Abstract—This paper presents the solution of a conventional Optimal Power Flow (OPF) model by using the optimization toolbox provided by Matlab. For the OPF model solution it is only required to be defined into two separated M files: i) the given objective function and ii) the equality and inequality constraints sets, without the need of defining associated Gradient and Hessian expressions required by gradient search-based algorithms. Therefore the complexity degree related to the proposed OPF model implementation is similar to that required by commercial specialized packages for solving optimization models, but with the advantage that Matlab is commonly the available software for research and academic purposes in electric engineering. This hugely reduces the time to the implementation, and for obtaining results, of a conventional OPF analysis of electric energy systems.

The prowess and numerical accuracy of the proposed implementation are demonstrated on the WSCC 3-machine, 9-bus test system and the Mexican 46-machine, 190-bus test system.

Index Terms— Electric energy systems, steady state, power flow, optimal power flow, optimization, Matlab.

I. INTRODUCTION onventional power flow has been the analysis tool that is routinely executed in control centers to assess the system

steady state operating condition [1]. However, opposite to Optimal Power Flow (OPF) analysis, power flow analysis cannot deal with economic and security aspects in a unified reference frame. The concept of optimal power flow, introduced by Dommel and Tinney in the early 1960's [2], has received great attention since its early application to power systems analysis [3]. OPF is a nonlinear optimization problem, where a specific objective function must be optimized while satisfying operational and physical constraints of the electric power system [4]. A large variety of optimization techniques have been employed to solve OPF problems, such as linear, nonlinear, quadratic, mixed integer programming, interior-point methods and Newton-based methods [2], [3] and [5].

From the power systems planning viewpoint the OPF model solution provides the optimal settings for the variables of a power network [2]. From the power system operation and control viewpoints, an OPF solution gives an answer to adjust

A. Pizano-Martínez and E.A. Zamora-Cárdenas are with the Department of Electromechanical Engineering, Instituto Tecnológico Superior de Irapuato, at Guanajuato, México. (e-mail: alpizano@itesi.edu.mx; eazamora@itesi.edu.mx).

C.R. Fuerte-Esquivel is with the Faculty of Electrical Engineering, Universidad Michoacana de San Nicolás de Hidalgo, 58030 Morelia, México (cfuerte@umich.mx).

J. Segundo-Ramírez is with Universidad Autónoma de San Luis Potosi, Centro de Investigación y Estudios de Posgrado de la Facultad de Ingeniería (CIEP-FI). (e-mail: j2ramirez@hotmail.com).

available controls in order to meet the energy demand in the most economically manner while keeping within bounds all the constraints imposed on the system. OPF studies are being used more and more by engineers, but further applications ranging from planning, operation and control of modern power systems are of great interest and must be investigated [6]. In order to speed up the research of such those applications, researchers have put attention on commercial software packages for solving a large variety of OPF models.

Nowadays the commercial software packages commonly used to solve OPF models are AMPL [7] and GAMS [8]. On the one hand, by way of example, AMPL software has been employed to; solve a OPF model with complementary constraints [9], identify and analyze saddle-node bifurcations and limit-induced bifurcations of power systems [10], solve different OPF models used to investigate the effects of reactive power limit modeling on Maximum System Loading and Active and Reactive Power Markets [11]. More recently, AMPL has been applied to solve a OPF model that considers voltage stability constraints in order to control voltage stability of power systems [12], among other applications. On the other hand, GAMS software has been employed to; solve a voltage stability constrained OPF from a market-clearing point of view [13], compute the solution of a OPF model of FACTS systems with static security constraints [14] and solve a transient stability constrained OPF model [15].

This paper presents implementation and solution of a conventional OPF model by using the optimization toolbox of Matlab [16]. Section II of the paper shows the general OPF formulation. The power system components modeling and the explicit OPF formulation are presented in Section III. The computational implementation of the OPF model is described in Section IV. The prowess of the proposed implementation is illustrated by means of numerical examples in Section V. The work conclusions are given in Section VI.

