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Interior point models for power system stability problems
European Journal of Operational Research 171 (2006) 1127–1138 www.elsevier.com/locate/ejor
Interior point models for power system stability problems William Rosehart *, Codruta Roman, Laleh Behjat Department of Electrical and Computer Engineering, University of Calgary, 2500 University Drive, N.W., Calgary, AB, Canada T2N 1N4 Available online 10 March 2005
Abstract In this paper, a detailed analysis of the use of optimization techniques in the study of voltage stability problems, leading to the incorporation of voltage stability criteria in traditional Optimal Power Flow (OPF) formulations is presented. Optimal power flow problems are highly nonlinear programming problems that are used to find the optimal control settings in electrical power systems. The relationship between the Lagrangian Multipliers of the OPF problem and the classification of the maximum loading point level of the system is given. Finally, the paper presents a sequential OPF technique to enhance voltage stability using reactive power and voltage rescheduling with no increase in real (active) generation cost. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Energy; Optimization; Voltage collapse; Optimal power flow
1. Introduction Electricity price volatility, and in some cases a trend towards higher prices, has resulted in uncertainly about the future of some electricity markets. This uncertainty, coupled with previous financial deterrents, has resulted in reduced capital investment in power system infrastructure relative to the growth in demand, particularly with respect to ancillary services such as stability margins. As *
Corresponding author. E-mail addresses: [email protected] (W. Rosehart), [email protected] (C. Roman), [email protected] (L. Behjat).
the overall stability limits can be closely associated with the voltage stability of the network, the incorporation of voltage collapse criteria and tools in the operation of power systems is becoming an essential part of new Energy Management Systems (EMS) [4]. Although, it is possible to directly incorporate voltage stability into Optimal Power Flow (OPF) formulations [16], this tends to result in increased operating costs. Typically, voltage collapse events can be related to a lack of a post contingency equilibrium point in the system, which in turn can be associated theoretically with either a saddle-node bifurcation (SNB) or a limit-induced bifurcation (LIB) [4]. Hence, analysis tools that are currently under use
0377-2217/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.01.021
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throughout the world are mostly based on bifurcation theory and continuation methods. However, it is possible to restate many voltage collapse problems as optimization problems. Although, power system analysis using bifurcation methods are numerically well developed, the use of optimization based techniques has many advantages, including their ability to incorporate limits. This issue becomes even more important when considering limit-induced voltage collapse [7], which can not be easily defined using some of the traditional bifurcation-based computational techniques. Limit-induced voltage collapse is generally the most likely form of collapse to be encountered in power systems [4]. New voltage stability analysis tools that use optimization methods to determine optimal control parameters that maximize load margins have been introduced in the last few years. Various uses of optimization methodologies applied to the voltage collapse problem can be found in the technical literature. In [20], a voltage-collapse point problem is formulated as a nonlinear optimization problem, allowing the use of optimization techniques and tools. In [14], the reactive power margin with respect to voltage collapse is determined using interior point methods; the authors used a barrier function to incorporate limits. In [2], the authors determine the closest bifurcation to the current operating point on the hyperspace of bifurcation points. In [12], the maximum loadability of a power system is examined using interior point methods. In [15], an interior point optimization technique is used to determine the optimal generator settings to maximize the distance to voltage collapse. Furthermore, the algorithm presented in [15] includes constraints on the present operating conditions. Lagrange multipliers also have a very practical applicability in the new deregulated electricity market. In some power systems operating with open access principles, the real power pricing is based on the locational (nodal) spot prices. In this way economical incentives are given to the generators to produce power efficiently and to the consumers to use power efficiently [1]. From the system operatorÕs perspective, the Lagrange multi-
pliers analyzed from the OPF problem gives information about the physical limitation that affects the active power transactions between the seller and the buyer. Electricity as a commodity has a very special behavior related to the way it is transported from the provider to the consumer. Thus, the Lagrange multipliers can be used in a detailed analysis for identifying the factors that affect the power flow and give those factors an economical quantification that can be used to control and improve the power system operation. The real power spot price can be decomposed into components corresponding to the generation, transmission losses, security, voltage control and spinning reserve [23]. The last three terms are very important for the emerging ancillary services in open markets. In open access power systems, a need for a reactive power market as an ancillary service is arising in many cases. Though the reactive power does not always represent a utility for the consumer, it is essential for ensuring real (active) power transmission and the proper range of voltage levels throughout the electrical network. Reactive power transmission and lack of adequate reactive power supplies are generally associated with voltage collapse. A close examination of the Lagrange multipliers associated with the equations involving reactive power dependent terms provides information about the way the reactive power flow affect the voltage levels and/or the secure operation of the system [23,6]. In [24] a Lagrange multiplier analysis is used for the implementation of a load curtailment program to enhance the voltage stability of the system. Based on the Lagrange multiplier values the quantity of load that has to be shed is determined and the level of financial compensation for those consumers can be established. The current paper uses general OPF problems that incorporates voltage stability margins. The main issue considered is how limits affect maximum loading point computations. Two different OPF problems that directly incorporate stability margins are presented. The paper also presents a sequential OPF technique to enhance voltage stability using reactive power and voltage rescheduling. In this approach, first a traditional OPF
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problem is solved. Then a Maximum Distance to Collapse problem is solved with the generator active power scheduling, from the OPF solution, being fixed. The paper is structured as follows. In Section 2, the basic background for the OPF problem and issues related to voltage collapse are discussed. In Section 3, the general formulation to combine OPF and voltage stability is given, including an analysis of the effect of limits on the results. Furthermore, two OPF formulations with voltage stability constraints are given along with a sequential OPF technique to enhance the stability margin. The results of testing the problems on three power system models are given in Section 4. Finally, conclusions are given in Section 5.
2. Background review 2.1. Optimal power flow and optimization techniques The OPF problem introduced in the early 1960s by Carpentier has grown into a powerful tool for power system operation and planning. In general, the OPF problem is a nonlinear programming (NLP) problem that is used to determine the ‘‘optimal’’ control parameter settings to minimize a desired objective function, subject to certain system constraints [8,11,19]. With the introduction of diverse objective functions, the OPF problem represents a variety of optimization problems [11], which includes, for example, active power cost optimization and active power loss minimization [9]. OPF problems are generally formulated as NLP problems as follows: min
GðyÞ
s:t:
F ðyÞ ¼ 0; H 6 H ðyÞ 6 H ; y 6 y 6 y ;
ð1Þ
where the mapping GðyÞ : Rn ! R is the function that is being minimized and can include, for example, total losses in the system or generator costs.
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The equality constraints in Eq. (1), F ðyÞ : Rn ! Rm generally represents the power flow equations. For each bus or node in the system, there are two power flow equations, i.e., active or real power flow and reactive power flow. Active power represents power that does work. For example, active power is used to model the power used to heat a stove, energize a light, or move an elevator. Reactive power is a term developed to represent the power that is being transferred in and out of inductive and capacitive elements in the system each cycle. The charging and discharging characteristic of inductive and capacitive elements is due to fundamental electrostatic and electromagnetic laws and the use of sinusoidal waveforms in electrical power systems. The power flow equations state that the active, P, and reactive, Q, power entering a bus or node must equal the active and reactive power leaving the bus or node. Power from generation directly connected to the bus or node is considered as entering and loads, such as residential or industrial loads, are consider as leaving. It should be noted, that for power system analysis, loads are generally aggregated together to reduce the number of nodes or buses in the problem. Power transfer through transmission lines or transformers is considered to be leaving the node or bus. The power flow equations can be generically written as: 0 ¼ P geni P branchi ðV ; dÞ P loadi for i ¼ 1; . . . ; m2 ;
ð2Þ
0 ¼ Qgeni Qbranchi ðV ; dÞ Qloadi for i ¼ 1; . . . ; m2 ;
ð3Þ
where the nonlinear functions Pbranchi(V, d) and Qbranchi(V, d) represent the power leaving the bus through transmission lines and transformers. These nonlinear functions are derived using Kirchoff Õs Laws and phasor theory [10]. The inequality constraints in Eq. (1), H ðyÞ : Rn ! Rp usually models transmission line and transformer limits, with lower and upper limits represented by H and H , respectively. These limits generally correspond to thermal limits, representing the maximum power flow in the transmission
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line or transformer. The greater the power flow, the higher the thermal losses. The equations to model the power flowing in the transmission lines are nonlinear. Finally, the vector of system variables in Eq. (1), denoted by y 2 Rn , typically includes voltage magnitudes and angles, generator power levels and transformer tap settings; their lower and upper limits are given by y and y , respectively. In most OPF problems, the power flow is modeled using phasor representations of the bus or node voltages. The phasor representations have a magnitude, normalized to the nominal voltage at the particular bus, and an angle defined as the phase difference between the voltage at a particular bus and a common reference bus voltage in the system. 2.2. Voltage collapse and bifurcation theory Several voltage collapse events throughout the world show that power systems are being operated close to their stability limits. Nonlinear phenomena, especially bifurcations, have been shown to be responsible for a variety of stability problems in power systems. In particular, the lack of post contingency equilibrium points, typically associated with saddle-node and limit-induced bifurcations, have been shown to be one of the main reasons for voltage collapse problems in power systems [4]. In general, bifurcation points can be defined as equilibrium points where changes in the ‘‘quantity’’ and/or ‘‘quality’’ of the equilibria associated with a nonlinear set of equations occur with respect to slowly varying parameters in the system [17]. Of interest is the point where the system goes from being stable to unstable with respect to a bifurcation parameter. In this paper, the bifurcation parameter models the loading level of the system, i.e. the power demand of the system. Bifurcations are mathematically characterized by one of the system eigenvalues becoming zero (saddlenode bifurcation) or by eigenvalues become positive when control limits are reached (limit-induced bifurcation). For limit-induced bifurcations, the change in the eigenvalues is due to changes in the
