TOTAL TRANSFER CAPABILITY COMPUTATION AND IMPROVEMENT IN INTERCONNECTED ELECTRIC POWER SYSTEMS F. M. Echavarren, E. Lobato, L. Rouco Universidad Pontificia Comillas Madrid, SPAIN
[email protected] Abstract – Total Transfer Capability (TTC) is an important index in power systems technical and economical analysis. The TTC computation in this paper is based on continuation techniques. In addition, a parameterization of the predictor step in the continuation process is used to speed up the TTC computation. The TTC computation is one of the most important steps in transfer capability analysis. However, additional information about TTC sensitivity is needed. A LP-based optimization algorithm is presented in this paper to improve the TTC in interconnected electric power systems employing active power generation redispatch as control variables. The performance of the methods is illustrated with a scenario of the Spanish electric power system.
Keywords: Power System Security, Total Transfer Capability, Sensitivity, Optimization, Active Power Generation Redispatch 1 INTRODUCTION Total Transfer Capability (TTC) is an important index in power systems technical and economical analysis. The TTC is defined as the amount of electric power that can be transferred over the interconnected transmission network under some established security system conditions [1, 2]. The security conditions considered are the maximum branch power flow, both the maximum and minimum bus voltages and distance to the point of voltage collapse. Other possible security conditions, such as transient stability constraints, are not considered in this paper. Therefore, TTC computation implies the definition of three main concepts [3]: 1) Base case: The base case definition constitutes an important task in TTC computation. The base case is assumed to be a secure and solved case, and must represent typical load and generation system patterns. 2) Transfer direction: A transfer is specified by changes in power injections at buses in the network. Thus generation of source areas must increase their generation pattern, whereas load buses of sink areas must increase their load pattern. All these generation and load increases are parameterized by a transfer capability parameterization factor. 3) System security limits: The energy transfer from source areas to sink areas is limited by system security limits. Hence, the definition of these
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system security limits determines the total transfer capability amount. The system security limits usually considered are line maximum flows and bus voltage limits. Many TTC computation methods are presented in the literature. These methods can be divided into two types: continuation techniques and non-linear optimization techniques. Continuation techniques are based on progressive power flow solutions as the transfer capability is increased from zero to its maximum value [4, 5]. The continuation techniques also provide special tools to speed up power flow equations re-convergence and to avoid new power flow solution oscillation or divergence. In the other hand, optimization methods formulate the problem as an optimization problem [3, 6], with a linear objective function (the transfer capability parameterization factor), non-linear equality constraints (the power flow equations) and non-linear inequality constraints (power system security limits). Thus, this optimization problem is solved using non-linear optimization techniques, such as interior point or reduced gradient. Both continuation and optimization techniques are based in the AC full system model and obtain the exact TTC amount. However, other techniques are presented in the literature, such as Power Transfer Distribution Factors (PTDF) and DC system model based approximations. On one hand, the PTDF are the sensitivities of the line flows with respect to the active power bus injection of the system generators [3], thus minimum increase in active power generation to reach line flow limits may be calculated. On the other hand, DC system model allows to obtain an approximation of the TTC formulating a linear optimization problem [7]. The TTC computation in this paper is based on an optimization problem formulation used for security margins computation [8, 9], and solved using continuation techniques. However, the main disadvantage of the continuation technique is its computational effort. In order to speed up the continuation technique applied to TTC computation, a parameterization of the predictor step in the continuation process is used in this paper [10]. The parameterization consists of resizing the predictor step in order to minimize the distance to the nearest system security condition. The TTC computation is one of the most important steps in transfer capability analysis. However, if TTC
