On the Solution of LargeScale Robust Transmission Network Expansion Planning under Uncertain Demand and Generation Capacity
Abstract
Twostage robust optimization has emerged as a relevant approach to deal with uncertain demand and generation capacity in the transmission network expansion planning problem. Unfortunately, available solution methodologies for the resulting trilevel robust counterpart are unsuitable for largescale problems. In order to overcome this shortcoming, this paper presents an alternative columnandconstraint generation algorithm wherein the maxmin problem associated with the second stage is solved by a novel coordinate descent method. As a major salient feature, the proposed approach does not rely on the transformation of the secondstage problem to a singlelevel equivalent. As a consequence, bilinear terms involving dual variables or Lagrange multipliers do not arise, thereby precluding the use of computationally expensive bigMbased linearization schemes. Thus, not only is the computational effort reduced, but also the typically overlooked nontrivial tuning of bounding parameters for dual variables or Lagrange multipliers is avoided. The practical applicability of the proposed methodology is confirmed by numerical testing on several benchmarks including a case based on the Polish 2383bus system, which is well beyond the capability of the robust methods available in the literature.
Nomenclature
This section lists the main notation used throughout the paper. Additional symbols with superscripts “” and “” are used to indicate the value of a specific variable at iterations and of the columnandconstraint generation algorithm, respectively. Similarly, superscript “” is used to denote results obtained at iteration of the coordinate descent method.
a Indices

[\setlabelwidthELLL\usemathlabelsep]

Generating unit index.

Load index.

Transmission line index.

Bus index.
B Sets

[\setlabelwidthELLL\usemathlabelsep]

Set of indexes of loads.

Set of indexes of loads connected to bus .

Set of indexes of generating units.

Set of indexes of generating units connected to bus .

Set of indexes of existing transmission lines.

Set of indexes of candidate transmission lines.

Set of indexes of buses.
C Constants

[\setlabelwidthELLL\usemathlabelsep]

Fraction of the demand of load that can be curtailed.

Conservativeness parameter for uncertain demands.

Conservativeness parameter for uncertain generation capacities.

Maximum deviation from the nominal or forecast demand of load .

Maximum deviation from the nominal or forecast generation capacity of unit .

Convergence tolerance for the inner loop.

Convergence tolerance for the outer loop.

Investment budget.

Weighting factor.

Construction cost of candidate line .

Production cost coefficient of unit .

Loadshedding cost coefficient of load .

Power flow capacity of line .

Sending or origin bus of line .

Nominal or forecast demand of load .

Nominal or forecast generation capacity of unit .

Receiving or destination bus of line .

Reactance of line .
D Decision Variables

[\setlabelwidthELLL\usemathlabelsep]

Approximation of the worstcase operating cost.

Phase angle at bus .

Phase angle at bus under the worst case identified at iteration .

Operating cost resulting from the inner loop.

Operating cost.

Operating cost computed along the iterations of the coordinate descent algorithm.

Worstcase operating cost.

Total cost resulting from the outer loop.

Uncertain demand of load .

Power output of unit .

Power output of unit under the worst case identified at iteration .

Uncertain generation capacity of unit .

Power flow through line .

Power flow through line under the worst case identified at iteration .

Unserved demand of load .

Unserved demand of load under the worst case identified at iteration .

Binary variable that is equal to 1 if candidate line is built, being 0 otherwise.

Binary variable that is equal to 1 if the worstcase demand of load is equal to its upper bound, being 0 otherwise.

Binary variable that is equal to 1 if the worstcase generation capacity of unit is equal to its lower bound, being 0 otherwise.
E Dual Variables

[\setlabelwidthELLL\usemathlabelsep]

Dual variable associated with the demand of load .

