Complementarity model for load tap changing transformers in stability based OPF problem

Electric Power Systems Research 76 (2006) 592–599

Complementarity model for load tap changing transformers in stability based OPF problem夽 Codruta Roman ∗ , William Rosehart Department Electrical and Computer Engineering, University of Calgary, Calgary, Alta., Canada T2N 1N4 Received 12 May 2005; accepted 25 September 2005 Available online 21 November 2005

Abstract The paper presents a stability optimal power flow (OPF) based problem that incorporates complementarity models for load tap changing transformers (LTCs). The complementarity formulation models the change of behavior of LTC controlled buses when the tap becomes blocked at one of its limits and the voltage level can no longer be regulated. The incorporation of proposed model in stability OPF based problem leads to a more accurate estimation of the maximum or critical loading level. © 2005 Elsevier B.V. All rights reserved. Keywords: Voltage stability; Load tap changing transformers; Optimal power flow; Mathematical programming with complementarity constraints

1. Introduction The increase of power demand at a higher rate than the expansion of generation and transmission facilities has resulted in power systems functioning closer to their operational and physical limits. In heavily loaded systems, voltage profile in the transmission systems is often maintained by generator reactive power injection or by adjusting the tap ratio of load tap changing transformers. The possible effects of the limits of reactive power and voltage control devices on the voltage stability related phenomena have been acknowledged in the literature, for example [1–7]. In [5,8,9], complementarity modeling of generator buses yields accurate loadability margins in stability margin based optimization problems. Load tap changing transformers (LTCs) can be found in power systems at the generation plants as step-up transformers, within the transmission system to control the reactive power flow and the voltage profile, and at substations feeding loading centers to keep the secondary voltage at an acceptable value when the primary voltage magnitudes undergo variations.

夽 ∗

This work is funded through an NSERC Discovery Grant (Canada). Corresponding author. Tel.: +1 403 210 5453. E-mail address: [email protected] (C. Roman).

0378-7796/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2005.09.015

Many papers analyze tap changer behavior at distribution points, where the interaction between the load and the system characteristics can lead to voltage collapse [2,10–12]. In the current paper, transmission LTC’s are considered not LTC’s at distribution points. In [13] an algorithm to determine the optimal steady-state transmission LTC setting based on the sensitivity with respect to the collapse point for a known pattern of active generation is proposed. By re-distributing reactive flow in the network, LTCs can improve the loadability of the system [4,10,13]. In [14], the continuation method has been employed to show the effect on the maximum loadability of the system when LTC taps are at limit. When the tap is strictly within the operational range, the voltage at the controlled terminal is kept constant. Once one of the limits is reached, the controlled bus voltage becomes a variable and the tap a constant. This paper focuses on the importance of proper LTC modeling, with respect to limits, in optimization based stability margin problems. The proposed complementarity model incorporates the behavior of LTC controlled buses when tap ratio limits are attained, assuming steady-state conditions. Even though most transmission LTC’s are on manual remote control, the models presented in this paper assume the operators will adjust the tap settings to regulate the controlled voltage. Therefore, to model the mapping between a base loading point and a maximum steady-state loading point, there is a need to incorporate this manual control.

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The paper is organized as follows: In Section 2, the maximum loading operating points are defined. The complementarity concepts are briefly reviewed in Section 3.1. Section 3.2 presents a mathematical model and the complementarity model for LTC transformers. In Section 4, four OPF-based formulations to determine the maximum or critical loading level of the system are presented. The four problems have been solved using the LOQO interior point solver. The numerical results and their discussion are presented in Section 5. Finally, conclusions are given in Section 6. Fig. 1. Voltage profile for an LTC controlled bus when the tap ratio is at the limit.

2. Static loadability and bifurcations in power systems The loadability of a power system is the capability of the system to transfer the amount of the power demand from generation locations to the load locations while the operating parameters are within admissible ranges. In static voltage stability studies, the maximum loading margin problem determines the amount of increase in demand such that the maximum operating point is stable and characterized by operating settings within accepted tolerances. The loading margin can be measured using the loading factor, λ, defined as follows: PL = λPL0 ,

