Improved representation of control adjustments into the Newton Raphson power flow

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1 Electrical Power and Energy Systems () 1 1 Review Improved representation of control adjustments into the Newton Raphson power flow Abilio Manuel Variz a, Vander Menengoy da Costa a, José Luiz R. Pereira a, *, Nelson Martins b a Faculdade de Engenharia, Campus da UFJF, Brazil b CEPEL, P.O. Box 8, CEP 19-9, Rio de Janeiro, RJ, Brazil Abstract This paper describes a sparse ðn nþ formulation for the solution of power flow problem, comprising n current injection equations written in rectangular coordinates plus the set of control equations. This formulation has the same convergence characteristics of the conventional Newton power flow problem, expressed in terms of power mismatches written in polar coordinates and can be reduced to a ðn nþ formulation plus the control equations. It is best suited to the incorporation of flexible AC transmission system devices and controls of any ind. Complex user-defined control functions, involving the participation of several regulating devices, can be directly introduced as power flow control data. The results presented validate the proposed method. q Elsevier Science Ltd. All rights reserved. Keywords: Power flow; Power flow controls; Flexible AC transmission system devices 1. Introduction The power flow problem deals with solving the set of non-linear algebraic equations which represent the networ under steady state conditions. Over the last years, many algorithms have been developed regarding voltage stability tools [1 ], more advanced solution techniques [] and the representation of both the more realistic modeling of power system components [] and the recent technology devices [,]. In Ref. [] is presented a static model for synchronous generators with voltage dependent reactive power limits due to the maximum stator current, maximum and minimum rotor current as well as maximum rotor angle limiters. This generator model is included in an ordinary power flow program. In Ref. [] is discussed the modeling of flexible AC transmission system (FACTS) devices for power flow studies and the role of that modeling in the study of FACTS devices for power flow control. Three generic types * Corresponding author. Tel.: þ--9-; fax: þ addresses: jluiz@ieee.org (J.L.R. Pereira), jluiz@lacee.ufjf.br (J.L.R. Pereira), abilio@lacee.ufjf.br (A.M. Variz), vandermcosta@uol. com.br (V.M. da Costa), nelson@cepel.br (N. Martins). of FACTS devices are suggested and the integration of those devices into power flow studies is illustrated. Generalized nodal admittance models are presented in Ref. [] for series compensators, phase shifters interphase power controllers and unified power flow controllers, regarding to the insertion in a Newton Raphson power flow program. In Ref. [8] is presented a new procedure for the solution of power flow problem, by using the current injection equations written in rectangular coordinates. From this formulation it is possible to obtain the same convergence characteristics of the conventional power flow expressed in terms of power mismatches and written in polar coordinates. The set of controls and devices studied in Ref. [8] includes the representation of the voltage dependent load, load tap changing transformer and phase shifter transformer. In Ref. [9] the static var compensator (SVC) is represented by adding a fictitious PV bus to the system with a fixed voltage equal to the SVC reference voltage. This fictitious bus is connected to the physical SVC-bus through a slope reactance. The objective of this paper is to incorporate other control devices, by using the augmented formulation presented in Ref. [8]. The set of control devices studied includes the representation of P (only active power specified) and PQV (load bus in which the voltage is remotely controlled) buses, 1-1//$ - see front matter q Elsevier Science Ltd. All rights reserved. PII: S1-1()1-