II. GENERAL OPTIMAL POWER FLOW MODEL Conventional OPF is considered as a non-linear

programming problem where the objective is to minimize a given cost function subjected to a finite set of equality and inequality constraints. The general OPF general formulation is given by,

Minimize ( )f y (1) Subject to ( ) = 0 h y (2)

( ) ≤ 0g y (3) ≤ ≤y y y (4)

where f(y) is a real-valued objective function in n (f: ℜn→ℜ1)

C

Conventional Optimal Power Flow Analysis Using the Matlab Optimization Toolbox

A. Pizano-Martínez, C. Fuerte-Esquivel, E. A. Zamora-Cárdenas, and J. Segundo-Ramírez

XII Reunión de Otoño de Potencia, Electrónica y Computación, INTERNACIONAL ROPEC'2010

ISBN: 978-607-95476-1-5

Artículo aceptado por el comité revisor de la ROPEC 2010 para ser presentado como ponencia oral 189

to be minimized, h(y) is a set of r real-valued functions in n (h: ℜn→ℜr) corresponding to equality constraints representing both power flow mismatch equations and control equations, g(y) is a set of m real-valued functions in n (g: ℜn→ℜm) representing the set of inequality constraint functions which prevents the dispatching of generation that otherwise will lead to violations of system limits. The vector of system variables y∈S⊆ℜn, in the search space S, includes the set of sv state variables x (x∈ℜsv) and the set of cv control variables u (u∈ℜcv), with lower and upper limits represented by y and y ,

respectively. A point y in S that satisfies the set of equality and inequality constraints is a feasible point yF, the infinite set of feasible points defines the feasible region F. A feasible point

Fy is the optimum point y* sought if there is a neighborhood N of y* such that f(y*) ≤ f(y) for any y∈N∈F. The set of inequality constraints consisting of those variables and functions explicitly enforced to specified values at any stage of the search of y*, and it is called active set A. The set of inequality constraints that are active at the optimum solution is called binding set B. All equality constraints are regarded as active at any feasible point since they must be satisfied unconditionally at the optimal solution y*of (1)-(4).

III. POWER SYSTEM AND OPTIMAL POWER FLOW EXPLICIT MODELING

In order to present the explicit structure of the OPF mathematical model, the previous general formulation (1)-(4) is developed according to the specific optimization model considered in this work. The explicit OPF formulation is given in polar coordinates and according to the model of each power system component, as follows.

A. Power system Modelling The most common components of a power system are

generators, loads, shunt compensation elements, transmission lines and transformers. These elements are modeled as given below [17].

1) Generators The generator is modeled as a controllable source of

complex power, gi gi giS P jQ= + (5)

where Pgi and Qgi are the active and reactive power generation levels, with lower and upper bounds giQ , giP and giP , giQ , respectively. Since most of the based load of a power system is provided by thermal units, they are considered in this work, such that the active power production cost is described by the following quadratic function,

2( ) ( ) ( )gi i i gi i giC P a b P c P= + + (6)

where ai, bi and ci are the cost curve coefficients for the generation bus i. The voltage magnitude Vi upper and lower bounds for the generation bus i are giV and giV , respectively. 2) Loads

The power demand is considered as constant complex power consumption,

li li liP jQ= +S (7) where Pli and Pli are the active and reactive power at each load bus i. The upper and lower bounds of the voltage magnitude Vi for the load substation are liV and liV , respectively.

3) Shunt Compensation Element Shunt elements are mathematically described by means of a

magnitude voltage dependent complex power,

inj injsh i i iP jQ= +S (8)

The terms injiP and inj

iQ are the active and reactive power injections, respectively, through the shunt compensation admittance connected at the compensated bus i,

2inji i iP V G= (9)

2inji i iQ V B=− (10)

where Vi is the voltage magnitude at the compensation node with upper and lower limits shiV and shiV , respectively. The compensation admittance is Y=G+jB, where G and B are the shunt conductance and suceptance of the shunt compensator.

