Solving Multi-Objective Economic Dispatch Problem Via Semidefinite ...

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Solving Multi-Objective Economic Dispatch Problem Via Semidefinite Programming Abimbola M. Jubril, Member, IEEE, Olusola A. Komolafe, Member, IEEE, and Kehinde O. Alawode, Student Member, IEEE

Index Terms—Economic dispatch, multi-objective optimization, semidefinite programming (SDP), weighted sum method.

I. INTRODUCTION

M

optimal points. The goal attainment method [4], [7] has also been reported to find weakly Pareto optimal points and, in some cases, inferior solutions [8]. The normal boundary intersection (NBI) method [9] generates very uniform approximate solution set but is unable to find globally (or ideal) Pareto-optimal points [10]. In [11], game theory is applied to solve the multi-objective power problem, and more contemporary techniques are based on evolutionary algorithms (for more details, see [12] and the references therein). In this paper, the weighted sum method is adopted because of its simplicity and applicability to vector optimization problems [13]. Furthermore, appropriate choice of weights can provide a better capture of the Pareto surface [14], [15]. Most optimization problems in power systems have polynomial objective functions and constraints [16], [17], which can be formulated into a semidefinite program. However, they are generally nonconvex, and finding a direct solution is somewhat difficult. The semidefinite programming (SDP) relaxation method simplifies and allows nonconvex problems to be solved by convex optimization techniques. It provides an easily computable lower bound of the minimum value [18]. There are examples of the application of the SDP formulation to the ED and optimal power flow problems [19]–[24].

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Abstract—This paper presents a solution for multi-objective economic dispatch problem with transmission losses semidefinite programming (SDP) formulation. The vector objective is reduced to an equivalent scalar objective through the weighted sum method. The resulting optimization problem is formulated as a convex optimization via SDP relaxation. The convex optimization problem was solved to obtain Pareto-optimal solutions. The diversity of the solution set was improved by a nonlinear selection of the weight factor. Simulations were performed on IEEE 30-bus, 57-bus, and 118-bus test systems to investigate the effectiveness of the proposed approach. Solutions were compared to those from one of the wellknown evolutionary methods. Results show that SDP has an inherently good convergence property and a lower but comparable diversity property.

ODERN electric power systems are large, geographically distributed, yet highly interconnected. In the process of transmitting the generated power, an estimated 4% of the total energy produced is lost [1]. The economic viability of power systems, especially in a competitive energy market [2], demands an optimum mix of all of the parameters that influence power generation and transmission. An approach to achieving this optimum is to include the transmission losses as one of the objectives in the economic dispatch (ED) problem. Thus, the economic dispatch problem becomes a multi-objective optimization in which the fuel cost and the transmission losses are minimized. In general, multi-objective optimization problems are solved by reducing them to a scalar equivalent. This is achieved by aggregating the objective functions into a single function. The weighted sum method [3], [4] is a well-known technique to reduce a multi-objective optimization problem to a scalar equivalent. Some shortcomings of this method include the inability to capture nonconvex portion of the Pareto surface and the nonuniform distribution of the solution point with even variation of the weight factors [5]. In [6], the -constrained method was developed to overcome the problem of finding points on the nonconvex portion of the Pareto front. However, this entails a high computation cost and has a tendency to find weakly Pareto

Manuscript received July 16, 2010; revised January 11, 2011, June 21, 2011, December 05, 2011, May 14, 2012; accepted February 03, 2013. Paper no. TPWRS-00570-2010. A. M. Jubril and O. A. Komolafe are with Obafemi Awolowo University, Ile-Ife, Nigeria (e-mail: [email protected]; [email protected]). K. O. Alavode is with the Department of Electronic and Electrical Engineering, Osun State University, Osogbo, Nigeria (e-mail: [email protected]). Digital Object Identifier 10.1109/TPWRS.2013.2245688

A. Motivation

The ideal Pareto front is not known a priori. Therefore, the Pareto solution set generated by any algorithm is considered as an estimate of the ideal Pareto set. Most multi-objective evolutionary algorithms are population-based and can generate the Pareto front estimate in a single run. Due to the stochastic nature of evolutionary algorithms, the attainment of the ideal Pareto front may be difficult or even impossible. In comparison, the SDP-based weighted sum is not a population-based algorithm and will require several runs to generate an estimate of the Pareto set. However, the SDP formulation is convex and will provide globally optimum Pareto solutions when the relaxation is tight. B. Contributions