Dual variable associated with the generation capacity of unit .
I Introduction
Transmission network expansion planning is a key decisionmaking problem under both noncompetitive and marketbased settings. The traditional transmission network expansion planning problem consists in determining the optimal investments in transmission facilities so that power is supplied to consumers in a reliable and economic fashion [1]. The growing penetration levels of renewablebased generation have confronted network planners with major challenges [2], [3]. First, the uncertainty in the nodal net power injections, which was traditionally associated with demand growth, has drastically increased. In addition, the installation of renewablebased generation facilities becomes a relevant uncertain aspect itself. Moreover, as a major complicating factor, accurate probability distributions for such new sources of uncertainty are unavailable within a planning horizon.
Such challenges have triggered significant research effort to effectively address transmission network expansion planning under uncertain demand and generation capacity. Relevant approaches rely on the use of scenarios [4], [5], intervals [6], and chanceconstrained programming [7]. In order to overcome the limitations of the methods described in [4]–[7], recent contributions [8]–[12] suggest the use of twostage adaptive or adjustable robust optimization (ARO) [13]. Unlike scenariobased methods [4], [5] and chanceconstrained programming [7], ARO neither requires accurate probabilistic information nor relies on a discrete set of uncertainty realizations requiring a tradeoff between tractability and accuracy that may be hard to attain. Rather, uncertainty is modeled by decision variables within an uncertainty set, thereby comprising an infinite number of uncertainty realizations. Hence, the size of the robust counterpart does not depend on the dimension of the space of uncertainty realizations belonging to the uncertainty set, which is beneficial for implementation purposes. The uncertainty set can be built using intervals defined by the lower and upper bounds for the uncertain parameters. Such information, which is similar to that required by the intervalbased method presented in [6], may be easier to derive than probability distributions [11]. Moreover, and in contrast to [6], the robust solution guards against all realizations of uncertainty within the uncertainty set. Such a worstcase setting is a particularly desirable feature in planning problems [10], [11].
Within an ARObased setting [8]–[12], the robust counterpart is formulated as an instance of trilevel programming wherein the first stage is associated with the upper level and the second stage corresponds to the maxmin problem characterizing the two lowermost optimization levels. The upper level determines the leastcost firststage decisions, namely the optimal investment plan. For a given upperlevel decision vector, the middle level identifies the worstcase values of uncertain demand and generation capacity leading to maximum operating cost. Finally, the lower level models the operator’s best reaction, by means of adjustable variables, that minimizes the operating cost for given upper and middlelevel decisions.
In [8], the resulting trilevel program was addressed by a greedy randomized adaptive search procedure combined with a modified branchandbound algorithm for the secondstage problem. Alternative approaches applied Benders decomposition [9] and the columnandconstraint generation algorithm [10]–[12], both involving the iterative solution of a master problem and a maxmin subproblem associated with the second stage. However, available methods feature limitations.
Similar to the heuristic applied in [8], the approaches presented in [9]–[12] are unable to acknowledge global optimality. This shortcoming results from the transformation of the subproblem into a mixedinteger linear equivalent relying on setting bounds for dual variables or Lagrange multipliers, which may be in general unbounded [14]. Such unboundedness precludes the development of effective selection procedures for those bounds, being heuristic trialanderror algorithms [11] the current strategy. Thus, there is no guarantee that the subproblem is solved to optimality, which is the requirement for the exactness of the abovementioned decompositionbased methods.
Moreover, most of the literature in this area has focused on showing the benefits of using robust optimization to deal with uncertainty, while acknowledging that there are still computational hurdles to be overcome when solving large reallife instances. In theory, existing robust approaches [8]–[12] can be successfully implemented through the use of offtheshelf software. As reported in [9], [12], however, only relatively small instances appear to be tractable with such approaches, being the solution described in [12] the most efficient. Such intractability results from the transformation of the maxmin second stage to an equivalent singlelevel albeit bilinear problem comprising highly nonconvex products of binary variables and dual variables or Lagrange multipliers, for which the specific tools devised in [8]–[12] are impractical. Thus, the intractability issue remains a challenging obstacle to the practical implementation of the methods presented in [8]–[12].
Motivated by both facts, the thrust of this paper is the proposal of a novel and computationally efficient methodology suitable for largescale instances of the robust transmission network expansion planning problem under uncertain demand and generation capacity addressed in [9]–[12]. The proposed approach is a modified columnandconstraint generation algorithm solely involving primal decision variables wherein the maxmin subproblem is solved by a novel coordinate descent algorithm particularly tailored to its bilevel structure. Thus, as a distinctive feature over previous dualitybased methods [8]–[12], which require the iterative and time consuming solution of a largescale mixedinteger bilinear subproblem or its linearized equivalent, two simpler primal problems respectively associated with the middle and lowerlevel problems are iteratively solved. As a consequence, the computational burden is substantially decreased since neither dualitybased cuts, nor a singlelevel transformation based on dual variables or Lagrange multipliers, nor a linearization scheme, nor a casedependent, nontrivial, and computationally expensive bounding parameter selection procedure are required. Although, similar to previous works [8]–[12], global optimality is not guaranteed, the proposed approach is capable of attaining highquality nearoptimal investment decisions for largescale instances that are unsolvable by available methods [8]–[12].
The main contributions of this paper are twofold:

