Transmission Systems Power Quality Monitors Allocation - IEEE Xplore

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Abstract — This work develops and tests a branch and bound algorithm for solving optimum allocation of power quality monitors in a transmission power system.
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Transmission Systems Power Quality Monitors Allocation D. C. S. Reis, Student Member, IEEE, P. R. C. Villela, C. A. Duque, Member, IEEE, and P.F. Ribeiro, Fellow, IEEE Abstract — This work develops and tests a branch and bound algorithm for solving optimum allocation of power quality monitors in a transmission power system. The optimization problem is solved by using 0-1 integer programming techniques and depends highly on network topology. The algorithm, which is implemented in Matlab software, minimizes the total cost of the monitoring system and found the optimum number and locations for monitors on the network studied, under a set of given network observability constraints. Case studies are presented for IEEE test networks and for CEMIG actual transmission power systems. Current and voltage values are estimated by using monitored variables to validate the obtained results. Keywords— power quality, power quality monitoring, power quality monitors allocation, combinatorial optimization, branchand-bound.

I. INTRODUCTION

O

One of the greatest challenges to guarantee power quality (PQ) at transmission and distribution power systems is the polluting sources identification and its levels identification. This information is essential for energy regulating companies to create corrective and punitive politics to avoid or to diminish the polluting agents. The problem of harmonic sources localization and identification has motivated some recent research, i.e. [1-5]. However, this problem is difficult to solve due to the system’s high level of interconnection. Its solution depends on systems distributed monitoring. The determination of the PQ monitors and communication channels’ cost is the biggest problem faced in the monitoring, thus motivating development of methodologies that minimize the total cost of system monitoring [6-14]. The present paper shows the methodology that was used to find the minimum number and the localization of the power quality monitors in a power system. In this work, the objective is to minimize monitoring cost and is guaranteed by total observability of the system with monitors installation in strategic buses. It becomes possible to study any PQ events, D. C. S. Reis is with Energy Department of UFJF, Juiz de Fora MG, Brazil, 36 036 330, (email: [email protected]). C. A. Duque is with Electrical Circuits Department of UFJF, Juiz de Fora MG, Brazil, 36 036 330, (email: [email protected]). P. R. C. Villela is with Electrical Circuits Department of UFJF, Juiz de Fora MG, Brazil, 36 036 330. (email: [email protected]). Paulo F. Ribeiro is with CAPS/FSU and Calvin College, (e-mail: [email protected]).

©2008 IEEE.

like harmonics, sags or swells. The analyzed bibliography points clearly to the use of combinatorial optimization techniques in a solution of this type of problem, however the presented solutions do not obtain all the possible optimum solutions. The use of commercial softwares, like GAMS, LINDO or TOMLAB, in most cases, makes it difficult to know the algorithm’s details in solution search and its adaptation to special cases, due to the impossibility of knowing or modifying the programming code. Therefore, the authors developed a branch and bound (B&B) algorithm to determine all of the accurate solutions at minimum cost [13, 14]. This paper is divided as follows: Section II presents problem modeling, Section III presents the developed algorithm, Section IV simulation results are shown, and in Section V the results are validated. The conclusions of this work are presented in Section VI. II. PROBLEM MODELING This section presents the modeling of the power quality monitors allocation problem. The problem, called P0, can be written as one of the classic combinatorial optimization problems, called set covering problem [15], which can be mathematically described as follows n

P0

min z =

∑ c ( j ) ⋅ x( j ) ⇒ c ⋅ x

(1)

j =1

n

Subject to

∑ d (i, j) ⋅ x( j) ≥ 1 ⇒ D ⋅ x ≥ 1, i = 1,K, m j =1

(2) 0 ≤ x( j ) ≤ 1

(3) (4) where c is the cost vector, x is the variable vector, and D is the density matrix of the set covering problem. The set covering problem (SCP) applied to the power quality monitors allocation problem can be seen as the necessity of installing the minimum number of measures in an electrical power system. This guarantees that all voltages and currents of the system will be observed. This is necessary for the analysis of PQ events. The proposal modeling is strongly supported in Eldery et al. [9, 10]. In these papers the authors had considered a modeling of PQ monitors allocation problem based on system topology. For this reason the presented methodology is only x( j ) integer to j = 0,1,