QL = λQL0

(1)

where PL0 and QL0 are the “base” active and reactive power demand, and PL and QL are the current active and reactive power demand, respectively. The loading factor, λ, defined as a vector, gives the directions of the demand increase for each bus. If the pattern of demand increase is known, λ can be replaced with a single parameter. In this paper, for all models and problems considered, it is assumed that the loading direction is known and therefore λ is a scalar. With slow increases of λ, the system will eventually reach an operating point where the stability of the system can change or one of the operating parameters is at its limit. This point is characterized by the critical or maximum loading level, λc . In static loadability assessment, the maximum (critical) loading level λc can be due to a saddle-node bifurcation (SNB), a limit induced bifurcation (LIB) or an operational limit, for example a voltage limit or a line thermal limit. Saddle-node bifurcations are generally characterized by a local loss of equilibrium and single purely real zero eigenvalue [15]. The term limit induced bifurcation is associated with a stable equilibrium that becomes unstable when a limit of a device in the system, typically a generator’s reactive power limit, is attained [16]. As the load of the system increases, a voltage controlling equipment might be forced to operate at its controlling limit and lose the ability to regulate the voltage. The operating point when the controlling limit is encountered is known as a breaking or switching point. At the generator buses, the voltage is controlled and kept to a specified magnitude by generation or absorption of reactive power. Another device used for controlling the voltage levels is the load tap changing transformer that maintains the voltage constant at one of its terminal buses by adjusting its tap ratio.

As in the case of the generator, the voltage can be controlled only if the tap ratio takes values within the limits. If one of the limits is reached, the tap ratio will have a fixed value and voltage at the controlled bus starts to change as shown in Fig. 1 [14]. Accordingly, the set of equations describing the system change. The new set of equations can have stable solutions and the load of the system can be still increased until an operational limit, a saddle-node or a limit induced bifurcation is reached. In Fig. 1, continuous control of the tap is assumed for illustrative purposes. In actual systems, a band on the voltage would be defined, and the taps would be adjusted in discrete steps to maintain the voltage in the band. The loadability of the system can be computed by using continuation methods when the load and generation increase pattern are known. The switch between the different sets of equations when one of the generation or LTC limits is attained can be easily implemented. Another technique used in loadability assessment is based on nonlinear mathematical programming. The advantage of the optimization formulation is that it gives the maximum possible loading level of the system along with the control variable settings of this operating point. The classical nonlinear optimization formulation does not include the change of behavior of voltage controlled buses when one of the limits is attained. The change of behavior of voltage controlled buses can be implemented by using complementarity problems. The complementarity condition may lead to solving problems when standard nonlinear techniques are applied. Nevertheless, some interior point methods have been adjusted to handle complementarity type constraints [17]. 3. Complementarity model for load tap changing transformers 3.1. Brief review on complementarity In engineering and economical applications, the complementarity condition depicts the change of behavior of a system depending on the way certain conditions are satisfied. Many practical situations can be expressed as nonlinear complementarity problems (NCP) and mixed complementarity problems (MCP) [18]. The term “complementarity” comes from geometry where it is said that two angles complement each other if their sum is π/2.

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In complementarity theory, two vectors are said to complement each other if they form a right angle. If these two vectors are defined on the nonnegative real orthant, the nonlinear complementarity problem (NCP) can be stated as [19] find y ∈ n

for F : n → n 0 ≤ y ⊥ F(y) ≥ 0

such that

(2)

In (2), the vectors y and F(y) are perpendicular to each other if their scalar product is zero y F(y) = T

n


yi Fi (y) = 0,

with yi ≥ 0 and Fi (y) ≥ 0

(3)

i=1

which implies yi Fi (y) = 0,

∀i = 1, . . . , n

(1) yi = 0 and Fi (y) = 0 (2) yi = 0 and Fi (y) = 0 (3) yi = 0 and Fi (y) = 0 The first two solutions are said to satisfy strict complementarity conditions. The third solution, in which both values are zero, exhibits nonstrict complementarity behavior. In most mixed complementarity problems, the vector y is lower and upper bounded and the function F(y) can take positive, negative and zero values. In this sense, the MCP can be defined as [20] for F : n → n

such that ymin ≤ y ≤ ymax ⊥ F(y)

(5)

The vector y complements the function F(y) in the following sense [20]: if yi = yimin ⇒ Fi (y) ≥ 0

(6a)

if yimin < yi < yimax ⇒ Fi (y) = 0

(6b)

if yi = yimax ⇒ Fi (y) ≤ 0

(6c)