From a methodological perspective, this paper presents a new primal columnandconstraint generation algorithm for robust transmission network expansion planning under uncertainty that allows effectively dealing with reallife instances of this problem.

For the first time in the literature on robust transmission network expansion planning under uncertainty, this paper reports successful numerical experience with a largescale test system comprising thousands of buses and lines, which is well beyond the capability of existing approaches.
The remainder of the paper is structured as follows. The ARObased framework for robust transmission network expansion planning under uncertainty is described in Section II, wherein both the uncertainty characterization and the formulation of the robust counterpart are provided. Section III is devoted to the proposed solution approach. In Section IV, numerical results are presented and analyzed. Finally, relevant conclusions are drawn in Section V.
Ii Robust Transmission Network Expansion Planning
The proposed application of adjustable or twostage robust optimization [13] relies on the uncertainty set and the robust counterpart described next.
Iia Uncertainty Characterization
Under an ARObased framework, uncertainty sources are characterized by the extremes of the respective fluctuation range. Such uncertainty characterization can be implemented by modeling uncertain demands and uncertain generation capacities as variables that can vary around their nominal values. The information on the uncertainty sources thus reduces to nominal values and fluctuation bounds. In addition, the conservativeness of this uncertainty modeling can be controlled by two userdefined parameters, denoted by uncertainty budgets or conservativeness parameters. Here, based on [9] and [12], the uncertainty budgets represent the maximum numbers of demands and generating units that simultaneously experience fluctuations, respectively. The resulting cardinality uncertainty set can be mathematically cast as:
(1)  
(2)  
(3)  
(4) 
where (1) and (2) model the demand and generationrelated intervals, respectively, whereas (3) and (4) respectively impose the uncertainty budgets for demand and generation capacity fluctuations.
In adjustable robust optimization with an uncertainty set defined by the fluctuation range of the parameters allowed to vary, the optimal values of the decision variables characterizing the uncertainty set, i.e., the worstcase values, are one of the extremes of the corresponding range. A proof can be found in [8], where adjustable robust optimization was applied to the transmission network expansion planning problem under uncertain demand. Moreover, according to [11], the worst case corresponds to generation capacities being as low as possible and demands being as high as possible. Thus, under the abovedefined demand uncertainty budget, the worstcase demands, which represent all possible realizations of uncertain demands, can take two values, namely the upper bound , or the forecast value . As a consequence, for each uncertain demand, all possible uncertainty realizations can be modeled by a binary variable. Analogously, under the abovedefined generation uncertainty budget, the worstcase generation capacities, which represent all possible realizations of uncertain generation capacities, can take two values, namely the lower bound , or the forecast value . Hence, for each uncertain generation capacity, all possible uncertainty realizations can be modeled by an additional binary variable.
Thus, as done in [12], the uncertainty set adopted here is formulated as follows:
(5)  
(6)  
(7)  
(8)  
(9)  
(10) 
Constraints (5) and (6) respectively express the worstcase levels of demand and generation capacity in terms of the corresponding nominal levels and fluctuation levels. To that end, binary variables and are used, the integrality of which is modeled in (7) and (8), respectively. The conservativeness of the uncertainty characterization is modeled in (9) and (10), where demand and generationrelated uncertainty budgets are respectively imposed.
IiB Problem Formulation
Based on the static models presented in [10] and [11], the twostage adaptive robust optimization model for the transmission network expansion planning problem under uncertainty can be formulated as the following mixedinteger trilevel program:
(11)  
subject to:  
(12)  
(13)  
(14)  
(15)  
(16)  
(17)  
(18)  
(19)  
(20)  
(21) 
(22) 
Problem (11)–(22) is a trilevel program comprising three optimization levels: 1) the upper level (11)–(13), which is associated with the identification of the leastcost expansion plan; 2) the middle level (14)–(15), characterizing the worstcase realization of uncertainty sources for a given upperlevel investment plan; and 3) the lower level (16)–(22), which is related to the optimal system operation for given upperlevel investment decisions and middlelevel uncertainty realizations.