system model, since different control approaches are used when some limits become active. Although, there are many other types of bifurcations [17], the above two are expected to occur in power system models. It is also possible for a system to reach a maximum loading level, that is defined based on operational limits, for example, minimum voltage magnitude limits. In these cases, the system does not go unstable, but the system operators must take action, such as load shedding to prevent problems associated with low voltages. 2.2.1. Saddle-node bifurcations Saddle-node bifurcations are characterized by two equilibrium points, typically one stable (s.e.p.) and one unstable (u.e.p.), merging at an equilibrium point for the parameter value k = k*. If the two merging equilibria co-exist for k < k*, the two equilibrium points locally disappear for k > k*, or vice versa. Saddle node bifurcations are local bifurcations, occurring at the point where the equilibrium locally vanishes for further values of the bifurcation parameter. In normal situations, saddle-node bifurcations will only occur in power systems when the loading levels are very high. The following conditions hold for saddle-node bifurcations: (1) The point is an equilibrium point, i.e., Fk(y) = 0. (2) The Jacobian of the function Fk(y) with respect to y at the bifurcation point (y*, k*), has a unique zero eigenvalue with an associated nonzero eigenvalue. (3) At the saddle-node point, two branches of equilibria intersect and ‘‘disappear’’ beyond the saddle-node [17]. 2.2.2. Limit-induced bifurcations Although saddle-node bifurcations can be shown to be generic in power systems, limits, especially generator reactive power limits, may restrict the space of feasible solutions and voltage collapse may be induced by limits as opposed to a saddlenode bifurcation [7,5]. This has a significant effect on ‘‘measuring’’ the distance to voltage collapse since the voltage collapse may occur by reactive power limits (limit-induced bifurcation) and not by a saddle-node. Limit induced bifurcations as analyzed in [7] occur in power flow equations when generator models are changed from constant volt-
W. Rosehart et al. / European Journal of Operational Research 171 (2006) 1127–1138
age and active power models, to constant active and reactive power models on encountering reactive power limits. The change in models corresponds to a different set of equations; and it is found, in some cases, that the new equations are unstable at the current operating point. Both the original model and the ‘‘limit induced’’ model have the same equilibrium point when the limit is encountered but have different bifurcation diagrams. In the case where the system becomes unstable at the limit, the equilibrium point is on the unstable portion of the ‘‘limit induced’’ bifurcation diagram as shown in Fig. 1. 2.2.3. Operating limit constrained maximum point It is also possible for the maximum loading point to be constrained by operating limits, i.e. the maximum point is not a point where the system goes from being stable to unstable. Power systems have to be maintained within operating limits, for example there are strict rules on the magnitude of the AC voltages. Generally, the voltage magnitudes at each bus or node in the system must be always within a certain percentage of the nominal system voltage. Changes in loads can result in situations where these limits are violated. If changes can not be made to the independent variables to correct these violations, then corrective action
V
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such as load curtailment must be taken. It should be noted, that these types of maxima are not bifurcation points. 2.2.4. Maximum distance to collapse problem In practice, for a given power system, of most interest is to determine and maximize the distance from the current operating point to the maximum operating point, usually represented by a saddlenode, limit induced bifurcation or operating limit constrained maximum. The point of collapse can be found by using direct and/or continuation methods [17]. The problem of voltage collapse given the current operating point also can be formulated as an optimization problem [16]: max s:t:
k kp F ðxp ; x ; q; q ; kp ; k Þ ¼ 0;
ð4Þ
H 6 H ðxp ; x ; q; q Þ 6 H ; where the subscripts p and * indicate the current and critical points, respectively. The parameter q* is used to map the control variables at the current operating point, defined by q, into the critical point, i.e., q* models the optimal change in generation to deal with increases in demand or load. Generators at the critical point are assumed to have the same terminal voltage set points as at the base loading point, and their power levels are represented based on the following model: P G ¼ P Gp þ DP G ;
Limited System
where PGp is the generation scheduling at the current point, DPG* sets the optimal increase in generated powers to reach the critical point, and PG* is the generation scheduling at the critical point. For this model, the dependent and independent (control) variables at the critical point can be grouped as
Unlimited System
x ¼ ½d V L QG
T
and Q limit Q Fig. 1. Illustration of limit induced instability, solid lines represent stable operating points, dashed lines represent unstable operating points. The bolded lines represent the set of operating points when modeling limits.
q ¼ ½P G ðP Gp ; DP G Þ; where d* and VL is the bus voltage phasor magnitude and angle, and QG* is the reactive power of the generators.
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The parameter k 2 R is a scalar bifurcation parameter that is typically known as the ‘‘loading factor’’, as it represents loading level in the system for a linearly increasing, constant power factor load model (for a given bus, the ratio of active/real and reactive power remains constant for all loading levels). The variable k* is a variable in the optimization problem, i.e. it is fully free to change during the solution process; on the other hand, kp is given a fixed value. The loading level for each bus at the current loading point is modeled as follows: P L ¼ kp P Lo ; QL ¼ kp QLo :
QL ¼ kp QLo þ ðk kp ÞQk ;
In general and for the power flow model, an OPF problem that incorporates voltage stability criteria can be generically written as min
Gðxp ; q; kp ; k Þ
s:t:
F ðxp ; q; kp Þ ¼ 0; F ðx ; q ; k Þ ¼ 0; H p 6 H ðxp Þ 6 H p ;
ð7Þ
H 6 H ðx Þ 6 H ; ; q6q6q
ð5Þ
q 6 q 6 q ;
The loading level for each bus at the critical loading is modeled as follows: P L ¼ kp P Lo þ ðk kp ÞP k ;
3. Voltage collapse and Optimal Power Flow (OPF)
ð6Þ
where PL and QL are loading levels for real or active and reactive power, PLo and QLo are the reference or base loading values, and Pk and Qk are the loading direction. The loading direction states how the load in the system will increase and it is assumed that this information is known, i.e. is determined using load profiles [15]. The choice of the loading direction, does have an impact on the maximum loading point as certain areas in the system, particularly areas further from generation, can not support high loading or demand levels. Therefore, care must be taken to use conservative loading directions. In [2], techniques are presented to determine the worse loading direction, i.e. the loading direction that has the shortest distance to instability. In the results presented in this paper, it is assumed that the loading direction equals the base loading values, i.e. Pk = PLo, and Qk = QLo. The Maximum Distance to Collapse problem given in Eq. (4), guarantees the feasibility of the current operating point as well as the feasibility of control and operating limits. Solutions of Eq. (4) correspond to a maximum loading point that can be a real collapse point or a point where operating limits are attained such as bus voltage, line current, or power flow limits.
where the variables and constraints are the same as those defined for Eq. (4) but are written separately for the current and critical point. G(xp, q, kp, k*) is the objective function to be minimized, which has an OPF component, i.e. production costs or losses, that may be dependent on (xp, q, kp), and a voltage stability component that is a function of k* and possibly of kp, as discussed below. Using a Logarithmic Barrier approach [19], the first order Karush–Kuhn–Tucker (KKT) optimality conditions to problem (7) are given to demonstrate when the maximum loading point is defined by a limit-induced bifurcation, a saddle-node bifurcation or an operating limit constrained maximum point. Using slack variables problem (7) can be rewritten as min
Gðxp ; q; kp ; k Þ
s:t:
F ðxp ; q; kp Þ ¼ 0; F ðx ; q; k Þ ¼ 0; H ðxp Þ H p s1 ¼ 0; H p H ðxp Þ s2 ¼ 0; H ðx Þ H s3 ¼ 0; H H ðx Þ s4 ¼ 0; q q s5 ¼ 0; q s6 ¼ 0; q s1 ; s2 ; s3 ; s4 ; s5 ; s6 P 0;
ð8Þ
W. Rosehart et al. / European Journal of Operational Research 171 (2006) 1127–1138
where s1 ; s2 ; s3 ; s4 2 RN and s5 ; s6 2 Rm are the primal non-negative slack variables used to transform the inequality constraints to equalities. The nonnegativity constraints are now incorporated into the objective function using a logarithmic barrier as follows: m X min Gðxp ; q; kp ; k Þ l ðlog s5 ½i þ log s6 ½iÞ i¼1
l
N X ðlog s1 ½i þ log s2 ½i þ log s3 ½i þ log s4 ½iÞ i¼1
s:t: F ðxp ; q; kp Þ ¼ 0; F ðx ; q; k Þ ¼ 0; H ðxp Þ H p s1 ¼ 0; H p H ðxp Þ s2 ¼ 0; H ðx Þ H s3 ¼ 0; H H ðx Þ s4 ¼ 0;
q s6 ¼ 0; q
ð9Þ
where l is the barrier parameter and s[i] represents the ith element of the vector s. The Lagrangian function of the modified barrier problem (9) is then defined as m X Luk ¼Gðxp ; q; kp ; k Þ l ðlog s1 ½i þ log s2 ½iÞ i¼1 N X ðlog s3 ½i þ log s4 ½iÞ
3
7 6 rxp L 7 6 7 6 7 6 r k L 7 6 7 6 rq L 7 6 7 6 6 lIeN þ S 1 m1 7 7 6 6 lIe þ S m 7 6 N 2 2 7 7 6 6 lIeN þ S 3 m3 7 7 6 6 lIe þ S m 7 6 N 4 4 7 7 6 6 lIem þ S 5 f1 7 7 6 ru L ¼ 6 7 ¼ 0; 6 lIem þ s6 f2 7 7 6 6 F ðxp ; q; kp Þ 7 7 6 7 6 6 F ðx ; q; k Þ 7 7 6 6 H ðxp Þ H p s1 7 7 6 7 6 6 H p H ðxp Þ s2 7 7 6 7 6 6 H ðx Þ H s3 7 7 6 6 H H ðx Þ s 7 47 6 7 6 4 q q s5 5
ð11Þ
where S1 through S6 are diagonal matrices with elements of the corresponding vector s1 through s6 on the diagonal, I is an identity matrix, e is a vectors of ones, and rx L ¼ cT2 rx F ðx ; q; k Þ ðmT3 þ mT4 Þrx H ðx Þ; rxp L ¼ rxp Gðxp ; q; kp ; k Þ þ cT1 rxp F ðxp ; q; kp Þ ðmT1 þ mT2 Þrxp H ðxp Þ; rk L ¼ rk Gðxp ; q; kp ; k Þ cT2 rk F ðx ; q; k Þ; rq L ¼ Gðxp ; q; kp ; k Þ cT1 rq F ðxp ; q; kp Þ
i¼1
cT1 ðF ðxp ; q; kp ÞÞ
þ ðlog s5 ½i þ log s6 ½iÞ cT2 ðF ðx ; q; k ÞÞ mT1 ðH ðxp Þ H p s1 Þ
cT2 rq F ðx ; q; k Þ f1 þ f2 :
mT2 ðH p H ðxp Þ s2 Þ mT3 ðH ðx Þ H s3 Þ mT4 ðH H ðx Þ s4 Þ fT1 ðq q s5 Þ fT2 ð q q s6 Þ;
rx L
q s6 q
q q s5 ¼ 0;
l
2
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ð10Þ
where c1 ; c2 2 RN , m1 ; m2 ; m3 ; m4 2 Rm and f1 ; f2 2 Rm are the Lagrange multipliers. The vector u = (x*, xp, k*, q, s1, s2, s3, s4, s5, s6, c1, c2, m1, m2, m3, m4, f1, f2) is introduced to simplify the expression. The KKT first-order necessary conditions are used to define the local minimum of Eq. (9).
