Power distribution network fault recovery method considering island division and fault reconstruction

Abstract

The invention discloses a power distribution network fault recovery method considering island division and fault reconstruction, and relates to the technical field of power distribution network fault recovery, and the method comprises the following steps: constructing a first constraint based on a network topology constraint, a load constraint and an island operation constraint; constructing a collaborative planning model by taking the minimum load outage loss and the system operation cost as objective functions; converting the collaborative programming model into a mixed integer second-order cone programming model through a second-order cone relaxation technology; and integrating the maintenance constraint to the mixed integer second-order cone programming model, and outputting a power distribution network fault recovery strategy. The method is used for solving the problems that in power distribution network fault recovery, collaborative optimization of network reconstruction and island operation is difficult, the calculation efficiency is low, and multi-fault scene coping is insufficient.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of power distribution network fault recovery, and more particularly to a power distribution network fault recovery method considering island division and fault reconstruction. BACKGROUND

[0002] As a key link of the power system, the operation reliability and voltage stability of the power distribution network directly affect industrial production and residential power consumption. Natural disasters, improper human operation, etc. can easily cause power distribution network faults and large-area power outages. The traditional power distribution network relies on the top-down power supply recovery mode, which has the problems of slow recovery and insufficient flexibility. Under the promotion of smart grid technology, the active distribution network (ADN) can quickly restore power supply due to its dynamic adjustable topology and the black start capability of distributed generators (DGs). The core strategies of its fault recovery are network reconstruction (switch optimization topology and power flow) and island operation (DGs independently supply power to local loads). However, the traditional strategies separate the two, do not exert a synergistic effect, and the current mainstream mixed integer nonlinear programming (MINLP) fault recovery model is complex to solve and difficult to quickly apply, so researchers have proposed various optimization methods.

[0003] For example, the power distribution network network reconstruction method based on a large-scale parallel computing architecture disclosed in the patent with the announcement number CN119834206A obtains the optimal solution of the power system network reconstruction by combining the power system network parameters with the pre-constructed power system network reconstruction model and using a binary particle swarm optimization algorithm, including constructing a first reconstruction optimization objective with the minimum line loss; constructing a second reconstruction optimization objective with the minimum number of switch actions and the minimum number of transformer voltage regulation; setting a first constraint condition for the first and second reconstruction optimization objectives; and constructing a power system network reconstruction model from the first and second reconstruction optimization objectives and the constraint condition.

[0004] For example, the medium and low voltage power distribution network fault reconstruction method based on a particle swarm algorithm disclosed in the patent with the announcement number CN119543106A includes obtaining a target power distribution network management data set; determining an island division range with each DG as the center and the capacity of the DG as the constraint condition; using an optimization algorithm based on a particle swarm to solve the island division of the DGs with the minimum network loss as the target and the capacity of the DG as the constraint condition according to the target power distribution network management data set, obtaining the load points contained in each island and the required disconnected switches; and performing fault reconstruction according to the island division solving result to obtain the network topology structure after reconstruction and the system fault reconstruction result.

[0005] The above disclosed technical solutions at least have the following technical problems: The network reconfiguration and island operation lack linkage mechanisms, such as some schemes only optimize network reconfiguration without involving island division, and some divide islands without coordinating with the switch operation and load distribution of network reconfiguration, so that the two strategies are difficult to form a combined force, and the problem that the network reconfiguration and island operation are difficult to be cooperatively optimized cannot be solved; the solving method has limitations, the mixed integer nonlinear programming model is complex to solve due to non-convexity, the intelligent algorithm such as particle swarm is easy to fall into local optimum and the time consumption increases dramatically in a large-scale scenario, and the problem of low computational efficiency cannot be solved; and no adaptive optimization logic is designed for a multi-fault scenario, neither the repair resource scheduling and recovery timing coordination under the multi-fault is considered, nor a global optimization model covering multiple periods and multiple regions is constructed, and the problem that the multi-fault scenario is insufficient cannot be solved.