The contributions of this paper can be summarized as follows. • The convex combination of the components of the vector objective reduces the multi-objective ED problem to the single case. Such single-objective ED problems were considered in [19], [21]–[24]. As SDP has not been applied to the multi-objective ED problem, this paper extends the SDP to solve such occurrences. • The weighted sum method has been limited to problems that are convex because of the difficulty in capturing solution points on the nonconvex portion of the Pareto

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. 2) Total Transmission Losses, : The total transmission losses of the system is expressed by Kron’s loss formula (4) Equation (4) can be written in matrix form as (5) where .. .

..

.

.. .

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front. The SDP relaxation of nonconvex problems results in a convex approximate. In this paper, the weighted sum method is applied to the resulting convex problem and guarantees the capture of the entire Pareto front. • The weighted sum method has not been able to produce a uniform spread of points on the convex Pareto front with an even spread of the weight factor [5]. A nonlinear weight selection method that controls and improve the spread of the Pareto points is presented. The remainder of this paper is organized as follows. The multi-objective dispatch problem is described in Section II. A brief description of the SDP and the SDP formulation of the problem through semidefinite relaxations are presented in Sections III and IV, respectively. In Section V, the analysis of the multi-objective optimization problem is considered, and some performance metrics used in this paper are briefly discussed. Section VI considers the reduction of the multi-objective problem into its scalar equivalent using the method of weighted sum of objectives. The nonlinear selection of the weight vector to achieve greater control of the distribution of the generated solutions is presented in Section VII. In Section VIII, the effectiveness of the proposed approach is evaluated, and concluding remarks are provided in Section IX.

are Kron’s loss coefficients.

II. PROBLEM STATEMENT

Let be the number of generating units in a power system, and denote the th generator by . The power generated by the th generator is . The multi-objective economic dispatch problem with transmission losses can be formulated as follows: (1)

, the decision variable, is the where vector of the generated power; is the fuel cost objective, is the total transmission loss objective, and and are the generation capacity and power balance, inequality and equality constraints, respectively. A. Problem Objectives

1) Total Fuel Cost, : The generator costs are represented by quadratic functions. The total fuel cost in running the generating units can be expressed as

(2)

is the real power output of the th plant, and , where and are the fuel cost coefficients of that plant. In matrix form, the total fuel cost can be written as (3)

B. Problem Constraints

1) Generation Capacity Constraints: The real power output of each generating unit is constrained between the upper and the lower limits as follows: (6)

This defines the inequality constraint . 2) Power Balance Constraint: The power balance constraint is given by (7)

and in matrix form as

(8)

where is the total load demand and is a -dimensional vector of all ones. This defines the equality constraint . III. SEMIDEFINITE PROGRAMMING

SDP is a convex optimization method that generalizes the LP by replacing the vector variables with matrix variables [18], [25], [26]. Furthermore, the elementwise nonnegativity condition is replaced with the positive semidefiniteness of the matrices. In the SDP formulation adopted in this paper, the primal SDP is defined as the optimization problem

where (9)

JUBRIL et al.: SOLVING MULTIOBJECTIVE ECONOMIC DISPATCH PROBLEM VIA SDP

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to find the solution. It is possible to write each of the quadratic expressions in their matrix quadratic equivalents:

and the associated dual SDP is

(14)

(10)

A. SDP Relaxation

(15) where ation is

. The dual of Shor’s SDP relax-

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is the decision variable, and where . In the notation adopted in this paper, denotes the set of all symmetric matrices in . The constraint in (10) is a linear matrix inequality (LMI) of the form . The vector is the decision variable and . In (9) and (10), the inner product between two vectors is defined as , and the inner product between two matrices is defined as . The symbol denotes positive semidefiniteness, i.e., if . Furthermore, and are the optimal values of (9) and (10), respectively. More details on SDP can be found in [27]. In the LP formulation, duality is very strong and both primal and dual programs always give the same result. On the other hand, the duality results in SDP are weaker, resulting in a duality gap. In this situation, the result of one program yields a bound on the optimal value of the other. However, if the problem satisfies Slater’s condition, the two programs give the same result , thus closing the duality gap [26]. An efficient interior point method exists for the solution of the primal/dual program [16], [28], [29].