ð12Þ
The issue of collapse due to limit-induced bifurcation versus saddle-node bifurcation can now be explained as follows: The first condition in (12), $x*L = 0, includes the Jacobian of the system model at the maximum loading point multiplied by c2, which can be considered to be equivalent to an eigenvector of a Jacobian. Therefore, the first condition corresponds to a singular Jacobian if ðmT3 þ mT4 Þrx H ðx Þ ¼ 0; this would imply that, if one assumes that $x*H(x*) is non-singular (which is typically the case when limits directly on x are
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being enforced, i.e. $x*H(x*) is an identity matrix), the dependent variables are not at their limits, since m3 and m4 are zero when their corresponding limits are not active. If dependent variables of the critical point are at their limits, then m3 and m4 may become non-negative, i.e., the load flow Jacobian may be non-singular. If m3 and m4 are non-negative, which implies that locally the objective function could be improved if the limits are not enforced, the system has reached a limitinduced bifurcation point. If limits are reached and m3 and m4 remain zero, this implies that the limits are not locally limiting an improvement to the objective function and hence the system has reached a saddle-node bifurcation point. Although the above discussion is applied to the solution of the associated barrier problem, the characteristics are the same when applied to the original problem directly. The above derivation demonstrates the maximum loading point may be a limit-induced point only when the inequality constraints on the dependent variables become active and the inequality constraints can be separated based on the dependent and independent variables of the load flow model. The independent variables, q, being at their limits, do not directly affect the type of bifurcation. A particular example of this optimization problem is the ‘‘Maximum Distance to Collapse’’ whose constraints on the current and critical loading point are given in Eq. (4). The problem maximizes the distance to a saddle-node, limit-induced bifurcation or an operating limit constrained maximum. Including the current loading point into the constraints ensures that, when independent variables are calculated to maximize the distance to voltage collapse, feasibility and inequality constraints at the current loading point are met. For example, increasing generator voltage magnitude settings generally increases the distance to collapse but, under lighter loading conditions, the increased levels may lead to over-voltages. Incorporating the current operating point into the optimization problem can eliminate this problem; however, it also reduces the space of feasible solutions. This formulation differs from existing formulations, for example [3,12], since constraints are placed in
the critical loading point and the current loading point. 3.1. Voltage stability constrained OPF With the current loading point included into the optimization problem, it is possible to incorporate voltage stability criteria into an OPF formulation at the ‘‘currentÕ operating point xp. As the operating point moves closer to a critical point, i.e. as xp approaches x*, more emphasis must be placed on maximizing voltage stability as opposed to minimizing operating costs. The following techniques, proposed in [15] are used in this paper: 3.1.1. Linear Combination In this formulation the maximum loading is directly incorporated into the following objective function: Gðxp ; q; kp ; k Þ ¼ x1 gðxp ; q; kp Þ x2 ðk kp Þ ð13Þ subject to the constraints in Eq. (7). Observe that this requires the introduction of two weighting factors x1 and x2 to balance the emphasis placed on maximizing stability, i.e. (k* kp), versus minimizing costs, which are represented by g(xp, q, kp) in Eq. (13). Generally, x2 must be significantly larger than x1, as the relative difference in the magnitudes of each term in the objective function is large; it is assumed that x1 + x2 = 1 to normalize their values. Values obtained from previous OPF and Maximum Loading Distance analysis can be used to determine reasonable values of x1 and x2 at different loading conditions. 3.1.2. Fixed Loading Margin An alternative approach to assigning a cost to voltage stability is to include a voltage stability inequality constraint. In this formulation, the objective function is a traditional OPF generator operating cost minimization with the following equality constraint added to Eq. (7): k kp P Dkmin ; where Dkmin represents the minimum acceptable margin of stability for the system and is defined by the system operator.
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3.2. Reactive power and voltage rescheduling
4.1. Limits and Lagrange multiplier analysis
Generation is normally scheduled based on the solution of an OPF problem, with an objective to minimize the total cost of the active power. Voltage collapse points tend to be due to reactive power limits [4,20]. Therefore, it is possible to improve the stability margin by changing both the reactive power and voltage levels at generator buses [13]. The proposed methodology, outlined below, uses sequential optimization techniques to enhance the stability margin:
The ‘‘Maximum Distance to Collapse’’ problem (Section 2.2.4) is used to examine how limits affect the maximum loading point and the relationship to Lagrange Multipliers for the 30-bus system, focusing on reactive power limits. With no reactive power limits in the model, the maximum loading point corresponds to a saddle-node bifurcation. If the reactive power limits on generators are incorporated such that they become active, the maximum loading points correspond to limitinduced bifurcations; i.e., the power flow Jacobian is non-singular in this case. Due to the generator reactive power limits, the maximum loading point is reduced, as the space of feasible solutions is ‘‘smaller’’. For active and non-active reactive power limits, a comparison of some generator variables, at k*, when kp = 0.9 is given in Table 1. The Lagrange multipliers indicate that the reactive generation at bus 13 has the greatest influence on the stability margin. This agrees with the physical characteristic of the reactive power that it can not be transported over long distances since the generator at bus 13 is electrically closest to the bulk of the system load. At bus 13, a reactive power price based on the Lagrange multipliers would be large, thereby becoming an incentive to install additional reactive power sources.
(i) The OPF problem is solved, minimizing the total active power generation cost. (ii) The Maximum Distance to Collapse problem, i.e. Eq. (7), is then solved. A constraint fixing the active power generation scheduling found in part (i) is added. The reactive power and the voltage magnitude at the generating buses are free to vary within their limits. The slack bus active power is free to vary to compensate for minor changes in system losses. This methodology has the advantage that it can determine optimal generator voltage and reactive power settings, to enhance the stability margin, while having almost no effect on the real (active) power cost in the system.
4.2. Voltage stability constrained OPF 4. Numerical simulations The maximum distance to collapse and OPF with voltage stability constraints problems presented in Section 3 are tested on three systems: a 30-bus, 6-generator system that is based on the IEEE 30-bus test system, a 57-bus, 7-generator system that is based on the IEEE 57 bus system and a 118-bus, 54-generator system that is based on the IEEE-118 bus system [22]. A nonlinear primal-dual predictor-corrector interior point method [18] written in MATLAB is used to perform the numerical analysis. The results were confirmed using the software package LOQO, which is based on interior point methods [21]. The stability margin for the OPF problems are determined using a continuation method [17].
The multi-objective Linear Combination formulation (Section 3.1.1) was applied to the test systems. Figs. 2 and 3 depict the results obtained Table 1 Results of reactive power limits on various system variables (at k* for kp = 0.9) Parameter
Without reactive power limits (p.u.)
With reactive power limits (p.u.)
Qgen1 Qgen2 Qgen3
2.7950 2.8373 3.9500
1.0000a 0.7000a 0.5000a
Vgen1 Vgen2 Vgen3
1.1000a 1.0700a 1.1000a
1.0962 1.0700a 1.0200
a
Indicates the parameter is at its limit.