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amount would be less than minimum system security criteria establishes, control actions may be considered to increase the TTC. Therefore, additional information about TTC sensitivity is needed. Many TTC sensitivity studies have been proposed in the literature to evaluate the effect of arbitrary parameters on the transfer capability. S. Greene, et al., develop in [11] a first order sensitivities of the transfer capability parameterization factor with respect to arbitrary parameters of the power system. These sensitivities are employed in [12] to identify the most effective control actions to improve the TTC. Other authors formulate the TTC improvement problem as a security constrained optimal power flow (SCOPF). A. M. Leite da Silva, et al., present in [13] a SCOPF formulation for new generation planning problem where the main objective function is to maximize the TTC improvement. Finally, there are some authors that propose to combine continuation techniques and SCOPF. Moyano, et al., expose in [14] an algorithm based on both continuation techniques and SCOPF to evaluate the demand buses impact on TTC value. This paper presents a LP-based optimization generation redispatch algorithm for TTC improvement. First order sensitivities of the TTC with respect to the active power generation are considered. The algorithm selects both the optimal location of generation buses, and their corresponding power redispatch. The objective function consists of minimizing the total active power generation redispatch. The problem constraints consist of a target improvement of the TTC, the power balance equation and the generation limits. The TTC improvement is approximated with a linear constraint built based on sensitivities with respect to the active power generation. The main problem of the TTC linear formulation is the loss of accuracy for large variations of generation. To overcome this difficulty, the algorithm has been designed as an iterative process that imposes an additional constraint of the total active power generation redispatch in each iteration. The main advantages of the algorithm presented in this paper with respect to other algorithms in the literature are: • The LP-based formulation allows to consider technical aspects of the network, such as power generation limits of the generators. • The iterative scheme allows to update the sensitivities gradually, avoiding the loss of accuracy of the linear approximation. • The convergence of the TTC updating is guaranteed due to the fact that the maximum generation redispatch allowed in each iteration is controlled. The paper is organized as follows: Section 2 contains the TTC computation method employed in this paper, based on continuation techniques. Section 3 presents the sensitivity analysis of the TTC with respect to any electric power system parameter. Section 4 describes the mathematical formulation of the proposed TTC
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improvement algorithm. Section 5 illustrates the performance of the proposed algorithm applied to a hourly scenario of the operation of the Spanish power system. Finally, Section 6 contains the conclusions of the paper. 2 TTC COMPUTATION METHODOLOGY The TTC computation problem has been formulated as an optimization problem: max λ
s.t.
g ( x, λ ) = 0 h ( x, λ ) ≤ 0
(1)
where: λ : Transfer capability parameterization factor x : State variables vector g : Augmented state equations h : Security constraints The transfer capability parameterization factor λ is employed to control the energy transfer from exporting to importing areas. The state variables vector x is formed by both bus voltages and angles. Vector g represents the augmented active and reactive power flow equations, where the power variation vector is parameterized by λ. Finally, vector h represents the security constraints of the system: line flows and voltage magnitudes. To solve problem in equation (1), continuation techniques are employed. To obtain a new solution {xi+1, λi+1} of the augmented power flow equations g(x,λ)=0 from another one {xi, λi}, the continuation power flow technique consists of two main steps: predictor and corrector. The predictor step uses the linearized augmented system state equations g(x,λ)=0 to obtain an approximate evolution of the state variables: g x ·∆xp + g λ ·∆λ p = 0
(2)
where: gx : Jacobian matrix of the state equations g with respect to the state variables x gλ : Gradient vector of the state equations g with respect to the transfer capability parameterization factor λ To obtain a (n+1)×(n+1) linear equations system, where n is the number of augmented system state equations, an additional equation must be added. Usually, this additional equation fixes the variation of one variable to a maximum variation allowed. However, this additional equation is trivial, due to the fact that (2) defines the direction of the predictor step, and the magnitude will be set with the parameterization presented further on. The corrector step converges the augmented system state equations g(x,λ)=0 using the Newton-Rhapson method, starting from the approximation obtained by predictor step. Each iteration υ of the Newton-Rhapson method uses the linearized augmented system state equations:
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g x ·∆xυc + g λ ·∆λcυ = −g(xυ , λ υ )
(3)
Once again, an additional equation is needed to obtain a (n+1)×(n+1) linear equations system. To give an optimal direction to the corrector step, it is set perpendicular to the direction of the predictor step, thus a complete iteration of the corrector step will be:
⎡ gx ⎢∆xT ⎣ p
g λ ⎤ ⎡ ∆xυc ⎤ ⎡ −g(xυ , λ υ ) ⎤ = · ⎥ ∆λ p ⎥⎦ ⎣⎢ ∆λcυ ⎦⎥ ⎢⎣ 0 ⎦
(4)
Corrector step will end when the augmented system state equations mismatches g(xυ,λυ) reach the minimum threshold allowed. In order to speed the continuation process up, a novel parameterization of the predictor step is employed in this paper [10]. The main objective of the transfer capability parameterization is to reach the security border defined by h. Therefore, the predictor step is resized to not violate the linear approximation of the security border defined by h. Thus, the next constraint is added: h 0 + ∆h ≈ h 0 + h x ·∆x p + h λ ·∆λ p ≤ 0
(5)
To satisfy equation (5), the next resizing factor γ is defined: ⎧
hi ⎫ ⎬ ∆hi > 0 ⎩ ∆hi ⎭
γ = min ⎨1,
(6)
and applied to the predictor step: ∆λ p ← γ ·∆λ p
; ∆x p ← γ ·∆x p
(7)
Thus, the resizing factor γ defined in (6) brings the predictor approximation so close to the security border defined by h, reducing the computation cost of the TTC calculation via continuation techniques. 3 TTC SENSITIVITIES ANALYSIS Once the total transfer capability has been calculated by the transfer capability parameterization factor maximum, its sensitivity with respect to any system parameter can be calculated. Let u be the system parameters vector. Assuming that any small system parameters vector variation ∆u will keep the same active constraint hk, then the next linear equations system can be formulated:
⎡ gx ⎢h ⎣ kx
gλ hk λ
⎡ ∆x ⎤ g u ⎤ ⎢ ⎥ ⎡0 ⎤ · ∆λ = h ku ⎥⎦ ⎢ ⎥ ⎢⎣0 ⎥⎦ ⎣⎢ ∆u ⎦⎥
(8)
where: gu : Jacobian matrix of the state equations g with respect to the system parameters considered u hkx : Gradient vector of the active security constraint hk with respect to the state variables x
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hkλ : Gradient vector of the active security constraint hk with respect to the transfer capability parameterization factor λ hku : Gradient vector of the active security constraint hk with respect to the system parameters considered u Assuming that, if the system parameter perturbation ∆u is small, then ∂λ/∂u ≈ ∆λ/∆u and ∂x/∂u ≈ ∆x/∆u. Therefore, solving (8) the next sensitivities are obtained: ⎡ gx ⎡∂λ /∂u ⎤ ⎢ ∂x/∂u ⎥ = − ⎢h ⎣ ⎦ ⎣ kx
gλ ⎤ hk λ ⎥⎦
−1
⎡ gu ⎤ ·⎢ ⎥ ⎣h ku ⎦
(9)
Thus, the sensitivities obtained in (9) represent the variation of the transfer capability parameterization factor λ and the state variables vector x with respect to the variation of any system parameter u, such as active power generation dispatch, voltage control actions (generator bus voltages, shunt reactors, etc.) or bus loads. 4 TTC IMPROVEMENT METHODOLOGY This section details the mathematical formulation of the optimal generation redispatch algorithm for TTC improvement proposed in this paper. Firstly an overview of the iterative algorithm is presented. Then the notation, the objective function and the constraints of the LP-based generation redispatch algorithm are provided. 4.1 Overview of the algorithm Figure 1 depicts a flowchart of the proposed algorithm: Initial TTC computation
Do TTC reach minimum security requirements?
YES
FIN
NO Transfer capability parameterization factor λ sensitivities computation SGPi = ∂λ/∂PGi
Solve LP optimization problem
New TTC computation
Figure 1: Flowchart of the iterative active power generation redispatch algorithm.
The algorithm starts from the initial TTC computation of the base case. The algorithm has been designed as an iterative algorithm, where limits on the generation redispatch have been imposed in each iteration to avoid the loss of accuracy caused by the linear approximation of the transfer capability parameterization factor. While the desired security TTC value is not reached, a new iteration is performed updating the sensitivities and
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solving a new LP optimization problem, obtaining the optimal location of generation buses and their corresponding power redispatch, and calculating the new TTC. Another stop criteria will be that the total generation redispatch reaches a maximum total generation redispatch allowed. Next subsections contain the LP optimization problem formulation of each iteration to obtain the minimum load shedding. 4.2 Notation The active power generation redispatch algorithm is formulated in term of active power parameters and variables. Therefore, the following notation has been adopted: Parameters PG0 : Initial active generation. SGP : Sensitivity of the transfer capability parameterization factor λ with respect to the initial active generation. PG+: Upper limit of generation. PG-: Lower limit of generation. c: Unfeasibility cost of the residual variable. λ0: Initial value of the transfer capability parameterization factor. λsec : Security value of the transfer capability parameterization factor to be reached. MGR : The maximum generation redispatch allowed in each iteration. Variables ∆PGp : Active power generation positive redispatch. ∆PGn : Active power generation negative redispatch. Hλ : Residual variable included to avoid infeasibilities. Objective function The main objective function is to minimize the total active power generation redispatch.