4) Transmission Lines The transmission lines are represented by the π equivalent

circuit of Fig. 1, where Ij and Ej are the injected current and voltage phasors at node i (i=k,m). R, L and Bc are the series resistance, series inductance and the shunt suceptance. The current-voltage relation of the equivalent circuit is given by,

k kk km k

m mk mm m

= I Y Y EI Y Y E (11)

where,

( )2kk mm km kk kk

Bcj G jB= = + = +Y Y y (12)

km mk km km kmG jB= = − = +Y Y y (13)

2 2( )kmRG

R Lω=

+; 2 2( )km

LBR L

ωω

= −+

(14)

The power injected at node i through the transmission element connected between nodes i and j , where i=k;m, j=k,m, i≠j, is mathematically described as follows,

* *( )i i i i i i ii i ij jP jQ= + = = +S E I E Y E Y E (15) Applying the Euler’s identity to the voltage phasors Ei and Ej, and then separating into real and imaginary parts, we obtain,

2 ( ) ( )inji i ii i j ij i j ij i jP V G VV G Cos B Sinθ θ θ θ = + − + − (16)

2 ( ) ( )inji i ii i j ij i j ij i jQ V B VV G Sin B Cosθ θ θ θ = − + − − − (17)

Fig. 1. Equivalent circuit of the transmission line

5) Conventional Transformers In order to be able of simulating networks with transformers

having changers at off nominal tap ratio position at either primary or secondary side, the two winding transformer is

XII Reunión de Otoño de Potencia, Electrónica y Computación, INTERNACIONAL ROPEC'2010

ISBN: 978-607-95476-1-5

Artículo aceptado por el comité revisor de la ROPEC 2010 para ser presentado como ponencia oral 190

modeled with complex taps on both primary and secondary windings. The magnetizing branch non-linearity under saturated conditions is also considered in the model to account for the core losses. The schematic equivalent circuit is shown in Fig 2.

The primary winding is represented as an ideal transformer having complex tap ratios Tv:1 and Ti:1 in series with the impedance Zp, where Tv=Ti*=Tv∠φtv. The * denotes the conjugate operation. Also, the secondary winding is represented as an ideal transformer having complex tap ratios Uv:1 and Ui:1 in series with the impedance Zs, where Uv=Ui*= Uv∠φuv. The transfer admittance matrix relating the primary voltage Vp and current Ip to the secondary voltage Vs and current Is in the two-winding transformer is given by [17],

P PP PS PP PS P

S SP SS SP SS S

G G B BjG G B B = +

I VI V

(18)

where 2

2 2

1( 1) 2 21 2

vPP

F U R F RG

F F+ +

=+

; 2

2 2

1 2 2( 1)1 2

vPP

F R F U RB

F F− +

=+

2

2 2

1( 3) 2 41 2

vSS

F T R F RG

F F+ +

=+

; 2

2 2

1 4 2( 3)1 2

vSS

F R F T RB

F F− +

=+

( ) 2 21 11cos( ) 2sin( ) 1 2PS v vG T U F F F Fφ φ=− + +

( ) 2 21 12cos( ) 1sin( ) 1 2PS v vB T U F F F Fφ φ= + +

( ) 2 22 21cos( ) 2sin( ) 1 2SP v vG T U F F F Fφ φ=− + +

( ) 2 22 22cos( ) 1sin( ) 1 2SP v vB T U F F F Fφ φ= + +

2 211 Rv S v P eqF T R U R= + + ; 2 2

12 v S v P eqF T X U X X= + +

1 0 0R ( ) ( )eq P S P S P S S PR R X X G R X R X B= − − −

1 0 0( ) ( )eq P S P S P S S PX R R X X B R X R X G= − + −

0 01 S SR R G X B= − ; 0 02 S SR R B X G= −

0 03 P PR R G X B= − ; 0 04 P PR R B X G= −

1 tv uvφ φ φ= − ; 2 uv tvφ φ φ= − The active and reactive power injections at node i through

the transformer connecting nodes i and j, where i=p,s, j=p,s, i≠j, are respectively computed by using (16) and (17), but using conductances and suceptances of (18).