Using Shor’s relaxation [18], a good lower bound can be computed for (13) by solving the SDP,

Given a minimization problem

(11)

for which an estimate or a bound on the optimal value suffices, the feasible set may be increased from to such that . The relaxed problem becomes

(12)

with . The reformulation of the problem (11) as (12) forms the basis of SDP relaxation. In essence, SDP relaxation consists of linearization and replacement of restrictive equality constraints with inequality constraints. As an illustration, consider the relaxation of the universal combinatorial problem (in the form of the quadratically constrained quadratic program [QCQP]) [30], [31]

(16) The last constraint in (16) represents the relaxation of the constraint as . IV. SDP RELAXATION OF THE PROBLEM

The vector optimization ED problem can be decomposed into a number of single-objective ED problems. The component objective functions are represented by quadratic functions, and each of these is minimized over the feasible set defined by the equality and the inequality constraints. Each of the nonlinear objective functions is linearized through the substitution of the SDP variable and the equality constraints. The first single-objective ED problem in (1) is the classical ED problem (17)

The second ED problem, which minimizes the total transmission losses, is given as (18)

By writing and substituting in (3), the ED problem in (17) can be reduced to the form

(13) with

and . Each of the quadratic functions is convex if ; however, if , the problem is nonconvex and it is difficult

(19)

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In the same manner, the second ED problem in (18) is reduced to the form

(20)

(25) When , all members of are weakly dominated by . However, implies that no member of is weakly dominated by . It is possible that , always. Hence, both and must be calculated to ascertain the members of that are dominated by and vice versa. A metric that estimates the extent or the maximum spread of the Pareto front is given in [32]. It is defined as the Euclidean distance between the extreme function values observed in the nondominated solution set and given by

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These two single-objective ED problems can be combined and converted back to a vector objective ED problem of the form

between the generated solution set and the ideal Pareto front. Diversity measures the extent of the coverage and the uniformity of distribution of the solutions [32]. The two properties are somewhat conflicting and are rarely satisfied simultaneously by the solution of any multi-objective optimization algorithm. The convergence of two solution sets can be estimated by superimposing their Pareto fronts and considering the dominance property of the generated Pareto sets. In [33], the set coverage metric was suggested to evaluate convergence. The set coverage metric between two sets of solution sets and , denoted by , calculates the proportion of solutions in , which are weakly dominated by solutions of :

(21)

V. MULTI-OBJECTIVE OPTIMIZATION

A general multi-objective optimization problem consists of a number of objectives to be optimized simultaneously and is associated with a number of equality and inequality constraints. It can be formulated as follows:

where

(26)

where is the number of objectives. The spacing metric (denoted as ) is a measure of diversity of the generated solution set. It considers how uniformly distributed or equidistantly spaced the solution points are along the Pareto front [34], [32]. It is given by

(22)

where is the feasible region in the decision space, is the th objective function, is a decision vector, is the number of the objectives, and and are the inequality and the equality constraints, respectively. In a single-objective optimization problem, the optimal solution is that value such that no other value gives a better objective value. However, in a multi-objective optimization problem, there is no unique optimal solution. Rather, there is a set of optimal solutions that are non inferior to any other in the set. A vector is said to dominate another vector (denoted by ) if and only if is partially less than . This can be expressed as

(23) (24)

A solution there is no

is said to be Pareto optimal if and only if for which .

(27)

where

(28)

and is the mean value of all . The Pareto front yielding a smaller value of spacing metric is considered to have a better diversity. A zero value implies that all members of the generated set are equidistantly spaced. VI. WEIGHTED SUM METHOD

Consider the weight vector vector objective function and the map method derives the scalar objective combination of the objectives

, the , . The weighted sum , through a convex . Thus

A. Performance Metrics Generally, two important properties are used when comparing different multi-objective algorithms, namely, convergence and diversity. Convergence is a measure of the relative closeness

(29)

JUBRIL et al.: SOLVING MULTIOBJECTIVE ECONOMIC DISPATCH PROBLEM VIA SDP

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TABLE I VARIATION OF SLOPE WITH

Fig. 1. Weighted sum method representation.

where

is the number of objectives and (30)

TABLE II VARIATION OF SLOPE SENSITIVITY WITH

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This transforms the vector optimization to a scalar of the form

(31)