eT ·( ∆PGp + ∆PGn )
(10)
where e represents a vector whose entries are all ones. Objective function in (10) only considers technical aspects of the redispatch. However, in case of a market rules the generation redispatch, the vector e could be substituted by another vector whose entries are the corresponding energy costs of different generators. Also a penalized residual variable Hλ is included in the objective function with an unfeasibility cost c >> 0 to prevent the unfeasibility of the optimization problem. c·H λ
(11)
Therefore, the complete objective function will be: min
{
c·H λ + eT ·( ∆PGp + ∆PGn )
}
(12)
4.3 Constraints The following constraints have been considered: • The linear approximation of the transfer
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capability parameterization factor, bounded by a security value λsec. A residual variable has been added to avoid infeasibility. SG TP ·( ∆PGp − ∆PGn ) + H λ ≥ λsec − λ0
(13)
• Total generation redispatch bounded by the maximum generation redispatch MGR, to avoid the loss of accuracy. The value of MGR must keep a balance between the loss of accuracy and an excessive number of iterations: eT ·( ∆PGp + ∆PGn ) ≤ MGR
(14)
• Generation balance equation: eT ·( ∆PGp − ∆PGn ) = 0
(15)
• Limits of generation redispatch: PG − − PG 0 ≤ ∆PGp − ∆PGn
(16)
∆PGp − ∆PGn ≤ PG + − PG 0
(17)
• The residual variable Hλ must be a positive variable:
Hλ ≥ 0
(18)
This LP-based formulation has employed active power generation redispatch as control variables. However, it is important to remark that other control variables, such as voltage control resources or load shedding, can be used to formulate TTC improvement algorithms, due to generic sensitivities analysis presented in section 3. 5 RESULTS In this section, the computation and improvement of total transfer capability between areas is illustrated for an hourly scenario of the Spanish power system. The section is organized as follows. Subsection 5.1 presents an overview of the Spanish power system and the base case considered. In subsection 5.2 the total transfer capability is obtained for an actual Spanish power system scenario. Finally, subsection 5.3 illustrates the performance of the LP-based TTC improvement algorithm proposed in this paper, applied to the Spanish power system scenario considered.
5.1 Overview The Spanish power system includes 1497 buses and 2257 branches, and includes representations of the French, Portuguese and Moroccan systems. The Spanish power system and the external systems are divided into eight areas. Figure 2 displays the approximate situation of each area. Northwest and North areas (area 1 and 2) are usually exporting areas, and East, Center and South areas (area 3, 4 and 5) are importing areas. Areas 6, 7 and 8 correspond to the areas representing the Portuguese, Moroccan and French systems respectively. Table 1 summarizes the active power balance in each area for the Spanish power system scenario considered.
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&
!
" $ #
%
Figure 2: Area division of the Spanish power system.
Table 1 includes active power generation, demand, losses and exports in the base case considered.
applied to the generation dispatch of Northwest, North and South areas, and to the demand dispatch of the Center area. The system security limits considered are the maximum flows of line and power transformers belonging to the 400 kV and 220 kV networks, and voltage limits of 400 kV and 220 kV buses. The TTC computation takes only two iterations, due to the novel predictor parameterization used in this paper. The system security limit reached at TTC point was the maximum power flow in the 400 kV/220 kV power transformer of BENEJAMA, equal to 518 MVA. This maximum power flow corresponds to an active power flow of 491 MW and a reactive power flow of 159 Mvar, from 400 kV bus to 220 kV bus. BENEJAMA belongs to the LEVANTE subsystem, in the Center area. Figure 4 depicts the reduced grid of the LEVANTE subsystem of the Spanish power system Center area. CATADAU 132kV
AREA 12345-
Northwest North East Center South TOTAL
Active Power (MW) GEN DEM LOSS
EXP
9032 5123 8636 9233 5895
4350 4173 7818 13760 6715
108 143 191 347 182
4574 807 627 -4874 -1002
37919
36816
971
132
Table 1: Active power balance in each area of the Spanish power system
Figure 3 depicts the area power interchange in the base case considered. Figure 3 shows that the main power transfer is from Northwest area to Center and South areas, whereas North and East areas are mainly passing trough areas.
