A Genetic Algorithm for Transmission Network Expansion Planning Considering Line Maintenance

This paper proposes a decimal codification genetic algorithm to solve the transmission network expansion planning (TNEP) problem considering the economic impact of line maintenance. The goal is to extend the lifespan of the time-worn lines in order to reduce the investment cost in the expansion of the transmission network and to improve the worth of the transmission system. To assess the economic impact of the maintenance on the deterioration of transmission lines and transformers, the sum of years digit method is implemented. The proposed algorithm is evaluated using the IEEE reliability test system, and the assessment of the results shows that by including the effect of line maintenance on the TNEP problem, significant savings can be made in the overall cost of the system.

The transmission network expansion planning (TNEP) is a hard, large-scale, and nonlinear programming problem that aims to install new transmission lines in the power system so that the future demand is met [1]. After the year 1970 extensive research has been conducted on the TNEP, which can be categorized as follows: (i) papers that are about the problem's solution method [2], [3]; (ii) papers that have included several characteristics, such as uncertainty [4], network security [5], reliability criteria [6], [7], and bundled lines [8], for solution of the TNEP problem; (iii) works that investigate the integrated transmission and generation expansion planning problem [9].
The aim of this paper is to use a decimal codification genetic algorithm (DCGA) to solve the TNEP problem with a new framework. Thus, only papers that have considered different characteristics in the TNEP are investigated. Table I shows references for the problem classified with different aspects. In [10], the objective function includes investment and operational costs related to the fuel supply requirement of power plants, the investment cost for the construction of transmission lines, and the network's power losses. In [11], the TNEP is formulated as an optimization problem that accounts for three objective functions, including the investment for constructing lines, congestion cost, and the cost of load shedding. In [12], a method based on risk/investment is proposed to solve the TNEP problem taking into account multiple future generation and load scenarios. In [13], the investment cost related to the construction of new lines is minimized considering two probabilistic reliability constraints and the uncertainties related to the forced outage rates of the elements of the network. ----- [7] ---- [10] ---- [11] --- [12] ----- [13] ---- [14] ----- [15] --- [16] -- [17] ---Later, [14] introduced two probabilistic reliability criteria called loss of load expectation (LOLE) and expected energy not supplied (EENS) for solving the problem of [13]. It was concluded that the LOLE and the EENS provide higher reliability for customers. Also, in [15], the reliability criteria of expected demand not supplied (EDNS) and expected generation not served (EGNS) are accounted for in the objective function of the TNEP problem. It is demonstrated that the capacity of the existing lines should be upgraded besides the traditional approach of constructing new transmission lines in order to achieve more economic and reliable expansion plans. Furthermore, in [16], the investment cost in new transmission lines, the expected operation cost, and the load shedding costs were minimized under load uncertainty and voltage security constraints. Finally, in [17], the Benders decomposition method was used to solve a mixed-integer linear programming (MILP) formulation for the TNEP problem considering generation reliability.
These studies, however, do not consider the economic impact of lines' maintenance on the solution of the TNEP problem. Several transmission system equipment, such as the lines and transformers, have been increasingly getting older. This fact causes the lines to reach the end of their usual lifetime earlier. Maintenance activities can increase the lifetime of the lines so that they satisfy the required level of the system's reliability. Even though an increase in the maintenance budget may cause an increase in the total cost, it can avoid the construction of new lines that can result in a costly expansion of the transmission system. Therefore, it is crucial to include a general age-dependent formulation that explicitly considers the economic influence of maintenance in the TNEP problem.

II. MATHEMATICAL MODEL OF THE PROBLEM
The proposed problem is formulated as shown in (1)- (13).
The objective function (1) minimizes the total cost of investing in the construction of new transmission lines, the cost of replacing existing lines, the cost of investing in substations, the cost of annual losses, the maintenance cost of transmission lines, and the total load shedding in the system while maximizing the worth of the transmission system. Constraint (9) specifies that the total generation in the network is equal to the total demand, (10) is the equation of the active power balance, (11) represents the capacity of the power flow for the lines in each corridor, (12) limits the number of lines that can be installed in each corridor, and (13) limits the load shedding at each bus in the system.