L, n

2

valid if the system topology will not be modified. This constraint does not compromise the use of the method in one of the main applications of PQ, polluting sources localization, and must be made with unmodified topology. It was considered that the possible PQ monitors localizations would have been system buses. It is necessary, therefore, to identify the cost associated with the installation of monitors in each bus. Thus, the problem becomes finding the minimum cost of, and in which buses, this installation will be made. Moreover, another characteristic of the SCP is that voltage in each bus and the current in each transmission line must "be covered" by at least one measurer. This means that at least one monitor must measure or estimate each one of them. Consider a system with n busesand L transmission lines. The total number of state variables will be equal to the sum of the number of buses and lines, m = n + L. It will present the necessary vectors to formulate the problem. A. Existence Vector The existence vector (6) is a binary vector of dimension (nx1), and represents the installation or not of the monitor in that bus. Each element of this vector is defined by (5), ⎧1, if the monitor is installed at bus j (5) x( j ) = ⎨ ⎩0, otherwise

x = [ x(1) x(2)

L

x (n) ]

t

(6)

B. Cost Vector The installation of each monitor has an associated cost that is represented in the problem by the vector of cost c, (8), its dimension is (1xn) representing the total number of possible allocations, or the total number of buses. Each element of this vector can be defined as (7). c (j) = installation cost of monitor at bus j. (7) (8) c = [ c(1) c(2) c (n )]

L

C. Objective Function The objective of this problem is to minimize the total cost of the monitoring system, which is given by the sum of installation costs of each one of the measurers, thus the equation that the objective function represents is described as in (1), repeated here for convenience. n (9) min z = c( j ) ⋅ x( j ) = c ⋅ x

∑ j =1

D. Constraints The constraints of this problem guarantee that all of the state variables are measured or calculated for at least one PQ monitor, and are made using Ohm’s Law. Eldery et al. [9, 10] first proposed a modeling similar to the one presented here. The connectivity matrix, A, can be defined as an auxiliary matrix in the construction of the SCP density matrix, D, and represents the observability of the voltage buses. Its dimension is defined by the total number of state variables and by the possible places of installation (m x n). Its column k represents the monitor installed in bus k and its row r represents the state variable, voltage in buses and current in

transmission lines. Each element of this matrix is defined in (10). ⎧1, if r is observed by monitor k (10) a (r , k ) = ⎨ ⎩0, otherwise The matrices of co-connectivity, Bj and Bk, representing the necessity to observe the voltage in buses j and k, and considering them interconnected, are used as an auxiliary in the construction of the density matrix and represent the state variable that corresponds to currents in transmission lines. Thus it is possible to guarantee that ijk will be observable. The dimension of the matrices Bj and Bk is also (m x n). Its column k represents the monitor installed in bus k and its row r represents state variable referred to as current ijk. Each row of these matrices is defined in (11) and (12). ⎧A( j ), if r represents i jk and buses k e j are conected, (11) B (r ) = ⎨ ⎩0, otherwise ⎧A(k ), if r represents i jk e buses k e j are conected, Bk ( r ) = ⎨ ⎩0, otherwise j

(12)

It is possible to construct the density matrix (13). It will have a dimension equal to the number of buses, n, and two times the transmission line number, 2L, to generically represent the current variable that depends on the tension in two bars, j and k, which means that the dimension of the density matrix will be ((n+2L)x n). ⎡ ⎤ A(1:n)xn ⎢ ⎥ D = ⎢ A(L:m) xn + Bj(L:m) xn ⎥ ⎢A + Bk(L:m)xn ⎥⎦ ⎣ (L:m) xn

(13)

where A (1:n ) xn is a part of the connectivity matrix for rows 1 to n, A( L:m ) xn is a part of the connectivity matrix for rows L to m, B j( L:m ) xn and Bk ( L:m ) xn is a part of the co-connectivity matrix for rows L to m. III. SOLUTION ALGORITHM To solve the monitors allocation problem, as shown in the previous item, we propose a specific branch and bound algorithm [13, 14, 16], which makes it possible to find all the optimum solutions, thus allowing to choose among those that satisfy other factors not considered in the modeling. This algorithm was implemented in MatLab®. A. Formulation Finding all optimum solutions of P0 requires inquiring all 2n possible solutions for the binary variable xj. Only a few of them will satisfy the constraints (3) and also the constraints (4) of P0 problem. These are the viable solutions. For example, in a P0 problem with only two binary variables x1 and x2, the number of possible integer solutions is 4. They are represented in Fig. 1 for the vertices (0,0), (1,0), (0,1) and (1,1). The shaded area is the space of viable solutions of relaxed problem P0 (denoted as P0), the one where variables x1 and x2 can assume any real value between 0 and 1, that is, 0 ≤ x1 ≤ 1

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and 0 ≤ x2 ≤ 1. The optimum solution of the P0 problem is, in this example, the vertex whose optimum value is Z0. The viable integer solutions of the P0 problem are the vertices (0,0), (1,0) and (0,1). The vertex (1,1) is not a viable solution for P0. The optimum solution of the P0 problem is Z0, in this example, the point (0,1).

Fig. 1. Solution Space of Problem P0.

The Z0 value is smaller than Z0, optimum solution of the relaxed problem P0. This is an important property that relates any integer linear programming (ILP) problem with its relaxed correspondent, where the integer conditions have been suppressed. In the minimization problems, the optimum value (Z0) of the original ILP problem is bigger than the optimum value (Z0) of the corresponding relaxed problem. Therefore by solving the relaxed problem it is possible to obtain an upper bound (Z0) for the optimum value of the original problem (Z0). The calculation of these upper bounds is a very important step in the construction of the algorithms of the type branch and bound. The technique used for this kind of algorithm consists of searching optimum solutions of the P0 problem using two procedures: branching and bound [15, 16, 17, 18, 19]. Branching is the division of the original problem solution space into smaller spaces, with a lesser number of possible solutions to be inquired. The logic of the branching is the hope that it is easier to solve two smaller combinatorial problems than to solve the original problem. For example, the original problem P0 with n variable can be partitioned in: P1 = {P0 | xi = 0} e P2 = {P0 | xi = 1} where xi is any variable of the P0 problem. Each one of the two problems generated from P0 has (n-1) variables; therefore, it has 2n-1 possible solutions to be investigated.

Fig. 2. Branching tree with P0, P1e P2; P3 e P4; e P5 e P6 problems.

Fig. 2 illustrates the branching tree of the original problem P0 in the problems P1 and P2 and in the problems P3 and P4,

and P5 and P6 respectively, fixing in each step of the branching one variable in 0 or 1. Each one of Pk problems will be solved or partitioned again, generating two other problems in the tree that will be solved or partitioned successively. If any problem Pk is solved, its solution will be denoted by k Z and the corresponding optimum solution for the vector Xk. It will be called ZL the smallest value obtained between the calculated Zk. Thus ZL and its corresponding solution XL will store the best solution obtained until a determined moment. The decision to partition each one of the Pk problems is preceded by the calculation of an upper bound of the optimum value Zk of this problem. The upper bound is the optimum value Zk of Pk relaxed problem. In a minimization problem Zk ≥ Zk, if a solution XL of value ZL and Zk > ZL, the upper bound of the optimum value of Pk is worse (bigger in this case) than the value of an already found solution , it would not make sense to find the optimum solution from the Pk problem, so it must be trimmed and not partitioned. IV. RESULTS The algorithm detailed in the previous item was implemented and tested in MatLab® [14] using the internal routines of linear programming in the calculation of upper bounds. The algorithm was tested in IEEE 14, 30, 57 and 118 test power system topology, available at Washington University [20] and in an actual Minas Gerais Energy Company (CEMIG) power system available at Brazilian System National Operator [21]. Different installation costs have been considered, proportional to the number of transmission lines that pass through each bus. This was made because a great part of installation cost refers to the number of voltage and current transformers necessary to acquire the signal. Thus, the larger the line number in each bar, the greater the number of transformers necessary. The results for this case are shown in Table I. TABLE I – RESULTS FOR DIFFERENT COST INSTALLATION

System

Buses

Lines

IEEE 14 IEEE 30 CEMIG

14 30 48

18 41 64

Solution s Number 14 4 44

IEEE 57 IEEE 118

57 118

80 186

5 3458

Monitors Number

4, 5 or 6 12 18, 19, 21 or 22 18 or 19 39,40,41, 42, 43, 44 or 45

Although the number of necessary monitors is generally different for each solution, the installation cost is minimum for any solution One of the characteristics found in linear programming problems is a great number of optimum solutions [15], for example, for CEMIG’s actual system it was found that 44 global optimum solutions existed, and for IEEE-118, 3458

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optimum solutions. These solutions have the same value for its objective function. The alternative used to help in the choice between those solutions was calculating the data redundancy factor (DRF) that indicates how many times in average the state variable of a transmission system will be measured or calculated for the PQ monitors. The DRF can mathematically be described as equation (14). DRF =

Sum of observed state variable . Total number of state variable

(14)

It is considered that the redundancy in the measurements is desirable, because it increases the reliability of the system. If some measure is lost, with extreme noise or has been measured wrongly, it could be corrected or reconstructed if it has been acquired for other monitors. Table II presents the result of data redundancy factor and solutions for IEEE 57 buses. TABLE II - DATA REDUNDANCY FACTOR RESULTS FOR IEEE 57 BUSES AND DIFFERENT COST INSTALLATION

Syste m

Monitors Installation Buses

IEEE 57

2, 6, 12, 19, 22, 26, 29, 30, 33, 35, 39, 43, 45, 46, 47, 50, 54 e 56. 2, 6, 12, 19, 22, 26, 29, 30, 33, 35, 39, 40, 42, 43, 45, 46, 47, 50 e 54. 2, 6, 12, 19, 22, 26, 29, 30, 33, 35, 39, 40, 41, 45, 46, 47, 50 e 54. 2, 6, 12, 19, 22, 26, 29, 30, 33, 34, 39, 40, 42, 43, 45, 46, 47, 50 e 54 2, 6, 12, 19, 22, 26, 29, 30, 33, 34, 39, 40, 41, 45, 46, 47, 50 e 54.

DRF

1,460 1,511

1,467

GENERATORS.

System IEEE 14 IEEE 30 CEMIG IEEE 57

1, 3, 8 e 12

Solution s 7 8

The system state variable can be estimated from the monitored ones through differential equations calculation, which relates voltage and current, if line parameters are known. It is necessary to emphasize that it is not necessary to know the load parameters and generation. This is guaranteed in the constraints formularization during the modeling of the optimization problem.

1,511

- RESULTS FOR DIFFERENT COST AND BUSES WITH Buses with Generators 1e2 1e2 1, 2, 3, 4, 5, 6, 9, 12, 15, 22, 29, 30, 34 e 47

V. RESULTS VALIDATION

1,467

Another scenario for transmission systems is the existence of monitors in generation buses, since generally they are monitored because of their great importance. In these cases it is necessary, considering the existence of these monitors to model the problem of monitors allocation. To achieve that objective, the variable referring to the bus that already has a monitor is fixed with value equal to 1. This assists problem solution, since tree branching will have one less expansion possibility. Table III presents the results for the same systems showed, but it considers previous monitors existence at generation buses and with the different installation costs. TABLE III

Comparing data in Table III with the ones in Table I, it is observed that the number of necessary monitors increases in general, even though it is better to use those which are already installed. For example, CEMIG’s transmission system had 14 previously installed monitors and it would be necessary to install 11 more. If the installed ones were not considered, it would be necessary to install 18 new monitors. The results were satisfactory. However from a computational point of view it can be improved. The algorithm uses internal routines of the MatLab® repeatedly to solve relaxed linear programming problems. Its performance can hardly be improved by constructing a specific program. This section showed examples where problem constraints can be easily modified in accordance with system under study varieties. This shows the flexibility of problem modeling and the ease of implementing it.

Monitors

5 or 6 13

16

25 or 26

30

20, 21 or 22

Fig. 3 – IEEE 14 buses test power system with PQ monitors represented by rhombus installed at buses 1, 2, 6, 8 e 9.

For example, for IEEE’s 14 bus system a possible allocation would be the installation of monitors in bars 1, 2, 6, 8 and 9, Fig. 3. The modeling of the problem guarantees that this allocation will measure the voltages v1, v2, v6, v8 e v9 and currents i12, i15, i23, i24, i25, i49, i610, i612, i613, i78, i79, i911, i914,. Considering known line impedances, other buses’ voltage and current lines can be calculated, even though loads or sources are not known. Some equations (19) to (22) representing these relationships are presented.

v 3 = ( L23 ⋅ p + R23 ) i 23 + v 2 v 4 = ( L24 ⋅ p + R24 ) i 24 + v 2

(19) (20)

5

v 5 = ( L15 ⋅ p + R15 ) i15 + v1 or

v 5 = ( L25 ⋅ p + R25 ) i 25 + v 2

(21) (22)

dx(t) p.x(t) = dt . where p is the differential operator:

It is necessary to use an integration method in equations (19) to (22) to recover voltage at non monitored buses. There are two forms to calculate voltage at bus 5, equations (21) and (22), so, this bus has a measurement redundancy. After calculating all of the system buses voltage, line currents are calculated in a similar manner, thus systems state variables are completely known and can be estimated at any time instant. This procedure is valid for any electrical power system.

Simulink Power Systems®, MatLab® package, was used to simulate IEEE’s 14 buses. This system data was found at the University of Washington’s website [20]. It was necessary to calculate system power flow to find the buses voltage value and reactive power injected by the synchronous condensers’ values in bars 3, 6 and 8 of this system. The condensers were replaced by capacitors, guaranteeing a voltage around 1 p.u. at all the buses. This simulation is presented in Fig. 4. The PQ monitors have been installed in buses 1, 2, 6, 8 and 9, considering generation in 1 and 2. A nonlinear load was inserted in bus 7, composed of a switch that modifies the load at 0,5s simulation time, causing a voltage sag at the system buses. In bus 3 a nonlinear load was inserted containing third, fifth and seventh order harmonic components. The integration method used is of third order, Bogacki-Shanpine, simulation step is fixed, with 10.000 samples for 1 second. To avoid rewriting all equations, it used the same system already simulated to estimate the other variables. This characteristic dismisses the differential equations calculation to estimate variables not monitored. It would be necessary for a program implementation to solve the discrete integration state variable. Currents and voltages that have been acquired by monitors in the simulation of Fig. 4 are saved in variables that store its instantaneous values and time instant of the signal. Thus, these values are injected as current controlled sources in a new system with the same load, and the synchronization is guaranteed for all system variables. Fig. 5 presents voltages measured and estimated in bus 14 during 0,45 and 0,55 seconds, to show the sag occurring in 0,50s. It is possible to observe clearly harmonic presence in the voltage signal, which occurs because this bar has a harmonic load inserted.

Fig. 5 – Voltages: Actual and estimated at bus 14 with non-linear loads at buses 3 and 7.

Fig. 4 – IEEE 14 buses power system simulated at Simulink Power Systems®.

The absolute difference between estimated and original signal was calculated, Fig. 6. This value has very small amplitude, guaranteeing that it is possible to rebuild the signals not monitored satisfactorily. It is also observed a small transition at the switching moment in the load of bus 7.

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The absolute difference between estimated and original signal was very small, about 0,1%, using this signal reconstruction methodology through Simulink. The disadvantage for this case was the necessity to know loads and sources value. The use of a specific transitory simulator that considers only current and voltage in the monitored points information allows obtaining systems variables without the necessity of loads knowledge.

The developed algorithm was tested using 14, 30, 57 and 118 buses IEEE power systems test case and the actual CEMIG system. Besides determining with precision all of the optimum solutions, and not just one, the algorithm seems sufficient to support studies of parameter variations such as: monitors cost, desired redundancy, previous existence of monitors, etc. Thus, it is a flexible tool in the treatment of actual cases of interest for electrical energy companies. VII. REFERENCES

Fig. 6 – The absolute difference between estimated and original signal at bus 14 with voltage sag.

VI.

CONCLUSION

The development of a branch and bound algorithm to solve the problem of the optimum power quality monitors allocation in electric systems, minimizing the total cost of the monitoring system, was detailed. The proposal modeling is based on system topology and guarantees its observability to power quality events with unchanged topology, such as the study of harmonic localization. This boarding does not demand load or generation knowledge at system buses. Different from works presented by some authors, commercial packages of combinatorial optimization had not been used to search optimum solutions. A bigger flexibility in the modeling, greater autonomy of computational programming and better knowledge of inherent mathematical procedures to the combinatorial problems was achieved with the developed algorithm. From the studies of 0-1 combinatorial optimization presented by Villela [16], that uses branch and bound technique, a new algorithm was developed. This is the biggest contribution of this work, not an innovative proposal but mainly for allowing the identification of every optimum solution, and different from what happens in the packages of combinatorial optimization found in the market. The algorithm was programmed in MatLab® using calculations of the upper bounds internal routine for resolution linear programming problems. It is desirable in the future to develop algorithm code integrally.

[1] Ma, H., Girgis, A. A., Identification and tracking of harmonic sources in a power system using a Kalman filter, IEEE Transactions on Power Delivery, vol. 11, no. 3, pp. 1659-1665, July, 1996. [2] Parsons, A. C., Grady, W. M., Powers, E. J., Soward, J. C., A Direction Finder for Power Quality Disturbances Based upon Disturbance Power and Energy, Proceedings of the 8th International Conference on Harmonics and Quality of Power, pp. 693 to 699, Outubro de 1998. [3] Szczupak, J., Electrical Power Network Pollution Estimation, T&D Latin America 2004, artigo 361, November, 2004. [4] TESHOME, A., Harmonic source and type identification in a radial distribution system, Proc. 1991 IEEE Industry Applications Society Annual Meeting, pp. 1605-1609, 1991. [5] Yu, K. K. C., Watson, N. R., Arrilaga, J., An Adaptive Kalman Filter for Dynamic Harmonic State Estimation and Harmonic Injection Tracking, IEEE Tran. On Power Delivery, pp.1577- 1584, abril de 2005. [6] Abur, A., Magnano, F. H., Optimum Meter Placement Against Contingencies, IEEE Power Engineering Society Summer Meeting, 2001. [7] Almeida, C.F. M., Metodologia para a Monitoração Eficiente de Variações de Tensão Curta Duração em Sistemas Elétricos de Potência, Masters dissertation, Electrical Engineering, Universidade de São Paulo, 2007. [8] Ammer, C., Renner, H., Determination of the Optimun Measuring positions for Power Quality Monitoring, 11th International Conference on Harmonics and Quality of Power, 2004. [9] Eldery, M. A., El-Saadany, E. F., Salama, M. M. A., Optimun Number and Location of Power Quality Monitors, IEEE International Conference on Harmonics and Quality of Power, 2004. [10] Eldery, M. A., El-Saadany, E. F., Salama, M. M. A., Vannelli, A., A Novel Power Quality Monitoring Allocation Algorithm, IEEE Transactions on Power Delivery, April, 2006. [11] Olguin, G., Vulnovich, F., Bollen, M. H. J., An Optimum Monitoring Program for Obtaining Voltage Sag System Indexes, IEEE Transactions on Power Systems, February, 2006. [12] Rakpenthai, C., Premrudeepreechacharn, S., Uatrongjit, S., Watson, N. R., An Optimum PMU Placement Method Against Measurement Loss and Branch Outage, IEEE Transaction on Power Delivery, January, 2007. [13] Reis, D. C. S., Duque, C. A., Villela, P. R. C., Algoritmo Branch and Bound para Solucionar o Problema da Alocação de Monitores de Qualidade de Energia Elétrica em Redes de Transmissão, VII Conferência Brasileira sobre Qualidade da Energia Elétrica, Santos, August, 2007. [14] Reis, D. C. S., Um Algoritmo Branch and Bound para o Problema da Alocação Ótima de Monitores de Qualidade de Energia Elétrica em Redes e Transmissão, Masters dissertation, Electrical Engineering, Universidade Federal de Juiz de Fora, August, 2007. [15] Goldbarg, M. C., Luna, H. P. C., Otimização combinatória e Programação Linear: Modelos e Algoritmos, 2 ed, Editora Elsevier, p. 409-423, 2005. [16] Villela, P. R. C., Instalação de Postos de Atendimento e Venda de Insumos numa Cooperativa Agrícola: Uma aplicação do Problema da Mochila 0-1, Masters dissertation, UFRJ, 1983. [17] Caprara, A., Fischett, M., Toth, P., Algorithms for the Set Covering Problem, Annals of Operations Research, 1998. [18] Kuzjurin, N. N., Combinatorial Problems of Packing and Covering and Related Problems of integer Linear Programming, Journal of Mathematical Sciences, 2002

7 [19] Li, X. Y., Stallman, M. F., Brglez, F., Effective Bounding techniques for Solving Unate and Binate covering problems, 42nd ACM IEEE Design Automation Conference, 2005. [20] Universidade Of Washington. IEEE Power Systems Test Case Archive. Available at http://www.ee.washington.edu/research/pstca/. March of 2007. [21] Operador Nacional Do Sistema (ONS). ONS Website– Brazilian Power System Technical Data. Avaiable at http://www.ons.org.br/conheca_sistema/dados_tecnicos.aspx. March of 2007.

VIII. BIOGRAPHIES Débora Costa Soares dos Reis (SM’ 02) was born in Niterói, Brazil, in 1979. She received her B. SC. and M.Sc degrees in Electrical Engineering from University of Juiz de Fora, in 2005 and 2007 respectively. Currently she is a temporary Professor in the Department of Energy in Electrical Engineering of Juiz de Fora University. Her research interests are power systems, power systems quality, artificial computer intelligence and optimization. Paulo Roberto de Castro Villela. was born in Caratinga, Brazil, in 1953. He received the B.S. degree in Electrical Engineering from the Instituto Tecnológico de Aeronáutica, Brazil, in 1977, and the M.Sc. and Ph.D. degree in Computer and Systems Enginnering from COPPE /UFRJ, Brazil, in 1981 and 1991, respectively. His pos-doctoral was in System Dynamics University of Illinois em Urbana - Champaing (UIUC), EUA, in 2001. He is a Professor in the Engineering Faculty at Federal University of Juiz de Fora (UFJF), in Brazil. His research interests are combinatorial optimization, and computer simulation. Carlos A. Duque (M’91) was born in Juiz de Fora, Brazil, in 1962. He received the B.S. degree in Electrical Engineering from the Federal University of Juiz de Fora, Brazil, in 1986, and the M.Sc. and Ph.D. degree from the Catholic University of Rio de Janeiro, in 1990 and 1997, respectively, in Electrical Engineering. Since 1989 he is a Professor in the Engineering Faculty at Federal University of Juiz de Fora (UFJF), in Brazil. His research interests are power quality analysis, digital instrumentation and digital signal processing. Paulo F. Ribeiro (M'78–SM'88–F'03) received a BS in Electrical Engineering from the Universidade Federal de Pernambuco, Recife, Brazil, completed the Electric Power Systems Engineering Course with Power Technologies, Inc. (PTI) 1, and received the Ph.D. from the University of Manchester - UMIST, Machester-UK. Presently, he is a Professor of Engineering at Calvin College, Grand Rapids, Michigan, on sabbatical at the Center for Advanced Power Systems(CAPS) at Florida State University, Tallahassee, Florida. Dr. Ribeiro is active in IEEE, CIGRE and IEC working groups on power quality.