NOMENCLATURE
𝑁𝑆𝐸𝐿: Non-Served Energy Level.
𝑇: Planning period to perform the vegetation maintenance. In this case, 4 subperiods are used for T = 1 year.
𝑁𝑆: Total number of network segments to be considered in the vegetation maintenance schedule.
𝑈(𝑖, 𝑡): Unavailability during subperiod t in network segment i.
𝐴𝐷(𝑖, 𝑡): Average demand in network segment i of subperiod t.
𝛾(𝑖, 𝑡): Binary decision variable that takes a value of “1” when a maintenance activity has to be performed in network segment i of subperiod t, and “0” otherwise.
𝛺𝐷𝑆: Set of network segments downstream network segment i.
𝐶𝑚𝑎𝑖𝑛 (𝑖, 𝑗, 𝑡): Vegetation maintenance cost in network segment i, of crew j, in subperiod t [$COP/m].
𝑙(𝑖): Length of network segment i [m].
𝐿𝑚𝑎𝑥(𝑡): Maximum length of pruning allowed in subperiod 𝑡 [m]
𝜆𝑓𝑎𝑖𝑙𝑢𝑟𝑒(𝑖,𝑡): Failure rate in network segment i of subperiod t [failures/year].
𝑁??(𝑖): Number of customers connected to the end node of network segment i.
𝑇𝑁𝐶: Total number of customers in the network.
𝑆𝐴𝐼𝐹𝐼𝑚𝑎𝑥: Maximum allowed quantity of power supply outages [failures/customer].
𝑁𝑃: Maximum number of pruning allowed in planning period T in a network segment.
𝑇𝑜(𝑖, 𝑡): Time from the last vegetation pruning in network segment i of subperiod t [years].
𝜆 𝑔𝑟𝑜𝑤𝑡h (𝑖, 𝑡): Vegetation growth rate in network segment i of subperiod t [m/year].
𝐷𝑚𝑖𝑛: Minimum distance allowed between the overhead power lines and the vegetation [m].
𝐼𝐴𝑃: Introduction average percentage of the vegetation to the security zone [%].
𝑁𝐶𝑟𝑒𝑤: Number of crews to perform the pruning works.
𝜒(𝑖,𝑐,𝑡): Binary decision variable that takes a value of “1” if crew c performs the maintenance in network segment i in subperiod t, and “0” otherwise.
𝐾: Proportion factor between the cost of NSEL and pruning implementation costs [$].
𝐶𝐸: Cost of energy [$COP/kWh]
1. INTRODUCTION
The emergence of utility services such as water supply, sewage, electricity, and telecommunications has brought to the cities a complex system of wires and pipe-lines that must live together with the urban vegetation. The latter provides relief to the environment of metropolitan centers, which are often immersed in visual, noise and air pollution.
From the point of view of power energy supply, utility companies have had to face different difficulties related with pre-serving a suitable level of reliability in the power network and performing vegetation maintenance without compromising the physical integrity of the vegetal species [1].
Due to their low cost in comparison to other configurations, power distribution networks are, in most cases, of the over-head type. However, because of external factors related with the weather, animals, and vegetation, their reliability indices are lower than those of underground networks. The contact of a plant with the energized wires can cause short circuits (in the worst cases, broken wires) and an imminent activation of the protection systems. This event represents one of the main causes of power supply interruptions [2] [3].
In general, the network operator in charge of the system performs vegetation maintenance. Traditionally, a group of workers carries out a visual inspection along the power network to determine which network segments are susceptible to vegetation contact. Subsequently, vegetal species are pruned considering the most suitable method to preserve the physical integrity of the plant [4]. These procedures can have satisfactory results from the point of view of the compliance with reliability goals; furthermore, better results can be obtained if the methodologies are derived from mathematical modeling, considering different objectives, reliability indices, and technical constraints.
In the specialized literature, few works deal with vegetation maintenance underneath overhead power distribution lines. Generally, they present the following aspects for an efficient pruning schedule: historical failures, evolution of a failure caused by vegetation, duration of a maintenance cycle, reliability indices, time from the last vegetation pruning, network operator budget, Markov chains, etc. [5] [6] [7]. A criticality analysis of overhead power lines’ assets based on decision support systems and risk management is proposed in [8] to improve preventive pruning programs. Additionally, the study in [9] presents the characteristics of leakage current of different vegetation flames and the effect of flame conductivity on wire-plate gap breakdown characteristics.
Other works are focused on developing ways to monitor how close the trees are to the power distribution lines, considering inspections from the air and the ground with thermal cameras and satellite images [10] [11] [12]. An overview of modern remote sensing methods in power line corridor surveys is presented in [13], including synthetic aperture radar images, optical satellite and aerial images, thermal images, airborne laser scanner data, and unmanned aerial vehicles. The latter is particularly important in [14], as multi-rotor solutions are a viable technology option that should be incorporated into both preventive maintenance and damage assessment. From the perspective of satellite imagery, the authors of [15] establish some of the benefits of this technology for power line maintenance in forested areas: identifying individual trees growing near power lines, evaluating the height of vegetation in high voltage power line corridors, and detecting changes in corridors and surrounding areas. Some publications point out the relevance of smart methods, such as neural networks, for predicting the failure rate of distribution feeders caused by vegetation contact [16] [17]. Novel approaches in the framework of image processing were developed in [18], with a conceptual model of a computer vision system capable of identifying overhead urban lines and their three dimensional positioning to aid a teleoperated robot in pruning vegetation. Although their results are accurate and novel alternative solutions are developed, these methods require, in most cases, a high computational cost as well as expensive devices and software for image processing.
In turn, mathematical models have also been used to represent the vegetation maintenance problem through objective functions and a set of constraints, thus obtaining a result in terms of pruning schedules, where the maintenance cost is minimized or the reliability of the distribution network is maximized [19] [20]. Along the same line of research, the work proposed by [21] presents a methodology for long term maintenance scheduling of overhead lines considering feeders ranking for high accuracy investment. Such ranking is defined by the type of costumers, loads, length, and failure rates.
Compared with the works mentioned above, this paper proposes a feasible option that can be implemented, since the input data correspond to historical failures due to vegetation contact and/or vegetation growth rates, most often available in the database of the utility company. This work proposes four solution approaches, as an extension of the works developed by [19] and [20] for the optimal management of vegetation maintenance in overhead power lines, by using mathematical modeling. Each approach aims at minimizing the following aspects:
Non served Energy Level (NSEL);
Cost of implementation (CI) of the maintenance plan, in conjunction with the NSEL, using proportion factors;
Cost of implementation of the maintenance plan; and
NSEL with investment constraint. In this proposal, a set of solutions is obtained for pruning plans under a multi-objective context.
The solution to this problem is given in terms of a schedule of vegetation pruning activities and the moment at which each activity must be performed over planning period T. The effectiveness of the mathematical model is assessed by simulating, in a real system, what has been studied by other authors.
This article is organized as follows. Section 3 formulates the problem mathematically within the framework of four solution approaches. Afterward, Section 4 presents the real test system used in the computational implementations. Section 5 introduces the results to the problem based on the construction of a non-dominated solutions front and an improvement strategy to obtain better quality solutions.
2. MATHEMATICAL FORMULATION OF THE PROBLEM
Scheduling vegetation maintenance for distribution systems can be represented through a non-convex non-linear integer mathematical model, considering reliability metrics, implementation costs of activity scheduling, vegetation failure rates and growth rates, and the technical and financial capacity of the utility company, among others.
Such problem is formulated adopting four different approaches, thus obtaining four mathematical models, which are similar regarding constraints but different in terms of objective functions.
2.1 Model 1
In this model, the NSEL of the distribution system is minimized, considering the activity schedule obtained from the solution of the mathematical model, which is described in equations (1) to (6).
The objective function in (1) describes the NSEL that the distribution system will have over the planning period of vegetation pruning activities. The software NEPLAN was used to compute, offline, the energy not served downstream network segment i. Equations (2) and (3) are operating constraints that represent the available resources of the utility company to implement the maintenance plan of vegetation pruning. Such resources are limited by the number of crews, which have a determined capacity of linear meters of network that can be attended for vegetation pruning. The multiplication of the terms 𝑙(𝑖). 𝛾(𝑖, 𝑡) implies an increase in the pruned length when the binary variable 𝛾(𝑖, 𝑡) = 1 (maintenance on the network segment). Otherwise, if 𝛾(𝑖, 𝑡) = 0, no linear meters are added to the expression. The term 𝐿𝑚𝑎𝑥(𝑡) limits the allowed quantity of linear meters subject to pruning for each subperiod t. This resource can vary depending on the time of the year and availability of economic resources. Equation (3) limits the number of pruning activities that can be performed on a network segment over the planning period.
Equation (4) is a regulatory constraint that determines the quality indices of the utility company regarding power supply continuity. This index must be monitored and controlled to avoid economic penal-ties.
Equation (5) is the constraint that controls the Introduction Average Percentage (IAP) of vegetation in regards with the security zone, as shown in Fig. 1. This value is determined via the product 𝑇𝑜 (𝑖, 𝑡) 𝜆 𝑔𝑟𝑜𝑤𝑡h (𝑖, 𝑡), which is normalized by dividing by 𝐷𝑚𝑖𝑛, thus obtaining a percentage [%]. The variable 𝑇𝑜 (𝑖, 𝑡), computed in (6), is controlled by 𝛾(𝑖, 𝑡), and it determines the time elapsed since the last vegetation pruning on each network segment. Such time duration is accumulated every time that 𝛾(𝑖, 𝑡) = 0. When 𝛾(𝑖, 𝑡) = 1, 𝑇𝑜 (𝑖, 𝑡) is reset. Table 1 proposes an example of the result of 𝑇𝑜 (𝑖, 𝑡) according to the value of 𝛾(𝑖, 𝑡).
After the pruning schedule is obtained with the mathematical model described in (1) to (6), a mathematical programming problem, known as “assignment problem”, is solved. Such problem delegates the activities to work teams (maintenance crews, workgroups, etc.) so that the cost of implementing these activities can be minimized (see (7)).
Equation (8) is an operating constraint, similar to (2), which limits the linear meters that each crew can prune. The multiplication of the terms 𝑙(𝑖).𝜒(𝑖, 𝑐, 𝑡) indicates an increase in pruned length, when 𝜒(𝑖, 𝑐, 𝑡) = 1; otherwise, when 𝜒(𝑖, 𝑐, 𝑡) = 0, there is no addition of linear meters, as this variable shows the network segments that will be pruned and the crew assigned to perform the task. The term 𝐿𝑚𝑎𝑥(??) limits the allowed quantity of linear meters to be pruned in each subperiod t, which can vary depending on the time of the year and the availability of economic resources.
In (9), the models are combined when the binary variables 𝜒(𝑖, 𝑐, 𝑡) and 𝛾(𝑖, 𝑡) take equal values for each network segment i of subperiod t, thus guaranteeing that the assignment of tasks is completed in accordance with the pruning plan and maintenance can only be assigned when required by the network segment.
2.2 Model 2
In this mathematical model, the costs associated with NSEL and the implementation of pruning activities are minimized, focusing on a mono-objective function, which is expressed in (10). The implementation costs consider the work teams in charge of the activities.
Note that (10) has two conflicting objectives. In general, if the NSEL cost is high, the cost of pruning implementation is low, and vice versa. The value of k represents the level of significance of the terms in the equation, which depends on the particular interests of the decision maker.
2.3 Model 3
In Model 3, only implementation costs of the pruning activities are minimized, subject to the same constraints in Model 2.
min (7)
Subject to:
Equations (2) to (6), (8), and (9).
Subject to:
Subject to:
Subject to:
2.4 Model 4
In this case, the NSEL is minimized considering the operation constraints in all the models above. In summary, this model is represented as follows:
min (1)
Subject to:
quations (2) to (6), (8), and (9).
In addition to its constraints, this model considers the implementation cost, limited by the availability of a resource, as shown in (11).
Where 𝐶𝐼𝑑𝑖𝑠𝑝 is the implementation resources the utility company has available to perform the pruning activities in the planning period. In this case, Model 4 is fed the solutions provided by Models 1 and 3. In Model 1, a minimum value of NSEL and a high value of CI are obtained. Conversely, in Model 3, the value of CI is minimum and that of NSEL is high. That way, the maximum and minimum values of NSEL and CI are obtained. Afterward, Model 4 is executed for different values of CI, which are within the range of maximum and minimum values provided by Models 1 and 3. Therefore, Model 4 represents a multi-objective approach through an iterative process that provides a set of non-dominated solutions that consider a conflict of interests between NSEL and CI. Such process is depicted in Fig. 2.
Subsequently, based on the non-dominated solution front of this model, it is possible to improve most solutions obtained in the front. Thus, better quality maintenance schedules can be obtained by reducing both implementation costs and NSEL or one of them.
The improvement stage takes each solution in the front obtained in Model 4, and it evaluates them in the model of equations (7), (8), and (9).
3. TEST SYSTEM
In order to evaluate the mathematical models explained above, a real 34.5-kV distribution system, which can be requested from the authors, was used. Tables 2 to 5 were considered for the implementation of the mathematical approach.
Vegetation maintenance is planned over a period of one year, divided into 4 quarters, being t=1,..,4. Table 2 presents failure rates associated with vegetation in each network segment of the distribution system, which have a growing behavior overtime. This trend represents a non-homogeneous Poisson process; nevertheless, the statistical procedure to find the failure rates is out of the scope of this work. In this case, the reliability analysis module of NEPLAN [22] was used to find both the failure rates and the average demand of each network segment. The latter is based on the N-1 contingency criterion. Additionally, the network segment lengths and number of customers connected to the end node are presented in Table 2.
Unavailability 𝑈 (𝑖, 𝑡) is computed from the failure rates using (12) to find the NSEL, given a maintenance proposal.
Where is the mean repair time of a failure, assumed to be 1.5 hours in accordance with [23]. Furthermore, it is necessary to include the average demand of each network segment (see Table 3) in the input data of the mathematical models. Moreover, Table 4 lists the vegetation growth rates [24] with variations depending on the time of the year, due to environment factors that significantly influence this parameter.
The maintenance costs detailed in Table 5 correspond to the cost of vegetation pruning by a maintenance crew, considering the length of each network segment. A maximum number of three crews 𝑐 = 1, 2, 3 and an invariant maintenance cost for each network segment over the planning period were assumed.
The following are other parameters considered in the development of the models:
𝑇𝑁𝐶 = 70 customers
𝑆𝐴𝐼𝐹𝐼𝑚𝑎𝑥 = 7 failures
𝑁𝑃 = 1 pruning
𝐿𝑚𝑎𝑥(𝑡) = 1000 m
𝐼𝐴𝑃 = 75%
𝐷𝑚𝑖𝑛 = 1.5 m
𝐶𝐸 = 320 COP/kWh
4. RESULTS
The results of the mathematical models are represented by vegetation pruning schedules considering when and where the maintenance activities are performed. In general terms, Model 4 presents a complete procedure that includes Models 1, 2, and 3 and presents the results in a non-dominated solution front. Therefore, it is necessary to run Models 1 and 3 to find the solutions that represent a maximum value of NSEL with a low cost of implementation CI, and a maximum value of CI with a minimum NSEL. Such solutions are the extremes of the non-dominated front. Fig. 3 is a graph of the non-dominated solution front after Model 4 is run in the test system described in Section 4. Each solution is described in Table 6.
After the solution front is obtained, the improvement stage is completed by submitting each solution to the mathematical model described by (7), (8), and (9). In that context, the implementation cost of some of the solutions is enhanced, as described in Fig. 4. Each solution is described in Table 7, marking with a double asterisk the solutions that were improved. In Fig. 4, the solutions in the zoom-in correspond to alternative 8 in Table 6, with the same value of NSEL but different cost of implementation and a better solution found after the improvement stage. Such alternatives (alternative 8 before and after the improvement stage) have the same activity schedules, as shown in Table 8.
Regarding the information in Table 8, pruning activities are more common in the last quarter. In the second and third quarters, such activities are less frequent, and the first subperiod presents no interventions. This mainly due to the failure rates caused by vegetation, which exhibit an increasing behavior, thus making pruning activities more common in the last quarter of the planning period. With respect to the crews or work teams 𝑐1, 𝑐2 𝑐3, the execution of the pruning plan is assigned as shown in Tables 9, 10, 11 and 12. Note that the assignment of activities for the fourth subperiod is different in the solution after implementing the improvement. This is demonstrated in Table 7, as the option of improved solution establishes a vegetation maintenance activity in network segment 11 for Crew 2, while in the solution in the first stage (before improvement), the same activity is carried out by Crew 1. The enhancement of the CI is performed in accordance with the minimization of the maintenance cost developed by the mathematical approach in (7), (8), and (9). The mathematical models in this work were run using the software GAMS (General Algebraic Modeling System) and the solver DICOPT, specifically designed for the solution of mixed integer non-linear mathematical models. The GAMS codes can be requested from the authors.
5. CONCLUSIONS
The vegetation maintenance problem underneath overhead power distribution network has been represented in this work through mathematical programming, encompassing several aspects of power systems reliability, the financial and technical capacity of the utility companies, and the biological characteristics of the vegetation species. Therefore, the results were obtained in terms of where and when the vegetation pruning activities must be performed along the power distribution system, minimizing the NSEL and the implementation costs associated with such tasks. Likewise, crews or work teams were also assigned to conduct those activities.
The sensitivity parameters represented by the failure rates caused by vegetation influence the distribution of the pruning activities over the planning period under study. This aspect is reflected in the increased frequency of maintenance works in the last quarters.
The different mathematical modeling approaches adopted in this study to address such problem provide a non-dominated solution front that contains different vegetation maintenance proposals, allowing the decision maker to select the solution that best represents the priorities of the utility company.
Furthermore, an improvement strategy was applied to the solution front to obtain better proposals from the point of view of cost of implementation.