Fig. 2. Equivalent circuit of the two winding transformer

B. OPF Explicit Model The OPF explicit formulation readily derives from the

previous power system model.

1) Objective Function The objective function f(y*) is the minimization of the total

active power generation cost. According with (6), the objective function is formulated as follows,

2

1( ) ( ) ( )

gN

i i g i i g ii

f a b P c P=

= + +∑y (19)

where ai, bi and ci are the cost curve coefficients for the generation bus i. Ng is the number of generators, whose individual generation power level is Pgi.

2) Equality Constraints In order to represent the steady state, the energy balance of

the power system must be unconditionally satisfied. This is enforced by means of the active and reactive power balance at each bus according to the following equality constraint set,

0,1,2,...,

( ) , 1,2,..., |0

gi li inj ij i b

b ggk lk inj kj k

P P Pi N

k N k NQ Q Q∈

∈

− − = = = = ∉− − =

∑

∑h y (20)

where Nb is the number of buses. The active and reactive power levels, Pgj and Qgj, respectively, are provided by the generation controllable sources at the generation bus j(j=i,k), as indicated by (5). The active Plj and reactive Qlj power loads are the complex power consumption, as given by (7). ∑j∈i,k is the set of nodes adjacent to node j, whilst Pinj j and Qinj j are active and reactive power flows injected at bus j through the network elements, according to (9), (10), (16) and (17) . It is very important to point out that the generated reactive power Qgi is a function of the system variables and does not have a scheduled value, therefore the reactive power balance constraint can be only stated for non generation buses (k∉Ng). However, the reactive power balance at generation buses is achieved according to the procedure applied to handle the reactive generation limits, as explained below.

3) Inequality Constraints Physical and operating limits constrain the practical steady

state operation of power systems components. The physical and operating limits of generators and substations are mathematically described by the following inequality sets,

1,2,...,, 1,2,...,gi gi gi g

bj j j

P P P i Nj NV V V

≤ ≤ == =≤ ≤ Y (21)

{ }( ) , 1,2,...,gi gi gi gQ Q Q i N= ≤ ≤ =g y (22)

It must be pointed out that the active power generation Pgi and all the bus magnitude Vj limits are simply inequality constraint on variables, whilst the generator reactive power limits are modeled as a set of functional inequality constraints. In other words, the reactive power generation level Qgi in (22) is from (17) and (20), as given by the following function,

2 ( ) ( )gi li i ii i j ij i j ij i jj i

Q Q V B V V G Sin B Cosθ θ θ θ∈

= + − + − − − ∑ (23)

The relation (23) means that the reactive power balance generation bus i is always achieved when the generator is inside its reactive generation limits. When the generator hits either its lower Qgi or upper giQ bound the inequality

constraint (22) is activated by the optimization algorithm, it

XII Reunión de Otoño de Potencia, Electrónica y Computación, INTERNACIONAL ROPEC'2010

ISBN: 978-607-95476-1-5

Artículo aceptado por el comité revisor de la ROPEC 2010 para ser presentado como ponencia oral 191

then automatically becomes into an equality constraint in order to enforce the reactive power generation level Qgi to be the violated limit v

giQ ,

0vgi giQ Q− = (24)

where Qgi is defined by (23). Note that constraint (24) not only avoids the violation of the reactive power generation limits, but also represents the reactive power balance equation.

IV. OPTIMAL POWER FLOW PRACTICAL IMPLEMENTATION The explicit OPF model (19)-(22) is implemented into a

computational algorithm in order to be solved by means of the optimization toolbox of Matlab [16]. From an optimization point of view, the OPF model represents a continuous non linear constrained optimization problem, which can be solved by using the function “fmincon” of that optimization toolbox. The function uses a Sequential Quadratic Programming optimization algorithm, started with input/output arguments to configure the optimization parameters, set the model to be optimized and display information.

A. Arguments of the fmincon Function As any Matlab function, the fmincon function deals with

both input IA and output OA arguments. The general form of this function is,

[ ] fmincon( )A A=O I (25) where IA and OA are sets of input and output arguments, respectively. Tables I and II briefly show and describe the main elements of these argument sets according to the order they must be provided to the fmincon function. Details of the parameters opt∈IA and output∈OA are given in [16].