The mixed complementarity problem (5) can be written as two correlated complementarity problems by introducing two new positive variables, x and z, as follows [21]: find

y ∈ [ymin , ymax ], x ∈ n+ , z ∈ n+

such that

0 ≤ y − ymin ⊥ x ≥ 0

(1) if yi = yimin , xi ≥ 0 and zi = 0 ⇒ Fi (y) ≥ 0 (2) if yi = yimax , xi = 0 and zi ≥ 0 ⇒ Fi (y) ≤ 0 (3) if yimin < yi < yimax , xi = 0 and zi = 0 ⇒ Fi (y) = 0

(4)

Eq. (4) holds only if one of the following situations is satisfied:

find y ∈ [ymin , ymax ]

Fig. 2. Equivalent scheme of the load tap changing transformer.

(7a)

0 ≤ ymax − y ⊥ z ≥ 0

(7b)

F(y) = x − z

(7c)

Since yi can only be at either its maximum or minimum value, only one of xi or zi can be nonzero for the complementarity conditions to be satisfied. The equivalence between the results in (7) and those in (6) can be done by evaluating the three scenarios in (7) as follows:

3.2. Complementarity model for load tap changing transformers The load tap changing transformer (LTC) is a transformer with a variable tap ratio that can be used to control the voltage of one of its terminals. In this paper, the LTC transformer is modeled as a π equivalent circuit with elements that depend on the tap ratio. In Fig. 2, the LTC connected between nodes i and j has the variable tap ratio a and the longitudinal admittance Yt . The admittance of the magnetizing branch has been neglected [22]. The tap ratio is determined to maintain the voltage at the controlled bus (for example, node j in Fig. 2) to a desired value, Vref . When the voltage magnitude, Vt , at the controlled bus is outside the range defined by the adjustable dead band of the regulator, control action is taken to bring the voltage Vt back to the desired value Vref [23]. The control action consists of changing the tap ratio of the transformer by changing the tap position. The tap position is moved in a finite number of steps. In mathematical problems involving power flow calculations or quasi-steady-state, the difference between two tap positions is assumed small so that the tap ratio of the transformer can be modeled as a continuous function, i.e. [2] da 1 = (Vref − Vt ) dt T

(8)

where T > 0 is the time constant of the controller. According to (8), when the voltage Vt tends to decrease, the tap ratio, a, will take a higher value such that Vt is brought back to the desired value Vref . When using a continuous model, the dead band of the transformer is often ignored. Once a final solution has been found, a rounding or discretization algorithm is applied to converse the value of a to a discrete setting. Since the models considered in this paper takes into account only two steady operating points, a base operating point and the maximum or critical operating point, there is no significant advantage of incorporating dead bands and time delays into the model. The tap ratio, a, can take values only between the lower limit, amin , and the upper limit, amax . Therefore, the voltage at the

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controlled bus will have the following behavior:  min   Vt ≥ Vref if a = a V =

 

Vref

Vt ≤ Vref

if amin < a < amax if a = amax

(9)

The behavior described in (9) can be mathematically formulated as a mixed complementarity problem a ∈ , V1 ∈ , V2 ∈ 

find such that 0 ≤ (a

max

0≤

(a − amin )

⊥ V1 ≥ 0

− a) ⊥ V2 ≥ 0

(10a) (10b)

Vt = Vref + V1 − V2

(10c)

where the quantity V1 is the increase in the voltage magnitude at the controlled bus if the lower limit of the tap ratio is reached, and V2 is the decrease in the voltage magnitude at the controlled bus if the upper tap limit is attained.If the tap ratio is at the upper limit, in (10) a − amin = amax − amin > 0 ⇒ V1 = 0, amax − a = 0 ⇒ V2 ≥ 0,

Vt = Vref − V2 ≤ Vref

(11)

Thus, if the tap ratio is at the lower limit, in (10) a − amin = 0 ⇒ V1 ≥ 0,

(12)

If the tap ratio takes values strictly within the limits a − amin > 0 ⇒ V1 = 0,

amax − a > 0 ⇒ V2 = 0,

Vt = Vref

(13)

Eqs. (10) can be compacted as the following mixed complementarity problem: find

a ∈ , Vt ∈ 

such that

amin

≤a≤

amax

⊥ Vt

Vt = Vref + Vt

traditional maximum loading problem is an optimization problem with the objective of maximizing critical loading level, λc , and constraints that model the critical operating point. For a known base loading level and known direction of load increase, the critical operating point is determined by solving the following problem: max λc