The objective of the upperlevel problem is the minimization of the total cost (11), which comprises two terms, namely the annualized investment cost and the worstcase operating cost. The weighting factor is used to make investment and worstcase operating costs comparable quantities. The upperlevel minimization is subject to an upper bound on the investment cost (12). In addition, the binary nature of investment variables is modeled in (13).
The middlelevel problem (14)–(15) identifies the worstcase uncertainty realizations yielding the largest operating cost (14) for the solution identified by the upper level. The uncertainty characterization described in Section IIA is modeled by (15).
In the lowerlevel problem (16)–(22), the operating cost associated with upper and middlelevel variables is minimized in (16). Expressions (17)–(20) model the effect of the network including nodal power balances (17), line flows through existing lines (18), line flows through candidate lines (19), and line flow limits (20). Constraints (21) set the generation limits. Finally, constraints (22) impose bounds on load shedding.
Iii Solution Approach
The proposed solution approach is a primal columnandconstraint generation algorithm, hereinafter referred to as PCC. Unlike previous applications of the columnandconstraint generation algorithm [10]–[12], which require dealing with highly nonconvex bilinear terms including dual lowerlevel variables, PCC solely relies on linear expressions involving primal decision variables, which is beneficial for computational purposes. The master problem and the subproblem that are solved along the iterations are described next followed by an outline of the proposed iterative process.
Iiia Master Problem
The master problem constitutes a relaxation for problem (11)–(22) where a set of valid operating constraints are iteratively added. The addition of such constraints, which are set up with information from the subproblem, allows obtaining a more robust expansion plan at each iteration. At iteration , the master problem is formulated as the following mixedinteger linear program:
(26)  
(27)  
(28)  
(29)  
(30)  
(31)  
(32) 
where the additional decision variables, , , , and , corresponding to , , , and , respectively, are associated with the demands and generation capacities identified by the subproblem at iteration through and .
The objective function (23) is identical to (11) except for the last term, where is replaced with , which represents the pointwise maximum within all linear approximations of . Expression (24) includes the upperlevel constraints. As modeled in (25), the operating cost corresponding to the uncertainty realizations identified at iteration represents a lower bound for . Constraints (26)–(31) respectively correspond to lowerlevel constraints (17)–(22). Finally, the nonnegativity of is imposed in (32).
IiiB Subproblem
At each iteration , the subproblem determines the worstcase uncertainty realizations yielding the maximum operating cost for a given upperlevel decision provided by the previous master problem. Mathematically, the subproblem is a mixedinteger linear maxmin problem comprising the two lowermost optimization levels (14)–(22) parameterized in terms of the given upperlevel decision variables .
Here we propose solving such bilevel problem through a block coordinate descent method [15] involving the iterative solution of two simple optimization problems. At iteration , both problems are formulated as follows:
(33)  
(34)  
(35) 
(36)  
(37)  
(38)  
(39)  
(40)  
(41) 
and
where dual variables and represent the sensitivities of the operating cost with respect to fixed values of demands and generation capacities, respectively.
Problem (33)–(41) corresponds to the lowerlevel problem (16)–(22) for fixed values of middlelevel variables and . Such values of middlelevel variables result from the optimal solution to problem (42)–(43), which corresponds to the middlelevel problem (14)–(15). Note that the operating cost in (14) is replaced in (42) with its firstorder Taylor series approximation around the uncertainty realizations identified at the previous iteration of the coordinate descent algorithm. Those terms are based on the sensitivities and and the optimal operating cost previously obtained from (33)–(41).
Once initial values of middlelevel variables and are selected, the coordinate descent method iteratively solves problems (33)–(41) and (42)–(43). The iterative process terminates when the operating cost remains unchanged within a userdefined tolerance. Admittedly, under the nonconvexity of the maxmin subproblem, the algorithm may converge to a local optimum and hence the proposed columnandconstraint generation algorithm does not guarantee global convergence to optimality. Nonetheless, our expectation was that the proposed approach would still be a useful heuristic for finding good solutions. This was borne out by our computational experience, as described in Section IV.
IiiC Algorithm
The proposed PCC comprises two nested loops, namely 1) an outer loop associated with the mastersubproblem iterations of the modified columnandconstraint generation algorithm, and 2) an inner loop related to the iterative process of the coordinate descent algorithm. The proposed methodology works as follows:

Initialization of the outer loop.

Set the iteration counter of the outer loop to 1.

Set the initial expansion plan , and the total cost associated with the outer loop to 0.


Initialization of the inner loop. Set the iteration counter of the inner loop to 1, select initial values for and , and set the operating cost associated with the inner loop to .

Update the iteration counter of the inner loop. Increase the iteration counter .

Update the iteration counter of the outer loop. Increase the iteration counter .

Outer loop stopping criterion. If a solution with a level of accuracy has been found, i.e., , the algorithm stops; otherwise, go to step 2.
Iv Numerical Results
The performance of the proposed approach is illustrated with three cases respectively based on the IEEE 24bus Reliability Test System (RTS) [16], the IEEE 118bus system [17], and the Polish 2383bus system [18]. For the sake of reproducibility, input data for all case studies can be downloaded from [19].
Numerical testing has been conducted for different values of the uncertainty budgets and including the case of maximum uncertainty wherein all uncertain demands and generation capacities are allowed to simultaneously fluctuate. Results from the proposed approach PCC have been compared with those provided by the dualitybased columnandconstraint generation algorithm presented in [12], which, for quick reference, is hereinafter denoted by DCC. Note that, to the best of the authors’ knowledge, DCC is the most computationally efficient method available in the literature on robust transmission network expansion planning under uncertainty. Simulations have been implemented on a Dell PowerEdge R920X64 with four Intel Xeon E74820 processors at 2.00 GHz and 768 GB of RAM using CPLEX 12.6 under GAMS 24.2 [20]. For all simulations, the optimality tolerance for the branchandcut algorithm of CPLEX was set at 10. In addition, was set at 10 whereas , which was also used as the convergence tolerance for DCC, was set at 10.
Iva RTSBased Case
The first case study is based on the modified version of the IEEE RTS analyzed in [12]. This test system comprises 24 buses, 10 generating units, 17 loads, 34 existing lines, and 85 candidate lines [19]. For illustration purposes, all demands and generation capacities are considered sources of uncertainty, which are respectively characterized by a 20 and a 50 fluctuation with respect to the corresponding nominal level.
Table I summarizes the best results provided by the proposed PCC and DCC for a $20million investment budget, for which the deterministic solution costs $219161.7 million. For all instances, both approaches converged after 3 iterations of the outer loop in less than 2.2 s, thereby behaving similarly from a computational perspective. As for solution quality, the proposed PCC attained the best total cost identified by DCC for all instances but one corresponding to and . For such an instance, PCC provided a lower bound for the optimal total cost that slightly differed by 0.0035% from that achieved by DCC. It is worth noting, however, that the best expansion decisions identified by both methods were identical for all combinations of uncertainty budgets. These results corroborate the effectiveness of the proposed approach to attain highquality nearoptimal solutions.
PCC  DCC  
Total  Computing  Total  Computing  
cost  time  cost  time  
(10 $)  (s)  (10 $)  (s)  
05  03  348929.5  0.9  348941.6  1.2 
09  05  412396.2  0.9  412396.2  1.2 
12  07  454917.7  1.6  454917.7  2.1 
17  10  500752.3  1.3  500752.3  1.9 
IvB IEEE 118Bus Test System