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1.4
OPF
λ
*
1.2
Max. Dist ω1 = 1e−4
1
ω1 = 2e−4 ω1 = 4e−4
0.8 0.8
ω1 = 10e−4 1
1.2
1.4
1.6
λ Fig. 2. Maximum loading point versus current loading point for the Linear Combination and Maximum Loading Distance formulations for the 57-bus test system.
9000 8000 7000
cost
6000 5000 OPF Max. Dist ω1 = 1e−4 ω1 = 2e−4 ω1 = 4e−4 ω1 = 10e−4
4000 3000 2000 1000 0.8
1
1.2
1.4
1.6
λ
p
Fig. 3. Operating costs versus current loading point for the Linear Combination formulation, Maximum Loading Distance and traditional OPF for the 57-bus test system.
from applying the Linear Combination formulation, the Maximum Distance to Collapse and the normal strictly economic-based OPF to the 57bus test system. As the factor x1 (note: x2 = 1 x1) is increased, more emphasis is placed on operating costs and less on loading margin. As expected, the solutions obtained from the Linear Combination formulation are bound by the solutions obtained from the Maximum Distance to
Collapse and the normal OPF. At lower values of x1, the Linear Combination solutions tend to the Maximum Distance to Collapse solutions (solid-line versus h-line in Fig. 2), whereas at higher values of x1 these solutions approach the standard OPF solutions (*-line versus v-line in Fig. 3). This characteristic is consistent with Eq. (13), since with x1 = 0 the problem is effectively the same as the Maximum Distance to Collapse Problem, and with x2 = 0 the problem is the same as the standard OPF. From the results obtained in these studies, it was observed that the generator powers were the variables most affected by the changes in the weighting factors, which is expected, as scheduling generation in an area with high loading levels can enhance stability but may result in increased costs. The disadvantage of the Linear Combination formulation is there is no direct control on the size of the stability margin, which is effectively essential since system collapses are generally considered unacceptable. For example, with x1 = 1e 4, it is shown in Fig. 2, that after the current loading point become greater than 0.9, i.e., k > 0.9 the difference between k* and kp, i.e., the stability margin, decreases to zero. As shown in Fig. 2, the stability margin varies as a function of the current loading level, kp and the weighting factors. To overcome this disadvantage, the Fixed Loading Margin formulation (Section 3.1.2) is used. Recall that this method is basically an OPF where a minimum loading margin is ensured. In the numerical analysis presented, a minimum loading margin Dkmin = 0.1 p.u. is used. In general, the algorithm found a solution that ensured this constraint; however, this results in higher operating costs. A comparison of the operating costs of the 118-bus system versus current loading point for the Fixed Loading Margin and normal OPF formulations is shown in Fig. 4. At lower loading levels, kp, there is no difference between the strictly economic-based OPF solutions and the Fixed Loading Margin formulation solutions. This is attributed to the solutions to the OPF problem meeting the stability constraint (note: the OPF problem does not model or incorporate the stability margin). Therefore, the optimal OPF solution lies in the feasible space of the Fixed Loading Margin formulation problem. At higher loading levels,
W. Rosehart et al. / European Journal of Operational Research 171 (2006) 1127–1138 7000
220 Fixed Stability Margin VSC-OPF OPF
200 180
Stability Margin (%)
6000 5000
cost
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4000 3000 2000
160 140 120 100 80 60 40
1000 0
20 0 1
λ
1.5
0.6
0.7
0.8
4.3. Reactive power and voltage rescheduling The results of the proposed reactive power and voltage rescheduling problem (Section 3.2) are compared with the solutions of both the Maximum Distance to Collapse problem and the OPF problem. The stability margin represented as a percentage of the current loading level for the three formulations is shown in Fig. 5. It can be seen from the figure that the stability margin is increased, relative to the OPF solution, when redispatching the reactive power generation and the voltage at the generating buses using the proposed formulation. This increase in stability margin, is obtained without changing active power settings of generators. At higher loading levels, the effect of the proposed formulation is less. This is attributed to the system being highly stressed and the space of feasible solutions is less, so possible changes in the reactive power and voltage settings is significantly less. Fig. 6 is a plot of the total
1.1
1.2
Fig. 5. Stability margin as a percentage of the current loading level for the 30-bus test system. The symbols , and v correspond to the solutions of OPF, Sequential and Maximum Distance formulations, respectively.
400 350
Total Cost
the solutions to the two problems separate, because the OPF solution does not meet the desire stability margin (which is to be expected since the stability margin is not modeled or a constraint in the OPF problem). Therefore, the OPF solution does not lie in feasible space of the Fixed Loading Margin formulation.
1
λP
p
Fig. 4. Cost versus current operating point for the Fixed Loading Margin and traditional OPF formulations for the 118bus test system.
0.9
300 250 200 150
0.6
0.7
0.8
0.9
1
1.1
1.2
λ
P
Fig. 6. The total cost of active power generation versus the current system loading level for the 30-bus test system. The symbols , and v correspond to the solutions of OPF, Sequential and Maximum Distance formulations, respectively.
active power cost for the three formulations at the different loading levels. As can be seen from the figure, the proposed formulation, as expected does not increase the operating cost of the system.
5. Conclusions This paper presents applications of optimization techniques in power system studies with an
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emphasis on solving stability problems. The conditions for saddle-node bifurcation versus limitinduced collapse points are shown. Two optimal power flow problems that incorporate voltage stability criteria are given and implemented on a test system. It is shown that incorporating voltage stability into a traditional OPF problem can result in higher operating costs. Furthermore, the paper presents a new methodology consisting of a sequential Optimal Power Flow to improve stability margins. The reactive power generation and voltage levels settings at the generation buses from the traditional OPF problem are rescheduled such that the stability margin is increased.
References [1] F. Alvarado, B. Borissov, L.D. Kirsch, Reactive Power as an Identified Ancillary Service, Transmission Administrator of Alberta, Ltd., 2001. [2] F. Alvarado, I. Dobson, Y. Hu, Computation of closest bifurcations in power systems, IEEE Transactions on Power Systems 9 (2) (1994) 918–928. [3] C.A. Can˜izares, Calculating optimal system parameters to maximize the distance to saddle node bifurcations, IEEETransactions on Circuits and Systems–I: Fundamental Theory and Applications 45 (3) (1998) 225–237. [4] C. Canizares, Voltage stability assessment: Concepts, practices and tools, Tech. Rep. SP101PSS, IEEE Power Engineering Society, Power System Stability Subcommittee, 2002. [5] C.A. Can˜izares, F.L. Alvarado, Point of collapse and continuation methods for large ac/dc systems, IEEE Transactions on Power Systems 8 (1) (1993) 1–8. [6] A.A. El-Keib, X. Ma, Calculating short-run marginal costs of active and reactive power production, IEEE Transactions on Power Systems 12 (2) (1997) 559–565. [7] I. Dobson, L. Lu, Voltage collapse precipitated by the immediate change in stability when generator reactive power limits are encountered, IEEE Transactions on Circuits and Systems–I: Fundamental Theory and Applications 39 (9) (1992) 762–766. [8] H.W. Dommel, W.F. Tinney, Optimal power flow solutions, IEEE Transactions on Power Apparatus and Systems 87 (10) (1965) 1866–1876. [9] M.E. El-Hawary, IEEE Tutorial Course: Optimal Power Flow: Solution Techniques, Requirements, and Challenges, IEEE Power Engineering Society, Piscataway, NJ, 1996.
[10] J.J. Grainger, J.W.D. Stevenson, Power System Analysis, McGraw-Hill, New York, 1994. [11] M. Huneault, F.D. Galiana, A survey of the optimal power flow literature, IEEE Transactions on Power Systems 6 (2) (1991) 762–770. [12] G. Irisarri, X. Wang, J. Tong, S. Mokhtari, Maximum loadability of power systems using interior point non-linear optimization method, IEEE Transactions on Power Systems 12 (1) (1997) 162–172. [13] T. Menezes, L.C.P. da Silva, C. Affonso, V.F. da Costa, S. Soares, Mvar management on the pre-dispatch problem for improving voltage stability margin, in: Proc. PowerCon 2002, International Conference on Power System Technology, IEEE, vol. 3, 2002, pp. 1690–1694. [14] C. Parker, I. Morrison, D. Sutanto, Application of an optimization method for determining the reactive margin from voltage collapse in reactive power planning, IEEE Transactions on Power Systems 11 (3) (1996) 1473– 1481. [15] W. Rosehart, C. Cazinares, V. Quintana, Multiobjective optimal power flows to evaluate voltage security costs in power networks, IEEE Transactions on Power Systems 18 (2) (2003) 578–587. [16] W. Rosehart, C. Canizares, V. Quintana, Cost of voltage security in electricity markets, in: Proc. of the 2000 IEEE/PES Summer Meeting, IEEE, 2000, pp. 2115– 2120. [17] R. Seydel, From Equilibrium to Chaos—practical Bifurcation and Stability Analysis, North-Holland, Amsterdam, 1988. [18] G.L. Torres, V.H. Quintana, Nonlinear optimal power flow in rectangular form via a primal-dual logarithmic barrier interior point method, Tech. Rep. 96-08, University of Waterloo, 1996. [19] G.L. Torres, V.H. Quintana, An interior-point method for nonlinear optimal power flow using voltage rectangular coordinates, IEEE Transactions on Power Systems (1998) 1211–1218. [20] T.V. Cutsem, A method to compute reactive power margins with respect to voltage collapse, IEEE Transactions on Power Systems 6 (1) (1991) 145–156. [21] R.J. Vanderbei, LOQO UserÕs Manual, Tech. Rep., Princeton University, Princeton, NJ, 2000. [22] University of Washington, Available from:
Interior point models for power system stability problems William Rosehart *, Codruta Roman, Laleh Behjat Department of Electrical and Computer Engineering, University of Calgary, 2500 University Drive, N.W., Calgary, AB, Canada T2N 1N4 Available online 10 March 2005
Abstract In this paper, a detailed analysis of the use of optimization techniques in the study of voltage stability problems, leading to the incorporation of voltage stability criteria in traditional Optimal Power Flow (OPF) formulations is presented. Optimal power flow problems are highly nonlinear programming problems that are used to find the optimal control settings in electrical power systems. The relationship between the Lagrangian Multipliers of the OPF problem and the classification of the maximum loading point level of the system is given. Finally, the paper presents a sequential OPF technique to enhance voltage stability using reactive power and voltage rescheduling with no increase in real (active) generation cost. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Energy; Optimization; Voltage collapse; Optimal power flow
1. Introduction Electricity price volatility, and in some cases a trend towards higher prices, has resulted in uncertainly about the future of some electricity markets. This uncertainty, coupled with previous financial deterrents, has resulted in reduced capital investment in power system infrastructure relative to the growth in demand, particularly with respect to ancillary services such as stability margins. As *
Corresponding author. E-mail addresses: [email protected] (W. Rosehart), [email protected] (C. Roman), [email protected] (L. Behjat).
