On the Solution of the Bilevel Programming Formulation ... - IEEE Xplore

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 2, MAY 2005

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On the Solution of the Bilevel Programming Formulation of the Terrorist Threat Problem José M. Arroyo, Member, IEEE, and Francisco D. Galiana, Fellow, IEEE

Abstract—This paper generalizes the “terrorist threat problem” first defined by Salmerón, Wood, and Baldick by formulating it as a bilevel programming problem. Specifically, the bilevel model allows one to define different objective functions for the terrorist and the system operator as well as permitting the imposition of constraints on the outer optimization that are functions of both the inner and outer variables. This degree of flexibility is not possible through existing max-min models. The bilevel formulation is investigated through a problem in which the goal of the destructive agent is to minimize the number of power system components that must be destroyed in order to cause a loss of load greater than or equal to a specified level. This goal is tempered by the logical assumption that, following a deliberate outage, the system operator will implement all feasible corrective actions to minimize the level of system load shed. The resulting nonlinear mixed-integer bilevel programming formulation is transformed into an equivalent single-level mixed-integer linear program by replacing the inner optimization by its Karush–Kuhn–Tucker optimality conditions and converting a number of nonlinearities to linear equivalents using some well-known integer algebra results. The equivalent formulation has been tested on two case studies, including the 24-bus IEEE Reliability Test System, through the use of commercially available software. Index Terms—Bilevel programming, deliberate outages, load shedding, mixed-integer linear programming (MILP), power system security and vulnerability, terrorist threat.

NOMENCLATURE A. Indices

Set of indices of buses. C. Functions Lagrangian function of the inner optimization problem, which depends on vector . D. Constants Element of the network incidence matrix that is equal to 1 if bus is the sending bus of line , if bus is the receiving bus of line , and 0 otherwise. Sending bus of line . Sufficiently large positive constant. Demand at bus (in megawatts). Power flow capacity of line (in megawatts). Capacity of generator (in megawatts). Minimum power output of generator (in megawatts). Receiving bus of line . Reactance of line ( ). Upper bound for the nodal phase angles (rad). Lower bound for the nodal phase angles (rad). Upper bound for . Lower bound for . Lower level of total load shed specified by the terrorist (in megawatts). E. Variables

Generator index. Transmission line index. Bus index. B. Sets Set of indices of generators. Set of indices of generators connected to bus . Set of indices of transmission lines. Manuscript received June 28, 2004; revised December 3, 2004. This work was supported by the Natural Sciences and Engineering Research Council of Canada and by the Fonds québécois de la recherche sur la nature et les technologies, Québec. The work of J. M. Arroyo was supported by McGill University under the R. H. Tomlinson Postdoctoral Fellowship, the Ministry of Science and Technology of Spain under Grant CICYT DPI2003-01362, and the Junta de Comunidades de Castilla—La Mancha, Spain, under Grant GC-02-006. Paper no. TPWRS-00337-2004. J. M. Arroyo is with the Departamento de Ingeniería Eléctrica, Electrónica y Automática, E.T.S.I. Industriales, Universidad de Castilla—La Mancha, Ciudad Real E-13071, Spain (e-mail: [email protected]). F. D. Galiana is with the Department of Electrical and Computer Engineering, McGill University, Montreal, QC H3A 2A7, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TPWRS.2005.846198

Slack variable used in the linear expression equivalent to the product of and . Vector of line power flows (in megawatts). Vector of generator power outputs (in megawatts). Slack variable used in the linear expression equivalent to the product of and (rad). Slack variable used in the linear expression equiva(rad). lent to the product of and Variable equal to the product of and . 0/1 variable that is equal to 0 if line is destroyed and otherwise is equal to 1. 0/1 variables used in the linear expressions of the complementary slackness conditions. Variable equal to the product of and (rad). Variable equal to the product of and (rad). Lagrange multiplier associated with the upper bound for the load shed at bus . Lagrange multiplier associated with the lower bound for the load shed at bus . vector of nodal phase angles (rad), Vector of nodal loads shed (in megawatts).