B. OPF Computational Algorithm Implementation In order to implement a general OPF program for digital computer, the OPF model is solved by using the fmincon function, which is executed according to the computational procedure described in Fig. 3. The proposed OPF computational implementation starts reading the power system data and convergence tolerance. The power system data are converted to pu in order to normalize the system quantities, but also to avoid optimization scaling problems. The system variables are initialized as follows, nodal voltage magnitudes V are set to 1pu and angles θ are set to 0 rad, as in conventional power flow analysis. The active power generation levels Pg are initialized according to a network lossless Economic Dispatch (ED) analysis. This analysis is formulated as a nonlinear programming problem, where the objective function (19) is the considered in the OPF model, the single equality constraint is the active power balance between the total generation and the total load, whilst the inequality constraints only correspond with the generation active power limits. The constraints and the objective function

TABLE I DESCRIPTION OF THE INPUT ARGUMENTS IA

Name Description

@fun The handle of the M-function file containing the objective function, in our case; @(X)objfun_OPF(X)

X The vector containing the numerical value of the initial condition of

system variables θ, V and Pg, i.e. X =[θ0, V0, Pg0]

A All these parameter are not of interest in this work, they refer to linear equality and inequality constraints. They simply are set as empty arguments ([ ]).

B Aeq Beq

Lb Vector of lower bounds value of variables, in this work applied to the sets V and Pg

Up Vector of upper bounds value of variables, in this work applied to the sets V and Pg

nlcon The handle of the M-function file containing the non linear equality and inequality functional constraints, here; @(X)constraints_OPF(X)

opt This structure provides optional parameters for the optimization process, which are set by means the optimset function of Matlab

TABLE II DESCRIPTION OF OUTPUT ARGUMENTS OA

Name Description

X This vector contains the numeric value of the system variables at the OPF model solution, and is the same vector considered in IA

(see Table 1) fval Value of the objective function at the solution X

eflag Describes the exit condition of fmincon; if eflag>0, the convergence was reached. If eflag=0, the maximum number of iteration was exceeded. If eflag <0, there was not convergence

output Structure that contains information of the optimization process results

λ Vector of Lagrangian multipliers at the solution X Gra Value of the Gradient of the objective function at the solution X Hess Value of the Hessian of the objective function at the solution X

Fig. 3. OPF computational algorithm

are written in two separated Matlab functions (M1 and M2), which are called by the fmincon function executed from the implemented ED function.

The vector of initial conditions X is then passed to the implemented OPF function, where the fmincon function is newly executed to obtain the solution of the OPF model. The OPF constraints (20)-(22) are written in another Matlab function (M3) called by fmincon, along with the objective function (19) written in M2, from the implemented OPF function. The optimal solution X is used to compute the network active and reactive power flows, which are finally reported in a text file and on the computer display.

V. NUMERICAL EXAMPLES In order to illustrate numerically the prowess of the

implementation proposed in Section IV to carry out the OPF analysis, the IEEE 3-machine 9-bus test system [18], the 10-

XII Reunión de Otoño de Potencia, Electrónica y Computación, INTERNACIONAL ROPEC'2010

ISBN: 978-607-95476-1-5

Artículo aceptado por el comité revisor de la ROPEC 2010 para ser presentado como ponencia oral 192

machine 39-bus New England system [19] and a 49-machine 190-bus model of the Mexican system are considered in the numerical examples [20]. The design of the numerical examples and the conventional OPF analysis results are given below.

