## Modeling and algorithmic considerations

Different mathematical optimization techniques are used for solving the network expansion planning. They can be classified as: classical and non-classical (metaheuristic) methods see references [2, 3].

Classical methods include linear, nonlinear and mixed integer programming methods. Linear optimization ignores the discrete nature of the investment decisions but still it can be useful is system is too large to be solved with discrete variables or a relaxed solution is good enough. A transportation or a direct-current (DC) load flow fit in this linear formulation. Nonlinear, in particular quadratic, models appear as a way to represent transmission losses. Finally, mixed integer optimization allows considering the integer nature of the decisions. If stochasticity in some parameters is included then models become stochastic and, therefore, decomposition techniques should be used for large-scale systems. Among them, Benders decomposition, Lagrangean relaxation or column generation are frequently used.

Non-classical metaheuristic methods include Greedy Randomized Adaptive Search Procedure (GRASP), tabu search, genetic algorithms, simulated annealing, swarm intelligence, ant colony optimization, differential evolution, ordinal optimization, etc.

Network expansion planning is per se a multicriteria decision problem although in many times a single objective function is used by combining and monetizing the multiple criteria under a single one with the weighted-sum method. The main criteria are usually: costs, environmental impact, market integration and other exogenous factors. Costs are measured by the attributes such as investment and operation costs of the transmission decisions but also operation costs of the system. In the cost criterion, it can also be introduced the reliability impact. Environmental impact is determined by attributes as amount of renewable integration or curtailment avoided at system level and impact of the line construction. Market integration is accounted as number of hours of market splitting. Social acceptance is an exogenous criterion and, nowadays, is a major concern of the current planning process and being the cause of many delays.

[1] G. Latorre, R. D. Cruz, J. M. Areiza and A. Villegas, "Classification of publications and models on transmission expansion planning," IEEE Transactions on Power Systems, vol. 18, pp. 938-946, 2003.

[2] R. Hemmati, R. Hooshmand, A. Khodabakhshian, "Comprehensive review of gen-eration and transmission expansion planning", IET Generation Transmission Distribution 7(2013) 955–964.