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Lagrange multiplier associated with the power balance equation at bus . Lagrange multiplier associated with the equation relating power flow and phase angles for line . Lagrange multiplier associated with the upper bound for the power output of generator . Lagrange multiplier associated with the lower bound for the power output of generator . Lagrange multiplier associated with the upper bound for the power flow of line . Lagrange multiplier associated with the lower bound for the power flow of line . I. INTRODUCTION

A

S A result of recent events, governments have shown growing interest in the protection of strategic infrastructures against sabotage and terrorist attacks [1], [2]. High-voltage electric power grids, in particular, are vulnerable because they cover large geographical areas and are difficult to protect [3], [4]. The vulnerability of power systems to deliberate outages could be assessed through the well-established methodology of power system security analysis [5]. However, in the new context where destructive agents come into play, the assessment of power system security must examine not only the typical credible contingencies associated with random failures, charcriterion, but acterizing what is commonly known as the also an additional set of plausible deliberate outages caused by terrorists. The general goal of the operation planner analyzing the impact of this extended set of contingencies is to identify those to which the system is most vulnerable in order to implement appropriate surveillance and protection measures. In contrast, the goal of the terrorists may be to maximize the level of total load interruption subject to some limit on their destructive resources, under the logical assumption that, following an outage, the system operator will implement all feasible corrective actions to minimize the level of system load shed. The identification of the most critical deliberate outage could be carried out using traditional approaches in which a prespecified set of contingencies is simulated, one at a time [6]–[10]. However, in selecting the set of credible contingencies in the new context of possible terrorist attacks, the required experience and engineering judgment may not be available. Moreover, since the extended set of contingencies is no longer limited to a narrow set of random events, the number of possible outage combinations can grow beyond the computational capability of today’s computers. As such, researchers have begun to look at new ways of addressing this security assessment problem, here called the terrorist threat problem. For example, in [11], a multi-agent system was proposed capable of assessing power system vulnerability, monitoring hidden failures of protection devices, and providing adaptive control actions to prevent catastrophic failures and cascading sequences of events. Salmerón et al. [12], [13] were the first to formulate the terrorist threat problem as a max-min problem in which the terrorist maximizes the load shed while the system operator mini-

mizes it. This important result permits the development of mathematical programming solutions to this complex problem. In [12] and [13], a heuristic method resembling Benders decomposition was proposed, the solution of which satisfied feasibility but, as stated by the authors, was suboptimal, meaning that the level of load interruption found through the most destructive strategy was optimistically low. In [14] and [15], the max-min formulation of the terrorist threat problem was solved by first transforming the existing nonlinear expressions into linear constraints. Subsequently, by replacing the inner minimization problem by its dual, the max-min problem became a max-max problem, i.e., a single-level maximization, which was solved through Benders decomposition and mixed-integer linear programming (MILP). The main contribution of this paper is to formulate a general terrorist threat problem as a bilevel problem [16], of which the max-min formulation in [12]–[15] is a special case. In addition, this paper presents a solution to the bilevel problem based on MILP that can be handled by efficient commercially available software. Unlike in [12]–[15] where the actions of both the terrorist and the system operator were guided by the same quantity, namely, the amount of load not served (one aiming to maximize it and the other to minimize it), in the general bilevel formulation presented here, the terrorist and the system operator may use different objective functions to meet their goals. For example, as analyzed in this paper, the terrorist may be interested in minimizing the number of system components that must be destroyed to achieve a minimum desired level of load shed, while the operator minimizes the level of load not served. Another example where the focus of the terrorist is not necessarily the same as that of the operator is to disrupt as much load as possible in a specific area of the network while the system operator minimizes the amount of load disrupted in a different area. These more general types of the terrorist threat problem cannot be analyzed with the max-min formulation of [12]–[15]. Another difference between the max-min and the general bilevel formulation is that the latter allows one to impose outer-level constraints that depend on both inner- and outer-level variables, as required by the problem analyzed in this paper. Such constraints cannot be imposed in the formulation of [14] and [15], which was solved in the dual variable space. An additional contribution of this paper is to convert the bilevel formulation, which is nonlinear, into an equivalent single-level mixed-integer linear program (S-MILP). This equivalence is obtained through two main steps: Step 1) the explicit characterization of the inner optimization problem by its Karush–Kuhn–Tucker (KKT) optimality conditions; Step 2) the use of integer algebra results due to Floudas [17] and to Fortuny-Amat and McCarl [18] to convert the nonlinear KKT relations into equivalent linear forms. The principal advantages of expressing the original bilevel optimization as an equivalent S-MILP is the guaranteed convergence to the optimal solution in a finite number of steps [19] and the ready availability of commercial software that can efficiently solve large-scale MILP problems [20].