A. Three- Machine, Nine-Bus System This section presents the results for the IEEE 3-machine 9-

bus test system obtained by the proposed OPF implementation model. For the study the convergence tolerance for the fmincon function was set to 1x10-6. The voltage magnitude limits for all nodes are set to 0.95≤Vi≤1.05 pu, the reactive power limits for all generators were set to -20≤Qgi≤20 MVAR, whilst the active power lower limits for generators connected to nodes 1, 2 and 3 are 0≤Pg1≤200MW, 0≤Pg2≤ 150MW and 0≤Pg3≤100MW, respectively. The cost functions of generators were taken from [21]. Taking into account parameters and system data from [18, pp.171-172], the optimum steady-state operating point OP is computed by conventional OPF analysis in a CPU time of 4.26sec. The optimal active and reactive power dispatches and cost are shown in Table III.

The Table III shows that all voltage magnitudes, active and reactive power generation levels are inside limits. However, it must be pointed out that because of the voltage magnitudes of nodes 1, 2 and 9 hit their upper limits, the corresponding inequality constraints were activated and belong to the binding set B, therefore they have a value equal to the violated limit. The total generation costs obtained by the proposed implementation are compared to those obtained using different OPF methods reported in the literature in Table IV. The generation cost obtained by the proposed implementation compare well with those reported in [22] and [15], where Newton’s method and GAMS are used to solve the OPF model, respectively. The cost reported in [23] is slightly higher; despite they use an evolutionary algorithm.

TABLE III OPERATING POINT COMPUTED BY OPF FOR THE NINE-BUS SYSTEM

Node V(pu) θ (deg) Pgi (MW) Qgi (MVAR) 1 1.05 0.000 105.92 17.30 2 1.05 2.613 113.05 4.77 3 1.04 2.539 99.25 -15.56 4 1.04 -3.196 5 1.02 -6.104 6 1.03 -5.098 7 1.05 -1.064 8 1.04 -3.004 9 1.05 -0.515 Total Generation Cost 1,132.18 ($/hr)

TABLE IV COMPARISON OF TOTAL GENERATION COSTS OF THE 9-BUS SYSTEM

OPF Approaches Proposed [22] [23] [15]

Cost ($/hr) 1, 132.18 1, 132.18 1,132.30 1,132.18

B. New England System The data of this system were taken from pp. 224-226 of

[18], whilst generator cost curves are as reported in Table III of [21]. The generators active and reactive power limits are set as indicated in columns 2, 3, 4 and 5 of Table V. The steady

state voltage magnitude limits for generation and load nodes are set to 0.95≤Vgi≤1.09 pu and 0.95≤Vli≤1.07 pu, respectively. The steady state operating point OP is computed by the implemented OPF in a CPU time of 14.89 sec, the optimum complex power dispatch and total generation cost corresponding to this optimal point are given in Table V.

The resulting active and reactive power dispatches are inside limits, however the reactive output powers of generators connected at nodes 30 and 37 were set to their corresponding violated limit. The total generation costs obtained by the proposed implementation are compared to those obtained using different OPF methods reported in the literature in Table VI. These comparisons indicate that the most economic dispatch is the one obtained by the proposed implementation.

TABLE V GENERATORS RATINGS AND OPF DISPATCH OF THE NEW ENGLAND SYSTEM

Node MW Limits MVAR Limits OPF Dispatch Lower Upper Lower Upper MW MVAR

30 0 350 140 400 240.13 140.00 31 0 650 -100 300 566.48 293.04 32 0 800 150 300 641.13 206.98 33 0 750 0 250 631.93 145.08 34 0 650 0 167 510.60 124.54 35 0 750 -100 300 654.03 195.00 36 0 750 0 240 560.07 115.69 37 0 700 0 250 535.88 0.00 38 0 900 -150 300 832.73 20.48 39 0 1200 -100 300 966.42 -28.71

Total Generation Cost 60,908.54 ($/hr)

TABLE VI COMPARISON OF TOTAL GENERATION COSTS OF THE NEW ENGLAND SYSTEM

OPF Approaches Proposed [23] [15]

Cost ($/hr) 6,0908.54 60,936.51 60,918.66

C. 46- Machine, 190-Bus Mexican Power System The implemented computational algorithm analysis is

applied to carry out the OPF analysis on a Mexican Interconnected Power System (MIS) equivalent model that consists of 190 buses, 46 generators, 90 loads and 265 transmission lines operating at voltage levels ranging from 400 kV to 115 kV [18]. The schematic diagram of this system is shown in Fig. 4.The voltage magnitude limits for all nodes are set to 0.94≤Vi≤1.07 pu, whilst the active and reactive power limits for all generators were set to 0≤Pgi≤1250 MW and -250≤Qgi≤350 MVAR, respectively. The OPF analysis computes an optimal steady state operating point with total generation cost of 21138.54$/hr. There is not active and reactive power limits violations, and only eight magnitude voltages hit their limits. The CPU time required to carry out the OPF solution was of 545sec.