The second case study is based on the modified version of the IEEE 118bus system examined in [12]. This test system comprises 118 buses, 54 generators, 91 loads, and 186 existing transmission lines, whereas a set of 61 candidate lines is available for expansion decisions [19] under a $100million investment budget. In addition, a 50 fluctuation level was considered for all demands and generation capacities.
PCC  DCC  
Total  Computing  Total  Computing  
cost  time  cost  time  
(10 $)  (s)  (10 $)  (s)  
10  05  19046.3  6.2  19279.2  27.4 
20  15  23286.7  5.1  23353.4  09.8 
60  35  30031.2  6.0  30033.9  25.1 
91  54  31995.0  3.9  31995.0  12.0 
The superior performance of the proposed PCC over DCC is illustrated in Table II. It is worth mentioning that the total costs of the best solutions attained by PCC slightly differed from those found by DCC by factors ranging between 0.0% for the most conservative instance to 1.2% for the least conservative solution, whereas the computational effort was significantly reduced by factors ranging between 77.4% for and and 48.0% for and .
The quality of the best investment plans identified by both approaches has been further verified by an outofsample assessment based on the simulation of the operation of the expanded system for a random collection of different uncertainty realizations. To that end, the lowerlevel problem (16)–(22) has been solved for the corresponding best expansion plan and each sampled uncertainty realization. Due to space limitations, a representative instance, namely and , has been selected to illustrate such an assessment. For this particular instance, out of the possible worstcase uncertainty realizations according to constraints (5)–(10), 100000 random samples were analyzed. Fig. 1 shows the histograms and normal fitted densities of the operating cost for both methods. As can be seen, both distributions are very similar. Although the curves yielded by PCC are displaced to the right, i.e., higher operating costs are incurred, the expected operating cost slightly increases by 0.39%, which is acceptable bearing in mind that the computational performance is significantly improved by 76.1%. Moreover, it is worth pointing out that no sampled operating cost exceeded the worstcase value for the best solution identified by PCC, thereby substantiating the robustness of the proposed approach.
IvC Polish 2383Bus Test System
The third case study is based on the Polish system described in [18]. This test system comprises 2383 buses, 327 generating units, 1500 loads, 2896 existing lines, and 124 candidate lines. Demand uncertainty is considered for the 100 largest loads, which are allowed to fluctuate by 20 around their nominal levels. Analogously, uncertainty in generation is modeled for the 212 generators with the lowest nominal generation capacities, for which their generation capacities are allowed to fluctuate between their nominal levels and zero. The interested reader is referred to [19] for a full description of the benchmark.
Table III lists the results obtained for a $100million investment budget. As can be seen, for all uncertainty budgets examined, the proposed PCC converged with acceptable computational effort for a planning setting. In contrast, for all those instances tested, DCC was unable to find a single feasible solution after one week. It is worth stressing that the performance bottleneck of DCC is the solution to the subproblem. As a matter of fact, for all simulations of this largescale case study, DCC exceeded the oneweek time limit while attempting to solve the subproblem corresponding to the first iteration of the columnandconstraint generation algorithm. This result clearly evidences the computational gain associated with the use of the proposed coordinate descent method.
Total cost  Computing time  

(10 $)  (s)  
040  020  18823.4  42855.8 
060  040  19514.2  11157.6 
080  050  19591.7  03364.4 
100  212  19689.9  03587.7 
As done with the 118bus system, the quality of the best investment plans provided by PCC has been assessed via randomly sampling 100000 uncertainty realizations and computing the corresponding operating costs through the solution of the lowerlevel problem (16)–(22). For expository purposes, results from the outofsample assessment for and are reported. Fig. 2 shows the histogram of the operating cost for this representative instance, for which possible worstcase uncertainty realizations exist. For such uncertainty budgets, the maximum and the average sampled operating costs are equal to $1.209 and $1.196 million, respectively, which are far below the worst operating cost identified by the proposed method, namely $2.236 million. These results confirm the robustness of the solution provided by the proposed PCC.
V Conclusions
Existing methods relying on robust optimization are unable to solve largescale instances of transmission network expansion planning under uncertain demand and generation capacity. This paper has presented a novel and computationally efficient primal columnandconstraint generation algorithm that overcomes such intractability issue. The strength of the proposed approach lies in its ability to address the subproblem associated with the maxmin secondstage problem without resorting to the customary dualitybased transformation to a singlelevel bilinear equivalent or its linearized version. To that end, a fast block coordinate descent algorithm is implemented on the space of primal decision variables, whereby two simple problems respectively associated with the middle and lowerlevel problems are iteratively solved. As major advantages over previously reported methods, the proposed approach does not require dualitybased cuts, a linearization scheme, or a casedependent, nontrivial, and computationally expensive bounding parameter selection procedure for dual variables or Lagrange multipliers, which may be unbounded.
The effective performance of the proposed approach has been demonstrated with several case studies of different dimensions. For relatively small and mediumsized benchmarks, the proposed approach is able to identify expansion decisions that are either identical or very close to the best known solutions with substantially lower computational effort than that required by the stateoftheart technique. Moreover, a practical test system with thousands of components is successfully handled by the proposed approach with moderate computing time, whereas the most efficient approach reported in the literature fails to provide a feasible solution within the allotted time limit.
References
 [1] X. Wang and J. R. McDonald, Modern Power System Planning. London, UK: McGrawHill, 1994.
 [2] R. Hemmati, R.A. Hooshmand, and A. Khodabakhshian, “Comprehensive review of generation and transmission expansion planning,” IET Gener. Transm. Distrib., vol. 7, no. 9, pp. 955–964, Sep. 2013.
 [3] S. Lumbreras and A. Ramos, “The new challenges to transmission expansion planning. Survey of recent practice and literature review,” Electr. Pow. Syst. Res., vol. 134, pp. 19–29, May 2016.
 [4] F. S. Reis, P. M. S. Carvalho, and L. A. F. M. Ferreira, “Reinforcement scheduling convergence in power systems transmission planning,” IEEE Trans. Power Syst., vol. 20, no. 2, pp. 1151–1157, May 2005.
 [5] H. Yu, C. Y. Chung, and K. P. Wong, “Robust transmission network expansion planning method with Taguchi’s orthogonal array testing,” IEEE Trans. Power Syst., vol. 26, no. 3, pp. 1573–1580, Aug. 2011.
 [6] A. H. Escobar, R. A. Romero, and R. A. Gallego, “Transmission network expansion planning considering uncertainty in generation and demand,” presented at the IEEE PES Latin America Transm. Distrib. Conf. Expo., Bogotá, Colombia, Aug. 2008.
 [7] H. Yu, C. Y. Chung, K. P. Wong, and J. H. Zhang, “A chance constrained transmission network expansion planning method with consideration of load and wind farm uncertainties,” IEEE Trans. Power Syst., vol. 24, no. 3, pp. 1568–1576, Aug. 2009.