ARROYO AND GALIANA: SOLUTION OF BILEVEL PROGRAMMING FORMULATION OF TERRORIST THREAT PROBLEM

The remainder of this paper is organized as follows. In Section II, the general bilevel formulation of the terrorist threat problem is presented. In Section III, the transformation to a S-MILP problem is described. Computational considerations are analyzed in Section IV. In Section V, results from two case studies are provided and analyzed. Some relevant conclusions are drawn in Section VI. Finally, the mathematics required to convert the original bilevel nonlinear problem to an equivalent S-MILP are explained in Appendixes A and B. II. GENERAL BILEVEL FORMULATION OF THE TERRORIST THREAT PROBLEM A general form of the terrorist threat problem can be posed as the following bilevel program [16]:

by repeated simulations. These modeling limitations notwithstanding, the solution of the terrorist threat problem based on the dc load flow provides the operation planner with a first estimate of the vulnerability of the system. In a power system, of the components vulnerable to attack (generators, substations, buses, lines, and transformers), transmission lines are the most unprotected and represent a relatively easy target for a destructive agent [21]. Although the method proposed here can include all types of component outages, for reasons of clarity and conciseness, this paper considers only intentional transmission line outages. For the interested reader, a model considering deliberate outages of other components such as power plants and substations can be found in [13]. The bilevel formulation for this type of terrorist threat problem is then

(1) subject to

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(5) subject to

(2)

(6)

(3) subject to

(7)

(4)

The above bilevel problem consists of an outer optimization (1) and (2) associated with the terrorist and an inner optimization (3) and (4) associated with the system operator. Note that superscripts and denote “inner” and “outer,” respectively. The terrorist controls the vector of variables representing the 0/1 variables, denoting whether a particular system component is to be destroyed or not. The decision variables , controlled by the system operator, are typically network variables, generation levels, and loads shed. is The objective function of the outer optimization maximized over in (1), subject to a set of constrained functions , depending on both the inner- and outer-level variables (2) and subject to the inner optimization in which a possibly is minimized over in (3), different objective function , with subject to another set of constrained functions as a parameter (4). The max-min problem formulation proposed in [12]–[15] is a particular case of (1)–(4) with the same outer and inner objective and with being a function only functions of the outer-level variables . To illustrate the above bilevel program framework, we now consider a specific terrorist threat problem in which the terrorist minimizes the number of power system components that must be destroyed in order to cause a level of loss of load greater than or equal to a specified amount. The objective of the system operator is to minimize the system load shed. As in [12]–[15], we make use of a dc load flow model to characterize the behavior of the network, recognizing that the use of such a simplified model leads to results that may be optimistic and that a complete study of the terrorist threat problem should also consider the effects of stability, primary regulation, and reactive power. This generalization would, however, render the problem essentially intractable through optimization and would have to be solved

subject to

(8) (9) (10) (11) (12) where the Lagrange multipliers associated with constraints (8)–(12) in the inner optimization problem are in parentheses. As can be noted, problem (5)–(12) constitutes an instance of the general bilevel formulation (1)–(4) with different outer- and and with inner-level objective functions being a function of the inner-level variables. The outer-level optimization consists of (5) and (6), whereas (7)–(12) represent the inner-level problem. The terrorist controls the vector of 0/1 is equal to 0 if line is destroyed and variables , where otherwise is equal to 1. The decision variables controlled by the , , and . system operator are , The terrorist’s objective (5) is to maximize the number of nonattacked lines (or equivalently to minimize the number of attacked lines) so that the damage, defined as the total load shed, is greater than or equal to a prespecified level (6). Note that constraints such as (6) cannot be modeled in a max-min formulation. In contrast, the system operator has a different objective (7), which is to minimize the total load shed under the combination of destroyed lines , chosen by the terrorist. To that end, the system operator solves a dc optimal power flow (OPF) modeled by (7)–(12). Constraints (8) express the line flows in terms

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of the nodal phase angles and the outer-level variables . Note that if line is destroyed, the power flow is set to 0 by (8). Constraints (9) represent the nodal power balance equations, while constraints (10) enforce the corresponding line flow capacity limits. Finally, constraints (11) and (12) set the limits of generation and load shedding, respectively. The limits of generation take into account fast spinning reserves and ramp rate limits, which are assumed to have been previously calculated by the system operator. Besides its intrinsic complexity due to the two levels of optimization, problem (5)–(12) is mixed integer (containing both continuous and binary variables) and nonlinear due to the product of the variables and in (8). However, as described next, by exploiting the fact that the inner optimization problem is linear for a given vector , problem (5)–(12) can be transformed into an equivalent S-MILP problem. III. EQUIVALENT S-MILP PROBLEM To convert the original bilevel formulation (5)–(12) into an equivalent S-MILP problem, the following steps are required. Step 1) The bilevel program is transformed into an equivalent single-level nonlinear optimization problem through the KKT optimality conditions of the inner-level problem. Step 2) The nonlinear products between binary variables and continuous variables are replaced by equivalent linear forms. Step 3) The nonlinear complementary slackness conditions are equivalently replaced by linear expressions. To execute step 1), replace the inner optimization problem (7)–(12), i.e., the dc OPF solved by the system operator, by its KKT necessary optimality conditions. The KKT conditions apply here since the inner optimization problem is convex in the , , and and since the binary continuous variables , variables , determined by the terrorist, can be considered as parameters by the system operator. A detailed derivation of the KKT conditions of the inner problem (7)–(12) can be found in Appendix A. The two types of nonlinearities contained in the single-level problem are the products of binary and continuous variables [see (8) and Appendix A] and the products of continuous variables, specifically Lagrange multipliers and constrained continuous functions appearing in the complementary slackness conditions (see Appendix A). These nonlinearities are transformed into equivalent linear expressions through steps 2) and 3) described in Appendix B. To sum up, the original bilevel problem (5)–(12) has been transformed into the equivalent S-MILP: (13)

(16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52)

subject

to

(53)

(14)

where (13) and (14) correspond to the outer optimization constraints and (15)–(53) equivalently replace the inner optimization problem (7)–(12). Constraints (15)–(25) are the primal feasibility constraints, (26)–(38) represent the dual

(15)


TABLE I DIMENSION OF THE S-MILP MODEL

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TABLE II CRITICAL COMBINATIONS OF DESTROYED LINES—FIVE-BUS SYSTEM

TABLE III DESTRUCTIVE PLANS—FIVE-BUS SYSTEM

Fig. 1.

Five-bus system.

feasibility constraints, and (39)–(53) express the complementary slackness conditions. Note that all the constraints are linear. IV. COMPUTATIONAL CONSIDERATIONS The equivalent problem formulated in the previous section is mixed integer and linear, allowing the use of well-established algorithms [20] capable of handling large-scale problems. In order to provide insight into the dimension of the resulting optimization problem, Table I shows the corresponding number , , and denote of constraints and variables where the number of buses, lines, and generators, respectively. For a power system such as the IEEE Reliability Test System (RTS) [22] with 24 buses, 38 lines, and 32 generators, the S-MILP model includes 1220 functional constraints, 377 bounds on variables, 226 binary variables, and 597 continuous variables. Experience shows that this size of problem is well within the capability of available commercial MILP software [20]. V. NUMERICAL RESULTS Results from two test cases are presented in this section. The first consists of five buses, while the second is based on the IEEE RTS [22]. The proposed model has been implemented on a Dell PowerEdge 6600 computer with 2 processors at 1.60 GHz and 2 Gb of RAM memory using CPLEX 9.0 under GAMS [20]. A. Five-Bus Example The topology of this test system and the bus demands are shown in Fig. 1, where line reactances, expressed on a base of 100 MVA and 138 kV, are also presented. Line capacities are all 100 MW. All generators have lower and upper generation bounds of 0 and 150 MW, respectively. This limited-scale case can be solved by enumeration for all ), the possible combinations of destroyed lines (

results of which are shown in Table II, including the number of lines destroyed, the corresponding actual system load shed, and the corresponding combination of lines that must be brought down. Note that with all lines down, the maximum attainable load shed is 170 MW, representing 26.56% of the system load. This relatively low vulnerability of the system is due to the fact that in this example, the bus demands can be largely supplied by local generation. However, as Table II shows, the terrorist need not bring down all the lines to achieve the maximum loss of load, since the destruction of lines 1–2, 2–3, 3–5, and 4–5 achieves the same goal. Applying now the proposed S-MILP model, Table III shows the destructive plans for the different ranges of specified loss of , as defined by Table II. Note that if the lower level of load load loss is specified over 150 MW, at least four lines have to be brought down. For this five-bus example, the CPU time required using to achieve the optimal solution for a given value of the equivalent S-MILP method is approximately 1 s. B. IEEE RTS-Based Case The case based on the 24-bus IEEE RTS comprises 38 lines, 32 generators, and 17 loads. The load profile corresponds to a winter weekday at 18:00 [22]. There are only two modifications with respect to the data listed in [22]. First, circuits sharing the same towers are treated as independent lines; for example, line 20–23 has two circuits: 20–23A and 20–23B. Additionally, all generator minimum power outputs are set to zero. The destruction of all 38 lines in this example incurs a maximum load shed of 1607 MW, representing 56% of the demand (2850 MW). The execution of the MILP algorithm [20] was stopped when the number of lines destroyed was below a specified threshold or when this number appeared to have reached a lower bound.

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TABLE IV DESTRUCTIVE PLANS—24-BUS SYSTEM

IMPACT

OF THE

TABLE V = 600 MW DESTRUCTIVE PLAN FOR NODAL GENERATION AND LOAD SHED

ON THE

It is interesting to note that if the destructive agents modify their plan so that line 14–16 is attacked instead of line 11–14, the system load shed would increase from 648 to 842 MW. This solution can be obtained from the S-MILP method by setting the specified load loss to 800 MW, as seen in Table IV. VI. CONCLUSION

Fig. 2. Destructive plan for

= 600 MW.

With these stopping criteria, Table IV shows the destructive , plans found for different levels of specified loss of load the actual system load shed, as well as the computing time. In all the cases tested, when allowed to run to optimality, the solution was the same as the one found when the stopping criterion was applied. For example, for a specified lower level of load shed of 600 MW (21% of the system load), the S-MILP model was stopped in 57.64 s with the number of destroyed lines at 5 (11–13, 11–14, 12–13, 12–23, and 15–24), identified in Fig. 2 with blackened circles. Note that even though the terrorist’s goal was to cause a loss of load greater than or equal to 600 MW, the actual system load shed is 648 MW. As a consequence of this destructive plan, the system is split into two islands: 1) the upper area with excess generation capacity and no load shed and 2) the lower area with a deficit of generation and all the loss of load. Table V shows the nodal generation, demand, and load shed corresponding to this destructive plan. Note that the loads at buses 3, 5, 6, and 9 are completely shed, while the demand at bus 8 is only partially interrupted.

This paper has formulated the terrorist threat problem as a general bilevel programming problem, offering flexibility in the problem definition not possible through existing max-min models. Specifically, the bilevel model allows one to define different objective functions for the terrorist and the system operator as well as permitting the imposition of constraints on the outer optimization that are functions of both the inner- and outer-level variables. This general bilevel formulation is applied to a version of the terrorist threat problem in which the destructive agent minimizes the number of lines that must be destroyed in order to cause a level of load loss greater than or equal to a specified level. This destructive strategy assumes that the system operator will implement all feasible emergency generation redispatch actions to minimize the loss of load. The resulting nonlinear mixed-integer bilevel programming formulation is transformed into a S-MILP by replacing the inner optimization by its equivalent KKT optimality conditions and converting a number of nonlinearities to linear equivalents using some well-known integer algebra results. The equivalent formulation has been tested on two case studies, including the 24-bus IEEE RTS, through the use of commercially available software. We are currently investigating new formulations of the S-MILP model with the goal of reducing the number of binary


variables needed to represent the complementary slackness conditions and the computation time. Another interesting avenue of research being examined is to include primary control actions and generator tripping in the inner problem formulation. APPENDIX A KKT OPTIMALITY CONDITIONS OF THE INNER OPTIMIZATION PROBLEM (7)–(12) For a given vector of binary variables , the Lagrangian function of the inner optimization problem (7)–(12) is

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(A17) where (A2)–(A11) represent the dual feasibility conditions, and (A12)–(A17) are the complementary slackness conditions [23]. APPENDIX B EQUIVALENT LINEAR EXPRESSIONS The nonlinearities in (8), (A2), and (A12)–(A17) are now expressed as equivalent linear expressions. A. Equivalent Linear Expressions of the Products of a Binary Variable and a Continuous Variable in (8) and (A2)

(A1)

, and are the multipliers where , , , , , , associated with the dc OPF constraints (8)–(12). In addition to the primal feasibility constraints (8)–(12), the KKT necessary optimality conditions of the inner optimization problem comprise the following:

Given the product of a binary variable and a con, a linear equivalent can be found as tinuous variable follows [17]: 1) Let be a new continuous variable that represents the product of and , 2) let be a new continuous vari, and 3) introduce new inequalities able satisfying and . Thus, if is equal to 0, is also equal to 0 (as desired), and is equal to . Conversely, if is equal to 1, is equal to 0, and then, is equal to (as desired). Note that in (8), there are two products of binary and continuous variables per line: 1) and the phase angle of the sending bus of line , denoted as , and 2) and the phase angle . Thus, by introof the receiving bus of line , denoted as , (representing ducing four new continuous variables the products , and , respectively), and and , the nonlinear constraints (8) are equivalently replaced by

(B1) (A2) (B2) (A3)

(A4) (A5) (A6) (A7) (A8) (A9) (A10) (A11) (A12) (A13) (A14) (A15) (A16)

(B3) (B4) (B5) (B6) (B7) Constraints (B1) are the new linear expressions of the line power flows. Expressions (B2) and (B3) relate the nodal phase , , and and , angles with the new variables , respectively. Finally, lower and upper bounds on variables , , and are imposed in (B4)–(B7), respectively. ), variables and are set If line is destroyed ( to 0 by (B4)–(B5), and consequently, the power flow through and line is equal to 0 by (B1). In addition, variables are, respectively, equal to the phase angles at the sending and and by (B2)–(B3). receiving buses Similarly, if line is not destroyed ( ), variables and are both equal to 0 by (B6)–(B7), and variables and are, respectively, equal to and by (B2)–(B3). Hence, the power flow is determined by the difference of phase angles at the sending and receiving buses (B1).

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Likewise, by introducing new continuous variables and , in the nonlinear constraints (A2) the products of the form can be expressed in the following equivalent linear form: (B8) (B9) (B10) (B11) where constraints (B8) are the linear expressions corresponding and with Lato (A2), constraints (B9) relate variables grange multipliers , and (B10)–(B11) set the bounds on the new variables and , respectively. B. Equivalent Linear Expressions of the Complementary Slackness Conditions (A12)–(A17) Fortuny-Amat and McCarl [18] first showed that a general , where is a complementary condition of the form ) and , a constrained nonnegative Lagrange multiplier ( continuous function ( ), can be equivalently replaced by and , where is a the inequalities sufficiently large positive constant, and is a new binary variable. If is equal to 1, the Lagrange multiplier is a nonnegative value bounded above by , and is equal to 0, meeting . Conversely, if is equal to 0, is set to the condition , and the condition is 0, is within limits, again satisfied. Thus, the nonlinear complementary slackness conditions (A12)–(A17) can be recast as (B12) (B13) (B14) (B15) (B16) (B17) (B18) (B19) (B20) (B21) (B22) (B23) Note that each nonlinear constraint (A12)–(A17) is replaced by the pairs of linear constraints (B12)–(B13), (B14)–(B15), (B16)–(B17), (B18)–(B19), (B20)–(B21), and (B22)–(B23), respectively. In practice, the sufficiently large positive constant can have different values for each constraint, which have to be conveniently tuned for computational purposes. As an example, the linear equivalent of the complementary slackness condition (A12) associated with the lower power flow limit of line is explained next. This condition states that if

is at its lower limit , then the associated Lagrange , which is modeled by the linear multiplier is nonnegative relations (B12)–(B13) when the associated binary variable is equal to 1. In addition, condition (A12) states that if is not is at its lower limit, the corresponding Lagrange multiplier equal to 0, which is modeled by (B12)–(B13) when is equal to 0. As a tighter search space can reduce the computational burden, we now derive additional constraints on the binary variables associated with the complementary slackness con, , ditions related to the limits on . As an example, in the case of , three and possible situations arise. 1) , implying that , , , . and 2) , implying that , , , . and 3) , implying that , and . Hence, variables and satisfy the condition (B24) which essentially states that the power flow through line cannot be its maximum and minimum simultaneously. and Similarly for (B25) (B26)

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José M. Arroyo (S’96–M’01) received the Ingeniero Industrial degree from the Universidad de Málaga, Málaga, Spain, in 1995 and the Ph.D. degree in power system operations planning from the Universidad de Castilla—La Mancha, Ciudad Real, Spain, in 2000. From June 2003 to July 2004, he held a Richard H. Tomlinson Postdoctoral Fellowship at the Department of Electrical and Computer Engineering, McGill University, Montreal, QC, Canada. He is currently an Associate Professor of Electrical Engineering at the Universidad de Castilla—La Mancha. His research interests include operations, planning, and economics of electric energy systems as well as optimization and parallel computation.

Francisco D. Galiana (F’91) received the B.Eng. (Hon.) degree from McGill University, Montreal, QC, Canada, in 1966 and the S.M. and Ph.D. degrees from the Massachusetts Institute of Technology, Cambridge, in 1968 and 1971, respectively. He spent some years at the Brown Boveri Research Center, Baden, Switzerland, and held a faculty position at the University of Michigan, Ann Arbor. He joined the Department of Electrical and Computer Engineering, McGill University, in 1977, where he is currently a Full Professor.