[kWh]	[kWh]	[kWh]	Average EENS
37-bus	N-2	21,512	21,705	21,413	21,413	21,413
	N-3	32,975	33,514	32,703	32,648	32,648
	N-4	45,231	46,235	44,735	44,532	44,532
	N-5	58,218	59,780	57,465	56,996	56,996
		[count]	[count]	[count]	[count]	[count]	Number of optimal scenarios
	N-2	606	638	666	666	666
	N-3	5963	6715	7635	7770	7770
	N-4	39,921	48,045	62,039	66,045	66,045
	N-5	194,447	247,219	378,416	435,897	435,897

.

When the branches have the same repair duration, the algorithm can apply both RS1 and RS2, which have a simple structure. However, if there are various types of components in the system or if the repair durations of branches are different, they are not applicable. The more diverse the repair durations of branches, the more conspicuous is the disadvantage of RS3; therefore, the application of RS4 should be accepted in this case.

6. Conclusions

In this paper, the following three techniques are proposed:

• A technique for identifying the unserved areas;
• Repair priority by RCI (=the increased amount of the available capacity of the entire system divided by the repair duration);
• Repair priority by comparing the RCIs due to the repair of two or more damaged branches.

In a natural disaster where failures occur in three or more components, the greedy algorithm cannot optimize the repair priority, and if the objective function is changed to power flow, a problem occurs in other scenarios due to the ignored repair duration. For a system consisting of various components with different repair durations, the power flow of the branch cannot be used as a criterion for deciding the repair priority. If the repair priority cannot be determined by the prime objective function, namely, the recovered capacity, it is possible to use power flow as a subobjective function. Applying the power flow increment to include both power flow and repair duration in the objective functions hinders the strengths of the strategy based on power flow (repair important branches first). Consequently, even in the process of selecting one branch to repair among multiple damaged branches, the recovered capacity by two or more repaired branches must be compared.

The real system is much more complex because there are multiple repair teams, the path including the travel time must be optimized, the system must be reconfigured to connect the repaired branches, and DERs are used. Even in other complex algorithms to improve resilience by optimizing the repair priority (or restoration path) of the distribution system, it is recommended to define the objective function by the recovered capacity increment rather than by the recovered capacity.

To evaluate the strategy to determine the repair priority for all possible disaster scenarios, a technique to identify the unserved areas of a radial distribution system was proposed. Although it was applied to a distribution system with a very simple structure in the present stage, there is a significant potential for expansion. If the unserved areas can be easily determined even in serious disasters, varied disaster scenarios could be simulated in the resilience research, therefore simplifying the complex repair and restoration strategy algorithms used by other researchers.