XII Reunión de Otoño de Potencia, Electrónica y Computación, INTERNACIONAL ROPEC'2010

ISBN: 978-607-95476-1-5

Artículo aceptado por el comité revisor de la ROPEC 2010 para ser presentado como ponencia oral 193

Fig.4. Schematic diagram of the Mexican interconnected power system [20].

Fig.5. Optimal Active Power Dispatches.

VI. CONCLUSIONS A new computational implementation of a conventional

OPF model has been presented. The prowess of this proposal has been demonstrated by means of numerical examples. The numerical results obtained by the proposed implementation compare well with those reported in the literature. As it would be expected, the fmincon function of Matlab does not run too fast, this might be because of the gradient vector and hessian matrix elements associated with the optimization model are obtained numerically, i.e., they are not provide by the user. However, this fact allows to undergraduate and graduate students, as well as researchers, speeding up the OPF models implementation and in getting of research results, since avoids the need of developing complex programming routines, but overall circumvents the difficulty of clearing hidden programming errors, which result in saving a lot of very valuable time.

VII. REFERENCES [1] B. Stott, “Power system dynamic respond calculations”. Proc. of the

IEEE, Special Issue on Computers in Power System Operations, Vol. 67, No. 2, pp. 219-241, Feb. 1979.

[2] M. Huneault and F. D. Galiana, "A survey of the optimal power flow literature". IEEE Trans. on Power Syst., Vol.6, No.2, pp.762-770, May 1991.

[3] J.A. Momoh, "Optimal Power Flow with Multiple Objective Functions". Proc. of the 1989 IEEE North American Power Symposium, pp.105-108.

[4] J. A. Momoh, M.E. El-Hawari, and R. Adapa, “A review of selected optimal power flow literature to 1993: part I: nonlinear and quadratic programming approaches”. IEEE Trans. Power Syst., Vol. 14, No. 1, pp. 96-104, February 1999.

[5] J. A. Momoh, M.E. El-Hawari, and R. Adapa, “A review of selected optimal power flow literature to 1993 part II: Newton, linear programming and interior point methods”. IEEE Trans. Power Syst., Vol. 14, No. 1, pp. 105-111, February 1999.

[6] J.A. Momoh, R.J. Koessler, M.S. Bond, B. Stott, and D. Sun, A. Papalexopoulos and P. Ristanovic, “Challenges to optimal power flow”. IEEE Trans. Power Syst. Vol. 12, No. 1, pp. 444-455, Feb. 1997.

[7] KNITRO. [Online]. Available: http://www.ziena.com.

[8] A. S. Drud, GAMS/CONOPT, ARKI Consulting and Development, Bagsvaerdvej 246A, DK-2880 Bagsvaerd, Denmark, 1996, available at http://www.gams.com/.

[9] W. Rosehart, C. Roman, and Antony Schellenberg, “Optimal Power Flow with Complementary Constraints”. IEEE Trans. Power Syst., Vol. 20, No. 2, pp. 813-822, May 2005.

[10] R. J. Avalos, C.A. Cañizares, F. Milano, and A. J. Conejo, “Equivalency of Continuation and Optimization Methods to Determine Saddle-Node and Limit-Induced Bifurcations in Power Systems”. IEEE Trans. on Circuits and Systems, Vol. 56, No. 1, pp. 210-223, January 2009.