This process maps the -dimensional objective space onto the positive real line , and all the optimal (non-dominated) points are mapped to the same point on the line. More specifically, consider the bi-objective problem with ; (29) and (30), respectively, reduce to

(32) (33)

If the vectors are expressed in polar coordinate forms as shown in Fig. 1, (32) can be written as

(34)

where , and . The minimum points (equivalently, the Pareto points) of the map can be found at points where the weight vector is orthogonal to the objective vector , i.e., or . Note that, for every weight and are determined, and the scalar optimization problem reduces to finding in the direction such that is minimum. If the weight vector is parameterized by , such that and , then the slope of is given as

The relative change of the slope of and the slope sensitivity with respect to change in the parameter for the standard weighted sum (SP) are shown in the second column of Tables I and II, respectively. Table I shows that a change in the value of from 0 to 0.1 results in a change in the value of slope from to 9.0. This implies that slope values between and 9.0 are missed; therefore, all solution points in that section of the tradeoff curve will not be captured. This change in slope can be considered as the sensitivity of the scalarization process to capture solution points on the Pareto surface. For values of close to 0, in Table II, the slope is very sensitive to very small change in , and for values of close to 1, the sensitivity is close to 1. This was noted in [35] to be the source of deficiencies of linear weight selection, resulting in the omission of Pareto points in the first and the clustering of Pareto points in the second. This motivates the idea of reducing these effects by controlling the slope sensitivity.

(35) VII. WEIGHT SELECTION ADAPTATION SUM METHOD

and the slope sensitivity as (36)

IN

WEIGHTED

The weight vector in the weighted sum method is constrained along the line (as SP), and as varies from 0 to 1,

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varies between 1 and . However, if the weight vector is constrained to the unit circle as (37) the slope becomes (38) and the slope sensitivity can be written as (39)

values around 1. This fast change leads to high spacing distance between the generated optimal points, while the very low sensitivity leads to the clustering effect in the resulting solutions. The nonlinear weight selection gives a higher sensitivity and provides for further sensitivity improvement through the free parameter . It can be observed that, as gets close to 1, the slope sensitivity tends to approach . The additional free parameter can be manipulated in the search of the solution points and facilitates the control of the slope of the weight factor such that clustered points can be spread out. There is also an improvement in the computational efficiency of the method. VIII. SIMULATION RESULTS AND DISCUSSIONS

(40) (41)

To provide greater control over the slope sensitivity, a weight selection such that is constrained on the surface of an ellipsoid (ELP) is proposed. Let (42)

Substituting in (33) gives the equation of the ellipsoid

(43)

For

number of objectives, an ellipsoid

is defined as

(44)

of the ellipsoid. Setting becomes

In this study, the standard IEEE 30-bus, 57-bus, and 118-bus test systems were considered to investigate the effectiveness of the proposed approach. The cost coefficients, power generation limits, and details of the bus and line data can be found in [12], [36] and, [37], respectively. The SDP problem was converted into the standard primal/dual form using YALMIP parser [38] and solved with SeDuMi [39], an SDP solver. The problems were initially solved for the ideal minimum points with each of the single objectives. These values at the minimum points were used in the determination of the maximum spread of the algorithm. The problem was then solved using the standard weighted sum method. Two cases were investigated: Case 1) investigation of the effect of on the quality of Pareto generation. Case 2) comparison of the solutions from SDP and the Nondominated Sorting Genetic Algorithm II (NSGA-II).

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A slight improvement over the standard weighted sum method can be observed when the sensitivities are compared (see the third column of Table I). This is due to the factor in (39), and more solution points can be found. Note that the weight constraint is not satisfied; hence, it cannot be considered as a weighted sum method. Consider setting and . This will constrain to a unit circle while satisfying the weight constraint. The slope and its sensitivity can now be written as

, where and

are the axes in (43), the slope

(45)

and the slope sensitivity is given by

(46) A comparison of results in Tables I and II shows an improvement in slope sensitivity as one changes from weights constrained along the line , (SP), to those constrained on a unit circle (UC) and on an ellipsoid (ELP). It is interesting to observe that the rate of change of the slope with weight parameter change is monotonically decreasing: a very fast change for initial values of and almost zero change for

