An Optimal Power Flow Function to Aid Restoration

5/21/2018 An Optimal Power Flow Function to Aid Restoration

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IEEE TRANSACTIONS ON POWER SYSTEMS 1

An Optimal Power Flow Function to Aid RestorationStudies of Long Transmission Segments

Eduardo Martins Viana, Edimar Jos de Oliveira, Nelson Martins, Fellow, IEEE,Jos Luiz R. Pereira, Senior Member, IEEE, and Leonardo Willer de Oliveira

AbstractThis paper presents an optimal power flow functionfor aiding restoration studies of subsystems having long transmis-sion segments. The method simultaneously computes an optimalset of variables which are critical for the subsystem restoration:the generation power plant high-side voltage, the minimum shuntreactor configuration and the maximum load to be safely ener-gized. A set of overvoltage restoration scenarios following load re-jection is constructed in such a way that the computed shunt re-actor compensation becomes distributed along the transmissioncorridor. The whole problem is formulated as a single optimiza-tion problem whose solution is obtained by an optimal power flow,

based on the Primal-Dual Interior Point Method. The describedresults for a restoration study on a system from practice confirmthe effectiveness of the proposed method.

Index TermsAutomatic voltage regulator, generator capabilitycurve, load rejection, long transmission segments, optimal powerflow, power system restoration, shunt reactors, sustained overvolt-ages.

I. INTRODUCTION

S YSTEM faults whose severity are above planning criteriacan cause electricity supply interruptions and, occasion-ally, lead to regional or widespread blackouts[1]. Following a

blackout, restorative actions are promptly started and efficiently

carried out in order to safely bring the system back to normal

operation in minimum time. Power system restoration is a com-

plex multi-stage process involving many issues[2] and whose

studies require the use of computer tools for power flow [3],

short-circuit, harmonic distortion, electromechanical stability

and electromagnetic transient analyses. Restoration drills have

become an important training resource, mainly when aided by

powerful operator training simulators[4].

Until the mid-1970s, all the Brazilian restorative actions

were centrally supervised[5] and the average time required to

completely restore the supply was excessively long. In order

to speed-up the process, the Brazilian Interconnected PowerSystem (BIPS) operating bodies (previously the Eletrobras

coordinating pool and now the system operator, ONS), divided

Manuscript received May 05, 2011; revised October 06, 2011, January 07,2012, April 02, 2012, and May 08, 2012; accepted May 11, 2012. This workwas supported by CAPES, CNPq/INERGE, FAPEMIG, and CEPEL. Paper no.TPWRS-00414-2011.

E. M. Viana is with Petrobrs (e-mail: [email protected]).E. J. de Oliveira, J. L. R. Pereira, and L. W. Oliveira are with the Elec-

trical Engineering Department, Federal University of Juiz de ForaUFJF, Juizde Fora, Brazil (e-mail: [email protected]; [email protected]; [email protected]).

N. Martins is with CEPEL, CP 68007, 21944-970 Rio de Janeiro, RJ, Brazil(e-mail: [email protected]).

Digital Object Identifier 10.1109/TPWRS.2012.2202924

the restoration process in two sequential stages, in which the

first stage, calledparallel phase [5], is followed by the second

stage, calledcoordinated phase[6],[7]. The former only deals

with the restoration of the individual subsystems, which can

be carried out independently, while the latter involves actions

requiring rigorous coordination among subsystems.

Parallel phase studies usually assume the generation-trans-

mission (GT) subsystem totally de-energized. The restoration

procedures for the subsystem are detailed in the operator in-

structions and the operating staff in the several stations/sub-stations belonging to this GT subsystem should conduct the

restoration with aminimum need of verbal communication be-

tween stations[8]. The transmission segments in these GT sub-

systems are energized in a radial configuration sequence and a

compatible amount of load is picked up at the end of the process

[9], evolving then to a generation-transmission-load (GTL) sub-

system, also known as a system restoration island.