the overall stability limits can be closely associated with the voltage stability of the network, the incorporation of voltage collapse criteria and tools in the operation of power systems is becoming an essential part of new Energy Management Systems (EMS) [4]. Although, it is possible to directly incorporate voltage stability into Optimal Power Flow (OPF) formulations [16], this tends to result in increased operating costs. Typically, voltage collapse events can be related to a lack of a post contingency equilibrium point in the system, which in turn can be associated theoretically with either a saddle-node bifurcation (SNB) or a limit-induced bifurcation (LIB) [4]. Hence, analysis tools that are currently under use
0377-2217/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.01.021
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throughout the world are mostly based on bifurcation theory and continuation methods. However, it is possible to restate many voltage collapse problems as optimization problems. Although, power system analysis using bifurcation methods are numerically well developed, the use of optimization based techniques has many advantages, including their ability to incorporate limits. This issue becomes even more important when considering limit-induced voltage collapse [7], which can not be easily defined using some of the traditional bifurcation-based computational techniques. Limit-induced voltage collapse is generally the most likely form of collapse to be encountered in power systems [4]. New voltage stability analysis tools that use optimization methods to determine optimal control parameters that maximize load margins have been introduced in the last few years. Various uses of optimization methodologies applied to the voltage collapse problem can be found in the technical literature. In [20], a voltage-collapse point problem is formulated as a nonlinear optimization problem, allowing the use of optimization techniques and tools. In [14], the reactive power margin with respect to voltage collapse is determined using interior point methods; the authors used a barrier function to incorporate limits. In [2], the authors determine the closest bifurcation to the current operating point on the hyperspace of bifurcation points. In [12], the maximum loadability of a power system is examined using interior point methods. In [15], an interior point optimization technique is used to determine the optimal generator settings to maximize the distance to voltage collapse. Furthermore, the algorithm presented in [15] includes constraints on the present operating conditions. Lagrange multipliers also have a very practical applicability in the new deregulated electricity market. In some power systems operating with open access principles, the real power pricing is based on the locational (nodal) spot prices. In this way economical incentives are given to the generators to produce power efficiently and to the consumers to use power efficiently [1]. From the system operatorÕs perspective, the Lagrange multi-
pliers analyzed from the OPF problem gives information about the physical limitation that affects the active power transactions between the seller and the buyer. Electricity as a commodity has a very special behavior related to the way it is transported from the provider to the consumer. Thus, the Lagrange multipliers can be used in a detailed analysis for identifying the factors that affect the power flow and give those factors an economical quantification that can be used to control and improve the power system operation. The real power spot price can be decomposed into components corresponding to the generation, transmission losses, security, voltage control and spinning reserve [23]. The last three terms are very important for the emerging ancillary services in open markets. In open access power systems, a need for a reactive power market as an ancillary service is arising in many cases. Though the reactive power does not always represent a utility for the consumer, it is essential for ensuring real (active) power transmission and the proper range of voltage levels throughout the electrical network. Reactive power transmission and lack of adequate reactive power supplies are generally associated with voltage collapse. A close examination of the Lagrange multipliers associated with the equations involving reactive power dependent terms provides information about the way the reactive power flow affect the voltage levels and/or the secure operation of the system [23,6]. In [24] a Lagrange multiplier analysis is used for the implementation of a load curtailment program to enhance the voltage stability of the system. Based on the Lagrange multiplier values the quantity of load that has to be shed is determined and the level of financial compensation for those consumers can be established. The current paper uses general OPF problems that incorporates voltage stability margins. The main issue considered is how limits affect maximum loading point computations. Two different OPF problems that directly incorporate stability margins are presented. The paper also presents a sequential OPF technique to enhance voltage stability using reactive power and voltage rescheduling. In this approach, first a traditional OPF
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problem is solved. Then a Maximum Distance to Collapse problem is solved with the generator active power scheduling, from the OPF solution, being fixed. The paper is structured as follows. In Section 2, the basic background for the OPF problem and issues related to voltage collapse are discussed. In Section 3, the general formulation to combine OPF and voltage stability is given, including an analysis of the effect of limits on the results. Furthermore, two OPF formulations with voltage stability constraints are given along with a sequential OPF technique to enhance the stability margin. The results of testing the problems on three power system models are given in Section 4. Finally, conclusions are given in Section 5.
2. Background review 2.1. Optimal power flow and optimization techniques The OPF problem introduced in the early 1960s by Carpentier has grown into a powerful tool for power system operation and planning. In general, the OPF problem is a nonlinear programming (NLP) problem that is used to determine the ‘‘optimal’’ control parameter settings to minimize a desired objective function, subject to certain system constraints [8,11,19]. With the introduction of diverse objective functions, the OPF problem represents a variety of optimization problems [11], which includes, for example, active power cost optimization and active power loss minimization [9]. OPF problems are generally formulated as NLP problems as follows: min
GðyÞ
s:t:
F ðyÞ ¼ 0; H 6 H ðyÞ 6 H ; y 6 y 6 y ;
ð1Þ
where the mapping GðyÞ : Rn ! R is the function that is being minimized and can include, for example, total losses in the system or generator costs.
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The equality constraints in Eq. (1), F ðyÞ : Rn ! Rm generally represents the power flow equations. For each bus or node in the system, there are two power flow equations, i.e., active or real power flow and reactive power flow. Active power represents power that does work. For example, active power is used to model the power used to heat a stove, energize a light, or move an elevator. Reactive power is a term developed to represent the power that is being transferred in and out of inductive and capacitive elements in the system each cycle. The charging and discharging characteristic of inductive and capacitive elements is due to fundamental electrostatic and electromagnetic laws and the use of sinusoidal waveforms in electrical power systems. The power flow equations state that the active, P, and reactive, Q, power entering a bus or node must equal the active and reactive power leaving the bus or node. Power from generation directly connected to the bus or node is considered as entering and loads, such as residential or industrial loads, are consider as leaving. It should be noted, that for power system analysis, loads are generally aggregated together to reduce the number of nodes or buses in the problem. Power transfer through transmission lines or transformers is considered to be leaving the node or bus. The power flow equations can be generically written as: 0 ¼ P geni P branchi ðV ; dÞ P loadi for i ¼ 1; . . . ; m2 ;
ð2Þ
0 ¼ Qgeni Qbranchi ðV ; dÞ Qloadi for i ¼ 1; . . . ; m2 ;
ð3Þ
where the nonlinear functions Pbranchi(V, d) and Qbranchi(V, d) represent the power leaving the bus through transmission lines and transformers. These nonlinear functions are derived using Kirchoff Õs Laws and phasor theory [10]. The inequality constraints in Eq. (1), H ðyÞ : Rn ! Rp usually models transmission line and transformer limits, with lower and upper limits represented by H and H , respectively. These limits generally correspond to thermal limits, representing the maximum power flow in the transmission
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line or transformer. The greater the power flow, the higher the thermal losses. The equations to model the power flowing in the transmission lines are nonlinear. Finally, the vector of system variables in Eq. (1), denoted by y 2 Rn , typically includes voltage magnitudes and angles, generator power levels and transformer tap settings; their lower and upper limits are given by y and y , respectively. In most OPF problems, the power flow is modeled using phasor representations of the bus or node voltages. The phasor representations have a magnitude, normalized to the nominal voltage at the particular bus, and an angle defined as the phase difference between the voltage at a particular bus and a common reference bus voltage in the system. 2.2. Voltage collapse and bifurcation theory Several voltage collapse events throughout the world show that power systems are being operated close to their stability limits. Nonlinear phenomena, especially bifurcations, have been shown to be responsible for a variety of stability problems in power systems. In particular, the lack of post contingency equilibrium points, typically associated with saddle-node and limit-induced bifurcations, have been shown to be one of the main reasons for voltage collapse problems in power systems [4]. In general, bifurcation points can be defined as equilibrium points where changes in the ‘‘quantity’’ and/or ‘‘quality’’ of the equilibria associated with a nonlinear set of equations occur with respect to slowly varying parameters in the system [17]. Of interest is the point where the system goes from being stable to unstable with respect to a bifurcation parameter. In this paper, the bifurcation parameter models the loading level of the system, i.e. the power demand of the system. Bifurcations are mathematically characterized by one of the system eigenvalues becoming zero (saddlenode bifurcation) or by eigenvalues become positive when control limits are reached (limit-induced bifurcation). For limit-induced bifurcations, the change in the eigenvalues is due to changes in the