A. Case 1

was Generation of Pareto points for different values of considered. To explore the effect of changes in and compare different cases, 50 runs were carried out for each parameter value. Detailed results for the 57-bus test system only are presented here due to space constraints. The Pareto curves for values of 1, 5, and 10 are shown in Fig. 2. With , only a few distinct points were generated from 50 runs. This implies that different runs with different values of gave very close values. This is a waste of computational effort. These points spread out as is increased. As the value of is further increased, a progressive improvement in the distribution of the Pareto points is observed, and the clustering of the solution points disappeared. However, it was also observed that the solutions near the lower extreme point are not captured. As the value of is increased from 1, more points are missed from the lower extreme point. This inability to capture the lower extreme point of the Pareto and other solution points in its neighborhood resulted in the increased power recorded for most of the generating units in the third and fourth columns of Tables III and IV. This is observed to be due to the expression for in (43) in terms of as (47) Therefore, and , implies that the extreme solution point defined by and those close to it are missed. This results in the reduction of the extent of the Pareto

JUBRIL et al.: SOLVING MULTIOBJECTIVE ECONOMIC DISPATCH PROBLEM VIA SDP

Fig. 2. Effect of variation of

on the Pareto front. (a)

. (b)

. (c)

.

B. Case 2 The multi-objective dispatch problem was also solved with the Nondominated Sorting Genetic Algorithm II (NSGA-II) using the three test systems. Fig. 3 gives a visual comparison of Pareto fronts obtained by NSGA-II (population size of 50) and SDP with different values of used (50 runs for each value) for the 57-bus test system. While SDP was able to capture the upper extreme point for the different values of , NSGA-II could not capture it. On the other hand, as the value of is increased, the inability of the SDP approach to capture the lower extreme point, and solution points close to it, can be noted. This leads to the reduction of the extent parameter, , in all the problems considered (e.g., when increased to 10). Tables III and IV show the best solutions obtained by both the NSGA-II and the SDP approaches for minimization of fuel cost and transmission losses, respectively, in bold numerals. The corresponding values in the other objective are also given. The movement of the lower extreme point from its ideal value as the value of increases is seen in the best fuel cost obtained by the SDP method that increases with in Table III. From Table IV, it can be seen that the SDP (S) approach returned a lower value for transmission losses than NSGA-II (NS). A summary of the results for the three systems considered is presented in Table VI. Tables VII–IX show the results of the set coverage, spacing, and extent metrics for comparing the performance of both the NSGA-II and the SDP approaches in the 30-bus, 57-bus, and 118-bus test cases, respectively. Values for the set coverage metric indicate that the SDP approach yielded solutions with better convergence properties than NSGA-II, with convergence improving as the value of is increased. The spacing metric values show that the solutions obtained by NSGA-II are more uniformly distributed than those of SDP. However, as the value of increases, the distribution of the SDP-obtained solution sets along the Pareto front improves. SDP also yielded solution sets with better extent metric values than NSGA-II, although a decrease in value is observed as is increased. Looking at Tables VII, VIII, and IX, an increase in the set coverage with is observed, but for , for the 118-bus system, a reduction is noted. This is explained as follows. By the definition of dominance, for a comparison of two solution points, one of their function values must remain the same. The relative values of the other function value determine which of

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TABLE III BEST SOLUTIONS FOR FUEL COST (57-BUS)

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TABLE IV BEST SOLUTIONS FOR TRANSMISSION LOSSES (57-BUS)

TABLE V SOLVER TIME AND ITERATIONS FOR DIFFERENT TEST SYSTEMS

set generated as the value of is increased. This effect can be removed by combining solution sets generated with two different values of , one large value and one small value close to 1. The time taken by the solver and the number of iterations for each run for various problem sizes are given in Table V. An exponential increase in the solver time with system size can be observed. This shows a weakness of SDP in handling large systems.

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Fig. 3. Case 2: Pareto front comparison for 57-bus system. (a) NSGA-II and

. (c) NSGA-II and

TABLE VII PERFORMANCE METRICS (30-BUS)

TABLE VIII PERFORMANCE METRICS (57-BUS)

TABLE IX PERFORMANCE METRICS (118-BUS)

.

the solutions dominates. In general, as increases, the spacing metric decreases. This increases the number of points that can be compared as defined in (25). Also, due to (47), increasing reduces the extent metric of the SDP-generated Pareto front. However, as the extent of the Pareto front of the NSGA-II is constant, the reduction of the SDP extent reduces the number of points that can be compared. The combination of these two effects on the number of points to be compared determines the effect of on the set coverage. Initially, an increase in the set coverage is observed as is increased. This is due to the dominating increasing effect of the spacing metric on the set coverage metric over the decreasing effect of the extent metric on the set coverage. As the value of is further increased, an optimal point is reached beyond which an increase in causes a decrease in the set coverage. Increasing beyond this point will cause a reduction in the set coverage. This is due to the dominating reducing effect of decreasing extent. The optimal point where the highest set coverage is achieved differs for different systems. We observed, for the test systems considered, that the value of around which this was attained for the 30-bus system is 17.5; for the 57-bus system, 10; and for the 118-bus, 7.5.