In order to face a multitude of possible regional blackouts, as

well as the very low probability event of a complete blackout

in the continental-sized Brazilian Interconnected Power System

(BIPS) the operator, ONS, has planned for more than 30 GT and

GTL subsystem restoration processes to take place in parallel

and simultaneously. The restoration of the GT/GTL subsystemsis based on hydroelectric black-start units and carried out under

the responsibility of numerous owners and by their teams of

operators, with ONS supervision[5]. The overvoltages during

the subsystem restoration process must be kept at safe levels at

all times by using the available shunt reactors and the reactive

power capability of the synchronous machines[10].

Reference[11]presents a method for the control of sustained

overvoltage during the early stages of restoration in which an

expert system and a nonlinear programming module were de-

signed taking into account the differences between the normal

condition and the early stages of restoration. The over-voltage

control scheme is solved by this module, using power fl

owcalculations and sensitivity analyses in order to determine the

target buses. System restoration voltage control strategies and

overall planning are discussed in [2], [11], and [12], which

provided important guidelines to the work reported in the

present paper.

The present paper proposes a new methodology to determine

the minimum number of generating units, in-service transmis-

sion circuits and shunt reactors configuration capable of safely

restoring the individual GT and GTL subsystems and picking up

a substantial amount of load. The proposed method utilizes an

optimal powerflow (OPF) function based on the primal-dual in-

terior point method[13]. Important phenomena like frequency

dips that limit the amount of load that could be picked up to a

0885-8950/$31.00 2012 IEEE
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2 IEEE TRANSACTIONS ON POWER SYSTEMS

Fig. 1. Restoration of transmission segments.

small fraction of the available generation capacity, as well as

electromagnetic transients resulting from line and transformer

energization, are not captured by the steady-state OPF tool of

this paper.

II. PROBLEMDESCRIPTION

A subsystem having a long transmission corridor is restored

by following a safe equipment switching sequence, in which

the number of shunt reactors connected to the system should be

minimized.Fig. 1shows a restoration subsystem in which theenergization initiates at bus 1, the generator bus, and the load is

picked up at bus 4, the remote end. There are 4 bus reactors in

this subsystem, which are pictured all energized inFig. 1.

During the restoration process of the GTL subsystem in

Fig. 1, the fundamental frequency overvoltages are controlled

by the reactors at buses 2, 3, and 4 and by the generator at bus

1 in such a way that the load picked up at bus 4 is maximized.

Picking up a remote load without resorting to shunt reactors

is not feasible when relatively long transmission segments are

involved. Also, a minimum set of bus reactors should not be

switched off as loading is increased since there is always the risk

of intempestive load rejection with associated overvoltages and

generator self-excitation. The restoration process can be studied

by establishing a sequence of subsystem scenarios. The sce-

narios differ by the number of transmission line sections con-

nected as well as the number of generating units. The voltage

profile, reactive shunt compensation level and remote load sup-

plied are solutions to the proposed OPF function. The number

of generation units should ideally be also varied in these sce-

narios, with less units connected when energizing the first line

sections but, in this first implementation, the number of units is

not a variable in the optimization process. If a heuristic search

optimization method were used for the system in Fig. 2, the

number of scenarios would be to cater for all the reactor con-

nection possibilities for the base case alone, and on top of theseoptimize to maintain generation within its reactive power capa-

bility and avoid overvoltages[14]. In our method this discrete

optimization is approximated as a continuous problem, which is

adjusted to the nearest discrete values, after convergence. Our

initial scenarios, should therefore include, as data set, those can-

didate scenarios and compensation alternatives that appear to be

of practical value (seeFig. 2), since this provides a better initial

condition for the optimization process.

In the proposed method, the bus voltages are variables

(unknowns) of the OPF problem and the various scenarios

are included as data for the global optimization problem. The

OPF method, considering the load rejection conceptual model

sketched inFig. 3, is designed to simulate the system state at the

three time instants, , and , which are related to the

Fig. 2. Restoration scenarios.