(15a)

s.t. : f (δc , Vtc , ac , Vc , Pgenc , Qgenc , λc ) = 0

(15b)

hmin ≤ h(δc , Vtc , ac , Vc , Pgenc , Qgenc , λc ) ≤ hmax

(15c)

where (15b) represents the standard ac power flow equations and (15c) represent the operational limits at the critical operating point. The variables characterizing the critical operating point are: the critical voltage magnitudes at the LTC controlled buses, Vtc , and the critical voltage magnitudes at the generator and load buses, Vc ; the critical voltage angles, δc ; the critical tap ratios, ac ; the critical active and reactive generation, Pgenc and Qgenc , respectively; and the critical loading level, λc . The critical loading level obtained from maximum loadability problem (15) can be due to an SNB, an LIB, or an operational limit. This problem cannot accurately incorporate the control at LTC controlled buses and there is no mechanism to implement the switch between controls. 4.2. Maximum loading distance problem for a given current operating point (problem P2)

amax − a = amax − amin > 0 ⇒ V2 = 0, Vt = Vref + V1 ≥ Vref

595

(14)

where Vt = V1 − V2 is the change in the voltage magnitude at the controlled bus. 4. Maximum loading margin for systems with load tap changing transformer In each of the following problems, the loading level at the maximum or critical loading point is defined according to (1), where λ is a scalar variable in the optimization problem. 4.1. Maximum loading problem (problem P1) Most of the initial optimization formulations used for computing the critical loading level were developed based on the equivalence between the saddle-node bifurcation equations and the first order optimality necessary conditions [1,24]. Thus, the

The maximum loading problem, (15), incorporates only the critical operating point, and consequently, the effect of given controls at the base operating point on the value of λc is not known. Therefore, an optimization problem that takes into account the control settings of the current operating point is introduced max λc − λ0

(16a)

s.t. : fλ0 (δc , Vtc , ac , Vc , Qgenc , Pgenc , λc ) = 0

(16b)

hmin ≤ h(δc , Vtc , ac , Vc , Qgenc , Pgenc ) ≤ hmax

(16c)

Vtc = Vt0

(16d)

Vgenc = Vgen0

(16e)

where Eq. (16b) represent the standard ac power flow equations and the operational limits at the critical operating point depending on the control variables at the current operating point. In this formulation, Vt0 , Vgen0 , represent the voltage levels at the LTC and generator controlled buses at the base operating point, and Vgenc the voltage level at the generator buses at the critical operating point. The settings at the base point, Vt0 , Vgen0 , are determined in advanced from an economical dispatch problem. The objective of the second problem is to maximize the difference between the unknown loading level at the critical operating point, λc , and the known loading level at the current (or base) operating point, λ0 . The second problem includes a mapping between the base operating point and the critical operating point. For this problem,

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this mapping is realized through the same control settings at the current and critical operating points for LTC and generator controlled buses, enforced by (16d) and (16e). In the numerical results section, an alternative of this problem, denoted P2 , i.e. (16), is considered where the constraint (16d) is replaced with ac = a0

(17)

i.e. the tap has the same ratio for both the current and critical operating points. These problems, (16) and (16) with (16d) replaced by (17), might yield a critical operating point that can correspond to a switching (breaking) point that represents a false maximum. For the original problem, i.e. (16), this point would correspond to the case where a constant voltage bus should be changed to a constant tap ratio for a LTC controlled bus which has reached a tap limit. It should be noted, that in the case of (16) with (16d) replaced by (17), it is unlikely that control action would be taken to switch the LTC from constant tap ratio to constant voltage in the event of a voltage limit becoming active. 4.3. Maximum loading distance problem for a given current operating point with complementarity models for LTCs (problem P3) A further enhancement of the maximum loading margin problem is the introduction of the change of behavior of LTC controlled buses. As discussed in Section 3, the LTC controlled buses are characterized by a constant voltage as long as the tap setting does not reach its limit. The system can still be working satisfactory, even if some LTCs have lost their control, resulting in a variable voltage and a constant tap ratio at the controlled buses. In this sense, the following optimization problem is defined: max λc − λ0