[11] B. Tamimi, C.A. Cañizares, and S. Vaez-Zadeh, “Effect of Reactive Power Limit Modeling on Maximum System Loading and Active and Reactive Power Markets” IEEE Trans. on Circuits and Systems, Vol. 25, No. 2, pp. 1106-1116, May 2010.

[12] V.J. Gutierrez-Martinez, C.A. Cañizares, C.R. Fuerte-Esquivel, A. Pizano-Martínez, and X. Gu, “Neural-network security-boundary constrained optimal power flow”. Transactions on Power Systems, 2010.

[13] F. Milano, C. A. Cañizares, and A. J. Conejo, "Sensitivity-Based Security-Constrained OPF Market Clearing Model". Proc. of the 2006 IEEE PES Power Systems Conference & Exhibition, pp.418-427.

[14] R. Zárate-Miñano, A. J. Conejo, and F. Milano, “OPF-based security redispatching including FACTS devices”. IET Generation, Transmission & Distribution, vol.2, no.6, pp. 821-833, 2008.

[15] R. Zárate-Miñano, T. Van Cutsem, F. Milano and A. J. Conejo, “Securing transient stability using time-domain simulations within an optimal power flow”. IEEE Trans. on Power Syst., vol.25, no.1, pp. 243-253, Feb. 2010.

[16] The MathWorks, Inc., “Matlab Optimization Toolbox,” Users Guide Version 2, available at http://www.mathworks.com.

[17] E. Acha, C. R. Fuerte Esquivel, H. Ambriz Pérez, C. Angeles Camacho: FACTS: Modelling and Simulation in Power Networks. John Wiley and Sons, 2004.

[18] P. W. Sauer and M. A. Pai. Power System Dynamics and Stability. Prentice-Hall, 1998.

[19] M. A. Pai, Energy Function Analysis for Power System Stability. Kluwer, 1989.

[20] A. R. Messina and V. Vittal. “Assessment of nonlinear interaction between nonlinearly coupled modes using higher order spectra”. IEEE Trans. on Power Syst., vol.20, no.1, pp. 375-383, Feb. 2005.

[21] T. B. Nguyen and M. A. Pai. “Dynamic security-constrained rescheduling of power systems using trajectory sensitivities”. IEEE Trans. on Power Syst., vol.18, no.2, pp. 848-854, May 2003.

[22] Pizano-Martínez A., Fuerte-Esquivel C.R., and Ruiz-Vega D., “Global Transient Stability-Constrained Optimal Power Flow using SIME Sensitivity analysis,” IEEE-PES General Meeting,, 25-29, July, 2010, Minneapolis, USA, 8 pages.

[23] H.R. Cai, C.Y. Chung, and K.P. Wong, “Application of differential evolution algorithm for transient stability constrained optimal power flow,” IEEE Trans. on Power Syst., Vol.23, No.2, May 2008.

VIII. BIOGRAPHIES Alejandro Pizano-Martinez received his Ph.D. degree from Universidad Michoacana, México in 2010. Currently, he is Associated Professor at Instituto Tecnológico de Estudios Superiores de Irapuato (ITESI), México.

Claudio R. Fuerte-Esquivel (M’1991, SM’04) received his Ph.D. degree from the University of Glasgow, Scotland, UK in 1997. Currently, he is Professor at Universidad Michoacana, México.

Enrique A. Zamora Cárdenas received his BEng degree (Hons) from University of Colima, México in 2001, and his M.Sc. degree from Universidad Michoacana, México, in 2004. He is currently working towards his Ph.D. degree at Universidad Michoacana. He is also Associated Professor at Instituto Tecnológico de Estudios Superiores de Irapuato (ITESI), México.

Juan Segundo-Ramirez recived his Ph.D from UMSNH in 2010. He is currently with Universidad Autónoma de San Luis Potosi, Centro de Investigación y Estudios de Posgrado de la Facultad de Ingeniería (CIEP-FI).

XII Reunión de Otoño de Potencia, Electrónica y Computación, INTERNACIONAL ROPEC'2010

ISBN: 978-607-95476-1-5

Artículo aceptado por el comité revisor de la ROPEC 2010 para ser presentado como ponencia oral 194