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TABLE VI SUMMARY OF BEST SOLUTION FOR THE TEST SYSTEMS

. (b) NSGA-II and

IX. CONCLUSION

This paper formulated a multi-objective economic dispatch problem with transmission losses as a convex optimization problem using SDP relaxation technique. The difficulties of the weighted sum method in handling a nonconvex Pareto front were obviated by a nonlinear selection of weight factors. The nonlinear weight selection helped to reduce the clustering of solution points on the Pareto surface, thus providing a more uniform distribution. There is the added improvement in the use of computational resource. A numerical example based on standard data set showed that the proposed formulation is efficient. Comparison with the NSGA-II showed that the SDP-based approach possesses comparable diversity property and better convergence property. The formulation provided in this paper can be extended to other power system problems, as polynomial objectives were considered.

JUBRIL et al.: SOLVING MULTIOBJECTIVE ECONOMIC DISPATCH PROBLEM VIA SDP

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[Pls provide the department for which this dissertation was prepared.].

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[1] A. M. L. da Silva and J. G. de Carvalho Costa, “Transmission loss allocation: Part I—Single energy market,” IEEE Trans. Power Syst., vol. 18, no. 4, pp. 1389–1394, Nov. 2003. [2] D. Dochain, W. Marquardt, S. C. Won, O. Malik, M. Kinnaert, and J. Lunze, “Monitoring and control of process and power systems: Adapting to environmental challenges, increasing competitivity and changing customer and consumer demands,” in Proc. 17th World Congress, Jul. 2008, pp. 7160–7171. [3] F. Milano, C. A. Canizares, and M. Invernizzi, “Multi-objective optimization for pricing system security in electricity markets,” IEEE Trans. Power Syst., vol. 18, no. 2, pp. 596–604, May 2003. [4] W. D. Rosehart, C. A. Canizares, and V. H. Quintana, “Multi-objective optimal power flow to evaluate voltage security costs in power networks,” IEEE Trans. Power Syst., vol. 18, no. 1, pp. 578–587, Mar. 2003. [5] I. Das and J. E. Dennis, “Closer look at drawbacks of minimizing weighted sums of objectives for pareto set generation in multicriteria optimization problems,” Structural Optimization, vol. 14, pp. 63–69, 1997. [6] Y. L. Chen and C. C. Liu, “Multiobjective var planning using the goal attainment method,” in Proc. IEEE Gen., Transm. Distrib., Mar. 1994, vol. 141, pp. 227–232. [7] A. Berizzi, C. Bovo, and M. Innorta, “Multi-objective optimization techniques applied to modern power systems,” in Proc. IEEEE Power Eng. Soc., 2001, vol. 3, pp. 1503–1508. [8] J. G. Lin, “On multiple-objective design optimization by goal methods,” in Proc. Amer. Control Conf., Boston, MA, USA, Jun. 1991, pp. 372–373. [9] C. Roman and W. Rosehart, “Evenly distributed pareto points in multiobjective optimal power flow,” IEEE Trans. Power Syst., vol. 21, no. 2, pp. 1011–1012, May 2006. [10] I. Das and J. E. Dennis, “Normal boundary intersection: A new method for generating the pareto surface in nonlinear multicriteria optimization problems,” SIAM J. Optim., vol. 8, no. 3, pp. 631–657, 1998. [11] A. Orths, A. Schmitt, and Z. A. Styczynski, “Multi-criteria optimization methods for planning and operation of electrical energy,” Electrical Engineering(Archiv for Elektrotechnik), vol. 83, no. 5–6, pp. 251–258, 2001. [12] M. A. Abido, “Multiobjective evolutionary algorithms for electric power dispatch problem,” IEEE Trans. Evol. Comput., vol. 10, no. 3, pp. 315–329, 2006. [13] L. A. Zadeh, “Optimality and non-scalar-valued performance criteria,” IEEE Trans. Automat. Control, vol. AC-8, pp. 59–60, 1963. [14] T. W. Athan and P. Y. Papalambros, “A note on weighted criteria methods for compromise solutions in multi-objective optimization,” Eng. Opt., vol. 27, pp. 155–176, 1996. [15] M. A. Gennert and A. L. Yuille, “Determining the optimal weights in multiple objective function optimization,” in Proc. 2nd Int. Conf. Comput. Vis., Los Alamos, CA, USA, Sep. 1998, pp. 87–89. [16] I. A. Farhat and M. E. El-Hawary, “Interior point methods application in optimum operational scheduling of electric power systems,” IET Gener. Transm. Distrb., vol. 3, no. 11, pp. 1020–1029, 2009. [17] J. B. Lasserre, “Global optimization with polynomials and the problem of moments,” SIAM J. Optim., vol. 11, no. 3, pp. 796–817, 2001. [18] S. Boyd and L. Vandenberghe, “Semidefinite programming relaxations of non-convex problems in control and combinatorial optimization,” in Communications, Computation, Control and Signal Processing: A Tribute to Thomas Kailath, A. Paulraj, V. Roychowdhuri, and C. Schaper, Eds. Boston, MA, USA: Kluwer, 1997, ch. 15, pp. 279–288. [19] M. Madrigal and H. Quintana, “Semidefinite programming relaxations for 0,1 power dispatch problems,” in Proc. IEEE Power Eng. Soc. Summer Meeting Conf., Edmonton, AB, Canada, July 1999, pp. 697–701. [20] R. Fuentes-Loyola and V. H. Quintana, “Medium-term hydothermal coordination by semidefinite programming,” IEEE Trans. Power Syst., vol. 18, no. 4, pp. 1515–1522, Nov. 2003. [21] X. Bai, H. Wei, K. Fujisawa, and Y. Wang, “Semidefinite programming for optimal power flow problem,” Int. J. Electr. Power Energy Syst., vol. 30, pp. 383–392, 2008. [22] B. C. Lesieutre, D. K. Molzahn, A. R. Borden, and C. L. DeMarco, “Examining the limits of the application of semidefinite programming to power flow problems,” in Proc. 49th Annual Allerton Conf., Allerton House, IL, USA, Sep. 2011, pp. 1492–1499.