(18a)

s.t. : fλ0 (δc , Vtc , ac , Vc , Qgenc , Pgenc , λc ) = 0

(18b)

hmin ≤ h(δc , Vtc , Vc , Qgenc , Pgenc ) ≤ hmax

(18c)

amin ≤ ac ≤ amax ⊥ Vt

(18d)

Vtc = Vt0 + Vt

(18e)

Vgenc = Vgen0

(18f)

where (18d) and (18e) represent the complementarity model for LTCs controlled buses. The variable Vt represent the change in voltage magnitude at an LTC controlled bus if the tap ratio has reached the lower or the upper limit. The variables of the third problem have the same meaning as those of the other two problems. Problem (18) does not include a complementarity model for generator buses because their reactive resources are not desired to be exhausted in steady-state. The fast response of the synchronous generators with respect to reactive generation makes their intervention more desirable in an emergency situation [25,26].

Fig. 3. IEEE 30 bus test system [28].

In this problem, the critical operating point corresponds to either an SNB, an LIB, or an operational limit. 5. Numerical results The three problems have been solved for a 30-bus system based on the IEEE 30 test system [27] shown in Fig. 3 using the modeling language AMPL and the interior point solver LOQO [17]. The base operating point has been obtained form an economical dispatch that considered standard ac power flow equations with the objective of minimizing the active power cost. The settings obtained using the economic dispatch are recalculated for each base loading point. In addition, the maximum loading problem for a known base operating point that considers fixed (constant) tap ratios and variable voltage levels at the LTC controlled buses have been solved. This problem is referred as the problem P2 and does not incorporate complementarity. Fig. 4 shows the variation of the critical loading level with respect to the current loading level when solving the four problems. The critical loading level obtained from the maximum distance problem P2, denoted by “
”, is significantly lower than the maximum critical loading level obtained from the problem P1, denoted by “*”. Keeping the tap ratio the same at the critical operating point as at the current operating point results in a larger loadability, denoted by “+”, in comparison to the loadability obtained from the problem with fixed voltage and variable tap, P2. The optimization problem with the complementarity model for the variable tap ratio determines a critical operating

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to bus 9). The 6–9 LTC tap is set at the maximum ratio by the economical dispatch. Therefore, as expected, the limiting constraint for the critical operating point is the maximum generator reactive limit at bus 11. The P2 critical operating point is due to the maximum reactive limit reached at bus 13. The maximum loading level obtained in this case is inversely proportional to the loading level for λ0 ≥ 1 because of conflicting objectives of the economic dispatch and the stability margin problem. In terms of economical dispatch, voltage support at bus 12 is better provided by the generator at bus 13. As the base loading level increases, the reactive generation at bus 13 increase while the tap ratio for the 4–12 LTC decreases. At the critical operating point, the tap ratio has a lower set-point providing less voltage support. Since the reactive reserve at bus 13 is also less available, the loading margin decreases. Fig. 4. Maximum loading level against the current loading level, where *,
, +, and represent the critical loading level solution of the P1 problem, the P2 problem, the P2 problem, and the P3 problem, respectively.

point with larger loadability margin, denoted by “ ”, than the P2 and the P2 problems. In the following paragraphs, the rationale for the characteristics summarized above are discussed in terms of the various limits that enforce the maximum point for each of the problems. It should be highlighted that the objective of the proposed complementarity modeling is not to find a higher maximum loading margin, but to more accurately incorporate how settings of LTC’s will be controlled as their limits are reached. 5.1. Maximum loading problem (problem P1) The maximum loading problem, P1, formulation does not model the current operating point and there is no relationship between the current loading level and the critical loading level. Therefore, in Fig. 4, the critical loading level, λc , does not change with the variation of current loading level, λ0 . The limiting constraint at the maximum loading level is the minimum voltage magnitude at bus 30. 5.2. Maximum loading distance problem for a given current operating point (problems P2 and P2 ) For the problem with variable tap ratio and fixed voltage at LTC controlled buses, P2, the voltage level is kept constant by reactive support provided by the generator at bus 11 (connected

5.3. Maximum loading distance problem for a given current operating point with complementarity models for LTCs (problem P3) This problem yields a significantly larger and more accurate loading margin than the problems without complementarity. The voltage level at bus 9 is allowed to change by the complementarity model such that the reactive generation at bus 11 is no longer brought to its maximum limit to keep the voltage at the value set by the economical dispatch. In the same time, the tap ratio of the 28–27 LTC reaches its maximum limit and the voltage at bus 27 decrease with respect to the initial set-point. The formulation gives a critical operating point characterized by higher tap ratios that leads to a better voltage profile in the network than in the case of the of the problem with fixed tap ratios. In the same time, the reactive dispatch pattern is better distributed among the generators. The binding constraints in this case are the maximum reactive limits at buses 5 and 8. 5.4. Summary of the results The reactive dispatch at the critical operating point for λ0 = 1 is presented in Table 1. It is observed that the generator at bus 1 operates in under-excitation mode for the problems P2 and P2 . The generators at buses 11 and 13 produce more reactive power in the case of the problem with fixed tap ratios than in the case of the problem with complementarity models. The voltage magnitudes for the LTC controlled buses at the maximum point are presented in Table 2 for the four problems

Table 1 Reactive generation at the critical operating point for λ0 = 1 Generator bus

Initial

Maximum (P1)

Without compl. (P2)

Fixed tap (P2 )

With compl. (P3)

1 2 5 8 11 13

−2.71 12.90 2.01 38.24 12.81 13.57

19.77 50.00 40.00 40.00 15.52 17.48

−2.73 16.41 40.00 17.68 24.00 13.57

−20.97 22.87 40.00 40.00 21.60 24.00

1.35 28.42 40.00 40.00 13.80 14.79

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Table 2 Voltage magnitudes at the LTC controlled buses at the critical operating point for λ0 = 1 LTC bus

Initial

Maximum (P1)

Without compl. (P2)

Fixed tap (P2 )

With compl. (P3)

9 10 12 27

1.025 1.035 1.032 1.036

1.028 1.013 1.030 1.013

1.025 1.035 1.032 1.036

1.010 1.007 1.020 1.004

1.025 1.019 1.032 1.030

Table 3 Solving time and number of iterations λ0

Maximum (P1)

Without compl. (P2)

Fixed tap (P2 )

With compl. (P3)

0.9 1.0 1.2

0.20 s/16 it 0.20 s/18 it 0.16 s/15 it

0.18 s/19 it 0.19 s/18 it 0.18 s/17 it

0.18 s/18 it 0.19 s/19 it 0.17 s/16 it

0.19 s/21 it 0.20 s/19 it 0.20 s/20 it

considered. In the P2 problem, the voltage level decreased significantly with respect to voltage levels at the initial operating point. In the case of the P3 problem, the voltages at buses 10 and 27 have decreased with respect to the economical (initial) set-point as the tap ratios are at the maximum limit and the complementarity model allowed the switch from a constant voltage controlled bus to a constant tap controlled bus. It should be noted, that in Fig. 4, the solution from the maximum loading problem (P1, denoted by “*”) is different than the solution obtained with the complementarity LTC model (P3, denoted by “ ”). This demonstrates the need for the complementarity approach, since the solution from the maximum loading problem, P1, does not necessarily correspond to a reasonable control settings for the current operating point. A general characteristic of the complementarity based LTC model was an optimal solution that both better modeled the behavior of the system and also allowed for control variable settings that corresponded to higher maximum loading values. 5.5. Remarks on the computational effort and efficiency In Table 3, the solving time and the number of iteration for the four problems applied to the 30 test system are presented. In general, incorporating the complementarity model did not correspond to significant increases in the solution efficiency. This characteristic was expected, because the number of complementarity constraints is relatively low because they only are introduced where there are LTCs. In the worst case situation, the number of iterations for the complementarity model was 3 additional iterations (20 iterations versus 17 iterations), corresponding to a 17.6% increase in computation time. For larger systems, the efficiency of the model should not significantly decrease because the number of LTC’s with respect to the total number of branches is more likely to decrease than increase. Therefore, the relative number of complementarity constraints should not increase. 6. Conclusions In general, the optimization based stability-margin problems do not model the change of behavior of the voltage controlled

buses when control limits are attained. This paper proposes an enhanced modeling approach capable of incorporating the change in behavior of LTC controlled buses by using complementarity techniques. The results for the 30 test system have been analyzed and compared with the results obtained from problems without complementarity models for LTC controlled buses. The new modeling leads to a better evaluation of the loadability of the system and more appropriate settings for the critical operating points. By incorporating the control actions associated with reaching limits, the complementarity LTC model allows the maximum loading problem to find better control settings to maximize the static stability margin of the system.

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