 [8] P. Wu, H. Cheng, and J. Xing, “The interval minimum load cutting problem in the process of transmission network expansion planning considering uncertainty in demand,” IEEE Trans. Power Syst., vol. 23, no. 3, pp. 1497–1506, Aug. 2008.
 [9] R. A. Jabr, “Robust transmission network expansion planning with uncertain renewable generation and loads,” IEEE Trans. Power Syst., vol. 28, no. 4, pp. 4558–4567, Nov. 2013.
 [10] B. Chen, J. Wang, L. Wang, Y. He, and Z. Wang, “Robust optimization for transmission expansion planning: Minimax cost vs. minimax regret,” IEEE Trans. Power Syst., vol. 29, no. 6, pp. 3069–3077, Nov. 2014.
 [11] C. Ruiz and A. J. Conejo, “Robust transmission expansion planning,” Eur. J. Oper. Res., vol. 242, no. 2, pp. 390–401, Apr. 2015.
 [12] R. Mínguez and R. GarcíaBertrand, “Robust transmission network expansion planning in energy systems: Improving computational performance,” Eur. J. Oper. Res., vol. 248, no. 1, pp. 21–32, Jan. 2016.
 [13] A. BenTal, A. Goryashko, E. Guslitzer, and A. Nemirovski, “Adjustable robust solutions of uncertain linear programs,” Math. Program., vol. 99, no. 2, pp. 351–376, Mar. 2004.
 [14] D. G. Luenberger, Linear and Nonlinear Programming, 2nd ed. Reading, MA, USA: AddisonWesley, 1989.
 [15] A. J. Conejo, E. Castillo, R. Mínguez, and R. GarcíaBertrand, Decomposition Techniques in Mathematical Programming. Engineering and Science Applications. New York, USA: Springer, 2006.
 [16] Reliability Test System Task Force, “The IEEE Reliability Test System1996,” IEEE Trans. Power Syst., vol. 14, no. 3, pp. 1010–1020, Aug. 1999.
 [17] IEEE 118Bus System. [Online]. Available: http://motor.ece.iit.edu/data/
 [18] Polish 2383Bus System. [Online]. Available: http://www.pserc.cornell.edu/matpower/docs/ref/matpower5.0/case2383wp.html
 [19] R. Mínguez, R. GarcíaBertrand, J. M. Arroyo, and N. Alguacil, “On the solution of largescale robust transmission network expansion planning under uncertain demand and generation capacity: Data for case studies,” [Online]. Available: https://drive.google.com/folderview?id=0B3ZNZXBazCeazBfN3VvbGpWNms&usp=sharing
 [20] The GAMS Development Corporation website, 2016. [Online]. Available: http://www.gams.com
Roberto Mínguez received the Civil Engineer degree and the Ph.D. degree from the Universidad de Cantabria, Santander, Spain, in 2000 and 2003, respectively. He is currently a research fellow at the company Hidralab Ingeniería y Desarrollo, S.L., spinoff from the Universidad de CastillaLa Mancha. His research interests include reliability engineering, sensitivity analysis, numerical methods, and optimization. 
Raquel GarcíaBertrand (S’02–M’06–SM’12) received the Ingeniera Industrial degree and the Ph.D. degree from the Universidad de CastillaLa Mancha, Ciudad Real, Spain, in 2001 and 2005, respectively. She is currently an Associate Professor of electrical engineering at the Universidad de CastillaLa Mancha. Her research interests include operations, planning, and economics of electric energy systems, as well as optimization and decomposition techniques. 
José M. Arroyo (S’96–M’01–SM’06) received the Ingeniero Industrial degree from the Universidad de Málaga, Málaga, Spain, in 1995, and the Ph.D. degree in power systems operations planning from the Universidad de CastillaLa Mancha, Ciudad Real, Spain, in 2000. From June 2003 through July 2004 he held a Richard H. Tomlinson Postdoctoral Fellowship at the Department of Electrical and Computer Engineering of McGill University, Montreal, QC, Canada. He is currently a Full Professor of electrical engineering at the Universidad de CastillaLa Mancha. His research interests include operations, planning, and economics of power systems, as well as optimization. 
Natalia Alguacil (S’97–M’01–SM’07) received the Ingeniero en Informática degree from the Universidad de Málaga, Málaga, Spain, in 1995, and the Ph.D. degree in power systems operations and planning from the Universidad de CastillaLa Mancha, Ciudad Real, Spain, in 2001. She is currently an Associate Professor of electrical engineering at the Universidad de CastillaLa Mancha. Her research interests include operations, planning, and economics of power systems, as well as optimization. 