5.2 TTC computation The areas selected as exporting areas for the TTC calculation are Northwest, North and South areas, whereas Center area has been selected as importing area. Thus, the transfer capability parameterization factor is
CATADAU 400kV
ALZIRA 220kV JUAN de URRUTIA 132kV
CATADAU 220kV
BENEJAMA 400kV
BENEJAMA 220kV
JIJONA 220kV
SAN VICENTE 220kV
ROCAMORA 400kV
NUEVA ESCOMBRERAS 400kV
FAUSITA 220kV
LITORAL 400kV ESCOMBRERAS 220kV
LA ASOMADA 400kV
ESCOMBRERAS 400kV
Figure 4: Reduced grid of the Spanish power system Center area: LEVANTE subsystem.
At this point, the energy transfer from Northwest, North and South areas to Center area is 558 MW. Figure 5 shows the area power interchange redistribution in TTC point.
Figure 5: Area power interchange redistribution in TTC point.
Figure 3: Area power interchange in base case.
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As shown in Figure 5, the main increase in energy export corresponds to the Northwest area (302 MW),
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whereas South area reduces its energy import in 195 MW. East area and the Portuguese and French systems exports keep constant and therefore remain as passing trough areas.
5.3 TTC improvement After TTC computation, the TTC improvement algorithm presented in this paper is applied using active generation power redispatch as control variable. The initial TTC is 558 MW and the maximum active generation power redispatch allowed is 500 MW. The Figure 6 depicts the transfer capability evolution during active power generation redispatch.
Transfer Capability (MW)
750
As Table 2 exposes, the most efficient generation redispatch is the increase of active power generation in JUAN DE URRUTIA and ESCOMBRERAS power plants. At this point, both of them reach their maximum active power generation limits, thus new less efficient generators must be redispatched. Due to this fact, the efficiency of the algorithm is saturated at this point, as Figure 6 shows. The system security limit reached at TTC point was again the maximum power flow in the 400 kV/220 kV power transformer of BENEJAMA, equal to 518 MVA. At this point, the new energy transfer from Northwest, North and South areas to Center area is 764 MW. Figure 7 shows the area power interchange redistribution in the new TTC point after the active power generation redispatch.
700
650
600
550 0
50
100
150 200 250 300 350 400 Active Power Generation Redispatch (MW)
450
500
Figure 6: Transfer capability evolution during active power generation redispatch.
As Figure 6 shows, the TTC improvement presents two different evolutions during active power generation redispatch. As the first 155 MW of active power generation redispatch is applied, the TTC grows linearly from its initial amount of 558 MW to a final amount of 764 MW. Therefore, the average efficiency of the TTC improvement algorithm is 1.33 MW for each MW of active power generation redispatch. However, from this point and during the next 345 MW of active power generation redispatch, the TTC improvement is saturated, thus efficiency of the TTC improvement algorithm is negligible. Hence, all the active power generation redispatches after this saturation point have been not considered. The generators redispatched by the algorithm are JUAN DE URRUTIA, ESCOMBRERAS and NUEVA ESCOMBRERAS, all inside LEVANTE subsystem. Table 2 shows the active power generation redispatches, and the TTC improvement and efficiency associated to each of them. Power Plant J. DE URRUTIA ESCOMBRERAS
Active Power (MW)
∆TTC ∂TTC
INI
FIN
INC (MW)
32.2
70.0
37.8
∂PG
49.0
1.296
419.7 537.0 117.3 133.5
1.138
N. ESCOMBRERAS 1170.0 1014.9 -155.1
24.0
-0.155
Table 2: Optimal active power generation redispatch
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Figure 7: Area power interchange redistribution in TTC point, after optimal generation redispatch.