[0006] In view of the above problems, the present application provides a solution. SUMMARY

[0007] In order to overcome the above-mentioned defects of the prior art, the embodiments of the present application provide a power distribution network fault recovery method considering island division and fault reconfiguration, a cooperative planning model supporting load shedding and island operation mode is constructed, a non-convex nonlinear problem is converted into a mixed integer second-order cone programming model by combining a second-order cone relaxation technology to improve the solving efficiency, and a fault repair constraint is introduced to form a cooperative optimization framework, so as to solve the problems that the network reconfiguration and island operation are difficult to be cooperatively optimized in the power distribution network fault recovery, the computational efficiency is low, and the multi-fault scenario is insufficient.

[0008] To achieve the above object, the present application provides the following technical scheme: A power distribution network fault recovery method considering island division and fault reconfiguration comprises the following steps: constructing a first constraint based on network topology constraints, load constraints and island operation constraints; constructing a cooperative planning model with the minimum load outage loss and system operation cost as an objective function; converting the cooperative planning model into a mixed integer second-order cone programming model through a second-order cone relaxation technology; integrating the repair constraint into the mixed integer second-order cone programming model to output a power distribution network fault recovery strategy.

[0009] In a preferred embodiment, the first constraint is constructed based on network topology constraints, load constraints and island operation constraints.

[0010] In a preferred embodiment, the cooperative planning model is constructed by discretizing the fault duration into a plurality of periods, formulating a corresponding recovery scheme for each period, constructing an objective function with the minimum load outage loss and system operation cost of the power distribution network, and constructing the cooperative planning model with the first constraint and the objective function.

[0011] In a preferred embodiment, the system operation cost includes switching action loss, distributed power operation cost and substation power supply cost.

[0012] In a preferred embodiment, the mixed integer second order cone programming model is specifically: the non-convex non-linear optimization problem in the collaborative planning model is converted into a mixed integer second order cone programming model through a second order cone relaxation technology; the non-convex non-linear problem includes network flow constraints and line constraints.

[0013] In a preferred embodiment, the maintenance constraint is specifically: the maintenance constraint is integrated into the mixed integer second order cone programming model to obtain a distribution network fault recovery strategy, and when a multi-loop fault occurs, the optimal fault recovery strategy can be found within the fault period.

[0014] The technical effects and advantages of the power distribution network fault recovery method considering island division and fault reconstruction of the application are as follows: The application realizes collaborative modeling of network topology and multiple operation modes by integrating the constraints of load shedding and island operation into a unified framework. The non-convex non-linear problem is converted into a mixed integer second order cone programming model through a second order cone relaxation technology, which significantly improves the solving speed and ensures the calculation accuracy. By introducing fault maintenance constraints and using a double-objective optimization framework and fault period discretization processing, the system operation economy is considered while ensuring the reliability of key load power supply, thereby realizing global collaborative decision-making of distribution network fault recovery. This technical solution effectively solves the problems of difficult collaborative optimization of network reconstruction and island operation, low calculation efficiency and insufficient response to multiple fault scenarios in distribution network fault recovery. BRIEF DESCRIPTION OF DRAWINGS

[0015] Figure 1 A power distribution network fault recovery method considering island division and fault reconstruction is provided for the embodiment of the application.

[0016] Figure 2 An IEEE33 node power distribution network system structure diagram is provided for the embodiment of the application.

[0017] Figure 3 A node voltage distribution characteristic diagram of each power supply area after using the optimization strategy provided for the embodiment of the application is provided. DETAILED DESCRIPTION

[0018] The technical solutions in the embodiments of the application will be described clearly and completely below with reference to the drawings in the embodiments of the application. Obviously, the described embodiments are only part of the embodiments of the application, rather than all the embodiments of the application. Based on the embodiments in the application, all other embodiments obtained by those skilled in the art without creative labor fall within the protection scope of the application.

[0019] Embodiment 1, Figure 1 A power distribution network fault recovery method considering island division and fault reconfiguration is given, comprising the following steps: S1, constructing a first constraint based on network topology constraints, load constraints and island operation constraints; S2, constructing a collaborative planning model with the objective function of minimizing load outage loss and system operation cost; S3, converting the collaborative planning model into a mixed integer second-order cone programming model through second-order cone relaxation technology; S4, integrating the maintenance constraints into the mixed integer second-order cone programming model, and outputting the power distribution network fault recovery strategy.

[0020] In this embodiment, the constraint conditions of load shedding and island operation are included in a unified framework to realize the collaborative modeling of network topology and multiple operation modes. Through the second-order cone relaxation technology, the non-convex nonlinear problem is converted into a mixed integer second-order cone programming model, which significantly improves the solving speed and ensures the calculation accuracy. By introducing fault maintenance constraints, using a double objective optimization framework and fault period discretization processing, the system operation economy is considered while ensuring the power supply reliability of key loads, thereby realizing the global collaborative decision of power distribution network fault recovery. This technical scheme effectively solves the problems of difficult collaborative optimization of network reconfiguration and island operation, low calculation efficiency and insufficient response to multiple fault scenarios in power distribution network fault recovery.

[0021] S1, constructing a first constraint based on network topology constraints, load constraints and island operation constraints.

[0022] In this embodiment, the network topology constraints are specifically:

[0023]

[0024] In the formula, denotes the total number of fault period time intervals, denotes the power flow variable from node to node through branch , denotes the power flow variable from node to node through branch , denotes the charging state of node , denotes the line connection state.

[0025] The present application is different from the conventional method which requires each node to have a unique parent node, that is, The present invention defines the node's charged state as always equal to 1. Defined as the sum of power flow variables flowing into this node, its value is extended to a binary state of 0 or 1. This improvement brings three advantages: it significantly simplifies the mathematical expression of power flow constraints; it combines the Big M method to handle line connection state variables; and it addresses these issues. This makes the trend model simpler; by allowing Taking a zero value allows for flexible removal of load nodes; when At that time, based on the maintenance constraints, the node will be automatically excluded from the power flow calculation, which satisfies the operational requirements and improves the calculation efficiency.

[0026] In this embodiment, the load constraints include dispatchable load constraints and non-dispatchable load constraints; The schedulable load constraints are as follows: ; Unschedulable load constraints are:

[0027] In the formula, For load exist Real-time value of active power at all times. for The upper limit constraint value, For load exist Real-time reactive power value at all times. for The upper limit constraint value, For binary decision variables, representing unschedulable loads in... The running status at any given moment, This indicates that the load has been removed. This indicates that the load is maintaining normal power supply.

[0028] In this embodiment, the islanding operation constraints are used to regulate the power balance and boundary isolation requirements between power sources and loads within the island. The inherent volatility of islanding operation is offset by constraining the output of the distributed generation (DG). The DG output constraints are as follows:

[0029] In the formula, , These represent the minimum and maximum active power output of the distributed generation, respectively. This refers to the actual active power output of the distributed power source. This refers to the actual reactive power output of the distributed power source. The power factor angle.

[0030] It should be noted that the DG output constraint refers to a limit condition of a distributed generation (DG) output power, and aims to ensure safe, stable and economic operation of a power system.

[0031] S2, taking the minimum load outage loss and system operation cost as an objective function, constructing a collaborative planning model.

[0032] In the embodiment, the construction method of the collaborative planning model is specifically: Discretize the fault duration into a plurality of time periods, and formulate a corresponding recovery scheme for each time period, to construct an objective function for minimizing the load outage loss and system operation cost of the distribution network; Construct a collaborative planning model with the first constraint and the objective function.

[0033] In the embodiment, the system operation cost includes a switch action loss, a distributed power supply operation cost and a substation power supply cost.

[0034] In the embodiment, the objective function is specifically: A dual-objective optimization framework is adopted, first, the load outage loss is minimized in priority, a higher weight is given in the objective function, and differentiated power supply protection is implemented according to the importance of the load; second, the system operation cost is optimized, including the switch action loss, the distributed power supply operation cost and the substation power supply cost, and the weight of this part is set to be relatively low

[0035] In the formula, total cost, the load shedding cost coefficient, the total number of time intervals of the fault period, the total number of nodes in the network, the length of each time interval of the fault period, the time period, the total amount of load shedding of the node, the load weight of the node, the photovoltaic operation cost coefficient, the switch operation cost coefficient, the branch, the switch state of the branch in the fault recovery period, which is a 0-1 variable, the total number of branches in the network, the distributed power supply operation cost coefficient, the time period the switch state of the branch in the fault recovery period, which is a 0-1 variable, the total number of branches in the network,Node distributed power output power, express Time period Output power of distributed power supply at nodes.

[0036] Figure 2 This is a structural diagram of an IEEE 33-node distribution network system. The test system is configured as follows: 33 distribution nodes, 32 normally closed sectionalizing switches, and 5 normally open tie switches. The system rated voltage is 12.66 kV, and the power baseline is 100 MW. The total load for the four time periods is 2775+1719kV·A, 3513+j2177kV·A, 3696+j2290kV·A, and 2954+j1831kV·A, respectively.

[0037] In this embodiment, Table 1 lists the priority of each load node and its corresponding weight coefficient, while Table 2 gives the access location and installed capacity configuration of DG.

[0038] Table 1

[0039] Table 2

[0040] Regarding the setting of objective function weights, considering the setting of load restoration priority, the following is taken: With the primary objective of prioritizing load recovery, the remaining optimization objectives ( , , All are set to 1 as auxiliary optimization items. System lines 6-7, 12-13, 2-19, 21-22, 24-25, and 32-33 are set to experience permanent faults at 9:00 AM. The fault recovery period is set from 9:00 AM to 1:00 PM, and a maximum of 2 fault lines can be processed per hour.

[0041] Experimental results show that, after adopting the collaborative recovery strategy proposed in this invention, 1012kW of power was restored in the first period (recovery rate of 36.8%); the total load recovery in the second period was 2548kW, with a recovery rate of 72.5%; the total load recovery in the third period was 3569kW, with a recovery rate of 96.6%; and the total load recovery in the fourth period was 2954kW, with a recovery rate of 100%. The detailed total recovery amounts for each load level are shown in Table 3. During the fault recovery process, priority is given to restoring power to loads with higher load levels. Therefore, the power recovery rate for first-level loads reached 100% in the second period (9:00-10:00 is the first period, and so on), while the power recovery rate for second-level loads also reached over 70%, effectively ensuring the reliability of power supply.

[0042] The fault recovery process has the following advantages: First, the three-stage recovery strategy of "regional autonomy - key protection - full network recovery" is adopted, which realizes the scientific and orderly handling of faults; second, through the coordinated cooperation of DG optimized scheduling and network reconstruction, the problem of low DG utilization in the traditional recovery process is effectively solved; finally, the decision-making method based on multi-objective optimization ensures the balance between economy and reliability in the recovery process.

[0043] Table 3

[0044] S3 uses the second-order cone relaxation technique to transform the collaborative programming model into a mixed-integer second-order cone programming model.

[0045] In this embodiment, the mixed-integer second-order cone programming model is specifically as follows: The non-convex nonlinear optimization problem in the collaborative programming model is transformed into a mixed-integer second-order cone programming model by using the second-order cone relaxation technique. The non-convex nonlinear problems include network power flow constraints and line constraints.

[0046] In this embodiment, the network power flow constraint specifically refers to:

[0047]

[0048]

[0049]

[0050]

[0051]

[0052] In the formula, and Indicates the line The active and reactive power transmission power, and Indicates the line The active and reactive power transmission power, and For the line Resistance and reactance parameters, and Represents a node The active and reactive power injected. and Represents a node The active and reactive power outputs of distributed power sources, and Represents a node The active and reactive power output of the load, Indicates the line The square term of the current, express Time period The square term of the node voltage, express Time period The square term of the node voltage, Indicates the line The square of the active power term, Indicates the line The square of reactive power.

[0053] It should be noted that network power flow refers to the flow state of electricity in network components such as transmission lines and transformers, including the distribution of parameters such as voltage, current, and power (active power and reactive power). It is defined by limiting the safety and economy of parameter balance within the framework of physical laws.

[0054] In this embodiment, the line constraints include line capacity constraints, line voltage constraints, and line current constraints; The line capacity constraint is as follows:

[0055] The formula specifies the transmission capacity limit for distribution network branches. and Indicates the line The active and reactive power transmission power, express Timetable The running status, For a constant with a large value, Indicates the line Maximum apparent power; The line voltage constraint is:

[0056] In the formula, the square of the rated voltage is used as the base. The reference defines the permissible range of node voltage fluctuations. The square term of the rated voltage. express Time period The squared term of the node voltage; The line current constraint is:

[0057] The formula employs a second-order cone relaxation technique to handle the quadratic constraint terms in order to address the nonlinear relationship between voltage, current, and power.

[0058] Regarding the handling of line voltage drop constraints, this invention constructs branch voltage equations based on the Big M method:

[0059] In this equation, the positive direction of the branch power is from the first node to the last node, and M is a large constant. express Timetable This method effectively solves the problem of power flow calculation when the distribution network topology changes dynamically, and enhances the model's adaptability to network reconfiguration and islanded operation.

[0060] It should be noted that the second-order cone relaxation technique is an important relaxation method in the field of convex optimization. It is mainly used to transform non-convex optimization problems into efficiently solvable second-order cone programming (SOCP) problems. Its core idea is to introduce second-order cone constraints to "relax" the non-convex constraints or objective function in the original problem, thereby reducing the difficulty of solving the problem while maintaining a certain level of accuracy.

[0061] S4 integrates maintenance constraints into a mixed-integer second-order cone programming model and outputs a power distribution network fault recovery strategy.

[0062] In this embodiment, the maintenance constraint specifically refers to:

[0063]

[0064] In the formula, F represents the set of phasors of the endpoint nodes of the faulty line. for Timetable The running state is a 0-1 variable. This is a limit on the maximum number of lines that can be inspected in parallel. This is the maintenance time for a single line. Before the fault is cleared, the relevant line status variables are forcibly set to 0.

[0065] When multiple line faults occur, it is necessary to find the optimal inspection sequence within the fault period. The method to optimize the fault inspection strategy is to add a series of new constraints to the switch state variables during network reconstruction and islanding, and integrate them into a collaborative planning model for network reconstruction and islanding. It is also stipulated that a maximum of h faulty lines can be restored.

[0066] Figure 3The paper demonstrates the node voltage distribution characteristics of each power supply area after adopting the optimization strategy proposed in this invention. Under strict voltage constraints, the system exhibits good voltage regulation performance. Specific analysis shows that in the main grid power supply area, thanks to the strong support of the power grid, the voltage of each node remains stable within the ideal range of 0.98-1.03 per unit, with a voltage fluctuation standard deviation of less than 0.015. While the DG islanded operation area is limited by the capacity and regulation capabilities of distributed power sources, the multi-objective collaborative optimization strategy proposed in this invention still achieves a stable voltage output of 0.96-1.04 pu, with its voltage fluctuation characteristics improved by approximately 23% compared to traditional methods. Throughout the entire fault recovery process, the voltage deviation of all nodes in the system is strictly controlled within the allowable range of ±0.05 per unit, achieving a 100% voltage qualification rate. This result not only verifies the robustness of the proposed strategy but also highlights its adaptability under complex operating conditions. Compared with existing recovery strategies, the method of this invention improves the voltage quality of the system while ensuring power supply reliability by introducing voltage sensitivity analysis and dynamic weight adjustment mechanism, and realizes the synergistic optimization of power supply reliability and power quality, providing a new technical approach for fault recovery of smart distribution networks.

[0067] The above formulas are all dimensionless calculations. The formulas are derived from software simulations based on a large amount of collected data to obtain the most recent real-world results. The preset parameters in the formulas are set by those skilled in the art according to the actual situation.

[0068] The above embodiments can be implemented, in whole or in part, by software, hardware, firmware, or any other combination thereof. When implemented using software, the above embodiments can be implemented, in whole or in part, in the form of a computer program product.

[0069] Those skilled in the art will recognize that the modules and algorithm steps of the various examples described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution. Those skilled in the art can use different methods to implement the described functions for each specific application, but such implementation should not be considered beyond the scope of this application.

[0070] In addition, the functional modules in the various embodiments of this application can be integrated into one processing module, or each module can exist physically separately, or two or more modules can be integrated into one module.

[0071] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

[0072] In conclusion, the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A distribution network fault recovery method considering islanding and fault reconfiguration, characterized in that, Includes the following steps: The first constraint is constructed based on network topology constraints, load constraints, and islanded operation constraints; A collaborative planning model is constructed with the objective function of minimizing load outage losses and system operating costs; The collaborative programming model is transformed into a mixed-integer second-order cone programming model using the second-order cone relaxation technique. The maintenance constraints are integrated into a mixed-integer second-order cone programming model to output a distribution network fault recovery strategy.

2. The distribution network fault recovery method considering islanding and fault reconfiguration according to claim 1, characterized in that, The network topology constraints are specifically as follows: In the formula, This indicates the total number of time intervals during the fault period. Indicates power from the node via branch road Flow to Node The direction of flow variable, Indicates power from the node via branch road Flow to Node The direction of flow variable, Represents a node The charged state, This indicates the connection status of the line.

3. The distribution network fault recovery method considering islanding and fault reconfiguration according to claim 2, characterized in that, The load constraints include dispatchable load constraints and non-dispatchable load constraints; The schedulable load constraints are as follows: ； Unschedulable load constraints are: In the formula, For load exist Real-time value of active power at all times. for The upper limit constraint value, For load exist Real-time reactive power value at all times. for The upper limit constraint value, For binary decision variables, representing unschedulable loads in... The operational status at any given moment.

4. The distribution network fault recovery method considering islanding and fault reconfiguration according to claim 3, characterized in that, The islanding operation constraints are used to regulate the power balance and boundary isolation requirements between power sources and loads within the island. They offset the inherent volatility of islanding operation by constraining the output of distributed generation (DG). The DG output constraints are as follows: In the formula, , These represent the minimum and maximum active power output of the distributed generation, respectively. This refers to the actual active power output of the distributed power source. This refers to the actual reactive power output of the distributed power source. The power factor angle.

5. The distribution network fault recovery method considering islanding and fault reconfiguration according to claim 4, characterized in that, The method for constructing the collaborative planning model is as follows: The duration of the fault is discretized into several time periods, and a corresponding recovery plan is formulated for each time period to construct an objective function that minimizes load outage losses and system operating costs in the distribution network. Construct a collaborative programming model using the first constraint and the objective function.

6. The distribution network fault recovery method considering islanding and fault reconfiguration according to claim 5, characterized in that, The system operating costs include switching operation losses, distributed power supply operating costs, and substation power supply costs.

7. The distribution network fault recovery method considering islanding and fault reconfiguration according to claim 6, characterized in that, The mixed-integer second-order cone programming model is specifically as follows: The non-convex nonlinear optimization problem in the collaborative programming model is transformed into a mixed-integer second-order cone programming model by using the second-order cone relaxation technique. The non-convex nonlinear problems include network power flow constraints and line constraints.

8. The distribution network fault recovery method considering islanding and fault reconfiguration according to claim 7, characterized in that, The network flow constraints are specifically as follows: In the formula, and Indicates the line The active and reactive power transmission power, and Indicates the line The active and reactive power transmission power, and For the line Resistance and reactance parameters, and Represents a node The active and reactive power injected. and Represents a node The active and reactive power outputs of distributed power sources, and Represents a node The active and reactive power output of the load, Indicates the line The square term of the current, express Time period The square term of the node voltage, express Time period The square term of the node voltage, Indicates the line The square of the active power term, Indicates the line The square of reactive power.

9. The distribution network fault recovery method considering islanding and fault reconfiguration according to claim 8, characterized in that, The line constraints include line capacity constraints, line voltage constraints, and line current constraints. The line capacity constraint is as follows: In the formula, and Indicates the line The active and reactive power transmission power, express Timetable The running status, For a constant with a large value, Indicates the line Maximum apparent power; The line voltage constraint is: In the formula, The square term of the rated voltage. express Time period The squared term of the node voltage; The line current constraint is: The formula employs a second-order cone relaxation technique to handle the quadratic constraint terms in order to address the nonlinear relationship between voltage, current, and power.

10. The distribution network fault recovery method considering islanding and fault reconfiguration according to claim 9, characterized in that, The maintenance constraints are specifically as follows: In the formula, F represents the set of phasors of the endpoint nodes of the faulty line. for Timetable The running state variables, This is a limit on the maximum number of lines that can be inspected in parallel. This refers to the maintenance time for a single line.

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