A. Description of the Maintenance Cost Coefficient Effect on the Life Coefficient
Transmission equipment, such as lines and transformers, have normal lifespans under normal operating conditions if the required maintenance actions are taken. Predefined or variable maintenance costs are required for these acts to be carried out. If the maintenance budget is more or less than that cost, the component age (life expectancy) will be longer or shorter than its usual lifetime [18]. This can be defined analytically using the age factor of element z (a z) as follows [18]: Rewriting (14) for the transmission lines and transformers of the TNEP problem, results in (15).
Where mz is a characteristic constant of element z. Larger or smaller mz correspond to the newer or older elements, respectively. The following equations arise from replacing    Therefore, from Fig. 1, above-mentioned results, and the nonlinear relation between m(ij) and a(ij) (18), the following equation can be defined: Fig. 1, it can be seen that the life expectancy of older lines increases more than that of newer ones when maintenance cost increases.

III. SOLUTION METHOD
The purpose of the presented version of the TNEP problem is to determine the number of new lines for network expansion while optimizing the costs of expansion, repair, reliability, and losses. There are several methods to solve this problem, such as classical and heuristic approaches [1]. Thanks to its versatility and simple implementation, the DCGA technique is used in this analysis to solve the TNEP problem. In this method the suggested chromosome codification is as follows:  (2) is calculated using a DC power flow, taking into consideration constraints (9) and (10). If (11) is fulfilled, the fifth term of the objective function (1) is solved using MATLAB's fmincon function, considering constraints (10) and (11). Next, (4)- (8) are computed, and the objective function (1) is determined accordingly.
The selection operator in the DCGA chooses the chromosomes in the population that are more suitable for reproduction. The reproduction operator reproduces each chromosome in proportion to the value of its cost function (1). Therefore, it is more probable that the chromosomes with better objective functions will be selected for the next population, rather than other chromosomes. After selecting the pairs of parent chromosomes, the crossover operator is applied to each of these pairs. In this method, the crossover can take place at the boundary of two integer numbers (between two variables). An even number of chromosomes is selected at random based on a predefined rate, known as the crossover probability (P C). Random positions (two positions) are chosen for each pair of selected chromosomes, followed by the two chromosomes of each pair swapping their genes (variables). In the final step of forming the new generation, each chromosome resulting from the crossover operation will be subjected to the mutation operator. This operator selects certain existing integer numbers (variables) in the chromosome and then randomly changes their values according to a small probability, defined as the mutation probability (PM).
The creation of the new generation is complete after the mutation operator, and the cycle will start again with the evaluation of objective function (1) for each chromosome. The process continues and is terminated either by setting a target value to be reached for the fitness function or by setting a certain number of generations to be formed. Because of the stochastic nature of the genetic algorithm, in this study, a more suitable termination criterion has been established: the production of a predefined number of generations after obtaining the best fitness and finding no better solution. The flowchart for the proposed method is shown in Fig. 2.

IV. CASE STUDIES AND RESULTS
The proposed DCGA was implemented in MATLAB, and the tests were carried out on a computer with a 3.6 GHz Intel® Core™ i7-7700 processor and 16 GB of RAM. The IEEE reliability test system (IEEE RTS) was used to evaluate the proposed approach. The data for this test system can be found in [19]. It should be mentioned that i n is 2 and ni u is 30 years. Also, ni l0 and VOLL of the existing lines are listed in Tables II and III, respectively.
The parameters of the algorithm were: the size of the population equal to five, crossover rate PC = 0.9, mutation rate PM = 0.1, number of generations equal to 10,000.  7  2200  15  5550  2  1700  8  3000  16  1750  3  3200  9  3100  18  5850  4  1300  10  3400  19  3250  5  1250  13  4200  20  2250  6  2400  14  3400 --In this section, the proposed approach was tested on the case study system in two scenarios as follows. A. Scenario 1 In this case, the TNEP problem was resolved considering network losses and the reliability of the transmission system. The proposed approach was tested on the case study system mentioned above, in which the proposed plan appears in Fig. 3. Furthermore, the network's expansion and operating costs are listed in Table IV.

B. Scenario 2
In this scenario, consideration is given to the economic effect of the maintenance of lines on the TNEP problem. The results are presented in Fig. 4 and Tables V and VI.
A comparison of Fig. 3 with Fig. 4 shows that proposed configurations are different for both scenarios. In other words, for Scenario 2, two 138 kV and 230 kV lines are installed in corridors 1-10, and 13-19, respectively. Also, the construction of a 230 kV line is not necessary between buses 15 and 23. Hence, the expansion cost of the lines is expected to be higher than in Scenario 1, but it is verified from Tables IV and VI that the expansion cost of the transmission system in Scenario 2 is $31.1 million less than in the other scenario.