As shown in Figure 7, the area power interchange redistribution in the new TTC point is qualitative equal to the presented in Figure 5. The main increase in energy export corresponds again to the Northwest area (480 MW), whereas the South area energy import reduction now is equal to 211 MW. East area and the Portuguese and French systems exports keep constant and therefore remain as passing trough areas. 6 CONCLUSIONS In this paper, the concept of Total Transfer Capability (TTC) and the influence of system parameters on its amount have been studied. A briefly technical literature revision of TTC computation and improvement methods is included in the paper. The TTC has been computed using continuation techniques. In order to speed up the TTC computation, a novel predictor parameterization has been used in this paper. This parameterization is based on first order approximation of the inequality constraints that represent system security limits. This paper has proposed an optimization algorithm to determine the amount and location of the most efficient control actions to improve the TTC. The algorithm has
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been formulated using active power generation redispatch as control variables The algorithm is based on LP optimization. First order sensitivities of the transfer capability parameterization factor with respect to the active power generation to be redispatched. The objective function consists of minimizing the total system generation redispatch. The problem constraints consist of a target improvement of the transfer capability parameterization factor, the power balance equation and the generation limits. To overcome the loss of accuracy of the linear approximation of the transfer capability parameterization factor, the algorithm has been designed as an iterative process that imposes an additional constraint of the total active power generation redispatch in each iteration. The performance of the proposed TTC improvement algorithm has been illustrated considering a hourly scenario of the operation of the Spanish power system.
[1]
[2]
[3]
[4]
REFERENCES NERC, "Available Transfer Capability Definitions and Determination", Princeton, New Jersey, U.S.A. Availability Transfer Capability Working Group (ATCWG) of the North American Electric Reliability Council (NERC) Task Force Report, Jun 1996. NERC, "Transmission Transfer Capability", Princeton, New Jersey, U.S.A. Availability Transfer Capability Working Group (ATCWG) of the North American Electric Reliability Council (NERC) Task Force Report, May 1995. I. Dobson, S. Greene, R. Rajaraman, C. L. DeMarco, F. L. Alvarado, and R. Zimmerman, "Electric Power Transfer Capability - Concepts, Applications, Sensitivity and Uncertainty", Power Systems Engineering Research Center publication 01-34, Cornell University, Nov 2001. A. J. Flueck, H. D. Chiang, and K. S. Shah, "Investigating the Installed Real Power Transfer Capability of a Large Scale Power System Under a Proposed Multiarea Interchange Schelude Using CPFLOW", IEEE Transactions on Power Systems, vol. 11, nº 2, pp. 883-889, May 1996.
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[5] R. Seydel, "4 - Principles of Continuation" in From Equilibrium to Chaos. Practical Bifurcation and Stability Analysis, Elsevier Science Publishing Co., 1988. [6] M. Shaaban, W. Li, Z. Yan, Y. Ni, and F. F. Wu, "Calculation of Total Transfer Capability incorporating the Effect of Reactive Power", Electric Power Systems Research, vol. 64, nº 3, pp. 181-188, Mar 2003. [7] P. Sandrin and L. Dubost, "Reliability of Transactions between Electric Utilities", presented at CIGRE Symposium - Electric Power Systems Reliability, Montreal, Canada, Sep 1991. [8] F. M. Echavarren, E. Lobato, and L. Rouco, "Contingency Analysis: A Feasibility Identification and Calculation Algorithm", IEE Proceedings Generation, Transmission & Distribution, vol. 152, nº 5, pp. 645-652, Sep 2005. [9] F. M. Echavarren, E. Lobato, and L. Rouco, "A Power Flow Solvability Identification and Calculation Algorithm", Electric Power Systems Research, vol. 76, nº 4, pp. 242-250, Jan 2006. [10] F. M. Echavarren, E. Lobato, and L. Rouco, "Computation and Sensitivity Analysis of the Total Transfer Capability in Interconnected Electric Power Systems", presented at 11th International Energy Conference & Exhibition (ENERGEX'06), Stavanger, Norway, Jun 2006. [11] S. Greene, I. Dobson, and F. L. Alvarado, "Initial Concepts for Applying Sensitivity to Transfer Capability", presented at NSF Workshop on Available Transfer Capability, Urbana, Illinois, U.S.A., Jun 1997. [12] S. Greene, I. Dobson, and F. L. Alvarado, "Sensitivity of Transfer Capability Margins with a Fast Formula", IEEE Transactions on Power Systems, vol. 17, nº 1, pp. 34-40, Feb 2002. [13] A. M. Leite da Silva, J. G. de Carvalho Costa, L. A. da Fonseca Manso, and G. J. Anders, "Transmission Capacity: Availability, Maximum Transfer and Reliability", IEEE Transactions on Power Systems, vol. 17, nº 3, pp. 843-849, Aug 2002. [14] C. F. Moyano, R. Salgado, and L. V. Barboza, "Calculating Participation Factors in the Maximum Loadability", presented at 2003 IEEE Bologna Power Tech Conference, Bologna, Italy, Jun 2003.
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