system model, since different control approaches are used when some limits become active. Although, there are many other types of bifurcations [17], the above two are expected to occur in power system models. It is also possible for a system to reach a maximum loading level, that is defined based on operational limits, for example, minimum voltage magnitude limits. In these cases, the system does not go unstable, but the system operators must take action, such as load shedding to prevent problems associated with low voltages. 2.2.1. Saddle-node bifurcations Saddle-node bifurcations are characterized by two equilibrium points, typically one stable (s.e.p.) and one unstable (u.e.p.), merging at an equilibrium point for the parameter value k = k*. If the two merging equilibria co-exist for k < k*, the two equilibrium points locally disappear for k > k*, or vice versa. Saddle node bifurcations are local bifurcations, occurring at the point where the equilibrium locally vanishes for further values of the bifurcation parameter. In normal situations, saddle-node bifurcations will only occur in power systems when the loading levels are very high. The following conditions hold for saddle-node bifurcations: (1) The point is an equilibrium point, i.e., Fk(y) = 0. (2) The Jacobian of the function Fk(y) with respect to y at the bifurcation point (y*, k*), has a unique zero eigenvalue with an associated nonzero eigenvalue. (3) At the saddle-node point, two branches of equilibria intersect and ‘‘disappear’’ beyond the saddle-node [17]. 2.2.2. Limit-induced bifurcations Although saddle-node bifurcations can be shown to be generic in power systems, limits, especially generator reactive power limits, may restrict the space of feasible solutions and voltage collapse may be induced by limits as opposed to a saddlenode bifurcation [7,5]. This has a significant effect on ‘‘measuring’’ the distance to voltage collapse since the voltage collapse may occur by reactive power limits (limit-induced bifurcation) and not by a saddle-node. Limit induced bifurcations as analyzed in [7] occur in power flow equations when generator models are changed from constant volt-
W. Rosehart et al. / European Journal of Operational Research 171 (2006) 1127–1138
age and active power models, to constant active and reactive power models on encountering reactive power limits. The change in models corresponds to a different set of equations; and it is found, in some cases, that the new equations are unstable at the current operating point. Both the original model and the ‘‘limit induced’’ model have the same equilibrium point when the limit is encountered but have different bifurcation diagrams. In the case where the system becomes unstable at the limit, the equilibrium point is on the unstable portion of the ‘‘limit induced’’ bifurcation diagram as shown in Fig. 1. 2.2.3. Operating limit constrained maximum point It is also possible for the maximum loading point to be constrained by operating limits, i.e. the maximum point is not a point where the system goes from being stable to unstable. Power systems have to be maintained within operating limits, for example there are strict rules on the magnitude of the AC voltages. Generally, the voltage magnitudes at each bus or node in the system must be always within a certain percentage of the nominal system voltage. Changes in loads can result in situations where these limits are violated. If changes can not be made to the independent variables to correct these violations, then corrective action
V
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such as load curtailment must be taken. It should be noted, that these types of maxima are not bifurcation points. 2.2.4. Maximum distance to collapse problem In practice, for a given power system, of most interest is to determine and maximize the distance from the current operating point to the maximum operating point, usually represented by a saddlenode, limit induced bifurcation or operating limit constrained maximum. The point of collapse can be found by using direct and/or continuation methods [17]. The problem of voltage collapse given the current operating point also can be formulated as an optimization problem [16]: max s:t:
k kp F ðxp ; x ; q; q ; kp ; k Þ ¼ 0;
ð4Þ
H 6 H ðxp ; x ; q; q Þ 6 H ; where the subscripts p and * indicate the current and critical points, respectively. The parameter q* is used to map the control variables at the current operating point, defined by q, into the critical point, i.e., q* models the optimal change in generation to deal with increases in demand or load. Generators at the critical point are assumed to have the same terminal voltage set points as at the base loading point, and their power levels are represented based on the following model: P G ¼ P Gp þ DP G ;
Limited System
where PGp is the generation scheduling at the current point, DPG* sets the optimal increase in generated powers to reach the critical point, and PG* is the generation scheduling at the critical point. For this model, the dependent and independent (control) variables at the critical point can be grouped as
Unlimited System
x ¼ ½d V L QG
T
and Q limit Q Fig. 1. Illustration of limit induced instability, solid lines represent stable operating points, dashed lines represent unstable operating points. The bolded lines represent the set of operating points when modeling limits.
q ¼ ½P G ðP Gp ; DP G Þ; where d* and VL is the bus voltage phasor magnitude and angle, and QG* is the reactive power of the generators.
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The parameter k 2 R is a scalar bifurcation parameter that is typically known as the ‘‘loading factor’’, as it represents loading level in the system for a linearly increasing, constant power factor load model (for a given bus, the ratio of active/real and reactive power remains constant for all loading levels). The variable k* is a variable in the optimization problem, i.e. it is fully free to change during the solution process; on the other hand, kp is given a fixed value. The loading level for each bus at the current loading point is modeled as follows: P L ¼ kp P Lo ; QL ¼ kp QLo :
QL ¼ kp QLo þ ðk kp ÞQk ;
In general and for the power flow model, an OPF problem that incorporates voltage stability criteria can be generically written as min
Gðxp ; q; kp ; k Þ
s:t:
F ðxp ; q; kp Þ ¼ 0; F ðx ; q ; k Þ ¼ 0; H p 6 H ðxp Þ 6 H p ;
ð7Þ
H 6 H ðx Þ 6 H ; ; q6q6q
ð5Þ
q 6 q 6 q ;
The loading level for each bus at the critical loading is modeled as follows: P L ¼ kp P Lo þ ðk kp ÞP k ;
3. Voltage collapse and Optimal Power Flow (OPF)
ð6Þ
where PL and QL are loading levels for real or active and reactive power, PLo and QLo are the reference or base loading values, and Pk and Qk are the loading direction. The loading direction states how the load in the system will increase and it is assumed that this information is known, i.e. is determined using load profiles [15]. The choice of the loading direction, does have an impact on the maximum loading point as certain areas in the system, particularly areas further from generation, can not support high loading or demand levels. Therefore, care must be taken to use conservative loading directions. In [2], techniques are presented to determine the worse loading direction, i.e. the loading direction that has the shortest distance to instability. In the results presented in this paper, it is assumed that the loading direction equals the base loading values, i.e. Pk = PLo, and Qk = QLo. The Maximum Distance to Collapse problem given in Eq. (4), guarantees the feasibility of the current operating point as well as the feasibility of control and operating limits. Solutions of Eq. (4) correspond to a maximum loading point that can be a real collapse point or a point where operating limits are attained such as bus voltage, line current, or power flow limits.
where the variables and constraints are the same as those defined for Eq. (4) but are written separately for the current and critical point. G(xp, q, kp, k*) is the objective function to be minimized, which has an OPF component, i.e. production costs or losses, that may be dependent on (xp, q, kp), and a voltage stability component that is a function of k* and possibly of kp, as discussed below. Using a Logarithmic Barrier approach [19], the first order Karush–Kuhn–Tucker (KKT) optimality conditions to problem (7) are given to demonstrate when the maximum loading point is defined by a limit-induced bifurcation, a saddle-node bifurcation or an operating limit constrained maximum point. Using slack variables problem (7) can be rewritten as min
Gðxp ; q; kp ; k Þ
s:t:
F ðxp ; q; kp Þ ¼ 0; F ðx ; q; k Þ ¼ 0; H ðxp Þ H p s1 ¼ 0; H p H ðxp Þ s2 ¼ 0; H ðx Þ H s3 ¼ 0; H H ðx Þ s4 ¼ 0; q q s5 ¼ 0; q s6 ¼ 0; q s1 ; s2 ; s3 ; s4 ; s5 ; s6 P 0;
ð8Þ
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where s1 ; s2 ; s3 ; s4 2 RN and s5 ; s6 2 Rm are the primal non-negative slack variables used to transform the inequality constraints to equalities. The nonnegativity constraints are now incorporated into the objective function using a logarithmic barrier as follows: m X min Gðxp ; q; kp ; k Þ l ðlog s5 ½i þ log s6 ½iÞ i¼1
l
N X ðlog s1 ½i þ log s2 ½i þ log s3 ½i þ log s4 ½iÞ i¼1
s:t: F ðxp ; q; kp Þ ¼ 0; F ðx ; q; k Þ ¼ 0; H ðxp Þ H p s1 ¼ 0; H p H ðxp Þ s2 ¼ 0; H ðx Þ H s3 ¼ 0; H H ðx Þ s4 ¼ 0;
q s6 ¼ 0; q
ð9Þ
where l is the barrier parameter and s[i] represents the ith element of the vector s. The Lagrangian function of the modified barrier problem (9) is then defined as m X Luk ¼Gðxp ; q; kp ; k Þ l ðlog s1 ½i þ log s2 ½iÞ i¼1 N X ðlog s3 ½i þ log s4 ½iÞ
3
7 6 rxp L 7 6 7 6 7 6 r k L 7 6 7 6 rq L 7 6 7 6 6 lIeN þ S 1 m1 7 7 6 6 lIe þ S m 7 6 N 2 2 7 7 6 6 lIeN þ S 3 m3 7 7 6 6 lIe þ S m 7 6 N 4 4 7 7 6 6 lIem þ S 5 f1 7 7 6 ru L ¼ 6 7 ¼ 0; 6 lIem þ s6 f2 7 7 6 6 F ðxp ; q; kp Þ 7 7 6 7 6 6 F ðx ; q; k Þ 7 7 6 6 H ðxp Þ H p s1 7 7 6 7 6 6 H p H ðxp Þ s2 7 7 6 7 6 6 H ðx Þ H s3 7 7 6 6 H H ðx Þ s 7 47 6 7 6 4 q q s5 5
ð11Þ
where S1 through S6 are diagonal matrices with elements of the corresponding vector s1 through s6 on the diagonal, I is an identity matrix, e is a vectors of ones, and rx L ¼ cT2 rx F ðx ; q; k Þ ðmT3 þ mT4 Þrx H ðx Þ; rxp L ¼ rxp Gðxp ; q; kp ; k Þ þ cT1 rxp F ðxp ; q; kp Þ ðmT1 þ mT2 Þrxp H ðxp Þ; rk L ¼ rk Gðxp ; q; kp ; k Þ cT2 rk F ðx ; q; k Þ; rq L ¼ Gðxp ; q; kp ; k Þ cT1 rq F ðxp ; q; kp Þ
i¼1
cT1 ðF ðxp ; q; kp ÞÞ
þ ðlog s5 ½i þ log s6 ½iÞ cT2 ðF ðx ; q; k ÞÞ mT1 ðH ðxp Þ H p s1 Þ
cT2 rq F ðx ; q; k Þ f1 þ f2 :
mT2 ðH p H ðxp Þ s2 Þ mT3 ðH ðx Þ H s3 Þ mT4 ðH H ðx Þ s4 Þ fT1 ðq q s5 Þ fT2 ð q q s6 Þ;
rx L
q s6 q
q q s5 ¼ 0;
l
2
1133
ð10Þ
where c1 ; c2 2 RN , m1 ; m2 ; m3 ; m4 2 Rm and f1 ; f2 2 Rm are the Lagrange multipliers. The vector u = (x*, xp, k*, q, s1, s2, s3, s4, s5, s6, c1, c2, m1, m2, m3, m4, f1, f2) is introduced to simplify the expression. The KKT first-order necessary conditions are used to define the local minimum of Eq. (9).