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[34] J. Schott, “Fault tolerant design using single and multi-criteria genetic algorithms,” Master’s thesis, Mass. Inst. Technol., Boston, MA, USA, 1995

[Pls provide the department for which this thesis was prepared.].

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Abimbola M. Jubril (M’12) received the B.Sc., M.Sc., and Ph.D. degrees in electronic and electrical engineering from the Obafemi Awolowo University, Ile-Ife, Nigeria, in 1996, 2002, and 2007, respectively. He is a Lecturer with the Department of Electronic and Electrical Engineering, Obafemi Awolowo University, Ile-Ife, Nigeria. His research interests include robust control, application of multi-objective optimization in control and power systems, and instrumentation systems.

Olusola A. Komolafe (M’87) received the B.Sc. (Hons.) degree in electronic and electrical engineering from the Obafemi Awolowo University, Ile-Ife, Nigeria, in 1979, the M.Sc. degree in electrical engineering from the University of New Brunswick, Fredericton, NB, Canada, and the Ph.D. degree from the University of Saskatchewan, Saskatoon, SK, Canada. He is a Senior Lecturer with the Department of Electronic and Electrical Engineering, Obafemi Awolowo University, Ile-Ife, Nigeria. His research interests include power system analysis, and control and application of FACTS in power systems.

Kehinde O. Alawode (S’XX) [pls provide your year of membership.] received the B.Tech. (Hons.) degree in electronic

and electrical engineering from Ladoke Akintola University of Technology (LAUTECH), Ogbomoso, Nigeria, in 2005. He is currently working toward

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IEEE TRANSACTIONS ON POWER SYSTEMS

IE E Pr E oo f

the M.Sc. degree in electronic and electrical engineering at Obafemi Awolowo University, Ile-Ife, Nigeria. He is with the Department of Electronic and Electrical Engineering, Osun State University, Osogbo, Nigeria. His current research interests include computational intelligence and its applications.