Fig. 3. Sketch of transient behavior of a system bus voltage magnitude fol-lowing load rejection in the GTL island.

conditions immediately prior to the load rejection, immediately

after it and at an anticipated steady-state condition.

At , the load picked up at the remote bus should be max-

imized for a given corridor, the reactive compensation by the

shunt reactors should be minimized and the generators must re-

main within their capability curves. At , the generators are

represented as constant voltage behind their subtransient reac-

tances, with thesubtransient voltage being a function of the gen-

erator dispatch at . At , the post-disturbance steady-state

is reached and the voltage regulators (AVR) are assumed to

have adjusted the power plant terminal voltage to the optimized

scheduled value, as long as its generating units do not violate

their reactive power limits. The equations used to model the gen-

erator capability limits are detailed inAppendix A.

III. PROPOSEDMETHOD

The proposed methodology is based on an OPF formulation

in which the AC network model for the GTL subsystem is

fully considered. As keeping voltage and Var limits during

the GT/GTL subsystem restoration process is vital, adequate

representation in steady-state and subtransient-state of the

synchronous machines, their capability curves [15][17] and

associated voltage regulator [18] steady-state regulation char-

acteristics, as well as the set of switchable shunt reactors placed

at the various subsystem buses, should be considered into the

optimization problem. These shunt reactors are considered to

be continuous in the OPF solution, being later adjusted to their

discrete values in practice which are closest to the obtained
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VIANAet al.: AN OPTIMAL POWER FLOW FUNCTION TO AID RESTORATION STUDIES OF LONG TRANSMISSION SEGMENTS 3

optimal solution. The proposed OPF problem is described as

follows:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

Equations(19)(23)ofAppendix Adefine the generator ca-

pability curve and are also considered as constraints of this OPF

problem, where

active power generation at bus

prior to load rejection ,

at the instant and final

steady state for scenario ;

reactive power generation at bus




active power load at bus prior

to load rejection , at the

instant and final steady

state for scenario ;

reactive power load at bus




generator internal voltages at

and ;

generator internal angles at

and ;

voltage at bus at the instants

and for the

scenario ;

generator terminal voltage at

the instants and for

scenario ;

steady state voltage limits at the

instant ;

voltage limits at the instant

;

steady state voltage limits at the

instant ;

angle at bus at the instants

and for the

scenario ;

load factor at bus having

existing load;

active powerflow at circuit -

at the instants and

for the scenario ;

Lagrange multipliers related to(2),(8), and(13);
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Lagrange multipliers related to

(3),(9), and(14);


(4),(10), and(15);


(5),(11), and(16);

reactive power consumed by

reactor at bus ;

maximum amount of MVAr

reactors at bus ;

reactive powerflow at circuit

- at the instants

and for the scenario ;

set of buses directly connected

to bus ;

complete set of buses;

set of shunt bus reactors;

subtransient and synchronous

reactances of generator;

total number of scenarios;

total number of system buses

for scenario ;

total number of load buses;

total number of shunt reactors;

total number of black-start units.

Equation(1)is the objective function that maximizes the sup-plied load and minimizes the reactive power drawn by the shunt

reactors. Note the maximization of supplied load implies in the

minimization of shunt reactors, indicating that there is no need

to consider weighting factors in the two terms of the objective

function. The load factor is an optimization variable which

enables determining the maximum load picked-up at the end of

the optimization process.

Equations (2)(7) and (18)(23) describe the operation

decision problem related to the amount of supplied load

and the number of switched-on reactors. Equations (4)(5)

and (19)(23) are defined only for the connected black-start

generating units, which are individually represented. If theoptimization problem considered only (2)(7) and (18)(23),

the optimal solution achieved would comprise a large amount

of supplied load and a very small amount of in-service shunt

reactors. However, a GTL island restoration process involves

many sequential stages of island component energizations as

well as load rejection possibilities, all of which must be checked

for sustained as well as temporary overvoltage problems.

Equations (8)(12) describe the constraints at the instant

associated with one generic restoration scenario. A set of

(8)(12)must be used to model each one of the scenarios

that are included in the OPF dataset. These scenarios data set

is intended to provide a better initial condition that will lead to

OPF solutions of more practical value rather than local minimaof lesser interest. The subproblem related to conditions at the

Fig. 4. Sparse Hessian structure.

instant of is connected to the subproblem , related

to conditions prior to load rejection, by the global variables:

and , as formulated in(8)(11). At the instant ,

the voltage regulators (AVRs) have not yet responded and then

, the generator subtransient voltage, remains unchanged. The

constraint (12) becomes active due to overvoltages detected

during the convergence process and then the decision in sub-

problem forces a reduction in the amount of picked-up

load and increases the reactive power consumption by the

shunt reactors .

Equations (13)(17)and (19)(23)describe the constraints

at the instant in which each equation must be written for

the restoration scenarios. Subproblem is connected

to the subproblem by the global variables: and

, as formulated in (13)(16). At the post-disturbance final

steady-state , the voltage regulators (AVR) have already

regulated the generation terminal voltages back to the optimized

scheduled values by adjusting to the steady state value

. The constraint(17)becomes active due to final steady-

state overvoltages detected during the convergence process, and

then the decision in subproblem is forced to decrease the

amount of supplied load and to increase . However, if

generator reactive power capability is insufficient to ensure that

bus voltage remains within limits, then only the reactive powerconsumption by the reactors, , is increased.

The distributed shunt compensation is a solution from the

continuous OPF method, and providing an adequate data set

of restoration scenarios as initial conditions, cause the OPF

problem to be more rapidly and effectively solved. As previ-

ously described, the continuous solution values obtained for

the shunt reactors are adjusted to the nearest discrete values,

adopting the following criteria:

1) The set of continuous reactive power values

is scaled by the MVAr rating of the reactor unit at that bus,

, say .

2) The bus with the maximum value of in this set is thenselected for the addition of a fixed reactor, which is then

modeled in the data set as a fixed impedance load.

3) A new OPF simulation, using the system data adjusted by

the addition of one fixed reactor as described in 2), is then

executed to obtain a new set of continuous reactive power

values absorbed by the system candidate reactors .

4) Repeat step 1) to 3) as many times as reactor additions are

needed, until a suitable tolerance is achieved.

This heuristics has worked well for the example system of

this paper as well as other simpler systems, but further testing

and research is needed.

The proposed solution for the global optimization problem

should take into consideration the data set of restoration sce-narios which are judged to be the most practical. As the GT/GTL
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Fig. 5. Diagram for the Rio de Janeiro restoration transmission segment.

subsystem contains a reduced number of buses, the overall OPF

problem to be solved is always of a relatively small size. In the

present case, the sparse Hessian matrix W has the structure pre-

sented inFig. 4, where its component blocks are described as

follows:

block matrix related to in which the elements

are obtained from (2)(7)and (18)(23)

describing the complete GTL subsystem being

considered;


are obtained from(8)(12). In this case there

exists one sub-matrix for each GT scenario;


are obtained from(13)(17)and(19)(23). In

this case there exists one sub-matrix for each

GT scenario;

coupling block matrix associated to the global

variables ( and ) of and ;

coupling block matrix associated to the global

variables ( and ) of and .

Note: It is worth mentioning a generation redispatch method de-

scribed in[6] and [7] that also uses OPF constraints related to

machine subtransient phenomena at and requires the mod-

eling of generators by voltages behind their subtransient reac-

tances.

IV. RESULTS

The proposed methodology was tested in the restoration study

of the main transmission corridor of the Rio de Janeiro area ofBIPS. This GTL subsystem is shown in Fig. 5 and the associated

system data is presented inAppendix B.

The main components of this GT/GTL subsystem are the

Marimbondo power plant, the 500-kV transmission system con-

necting this plant to the Rio de Janeiro area and the 138-kV

LIGHT distribution system and load. The Marimbondo hydro-

electric plant comprises eight 186-MW generating units, five of

which are needed for safe restoration according to[5], a refer-

ence describing current BIPS restoration practices. In order to

better test the proposed method, the line reactors in Fig. 5were

also assumed to be bus reactors and maneuverable in all OPF

simulations reported in this paper.

Careful OPF data preparation, including an adequate restora-

tion scenarios data set, is fundamental to obtain useful solutions.

This data set should reflect the transmission corridor sequential

build up as well as subsystem configurations following load re-

jection and split-up. Considering all these scenarios help to en-

force an even distribution of the shunt reactors along the trans-

mission corridor.Table Idescribes the chosen set of scenarios

for the test system, which are identified by their disconnected

lines, reactors and buses, as well as through the number of en-

ergized buses they contain. Scenario 1 corresponds to the GTL

case . Scenarios numbered 2, 3, 4, 5, and 6 are included

to ensure a more or less even distribution of shunt reactors, as

previously explained. Scenario 7 is included for being the mostimportant considering overvoltages and generator reactive ab-

sorption (capability curve). Scenarios 2 to 7 correspond to GT

subsystem cases ( and ).

The data set scenarios involve system topologies that may

occur after load rejection and consider both and con-

ditions (which have different equations and constraints), as

previously described. One should note that many scenarios

related to stages of line section energization are omitted, since

the only critical phenomena at this stage of transmission cor-

ridor build-up would be electromagnetic transients, which are

outside the scope of this work and can only be captured in

electromagnetic transient simulations.Theadopted scenarios for the test case are listed inTable I,

which have a reduced number of network buses (last column
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TABLE IRESTORATIONSCENARIOS

TABLE IICASE A: BUSREACTORS

TABLE IIICASE A: SHUNT LINE REACTORS

of table). The global OPF problem, for the test system of this

paper with its and scenarios included, has 959 variables.

A detailed description of the global OPF dimensions is given in

Appendix C.

Table II lists the shunt reactors that should be switched-on

according to the results from the proposed methodology and [5],

for a case referred to as CASE A. It is seen from Table IIthat

the bus reactor results from the two methodologies are identical.

Table IIIshows that in the proposed technique, a larger amount

of line reactors were connected. Note the reactive powers of

the shunt reactors are modeled as continuous variables in the

OPF problem, a frequently used approximation, and the discretenature of these equipment is taken into account by setting the

continuous solution to the nearest discrete reactor value once

the OPF process has converged[19].

Table IV presents the recommended maximum load to be

supplied and the generator voltage settings computed by the

proposed OPF method and[5]. The optimal generation voltage

, obtained from the proposed methodology, was 0.98 pu, a

value considerably higher than that reported in[5]. As a conse-

quence, the maximum load in the GTL subsystem has increased

to 234 MW, representing a gain of 44 MW.

In practice, synchronous condensers (SCs) are not used in the

early stages of restoration [20], due to transient stability con-straints and risk of self-excitation [21]. Therefore, the SC of

Fig.5 was assumed disconnected in all simulations of this paper.

TABLE IVGENERATIONVOLTAGE ANDMAXIMUMSUPPLIED LOAD

TABLE VCASE B: BUS REACTORS

TABLE VICASEB: SHUNT LINE REACTORS

TABLE VIICASE C: BUS REACTORS

TABLE VIIICASEC: SHUNT LINE REACTORS

Results are also included for a case, referred to as CASE B,

where the generation high-side voltage was fixed at 0.90 pu, so

as to allow a better comparison with the recommendations from

[5]. The CASE B results, in Tables V andVI, show that the

proposed optimization method required 10.8 MVAr less reactors

than the results reported in[5], while the supplied remote load

was increased by 5%.Results are also included for Case C, a test case in which the

temporary overvoltage limit was set at a reduced value, 1.15 pu

at , leading to the adjustment of the generation voltage at

0.95 pu and an amount of 231 MW as the recommended max-

imum for the load to be picked up. It is seen from Tables VII and

VIIIthat an extra end line reactor of 73.4 MVAr was switched

on to keep the overvoltage below the specified limit.

As expected, the constraints associated with the machine

under-excitation capability curve were active in most cases.

The CPU timings provided are for Matlab 6.5 on an Intel

Pentium IV processor with a 2.66-Ghz clock and 2 Gb of RAM.

The OPF solutions for cases A, B, and C took 82.14 s, 44.13s, and 118.8 s of CPU time, respectively. CASE B solved in

less cpu time because the high-side voltage was kept fixed,
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TABLE IXCASE D: SENSITIVITY ANALYSIS

eliminating therefore the need for the AVR regulating function.

On the other hand, CASE C requires more cpu time to solve

because the temporary overvoltage limit was reduced to 1.15

turning the optimization problem more restrictive.

V. SENSITIVITY ANALYSIS

A sensitivity analysis was carried out to determine the op-

timal number of black-start generating units needed to safely

restore the subsystem. Ideally, the number of units should also

be a decision variable of the OPF, but this would involve adding

another term to the objective function in(1)and facing yet more

problems related to the approximate modeling, by continuous

variables, of some system variables which are actually of dis-

crete nature.Table IXshows a summary of the results obtained

when considering cases with 3, 4, 5, or 6 generating units (in a

total of 8 unitssee data in theAppendix B) participating in the

restoration process.

The obtained results indicated that, for three generating units,

the OPF did not converge even when all the shunt reactors were

switched on. With 4 units dispatched and all shunt reactors

switched on, the amount of picked up load turned out to be verylow. With 5 units dispatched, the number of shunt reactors was

reduced and a considerable amount of load could be supplied.

The last simulation considered 6 units dispatched and yielded

a larger safety margin regarding the system voltages, with the

results indicating a slightly smaller value for the maximum

supplied load, since the generator voltage and the number of re-

actors were reduced. However, since the paralleling of a larger

number of units is always more time consuming in practice, the

6-unit alternative appears to be excessive. Then, this sensitivity

analysis showed that 5 is the most adequate number of gener-

ation units to be paralleled in the present restoration example.

This result is in agreement with operator instructions from

practice[5], which of course were also based on results from

dynamic simulations and electromagnetic transient studies.

VI. CONCLUDINGREMARKS

This paper presented an OPF-based methodology to deter-

mine the optimal generation voltages and minimum shunt re-

actor configuration for the safe restoration of a GT/GTL sub-

system containing a long transmission segment. The recom-

mended maximum value of remote-end load to be supplied is

also determined. The shunt reactors are required so as to keep

generators operating within their capability curves and limit the

temporary as well as steady-state overvoltages at fundamental

frequency for the chosen set of restoration scenarios.

The OPF restoration function has specific modeling aspects

and data requirements, such as the following.

The generation high-side voltage is adjusted in the opti-

mization process in such a way as to increase the recom-

mended maximum value of the remote load to be supplied

while minimizing the amount of shunt reactor compensa-

tion.

The set of chosen scenarios should reflect the sequential

build-up of the GT subsystem in order to ensure a bal-

anced distribution of shunt reactor compensation. Those

subsystem configurations prone to the highest overvolt-

ages following load rejection should be modeled as well.

The maximum temporary overvoltages at fundamental fre-

quency, occurring immediately following load rejection,

are better estimated with the adoption of a voltage behind

subtransient reactance model for the generator.

The optimal results can be obtained from just a few sim-

ulations. As a consequence, the operation planning staff

saves valuable time that can be used to perform other vital

analyses such as electromagnetic transient studies. A nextstep in this research will be assessing the impact of OPF

tools in the reduction of the overall restoration time.

The described results indicate the proposed methodology

may become a valuable tool for aiding system restoration

studies. Note that the OPF program optimizes the GTL

problem subjected to GT and GTL constraints.

APPENDIXA

This Appendix presents the generator capability curves con-

straints which were included in the proposed OPF formulation

for subproblems and .Fig. 6shows the parts A, B, C, D,

and E of capability curve, which are detailed as follows:

Part A: Mechanical Source Limit:

(19)

Part B: Stator Current Limit:

(20)

Part C: Over Excitation Limit:

(21)

Part D: Under Excitation Limit:

(22)
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Fig. 6. Generator capability curves.

TABLE XGENERATORDATA

TABLE XITRANSMISSION SYSTEM DATA

Part E: Stability Limit:

(23)

APPENDIXB

This Appendix presents the Restoration Test System data.

Table Xshows the generator data. Generator MVA base is as-

sumed to be equal to the nominal capacity. Finally, Table XI

shows the transmission line data on a 100-MVA base.

APPENDIXC

This Appendix describes the dimension of the global OPF

problem for the proposed restoration study. The dimension of

the matrix W is obtained from the determination of all optimiza-

tion variables for the test case presented inSection IV, as fol-

lows:

1) Instant

Variables and : ,

where :

Variables and , where

:

Variables and :

Variable , where :

Variable , where :

Generator capability variables [slack variable and

for(19),(21)(23); and for (20)]:

2) Instant

Variables and

, where

and :

Variables and

:

The global variables computed in and , de-

scribed for the subproblem , are also included in the sub-

problem .

3) Instant

Variables and

:
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Variables and

:

Variables and

:

Generator capability variables [slack variable

and , for (19),(21)(23); and

, for(20)]:

The global variables computed in and , describedfor the subproblem , are also included in the subproblem

. Then, the total number of variables (nv) of the global OPF

problem can be calculated as .

REFERENCES

[1] F. F. Wu and A. J. Monticelli, Analytical tools for power systemrestorationConceptual design,IEEE Trans. Power Syst., vol. 3, no.1, pp. 1026, Feb. 1988.

[2] M. M. Adibi, Power System Restoration: Methodologies and Imple-mentation Strategies, 1st ed. New York: Wiley/IEEE Press, 2000.

[3] F. R. M. Alves, A. P. Guarini, R. M. Henriques, J. A. P. Filho, N. Mar-tins, D. M. Falco, and P. Gomes, Changing paradigms for increasedproductivity in power system restoration studies: The Brazilian ISO

experience, inCigre Session 42, Paris, France, 2008, C2-109 paper.[4] R. Podmore, J. C. Giri, M. P. Gorenberg, J. P. Britton, and N. M.

Peterson, An advanced dispatcher training simulator, IEEE Trans.Power App. Syst., vol. PAS-101, no. 1, pp. 1725, Jan. 1982.

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[6] S. Wunderlich, M. M. Adibi, R. Fischl, and C. O. D. Nwankpa, Anapproach to standing phase angle reduction,IEEE Trans. Power Syst.,vol. 9, no. 1, pp. 470476, Feb. 1994.

[7] N. Martins, E. J. Oliveira, W. C. Moreira, J. L. R. Pereira, and R. M.Fontoura, Redispatch to reduce rotor shaft impacts upon transmissionloop closure, IEEE Trans. Power Syst., vol. 23, no. 2, pp. 592600,May 2008.

[8] ONSBrazilian System Operator, Guidelines and Criteria for Elec-trical Studies, item 10, sub-module 10.11 (in Portuguese). [Online].

Available: http://www.ons.org.br/ons/procedimentos/index.htm.[9] ONSBrazilian System Operator, Guidelines and Criteria for Elec-trical Studies, item 8.5, sub-module 23.3 (in Portuguese). [Online].Available: http://www.ons.org.br/ons/procedimentos/index.htm.

[10] Y. Ruan, R. Yuan, X. Tang, Z. Zhang, and B. Song, A new methodfor control of sustained over-voltage during the early stages of powersystem restoration, in Proc. Power and Energy Engineering Conf.,2009, pp. 15, Asia-Pacific.

[11] M. M. Adibi, R. W. Alexander, and B. Avramovic, Overvoltage con-trol during restoration, IEEE Trans. Power Syst., vol. 7, no. 4, pp.14641470, Nov. 1992.

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methods,IEEE Trans. Power Syst., vol. 9, no. 1, pp. 136146, Feb.1994.

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[15] N. G. Bretas, A. C. P. Martins, L. F. C. Alberto, and R. B. L. Guedes,Static simulationof voltage collapse considering the operational limitsof the generators, in Proc. IEEE Power Engineering Society General

Meeting, Toronto, ON, Canada, 2003, vol. 4, pp. 26522658.[16] P.-A. Lof, G. Anderson, and D. J. Hill, Voltage dependent reactive

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IEEE P ES Po wer S ystems Conf. Expo.PSCE, New York, 2004.[19] P. J. Macfie, G. A. Taylor, M. R. Irving, P. Hurlock, and H. B. Wan,

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Eduardo Martins Viana received the B.Sc. degree from the Federal Univer-sity of Viosa, Viosa, Brazil, in 2005 and the M.Sc. d egree from the FederalUniversity of Juiz de Fora, (UFJF), Juiz de Fora, Brazil, in 2008.

His research interests include power system optimization and control of elec-trical power systems.

Edimar Jos de Oliveira (M11) received the B.Sc. degree from the FederalUniversity of Juiz de Fora (UFJF), Juiz de Fora, Brazil, in 1984, the M.Sc. de-gree from the Federal University of Uberlndia, Uberlndia, Brazil, in 1993,and the D.Sc. degree from the Federal Univers ity of Itajub, Itajub, Brazil, in1998.

Since 1989, he has been with the Electrical Engineering Department of UFJF.His main interests include stability analysis, power economics, optimization,

and control of electrical power systems.

Nelson Martins (SM91F98) received the B.Sc. degree in electrical engi-neering from the University of Brasilia, Brazil, in 1972 and the M.Sc. and Ph.D.degrees from the University of Manchester Institute of Science a nd Technology,Manchester, U.K., in 1974 and 1978, respectively.

He works at CEPEL, Rio de Janeiro, Brazil, in the development of methodsand computer tools for power system analysis, with emphasis on dynamics andcontrol.

Jos Luiz R. Pereira(M85) received the B.Sc. degree from the Federal Uni-versity of Juiz de Fora, Juiz de Fora, Brazil, in 1975, the M.Sc. degree fromCOPPEFederal University of Rio de Janeiro, Rio de Janeiro, Brazil, in 1978,and the Ph.D. degree from the University of Manchester Institute of Science andTechnology, Manchester, U.K., in 1988.

From 1977 to 1992, he was with the Federal University of Rio de Janeiro.Since 1993, he has been with the Electrical Engineering Department of the Fed-eral University of Juiz de Fora. His research interests include online security,optimization, and control of electrical power systems.

Leonardo Willer de Oliveirareceived the B.Sc. and M.Sc. degrees from theFederal University of Juiz de Fora, Juiz de Fora, Brazil, in 2003 and 2005, re-spectively, and the the D.Sc. degree from COPPEFederal University of Riode Janeiro, Rio de Janeiro, Brazil, in 2009.

He is currently with the Electrical Engineering Department of the FederalUniversity of Juiz de Fora. His research interests include the development oftools for optimization, planning, and operation of energy and power systemsand stability analysis.
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An Optimal Power Flow Function to Aid Restoration

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