ð12Þ
The issue of collapse due to limit-induced bifurcation versus saddle-node bifurcation can now be explained as follows: The first condition in (12), $x*L = 0, includes the Jacobian of the system model at the maximum loading point multiplied by c2, which can be considered to be equivalent to an eigenvector of a Jacobian. Therefore, the first condition corresponds to a singular Jacobian if ðmT3 þ mT4 Þrx H ðx Þ ¼ 0; this would imply that, if one assumes that $x*H(x*) is non-singular (which is typically the case when limits directly on x are
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being enforced, i.e. $x*H(x*) is an identity matrix), the dependent variables are not at their limits, since m3 and m4 are zero when their corresponding limits are not active. If dependent variables of the critical point are at their limits, then m3 and m4 may become non-negative, i.e., the load flow Jacobian may be non-singular. If m3 and m4 are non-negative, which implies that locally the objective function could be improved if the limits are not enforced, the system has reached a limitinduced bifurcation point. If limits are reached and m3 and m4 remain zero, this implies that the limits are not locally limiting an improvement to the objective function and hence the system has reached a saddle-node bifurcation point. Although the above discussion is applied to the solution of the associated barrier problem, the characteristics are the same when applied to the original problem directly. The above derivation demonstrates the maximum loading point may be a limit-induced point only when the inequality constraints on the dependent variables become active and the inequality constraints can be separated based on the dependent and independent variables of the load flow model. The independent variables, q, being at their limits, do not directly affect the type of bifurcation. A particular example of this optimization problem is the ‘‘Maximum Distance to Collapse’’ whose constraints on the current and critical loading point are given in Eq. (4). The problem maximizes the distance to a saddle-node, limit-induced bifurcation or an operating limit constrained maximum. Including the current loading point into the constraints ensures that, when independent variables are calculated to maximize the distance to voltage collapse, feasibility and inequality constraints at the current loading point are met. For example, increasing generator voltage magnitude settings generally increases the distance to collapse but, under lighter loading conditions, the increased levels may lead to over-voltages. Incorporating the current operating point into the optimization problem can eliminate this problem; however, it also reduces the space of feasible solutions. This formulation differs from existing formulations, for example [3,12], since constraints are placed in
the critical loading point and the current loading point. 3.1. Voltage stability constrained OPF With the current loading point included into the optimization problem, it is possible to incorporate voltage stability criteria into an OPF formulation at the ‘‘currentÕ operating point xp. As the operating point moves closer to a critical point, i.e. as xp approaches x*, more emphasis must be placed on maximizing voltage stability as opposed to minimizing operating costs. The following techniques, proposed in [15] are used in this paper: 3.1.1. Linear Combination In this formulation the maximum loading is directly incorporated into the following objective function: Gðxp ; q; kp ; k Þ ¼ x1 gðxp ; q; kp Þ x2 ðk kp Þ ð13Þ subject to the constraints in Eq. (7). Observe that this requires the introduction of two weighting factors x1 and x2 to balance the emphasis placed on maximizing stability, i.e. (k* kp), versus minimizing costs, which are represented by g(xp, q, kp) in Eq. (13). Generally, x2 must be significantly larger than x1, as the relative difference in the magnitudes of each term in the objective function is large; it is assumed that x1 + x2 = 1 to normalize their values. Values obtained from previous OPF and Maximum Loading Distance analysis can be used to determine reasonable values of x1 and x2 at different loading conditions. 3.1.2. Fixed Loading Margin An alternative approach to assigning a cost to voltage stability is to include a voltage stability inequality constraint. In this formulation, the objective function is a traditional OPF generator operating cost minimization with the following equality constraint added to Eq. (7): k kp P Dkmin ; where Dkmin represents the minimum acceptable margin of stability for the system and is defined by the system operator.
W. Rosehart et al. / European Journal of Operational Research 171 (2006) 1127–1138
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3.2. Reactive power and voltage rescheduling
4.1. Limits and Lagrange multiplier analysis
Generation is normally scheduled based on the solution of an OPF problem, with an objective to minimize the total cost of the active power. Voltage collapse points tend to be due to reactive power limits [4,20]. Therefore, it is possible to improve the stability margin by changing both the reactive power and voltage levels at generator buses [13]. The proposed methodology, outlined below, uses sequential optimization techniques to enhance the stability margin:
The ‘‘Maximum Distance to Collapse’’ problem (Section 2.2.4) is used to examine how limits affect the maximum loading point and the relationship to Lagrange Multipliers for the 30-bus system, focusing on reactive power limits. With no reactive power limits in the model, the maximum loading point corresponds to a saddle-node bifurcation. If the reactive power limits on generators are incorporated such that they become active, the maximum loading points correspond to limitinduced bifurcations; i.e., the power flow Jacobian is non-singular in this case. Due to the generator reactive power limits, the maximum loading point is reduced, as the space of feasible solutions is ‘‘smaller’’. For active and non-active reactive power limits, a comparison of some generator variables, at k*, when kp = 0.9 is given in Table 1. The Lagrange multipliers indicate that the reactive generation at bus 13 has the greatest influence on the stability margin. This agrees with the physical characteristic of the reactive power that it can not be transported over long distances since the generator at bus 13 is electrically closest to the bulk of the system load. At bus 13, a reactive power price based on the Lagrange multipliers would be large, thereby becoming an incentive to install additional reactive power sources.
(i) The OPF problem is solved, minimizing the total active power generation cost. (ii) The Maximum Distance to Collapse problem, i.e. Eq. (7), is then solved. A constraint fixing the active power generation scheduling found in part (i) is added. The reactive power and the voltage magnitude at the generating buses are free to vary within their limits. The slack bus active power is free to vary to compensate for minor changes in system losses. This methodology has the advantage that it can determine optimal generator voltage and reactive power settings, to enhance the stability margin, while having almost no effect on the real (active) power cost in the system.
4.2. Voltage stability constrained OPF 4. Numerical simulations The maximum distance to collapse and OPF with voltage stability constraints problems presented in Section 3 are tested on three systems: a 30-bus, 6-generator system that is based on the IEEE 30-bus test system, a 57-bus, 7-generator system that is based on the IEEE 57 bus system and a 118-bus, 54-generator system that is based on the IEEE-118 bus system [22]. A nonlinear primal-dual predictor-corrector interior point method [18] written in MATLAB is used to perform the numerical analysis. The results were confirmed using the software package LOQO, which is based on interior point methods [21]. The stability margin for the OPF problems are determined using a continuation method [17].
The multi-objective Linear Combination formulation (Section 3.1.1) was applied to the test systems. Figs. 2 and 3 depict the results obtained Table 1 Results of reactive power limits on various system variables (at k* for kp = 0.9) Parameter
Without reactive power limits (p.u.)
With reactive power limits (p.u.)
Qgen1 Qgen2 Qgen3
2.7950 2.8373 3.9500
1.0000a 0.7000a 0.5000a
Vgen1 Vgen2 Vgen3
1.1000a 1.0700a 1.1000a
1.0962 1.0700a 1.0200
a
Indicates the parameter is at its limit.
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1.4
OPF
λ
*
1.2
Max. Dist ω1 = 1e−4
1
ω1 = 2e−4 ω1 = 4e−4
0.8 0.8
ω1 = 10e−4 1
1.2
1.4
1.6
λ Fig. 2. Maximum loading point versus current loading point for the Linear Combination and Maximum Loading Distance formulations for the 57-bus test system.