2 A.M. Variz et al. / Electrical Power and Energy Systems () 1 1 Nomenclature n number of buses h iteration counter DP þ jdq complex power mismatch at bus P GðÞ þ jq GðÞ generated complex power at bus P LðÞ þ jq LðÞ complex load at bus P þ jq net complex injected power at bus P calc þ jq calc calculated complex power at bus V r þ jv m complex voltage at bus u ; V G j þ jb j Du; DV r j þ jx j a j y j d x q voltage angle and magnitude at bus ð; jþth element of bus admittance matrix voltage angle and magnitude corrections series impedance of line ð jþ transformer tap from bus to bus j series admittance of line ð jþ load angle associated to the bus quadrature-axis synchronous reactance static var compensators, thyristor controlled series compensation (TCSC) and reactive limits at a generation bus. Userdefined control functions involving the participation of a combination of these devices can be incorporated into the power flow model.. The augmented formulation The augmented formulation uses current injection equations expressed in terms of rectangular coordinates of voltage buses, for both PQ and PV buses [8]. The calculation of real and imaginary current mismatches is straightforward for PQ buses, because of real and reactive power mismatches are nown. For PV buses the reactive power mismatch is unnown and it is treated in this formulation as a dependent variable. Then an additional equation is introduced in order to set the over-determination of the system of equations as follows: The mathematical model is then written in the following matrix form [8]: D uv Y p B C DV rm D PQ The Y p matrix blocs have the following structure: " Y p B G # G B " Y p m B # m G m G m B m ðþ V V r þ V m ð1þ B B a ðþ By linearizing Eq. (1) yields: G G b ðþ DV V r V DV r þ V m V DV m ðþ G G c ð8þ Thus, for each PV bus there are three equations and variables DV r ; DV rm and DQ : The voltage angle at a bus can be expressed by: u tg 1 V m V r Whose linearized form is: Du V r V DV m V m V DV r ðþ ðþ B B d The parameters a ; b ; c and d are presented in Appendix AofRef. [8]. The matrices B and C have a bloc-diagonal structure B 1 C 1 B C B... B n C... C n ð9þ

3 A.M. Variz et al. / Electrical Power and Energy Systems () 1 1 B i V mi V i V ri V i V ri Vi V mi V i C i V mi V i V ri V i V ri Vi V mi V i D uv ½Du 1 DV 1 Du DV Du n DV n Š t D PQ ½DP 1 DQ 1 DP DQ DP n DQ n Š t DV rm ½DV r1 DV m1 DV r DV m DV rn DV mn Š t The voltage buses updates are given by: D ðhþ uv CðhÞ DVrm ðhþ The new solution is given by: V ðhþ1þ V ðhþ þ DV ðhþ u ðhþ1þ u ðhþ þ Du ðhþ ð1þ ð11þ ð1þ As shown in Ref. [8], the step 1 of the solution algorithm for ðn nþ augmented formulation is equivalent to ðn nþ formulation plus control equations.. Remote voltage control and secondary voltage control models A generation bus (P bus) can be used to control the voltage at a remote bus (PQV bus) by assuming the voltage at the P bus as the unnown and specifying the voltage at the PQV bus. In this case, the voltage constraint equation related to the PQV bus must be included into the Jacobian matrix. On the other hand, if various P buses (say NP ) are assumed to control a single PQV bus voltage, then (NP 1) additional equations are required to yield a unique solution, where each equation describes the MVAr participation factors among the P buses, as follows Q GðiÞ a Q GðjÞ ð1þ where i and j denote P buses, 1; ; ; ðnp 1Þ Q GðiÞ Q i þ Q LðiÞ ð1þ equation for the PQV bus must be introduced Y p B DV rm D uv C D D PQ DQ D D 1 D. D n For all PQ, PV and PQV buses: D ½Š ð1þ For remote voltage control (only one P bus i ) one has: V mi V ri Vi Vi " # C i D i V rl V ml V l V l For secondary voltage control (NP 1 P buses j ) one has V mj V rj " # C j Vj Vj D j 1 a where l denotes PQV bus and the non-zero values in matrix D j occur in the columns i and j..1. Illustration example.1.1. Remote voltage control by a single reactive source In Fig. 1 suppose the voltage at bus is remotely controlled by reactive power at bus, and bus is assumed to be a PQ bus type. Bus 1 is the slac bus. The linear system of equations related to the solution algorithm step 1, Thus, Eq. (1) is linearized as shown below: DQ DQ ðiþ a DQ ðjþ ð1þ DQ Q LðiÞ þ a Q LðjÞ ðq calc i a Q calc j Þ ð1þ The augmented system of equations is then modified to incorporate Eq. (1). In addition, the voltage constraint Fig. 1. Four-bus test system.

4 presented in Ref. [8], is given by Eq. (18)..1.. Remote voltage control by multiple reactive sources Now, assume the reactive power sources at buses and are supposed to control the voltage at bus. In this case, the participation factor, which establishes the relationship between the reactive power sources must be specified, leading to: Q GðÞ a 1 Q GðÞ ð19þ The linear system of equations related to the solution algorithm step 1, presented in Ref. [8], is given by Eq. (1), DQ 1 Q LðÞ þ a 1 Q LðÞ ðþ DP DQ DP DQ DP B G V m V V r V B G B G G B V r V V m V G B G B 1 1 B G B G V m V V r V G B G B V r V V m V 1 1 B G B G V m V V r V G B G B V r V V m V 1 V r V V m V DV r DV m DP DQ DV r DV m DP DQ DV r DV m DP DQ ð18þ DP DQ DP DP DQ 1 B G V m V V r V B G B G G B V r V V m V G B G B 1 1 B G B G V m V V r V G B G B V r V V m V 1 V r V V m V B G B G V m V V r V G B G B V r V V m V 1 1 a 1 DV r DV m DP DQ DV r DV m DP DQ DV r DV m DP DQ ð1þ A.M. Variz et al. / Electrical Power and Energy Systems () 1 1

5 A.M. Variz et al. / Electrical Power and Energy Systems () 1 1 Eqs. (), () and () are linearized and the voltages are expressed in rectangular coordinates. Thus, Eq. () leads to DV V r V DV r þ V m V DV m þ r DQ ð8þ DV V þ r ðq LðÞ þ Q Þ V ð9þ. The static var compensator model The steady state voltage reactive power characteristics for an SVC is shown in Fig.. The linear control range lies within the limits determined by the maximum susceptance of the reactor and the total susceptance determined by the capacitor bans in service and the filter capacitance [1,11]. Within the linear control range the SVC is equivalent to a voltage source in series with a slope reactance X s as follows V V þ X s I where denotes the bus at which the SVC is installed. From Fig. V V þ r Q GðÞ V Vmax Q max Fig.. Voltage reactive power characteristic. Q max V min Q min Q min ðþ ðþ ðþ Eq. () leads to DQ max Qmax ðv min DQ max Q LðÞ þ Q Eq. () leads to DQ min Qmin ðv max Þ V rdv r þ Q max ðv min Þ V m DV m DQ ðþ Qmax ðv min Þ V ð1þ Þ V r DV r þ Qmin ðv max Þ V m DV m DQ ðþ DQ min Q LðÞ þ Q Qmin ðv max Þ V ðþ The implementation procedure is similar to that used for the PV bus. Instead of using the voltage constraint equation, either one of Eqs. (8), () and () is used depending on the SVC operating point..1. Illustration example r Vmin Q max V max Q min ðþ In Fig., suppose a SVC connected at bus to control the voltage at bus, say V sp : The linear system of equations For Q GðÞ. Q max the SVC behaves as a capacitor and the corresponding reactive power is given by: Q GðÞ Qmax ðv min Þ V ðþ For Q GðÞ, Q min the SVC behaves as a reactor and the corresponding reactive power is given by: Q GðÞ Qmin ðv max Þ V ðþ Fig.. Topology for incorporation of SVC.

6 related to the solution algorithm step 1, presented in Ref. [8], is given by: If the SVC is operating as a capacitor, then the linear system of equations becomes DP DQ DP DV DP B G V m V V r V B G B G G B V r V V m V G B G B 1 1 B G B G V m V V r V G B G B V r V V m V 1 V r V V m V r B G B G V m V V r V G B G B V r V V m V 1 V r V V m V DV r DV m DP DQ DV r DV m DP DQ DV r DV m DP DQ ðþ DP DQ DP DQ max DP B G V m V V r V B G B G G B V r V V m V G B G B 1 1 B G B G V m V V r V G B G B V r V V m V 1 f V r f V m 1 B G B G V m V V r V G B G B V r V V m V 1 V r V V m V DV r DV m DP DQ DV r DV m DP DQ DV r DV m DP DQ ðþ A.M. Variz et al. / Electrical Power and Energy Systems () 1 1

7 A.M. Variz et al. / Electrical Power and Energy Systems () 1 1 f Qmax ðv min Þ ðþ. Thyristor controlled series compensation model The TCSC has both electromechanical stability and power flow control duties. In the latter case, it is used to regulate the power flow through a specified line. The TCSC control model studied here is referred to in the literature as constant line power control. Assume the TCSC is connected between buses and j to control the active power flow P j through changes in the line reactance x j : Thus, the linearized equation of P j needs be introduced into the Jacobian matrix, as follows D uv DP j where Y p B E C D t t F DV rm D PQ Dx j " # P j P j P j P t j D V r V m V rj V mj ðþ DI r x j r jx j ðr j þ x j Þ ½a j cosðw j ÞV rj a j sinðw j ÞV mj þ a jv r Š r j x j ðr j þ x j Þ ½a j sinðw j ÞV rj a j cosðw j ÞV mj þ a jv m Š ð9þ DI mj x j r jx j ðr j þx j Þ ½a j cosðw j ÞV m a j sinðw j ÞV r þv mj Š r jx j ðr j þx j Þ ½a j sinðw j ÞV m þa j cosðw j ÞV r V rj Š DI rj x j r jx j ðr j þx j Þ ½a jsinðw j ÞV m a j cosðw j ÞV r þv rj Š ðþ r jx j ðr j þx j Þ ½a jcosðw j ÞV m a j sinðw j ÞV r þv mj Š ð1þ " # DI m DI r DI mj DI t rj E x j x j x j x j F P j x j with DP j x j r jx j a j ðr j þx j Þ ½cosðw jþðv r V rj þv m V mj Þsinðw j Þ r jx j ðv m V rj V r V mj ÞŠ ðrj ½a jðv þx j Þ r þvm ÞŠ þ a jðr jx jþ ðr j þx j Þ ½cosðw jþðv m V rj V r V mj Þ DI m x j r jx j ðr j þ x j Þ ½a j cosðw j ÞV mj þ a j sinðw j ÞV rj þsinðw j ÞðV r V rj þv m V mj ÞŠ ðþ þ a jv m Š r j x j ðr j þ x j Þ ½a j sinðw j ÞV mj The line reactance value for the next iteration is given by: þ a j cosðw j ÞV rj a jv r Š ð8þ x ðhþ1þ j x ðhþ j þdx ðhþ j ðþ

8 8 A.M. Variz et al. / Electrical Power and Energy Systems () Illustration example B G V m V r DV V V B G r G B V r V m DV V V G B m DP 1 DP V r V m DQ V V B G V m V r V V B G DI m DV r x G B V r V m DP V V G B DI r DV m x 1 DP DQ 1 DQ B G B G B G V m V r V V DI m x DV r G B G B G B V r V m V V DI r x DV m DP 1 DP DQ 1 DQ DP DP DP DP DP DP V r V m V r V m x Dx ðþ In Fig., suppose a TCSC connected between buses and to control the active power flow P at say, P sp : The linear system of equations related to the solution algorithm step 1, presented in Ref. [8], is given by Eq. () DP P sp Pcalc ðþ. Reactive limits at a generation bus model The model commonly used for power flow studies considers fixed reactive power limits for a synchronous generator. This is an approximation since the reactive power limits depend on the active power dispatch and the generator operating voltage. A realistic representation of the generator limits requires the determination of the capability diagram for the synchronous generator []. In Ref. [] is represented the maximum and minimum limits for the reactive power generation taing into account the maximum stator current, the rotor current limiter and underexcitation limiter. Both salient pole and round rotor generators can be represented. The proposal of this paper is to model the generation bus into a power flow program considering that the limits obtained for the reactive power generation are voltage dependent due to all the equipment limits modeled in Ref. []. When the reactive power generation of a machine connected at a bus is within the range defined by the maximum and minimum limits, the following equation is introduced into the power flow program: V V nom ðþ On the other hand, if the reactive power generation is outside the limits, the following equation is introduced: Q GðÞ Q lim ðþ Fig.. Topology for TCSC control. where Q lim is the maximum or minimum reactive power generation limits which is violated in a given iteration.

9 This value for a generation bus is determined from the reactive power limits imposed by stator, rotor and underexcitation, given by Eqs. (8) (), respectively []: Q ðmax;minþ est Q max rot Q min exc qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ ðs max Þ P vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u V ðeq max þ t ðþ Þ P V x qðþ P x q ðþ tgðd max Þ V x qðþ ð8þ ð9þ ðþ Eqs. () and () are linearized and the voltages are expressed in rectangular coordinates. Thus, Eq. () leads to: DV V nom V V r DV V r þ V m DV V m Eq. () leads to: DQ Q LðÞ þ Q Q lim DQ A.M. Variz et al. / Electrical Power and Energy Systems () ð1þ ðþ If the reactive power generation at bus violates either the maximum or minimum limits, the linear system of equations related to the solution algorithm step 1, presented in Ref. [8], is given by Eq. ().. Computational aspects The proposed formulation, by using the current injection method written in rectangular coordinates, has the same convergence characteristics of the conventional power flow expressed in terms of power mismatches and written in polar coordinates. This formulation has the advantage of presenting a highly sparse structure (augmented Jacobian) suitable to the incorporation of FACTS devices and control of any ind [8]. It is also very important to notice that in the absence of control devices this formulation can be reduced to a ðn nþ formulation as presented in Ref. [1]. As shown in Ref. [1], the saving cpu time for the current injection method is about % when compared to the conventional power flow formulation. B G V m V r DV V V B G B G r G B V r V m DV m V V G B G B DP DP 1 DQ 1 DQ B G B G V m V r V V DV r G B G B V r V m DV V V m DP 1 DP DQ 1 DQ B G B G V m V r V V DV r G B G B V r V m V V DV m DP 1 DP V r V m DV V V DQ ðþ The implementation procedure is similar to that used for the conventional PV bus. Instead of using the voltage constraint equation either one of Eqs. (1) and () is used depending on the synchronous machine operating point..1. Illustration example In Fig. 1, suppose the reactive power generation limits are considered for the synchronous machine connected at bus, by assuming that the bus is PQ type. The linear system of equations related to the solution algorithm step 1, presented in Ref. [8], is given by Eq. (). The main advantage of this formulation lies on the calculation of the matrix Y p, because its off-diagonal elements are exactly the terms of admittance matrix bus and the diagonal elements are calculated using nontranscendental functions, even if load models other than constant power are included. 8. Results The proposed power flow control models were validated through tests with the IEEE-118 buses and the Brazilian South Southeastern system. These systems have 118 buses

10 1 A.M. Variz et al. / Electrical Power and Energy Systems () 1 1 and 18 circuits, and 18 buses and circuits, respectively IEEE-118 buses system In order to validate the secondary voltage control model, the generators at buses and were set to control the voltage at the bus, in such a way that Q GðÞ :Q GðÞ : The power flow solution was achieved in three iterations with power mismatch less than p.u. The results of this simulation are shown in Table 1. reactive generation limits equal to. MVAr the SVC operates as a capacitor. For validating the TCSC model, a TCSC was placed between the buses and with the objective of controlling the active power flow between these buses. Table shows the results of three different situations, without TCSC (base case), and with TCSC controlling the power flow between the buses and in. and. MW. In all simulations the initial reactance was 9.% (equal to the base case). The convergence without TCSC was obtained in two iterations and with TCSC was achieved B G V m V r V V B G B G G B V r V m V DP V G B G B 1 DQ 1 B G B G V m V r V V G B G B V r V m V V DP 1 DQ 1 B G B G V m V r V V G B G B V r V m V V DP 1 1 DQ DV r DV m DP DQ DV r DV m DP DQ DV r DV m DP DQ ðþ A SVC was included at the bus 9 in order to control the voltage at the bus 1. This SVC was modeled by the reference voltage (V ) equal to 1. p.u. and reactance slope equal to %. Two ind of simulations were made, in the first one, the maximum and minimum reactive generation ðq max G ; Q min G Þ were set to be, respectively,. and. MVAr, and in the second simulation, the limits of the reactive generation were set to be and MVAr. Table illustrates the results of these simulations, in both cases the convergence was obtained in two iterations with power mismatch less than 1. 1 p.u. Note that with Table 1 Power flow solution, IEEE-118: two P buses controlling one PQV bus P buses Number Generation (MVAr) Generation (%) PQV bus Number Voltage magnitude Specified voltage in four iterations with power mismatch less than 1. 1 p.u. in all simulations. 8.. Brazilian South Southeastern system It was considered a set of twelve P buses controlling eleven PQV buses, in order to validate the remote voltage control and secondary voltage control models. Each PQV bus has its voltage magnitude controlled by a unique P bus, except for the 8 bus that was controlled by and buses, in such a way that Q GðÞ 1:Q GðÞ : The power flow solution obtained in five iterations and with power mismatch less than p.u. is shown in Tables and. A TCSC was placed between buses 98 and with the objective of controlling the active power flow between these buses in.9 MW. For this control the initial line reactance was the same of the base case and the convergence characteristics was achieved in five iterations when the absolute value of the difference between the specified and calculated active power flow was less than

11 A.M. Variz et al. / Electrical Power and Energy Systems () Table Power flow solution, IEEE-118: SVC SVC bus PQV Bus Num. Slope (%) Limits (MVAr) Generation (MVAr) Operation range Num. Calculated voltage Reference voltage (V ) 9... Capacitive Linear Table Power flow solution, IEEE-118: TCSC Case Bus Reactance (%) Active power flow To From Initial Final Compensation MW Specified MW No TCSC With TCSC With TCSC p.u. The results from the solution of the power flow, with TCSC and with no TCSC are shown in Table. It was included a SVC at bus 8 for controlling the voltage magnitude at bus 1, to validate the proposed model for the static voltage compensator. This SVC was modeled by the reference voltage (V ) equal to 1. p.u., and the maximum and minimum reactive generation ðq max G ; Q min G Þ equal to. and. p.u., respectively. Two different situations were simulated corresponding to the slope reactance (r ) equal to % (simulation I) and % (simulation II). The results related to these simulations are shown in Table, where the convergence characteristics were achieved in five iterations, with the active and reactive power mismatch less than 1. 1 p.u. in both simulations. Table Power flow solution, Brazilian South Southeastern system: P and PQV buses P buses Number Generation (MVAr) PQV buses Number Voltage magnitude Specified voltage Table Power flow solution, Brazilian South Southeastern system: two P buses controlling one PQV bus P buses PQV bus Number Generation (MVAr) Generation (%) Number Voltage magnitude Specified voltage

12 1 A.M. Variz et al. / Electrical Power and Energy Systems () 1 1 Table Power flow solution, Brazilian South Southeastern system: TCSC Case Bus Reactance (%) Active power flow To From Initial Final Compensation MW Specified MW No TCSC With TCSC Table Power flow solution, Brazilian South Southeastern system: SVC SVC bus PQV bus Number Slope (%) Generation (MVAr) Number Calculated voltage Reference voltage (V ) Table 8 Power flow solution, Brazilian South Southeastern system: reactive limits at a generation bus PV bus Reactive generation (MVAr) Voltage magnitude (p.u.) Reactive power load (MVAr) Proposed formulation Conventional formulation No reactive power limits Proposed formulation Conventional formulation No reactive power limits. 1, 1, 9, 1, 1, 1,1 1. 1,19 1, 9, 1,1 1,8 1,1. 1, 1, 19, 1,1 1,19 1,1 1. 9, 9, 9, 1,1 1,1 1,1.,,, 1,1 1,1 1,1. 1,8 1, 1,8 1,1,999 1,1. 1, 1,,8 1,1,989 1,1. 1,1 1,,8,99,9 1,1. 1,9 1,,8,98,9 1,1 The objective of these last simulations was to compare the results obtained by using the conventional procedure for the PV bus with those obtained by considering the proposed formulation in which the reactive power limits are voltage dependent. For this purpose the generator connected at bus is studied. The nominal values adopted for the power factor and the voltage at this generator bus are.8 and 1.1 p.u., respectively. The active power generation is adopted equal to zero, while the apparent power generation is.1 p.u. The value of x q was set to % and the resistance set to zero. The maximum continuous stator and rotor currents were set and 1% above respective nominal value and the maximum load angle was set to 88. In the conventional procedure the minimum and maximum reactive power limits for the bus were fixed with the values set to 1. and þ1. MVAr. The results for several values of the reactive power load for each formulation plus the simulations with no reactive power limits are shown in Table 8. These results shown that the proposed formulation presents violated voltages closer of the specified for the problem than the conventional formulation. The convergence characteristics in the both formulation were achieved in six or seven iterations with absolute power flow mismatch less than 1. 1 p.u. 9. Conclusions The augmented formulation is equivalent to the conventional Newton Raphson power flow regarding convergence characteristics, but allows an easier incorporation of control device models and power flow controls of any ind. This formulation also directly incorporates more realistic modeling of power system components, such as, static var compensators, TCSC and voltage control through multiple reactive sources. The studies performed so far indicate the proposed formulation may become a valuable tool for solving present

13 A.M. Variz et al. / Electrical Power and Energy Systems () day power flow problems, where the proper consideration of controls is becoming a ey issue. References [1] Ajjarapu V, Christy C. The continuation power flow: a tool for steady state voltage stability analysis. IEEE Trans Power Syst 199;(1): 1. [] Cañizares CA, Alvarado FL. Point of collapse and continuation methods for large ac/dc systems. IEEE Trans Power Syst 199;8(1): 1 8. [] Overbye TJ, Klump RK. Effective calculation of power system lowvoltage solutions. IEEE Trans Power Syst 199;11(1): 8. [] Semlyen A. Fundamental concepts of a Krylov subspace power flow methodology. IEEE Trans Power Syst 199;11():18. [] Lof PA, Andersson G, Hill DJ. Voltage dependent reactive power limits for voltage stability studies. IEEE Trans Power Syst 199; 1(1): 8. [] Gotham DJ, Heydt GT. Power flow control and power flow studies for systems with FACTS devices. IEEE Trans Power Syst 1998;1(1):. [] Fuerte-Esquivel CR, Acha E. A Newton-type algorithm for the control of power flow in electrical power networs. IEEE Trans Power Syst 199;1():1 8. [8] Da Costa VM, Pereira JLR, Martins N. An augmented Newton Raphson power flow formulation based on current injections. Int J Electr Power Energy Syst 1;(): 1. [9] Alvarado FL, DeMarco CL. Computational tools for planning and operation involving FACTS devices. Proceedings of Worshop on FACTS, Rio de Janeiro, Brazil; November 199. [1] Kundur K. Power system stability and control. New Yor: McGraw- Hill; 199. [11] Taylor CW. Power system voltage stability. New Yor: McGraw-Hill; 199. [1] Da Costa VM, Pereira JLR, Martins N. Developments in the Newton Raphson power flow formulation based on current injections. IEEE Trans Power Syst 1999;1():1.