Flexible resource optimal allocation strategy for distribution system resilience improvement under failure uncertainty

Extreme disasters have led to huge economical losses to the power system and posed a huge threat to the stable supply of electric energy. Especially for traditional passive distribution network systems, failures of transmission lines, towers, and other equipment will cause large-scale power outages and huge economic losses. In order to improve the ability of the power system to deal with extreme weather disasters, this paper proposes an optimal configuration strategy for the distribution network backup power resource oriented to resilience improvement, considering the uncertainty of equipment failure caused by disasters in the case of a known disaster range. The location and capacity of backup resources can be properly configured to reduce the load shedding of the power distribution network during the disaster process, thereby improving the resilience of the distribution network. The method proposed in this paper is modeled as a two-stage stochastic programming, which includes mix-integer constraints. In the first stage, the optimization problem confirms the capacity and location of the flexible resources. In the second stage, the operational constraints and costs of load shedding are considered. The resilience enhancement method is verified on the improved IEEE-33 node system, and the result shows that the method proposed in this paper can effectively enhance the resilience of the urban power network.


Introduction
In recent years, global warming and frequent extreme disasters have brought serious threats to the safe operation of power networks.From 2003 to 2012, extreme natural disasters led to more than six hundred power supply interruptions, affecting more than fifty thousand customers each, during which over 80% resulted from the failure of distribution networks in the USA [1].Further, in 2021, there were 20 natural disasters with losses of more than $20 billion [2].
To enhance the power network's ability to deal with natural disasters, experts and scholars have proposed the concept of a resilient power network [3].Power network resilience can be described as the ability to predict, withstand or absorb, adapt to, and quickly recover when facing a low-probability event [4].At present, according to the implementation term of the strategy, the strategy for improving the power network resilience can be divided into resilience-based planning [5], response [6], and restoration [7].The planning emphasizes that before disasters occur, the reasonable allocation of resources enables the power distribution system to have better coping capabilities in the face of extreme disasters.Response emphasizes that during the process of disaster impact, through network structure reconstruction and unit scheduling, the resilience reduction of the system can be mitigated.The restoration process emphasizes that after the disaster, through reasonable scheduling of maintenance equipment, the restoration of the power system can be completed quickly and efficiently, and the loss of resilience during the restoration process can be reduced.
Resilience planning is an important procedure to strengthen the ability to cope with disasters.Reasonable planning measures can mitigate the impact of disasters and make the system more resilient to disasters.Methods to enhance the resilience of distribution power networks can be divided into short-term and long-term planning [8].The short-term planning targets to make the best use of existing resources to minimize the impacts of the disaster [9].While long-term planning aims at optimally deploying new resources to improve the system resilience against disasters [10].At present, there has been a lot of research on resilience-based planning.In [11], a pre-disaster resource configuration method is employed to obtain a resource allocation plan, the resource includes batteries and electric buses.In [12], a strategy for resilience enhancement to make the distribution network more resilient to weather disasters is proposed.The proposed strategy includes reverse generator installation, hardening the transmission lines, and switches installation.In [13], the preparation and resource allocation process is modeled as a stochastic mixed-integer linear programming, which includes two stages.The optimized results can make the network more resilient to upcoming extreme weather events.
Although current scholars have studied the allocation schemes of backup power and other resources, the current configuration strategy cannot comprehensively consider the uncertainty of failures, and the planning of distributed combined heat and power is ignored.Therefore, in this paper, an optimal allocation strategy for backup power and distributed combined heat and power in power distribution systems oriented to improving system resilience is proposed, taking into account the uncertainty of equipment failure caused by disasters.In the case of knowing the affected area of the disasters, through the reasonable allocation of the location and capacity of flexible configuration resources, the load shedding during the disaster process can be reduced, thereby improving the system resilience.
The rest of this article is as follows: the next chapter introduces the modeling and analysis of flexible resource configuration in power systems, and the third chapter introduces the two-stage stochastic optimization model for resilience improvement.The case is verified in the fourth chapter, and the final chapter is the conclusion.

Modeling and analysis of flexible resource configuration in power system
Recently, with the introduction of the concept of an active distribution network, the distribution network has gradually changed to be active.The distribution network is no longer in a passive state but can provide active power for the internal load of the distribution network, so as to realize selfproduction and self-sale and reduce interaction costs.Especially in the current context of frequent extreme events, a reasonable allocation of distributed resources can enable the distribution network to achieve island operation and reduce the loss of resilience under the influence of disasters.The flexible allocation resources selected in this paper include backup generators and combined heat and power generators.

Backup generators
The structure of the standby generator is the same as that of the traditional distributed generator.The capacity is usually smaller than that of the traditional distributed generator, but it is different from the traditional generator in terms of usage.The backup power system is connected to the distribution network only when facing the disaster.The capacity of the backup power supply and the discharge power can be modeled as: ,  It is obvious that the power generation interval of the generator is closely associated with the capacity of the generator, so the capacity of the generator should be determined during configuration.

Combined heat and power generators
In addition, the current flexible allocation of resources in the power system also includes distributed heat and power cogeneration units.Under the background of the current distributed integrated energy network, the distributed heat network is associated with the traditional distribution network.The location of the combined heat and power (CHP) units can affect the ability of the energy network to cope with extreme disasters.The combined heat and power unit can be modeled as: , 0 where , P it CH P is the active power of the CHP generator on i at period t; CHP C denotes the capacity of the combined heat and power generator.
At present, some studies consider that the power grid and the heating network are different operating entities, and the unified scheduling of the electric heating system will have a large amount of information synchronous interaction, which will bring difficulties to scheduling.At the same time, the heating network itself has a large number of controllable flexible resources (such as temperaturecontrollable buildings), the heating network pipes also have heat storage and time delay characteristics, and the stable operation model of the heating network contains a large number of operational constraints [14].Therefore, the heating network resources can be aggregated.After considering the actual operating status, the actual model of CHP can be modified as follows: , where low r and high r represent the limit correction coefficient.The values of the two coefficients are related to the structure of the heating network and the internal equipment like storage.Similar to the backup generator, the generating power of CHP is also closely related to the capacity, so the capacity should be set properly when planning.

The two-stage stochastic optimization model for resilience improvement
By obtaining the historical data of the planning area, the lines that are prone to failure and the probability of line failure can be obtained.Therefore, this article assumes that the lines that are prone to disasters are known, and the probability of line failure of the distributed lines can be analyzed.In the case of line failure probability, the planning problem of flexible resource allocation can be modeled as a two-stage optimization model, and the objective function is:

Ax b =
(9) where ( , ) : min T y Q x q y = E (10) where x denotes decision variables relevant to planning; y denotes decision variables relevant to the operation.In the model of this article, the objective function is to minimize the economic cost including investment and operation.The investment cost can be modeled as: where cos exc t denotes interaction cost with the upper grid; cos ls t is the lost shedding cost.The two stages are connected by random fault generation.After the planning variables of the first stage are determined, the fault scenarios are randomly generated through the known line fault probability and maximum fault time, and the optimal configuration strategy in multiple scenarios is obtained.

Planning Related Constraints
The planning decision variables mainly include the backup power supply and the site selection and capacity of cogeneration unit equipment.The positional constraints can be described as: where CHP i n and BUG i n denote the location of CHP and BUG, respectively.Equation ( 13) and Equation ( 14) limit the number for planning, while Equation (15) restricts that a node cannot install CHP and BUG at the same time.For the capacity, the constraints can be modeled as:

Disaster Impact Constraints
According to the above, the disaster impact assumed in this paper is mainly reflected in the line fault, and the easily damaged fault line and the corresponding failure probability are known.Based on this, the line fault generation constraint is: where binary variable , ij t l indicates the line state, 1 means on, and 0 means off; a represents a randomly generated random number conforming to the 0~1 uniform distribution; ij p is the broken probability of line ij.In addition, considering the maintenance time of the line, this paper assumes that there is a maximum failure time, and the failure time of each faulty line is randomly generated:

Operation Related Constraints
The formulation of the configuration plan also needs to comprehensively consider the operating expense of the whole system, so as to ensure the rationality of the site selection and capacity determination strategy.The corresponding constraints in this part correspond to the proposed secondstage constraints.The first is the power system operating constraints: where j  is the lines from node j; j I is a set of lines on node j;

Q
are the lowest and highest output of the reactive power, respectively.Under the influence of extreme natural disasters, network lines and equipment components will have the risk of failure, so the constraints considering the working status of equipment components can be described as: , , , , where binary variable , ij t l indicates the line state, 1 means on, and 0 means off.

System description
The IEEE-33 node system is adopted to study the effectiveness of all planning strategies oriented to resilience.The modified network can be seen in Figure 1.Two DGs are connected to nodes 8 and 30.The parameter of the DGs is demonstrated in Table 1.

Planning results
Based on the optimal collaborative allocation scheme of flexible allocation resources for resilience improvement proposed in this paper, the optimal location and capacity of the two flexible resources can be obtained:

System operation result
On the basis of the planning results above, we can get the operation strategy of the system in a scenario.The generation of two DGs, the BUG and CHP can be seen in Figure 2.

Resilience enhancement results
To describe the resilience improvement effect of the system, this paper compares and analyzes the impact of whether resilience planning is performed on the total operating cost and total load shedding of the system in the scenario of 50 random failures.The results are shown in Table 4.

Conclusions
This article proposes an optimal allocation strategy for flexible resource allocation for resilience improvement, considering the optimal location and capacity of backup power and combined heat and power units, and taking into account a variety of fault scenarios.The test results on the IEEE33 node system show that the planning scheme can effectively mitigate the load shedding of the distribution network under disasters so that the distribution network has better resilience.
and up ramping constraint, respectively; BUG C denotes the capability of a backup generator.Equations (1) and (2) limit the upper and lower output of the backup generation, while Equation (3) indicates the ramp constraint of the generator.
installation capacity of CHP and BUG, respectively.
where ij TTR is the time to repair for line ij; max TTR denotes the maximum time to repair.20235th International Conference on Energy, Power and Grid (ICEPG 2023) Journal of Physics: Conference Series 2703 (2024) 012030 IOP Publishing doi:10.1088/1742-6596/2703/1/0120305
In addition, the number of BUG and CHP planned this time and the maximum capacity table are as follows:

Table 4
Planning results.