9000 8000 7000
cost
6000 5000 OPF Max. Dist ω1 = 1e−4 ω1 = 2e−4 ω1 = 4e−4 ω1 = 10e−4
4000 3000 2000 1000 0.8
1
1.2
1.4
1.6
λ
p
Fig. 3. Operating costs versus current loading point for the Linear Combination formulation, Maximum Loading Distance and traditional OPF for the 57-bus test system.
from applying the Linear Combination formulation, the Maximum Distance to Collapse and the normal strictly economic-based OPF to the 57bus test system. As the factor x1 (note: x2 = 1 x1) is increased, more emphasis is placed on operating costs and less on loading margin. As expected, the solutions obtained from the Linear Combination formulation are bound by the solutions obtained from the Maximum Distance to
Collapse and the normal OPF. At lower values of x1, the Linear Combination solutions tend to the Maximum Distance to Collapse solutions (solid-line versus h-line in Fig. 2), whereas at higher values of x1 these solutions approach the standard OPF solutions (*-line versus v-line in Fig. 3). This characteristic is consistent with Eq. (13), since with x1 = 0 the problem is effectively the same as the Maximum Distance to Collapse Problem, and with x2 = 0 the problem is the same as the standard OPF. From the results obtained in these studies, it was observed that the generator powers were the variables most affected by the changes in the weighting factors, which is expected, as scheduling generation in an area with high loading levels can enhance stability but may result in increased costs. The disadvantage of the Linear Combination formulation is there is no direct control on the size of the stability margin, which is effectively essential since system collapses are generally considered unacceptable. For example, with x1 = 1e 4, it is shown in Fig. 2, that after the current loading point become greater than 0.9, i.e., k > 0.9 the difference between k* and kp, i.e., the stability margin, decreases to zero. As shown in Fig. 2, the stability margin varies as a function of the current loading level, kp and the weighting factors. To overcome this disadvantage, the Fixed Loading Margin formulation (Section 3.1.2) is used. Recall that this method is basically an OPF where a minimum loading margin is ensured. In the numerical analysis presented, a minimum loading margin Dkmin = 0.1 p.u. is used. In general, the algorithm found a solution that ensured this constraint; however, this results in higher operating costs. A comparison of the operating costs of the 118-bus system versus current loading point for the Fixed Loading Margin and normal OPF formulations is shown in Fig. 4. At lower loading levels, kp, there is no difference between the strictly economic-based OPF solutions and the Fixed Loading Margin formulation solutions. This is attributed to the solutions to the OPF problem meeting the stability constraint (note: the OPF problem does not model or incorporate the stability margin). Therefore, the optimal OPF solution lies in the feasible space of the Fixed Loading Margin formulation problem. At higher loading levels,
W. Rosehart et al. / European Journal of Operational Research 171 (2006) 1127–1138 7000
220 Fixed Stability Margin VSC-OPF OPF
200 180
Stability Margin (%)
6000 5000
cost
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4000 3000 2000
160 140 120 100 80 60 40
1000 0
20 0 1
λ
1.5
0.6
0.7
0.8
4.3. Reactive power and voltage rescheduling The results of the proposed reactive power and voltage rescheduling problem (Section 3.2) are compared with the solutions of both the Maximum Distance to Collapse problem and the OPF problem. The stability margin represented as a percentage of the current loading level for the three formulations is shown in Fig. 5. It can be seen from the figure that the stability margin is increased, relative to the OPF solution, when redispatching the reactive power generation and the voltage at the generating buses using the proposed formulation. This increase in stability margin, is obtained without changing active power settings of generators. At higher loading levels, the effect of the proposed formulation is less. This is attributed to the system being highly stressed and the space of feasible solutions is less, so possible changes in the reactive power and voltage settings is significantly less. Fig. 6 is a plot of the total
1.1
1.2
Fig. 5. Stability margin as a percentage of the current loading level for the 30-bus test system. The symbols , and v correspond to the solutions of OPF, Sequential and Maximum Distance formulations, respectively.
400 350
Total Cost
the solutions to the two problems separate, because the OPF solution does not meet the desire stability margin (which is to be expected since the stability margin is not modeled or a constraint in the OPF problem). Therefore, the OPF solution does not lie in feasible space of the Fixed Loading Margin formulation.
1
λP
p
Fig. 4. Cost versus current operating point for the Fixed Loading Margin and traditional OPF formulations for the 118bus test system.
0.9
300 250 200 150
0.6
0.7
0.8
0.9
1
1.1
1.2
λ
P
Fig. 6. The total cost of active power generation versus the current system loading level for the 30-bus test system. The symbols , and v correspond to the solutions of OPF, Sequential and Maximum Distance formulations, respectively.
active power cost for the three formulations at the different loading levels. As can be seen from the figure, the proposed formulation, as expected does not increase the operating cost of the system.
5. Conclusions This paper presents applications of optimization techniques in power system studies with an
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emphasis on solving stability problems. The conditions for saddle-node bifurcation versus limitinduced collapse points are shown. Two optimal power flow problems that incorporate voltage stability criteria are given and implemented on a test system. It is shown that incorporating voltage stability into a traditional OPF problem can result in higher operating costs. Furthermore, the paper presents a new methodology consisting of a sequential Optimal Power Flow to improve stability margins. The reactive power generation and voltage levels settings at the generation buses from the traditional OPF problem are rescheduled such that the stability margin is increased.
References [1] F. Alvarado, B. Borissov, L.D. Kirsch, Reactive Power as an Identified Ancillary Service, Transmission Administrator of Alberta, Ltd., 2001. [2] F. Alvarado, I. Dobson, Y. Hu, Computation of closest bifurcations in power systems, IEEE Transactions on Power Systems 9 (2) (1994) 918–928. [3] C.A. Can˜izares, Calculating optimal system parameters to maximize the distance to saddle node bifurcations, IEEETransactions on Circuits and Systems–I: Fundamental Theory and Applications 45 (3) (1998) 225–237. [4] C. Canizares, Voltage stability assessment: Concepts, practices and tools, Tech. Rep. SP101PSS, IEEE Power Engineering Society, Power System Stability Subcommittee, 2002. [5] C.A. Can˜izares, F.L. Alvarado, Point of collapse and continuation methods for large ac/dc systems, IEEE Transactions on Power Systems 8 (1) (1993) 1–8. [6] A.A. El-Keib, X. Ma, Calculating short-run marginal costs of active and reactive power production, IEEE Transactions on Power Systems 12 (2) (1997) 559–565. [7] I. Dobson, L. Lu, Voltage collapse precipitated by the immediate change in stability when generator reactive power limits are encountered, IEEE Transactions on Circuits and Systems–I: Fundamental Theory and Applications 39 (9) (1992) 762–766. [8] H.W. Dommel, W.F. Tinney, Optimal power flow solutions, IEEE Transactions on Power Apparatus and Systems 87 (10) (1965) 1866–1876. [9] M.E. El-Hawary, IEEE Tutorial Course: Optimal Power Flow: Solution Techniques, Requirements, and Challenges, IEEE Power Engineering Society, Piscataway, NJ, 1996.
[10] J.J. Grainger, J.W.D. Stevenson, Power System Analysis, McGraw-Hill, New York, 1994. [11] M. Huneault, F.D. Galiana, A survey of the optimal power flow literature, IEEE Transactions on Power Systems 6 (2) (1991) 762–770. [12] G. Irisarri, X. Wang, J. Tong, S. Mokhtari, Maximum loadability of power systems using interior point non-linear optimization method, IEEE Transactions on Power Systems 12 (1) (1997) 162–172. [13] T. Menezes, L.C.P. da Silva, C. Affonso, V.F. da Costa, S. Soares, Mvar management on the pre-dispatch problem for improving voltage stability margin, in: Proc. PowerCon 2002, International Conference on Power System Technology, IEEE, vol. 3, 2002, pp. 1690–1694. [14] C. Parker, I. Morrison, D. Sutanto, Application of an optimization method for determining the reactive margin from voltage collapse in reactive power planning, IEEE Transactions on Power Systems 11 (3) (1996) 1473– 1481. [15] W. Rosehart, C. Cazinares, V. Quintana, Multiobjective optimal power flows to evaluate voltage security costs in power networks, IEEE Transactions on Power Systems 18 (2) (2003) 578–587. [16] W. Rosehart, C. Canizares, V. Quintana, Cost of voltage security in electricity markets, in: Proc. of the 2000 IEEE/PES Summer Meeting, IEEE, 2000, pp. 2115– 2120. [17] R. Seydel, From Equilibrium to Chaos—practical Bifurcation and Stability Analysis, North-Holland, Amsterdam, 1988. [18] G.L. Torres, V.H. Quintana, Nonlinear optimal power flow in rectangular form via a primal-dual logarithmic barrier interior point method, Tech. Rep. 96-08, University of Waterloo, 1996. [19] G.L. Torres, V.H. Quintana, An interior-point method for nonlinear optimal power flow using voltage rectangular coordinates, IEEE Transactions on Power Systems (1998) 1211–1218. [20] T.V. Cutsem, A method to compute reactive power margins with respect to voltage collapse, IEEE Transactions on Power Systems 6 (1) (1991) 145–156. [21] R.J. Vanderbei, LOQO UserÕs Manual, Tech. Rep., Princeton University, Princeton, NJ, 2000. [22] University of Washington, Available from: