Reactive power dispatch and planning using a non-linear branch-and-bound algorithm

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Published in IET Generation, Transmission & DistributionReceived on 3rd April 2009Revised on 19th November 2009doi: 10.1049/iet-gtd.2009.0422

ISSN 1751-8687

Reactive power dispatch and planning using anon-linear branch-and-bound algorithmC.R.N. Estevam1 M.J. Rider1 E. Amorim2 J.R.S. Mantovani11Departamento de Engenharia Eletrica, Faculdade de Engenharia de Ilha Solteira, UNESP – Universidade Estadual Paulista,Ilha Solteira – SP, Brazil2Departamento de Engenharia Eletrica da, Universidade Federal de Mato Grasso do Sul-UFMS, Campus de Campo Grande,Campo Grande – MS, BrazilE-mail: [email protected]

Abstract: This study proposes the use of a non-linear branch-and-bound (B&B) algorithm to solve the reactivepower dispatch and planning problem of an electrical power system. The problem is formulated as a mixedinteger non-linear programming (MINLP) problem. The MINLP is relaxed resulting in a set of non-linearprogramming (NLP) problems, which are solved at each node of the B&B tree through a primal dual-interiorpoint algorithm. The non-linear B&B algorithm proposed has special fathoming criteria to deal with non-linearand multimodal optimisation models. The fathoming tests are redefined, adding a safety margin value to theobjective function of each NLP problem before they are fathomed through the objective function criteria,avoiding convergence to local optimum solutions. The results are presented using three test systems from thespecialised literature. The B&B algorithm found several optimum local solutions and the best solution wasfound after solving some NLP problems, with little computational effort.

NotationThe notation used throughout this paper is reproduced belowfor quick reference.

NB Set of system buses.

M and N Set of candidate buses where continuousand discreet reactive power sources arealready installed or where new ones arepermitted.

I Set of buses where reactive sources arealready installed or where theinstallation of new ones is permitted(I = M < N ).

G, L Set of generation and load buses of thesystem, respectively.

J Set of candidate buses with fictitiousinjection of reactive power sources.

NT Set of transformers with automatic tapcontrol.

U Set of buses with installed discretecontrollable reactive power sources.

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w Objective function.

CFXi Fixed cost for allocation of reactive powersources in bus i.

Cci, Cri Varying costs due to the allocation of newreactive capacitive and inductive reactivepower sources, respectively, in bus i.

qci, qri (i [ I ) Investment variables: Capacitive andinductive reactive power source,respectively, either existing in the systemor to be installed in the chosen bus i.

y1j, y2j ( j [ J ) Fictitious injection of capacitive andinductive reactive power, respectively, inbus j.

ri Binary decision variable: if ri = 1 thenallocate the reactive power sources inbus i; if ri = 0 then do not allocate thereactive power source in bus i.

g Penalty factor.

(V, u, t) Voltage magnitude, phase angle of bus-bar voltage and transformer tap ratio,respectively.

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Pk (u, V, t ) Active power injection in bus k.

Qk (u, V, t) Reactive power injection in bus k.

IG G:RNB � RG Results in a sub-vector G of G, with Gcomponents

IL L:RNB � RL Results in a sub-vector L of L, with Lcomponents

Iq I:RNB � RI Results in a sub-vector I q of I, with Icomponents

Iu U:RNB � RU Results in a sub-vector U of U, with Ucomponents

ID J:RNB � RJ Results in a sub-vector J of J, with Jcomponents

PGg, QGg Active and reactive power generated inbus g.

PCl, QCl Active and reactive power load in bus l.

qcu0, qru

0 Capacity of capacitive and inductivereactive power, respectively, in bus u [ U .

QGglower, QGg

upper Lower and upper limits of reactive powerbus g.

Vklower, Vk

upper Lower and upper limits of voltagemagnitude in bus k.

tl Transformer tap ratio at branch l.

QGg Reactive power generation in bus g.

Vk Voltage magnitude in bus k.

qcmupper, qrm

upper Upper limits allowed for continuousreactive power compensation source inbus m [ M .

Sd Maximum number of steps allowed forthe discrete sources of reactive power.

T Set of discrete values of the transformertap ratio.

Dn Step size for discrete variables.

1 IntroductionThe reactive power dispatch is a large-scale non-convex, non-linear programming (NLP) problem with real and discretevariables that presents a high degree of complexity forapplication in real electric power systems (EPS). Thesolution of the problem consists in obtaining adjustmentsof the voltage control devices available in EPS, likegenerator voltage magnitudes, transformer tap ratios andreactive power generation of volt–Ampere reactive (VAr)sources (banks of capacitors and reactors), while satisfyingthe security and quality constraints of a steady-stateoperation [1]. There are several classical techniques forsolving the reactive power dispatch (e.g. gradient method,successive linear and quadratic programming, Bender’sdecomposition and the Newton-like methods). Among themore recent techniques, the interior-point method (IPM)[2–4] is considered. When the reactive power reservesavailable in EPS cannot meet the constraints imposed bysteady-state operation, a reactive power source expansion

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must be considered, taking into account the physicalcharacteristics and the operational conditions of the EPS.The objective of the reactive power expansion planningproblem is to determine ‘where’ and ‘how many’ new VArsources must be added to a network in order to make theoperation viable and secure, considering several operationalscenarios (contingencies and multiple load scenarios), andat a minimum cost.

There are many publications on the formulation andsolution of a reactive power dispatch and planning problem[5–11]. The authors of [5, 6] proposed a solution for thereactive power planning problem using the generalisedBenders’ decomposition. The reactive planning problem isdecomposed into two sub-problems (or levels). The VArdispatch problem is solved at level 1 and the VAr allocationproblem, which is formulated as a mixed integer linearprogramming problem, is solved at level 2, using a classicallinear branch-and-bound (B&B) algorithm. In [7], amethodology was presented to solve the reactive powerplanning, considering multiple load levels and multiplecontingencies. The methodology extends and generalisesthe two-level hierarchical approach proposed in [5, 6], bydeveloping a three-level optimisation hierarchical structure.A linear programming-based method of finding theoptimal solution for practical reactive power planning waspresented in [8]. In [9], a two-level decompositiontechnique for VAr source planning in EPS was presented.Several operating system conditions (different base caseswith associated contingencies) can be consideredsimultaneously in the proposed methodology. A heuristicapproach was used to obtain the solution of the reactivepower planning problem in [10]. In [11], the reactivepower planning problem was solved using a geneticalgorithm (GA). The optimal search is achieved throughmultiple paths, leading to a better solution. Furthermore,the approach considered the discrete variables included inthe reactive power dispatch and several objective functions,at the same time. Despite being an easy and simple methodthat shows good results, the GA presents various problemssuch as a high processing demand, difficulty to specify thestop criteria in an efficient way and its inability to identifythe optimum solution. In [12], a methodology based on abinary search tree technique and a heuristic method forobtaining discrete solutions related to existing or plannedreactive banks were proposed. The authors of [13]proposed a methodology combining a successive linearprogramming (SLP) method and a GA. The SLP is usedto solve the reactive power dispatch while the GA is usedto determine the necessary adjustments of both binary anddiscrete variables of the mathematical model. In [14], anefficient heuristic algorithm for optimal capacitor placementin distribution systems was proposed.

This work seeks to develop a new methodology to achieveoptimal reactive power dispatch and, if necessary, todetermine the minimal investments in new reactive powersources to thoroughly meet the operational constraints of the

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EPS in different operational scenarios. Owing to the discretenature of some reactive power compensation devices, themathematical model of the reactive power dispatch andplanning proposed in this work leads to a mixed integerNLP (MINLP) problem with binary, discrete andcontinuous variables. To solve this problem, a B&Balgorithm specialised in non-linear and multimodaloptimisation problems is proposed. The B&B is anenumerative algorithm whose solution method is based onbuilding a tree where nodes represent sub-problems (in ourcase, optimal reactive power dispatch problems) and branchesrepresent new restrictions on the discrete variables to beconsidered. The main feature of this work is a proposal forsolving MINLP problems directly without using Benders’decomposition techniques [5–7]. An NLP problem is solvedat each node of the B&B tree, which is the optimal reactivepower dispatch considering the discretisation of theoperation and the planning variables relaxed. These NLPproblems are solved using the IPM [2–4].

The difference between this proposal and those presentedin [2, 4] is the objective function model and some additionalconstraints that involve the discrete nature of reactive sourcesexisting in the system, not only regarding the dispatch, butalso the optimal reactive planning. There are studies in thespecialised literature that propose the use of metaheuristicsto solve optimal reactive power dispatch problems as, forinstance, [11, 13]. A metaheuristic technique is robust andeasy to implement, but requires parameter adjustments thatare difficult to make for medium and large power systemswhich operate under heavy loads or contingencies. Thelong computational time demanded by metaheuristics, evenin the solution of small problems, is the major constraintthat has been considered up to now regarding itsapplication in solving classic electrical engineering problemswith well-defined mathematical models that present a largenumber of constraints. Metaheuristics techniques arerecommended for solving problems that cannot establish aformal mathematical model, with variables of differentnatures, among other issues, which complicate or evenprevent the use of classical optimisation techniques. Thus,the study and development of classical optimisationtechniques to solve those problems as proposed in thispaper are considered to be an important research line.

Recently, the application of non-linear B&B in order tosolve MINLP problems directly, without the support ofdecomposition methods, has been shown to be efficient. In[15], the first application of a non-linear B&B algorithm tosolve the transmission expansion planning problem waspresented. In [16], it was shown that a B&B algorithm isfar superior to the one that uses the Benders’decomposition to solve a mixed integer linear programming(MILP) problem. This comparison is important, sincethere are two algorithms to solve the same MILP problemwith assured convergence to the global optimum formedium-size systems.

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The non-linear B&B algorithm proposed to solve thereactive power planning problem is based on the algorithmproposed in [15] and uses deep search last in first out(LIFO) [17]. The non-linear characteristic of the NLPproblems is not predicted in classical B&B algorithmsdesigned to solve the MILP problem. As for the classicalB&B algorithm to solve MILP problems, the objectivefunction of a successive problem is always greater than orequal to the previous one. However, for MINLP problems,this characteristic is not assured, as it is possible to fathomsome nodes that can find better solutions than the currentincumbent. A way to avoid that is to add a safety margin toeach sub-problem before it is fathomed. Thus, two changesto the classical B&B algorithm are proposed. The first oneis the solution of NLP problems at each node of the B&Btree. The second one is related to the fathoming test; amodification is made to avoid the convergence of non-linear B&B algorithm to sub-optimal solutions of thereactive power dispatch and planning problem caused bythe local minimums of the NLP problem solution at eachnode of the B&B tree.

In order to validate the mathematical model and theproposed solution technique, results and analyses with theIEEE-30, IEEE-118 and IEEE-300 bus test systems arepresented. The non-linear B&B algorithm presented doesnot guarantee convergence to the planning problem globaloptimum, as it is a non-convex, multimodal and highlycomplex MINLP problem. In this case, no algorithmpreviously proposed to solve this type of problem canguarantee convergence to global optimum in real systemtests. However, in the tests carried out, the B&Balgorithm found several optimum locations and convergeseasily towards the best solutions of the tested systems aftersolving some NLP problems with little computationaleffort.

2 Mathematical modelThe reactive power dispatch and planning problem, for eachoperational scenario, is formulated as an MINLP problemand the mathematical model is shown in (1)–(15).

min w =∑

i[I

[CFXi· ri + Cci· qci + Cri· qri] (1)

such that

Pk(u, V , t )− I GPGg + I LPCl = 0, k [ NB, g [ G, l [ L

(2)

Qk(u, V , t ) − I GQGg + I LQCl + I q(qci − qri)

+ I u(qc0u − qr0

u ) = 0, k [ NB, g [ G, l [ L,

i [ I , u [ U , j [ J (3)

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QGlowerg ≤ QGg ≤ QGupper

g , g [ G (4)

V lowerk ≤ Vk ≤ V

upperk , k [ NB (5)

y1j ≥ 0, j [ J (6)

y2j ≥ 0, j [ J (7)

tl [ T l [ NT (8)

0 ≤ qcm ≤ qcupperm rm, m [ M (9)

0 ≤ qrm ≤ qrupperm rm, m [ M (10)

qcn [ Sdrm, n [ N (11)

qrn [ Sdrm, n [ N (12)

qc0u [ Sd, u [ U (13)

qr0u [ Sd, u [ U (14)

ri [ {0, 1}, i [ I (15)

The objective function (1) considers the expansion cost of anew reactive power source (installation and operation costs).Equations (2) and (3) are the static power flow equationsthat represent the active and reactive power balance of theEPS. Equation (4) represents the constraints of the reactivegeneration capacity of the continuous reactive sourcespresent in the EPS. Equation (5) assures the operationallimits of the voltage magnitude levels. Equations (6) and(7) represent the non-negative conditions of the artificialinjections of the reactive power sources. Equation (8)provides the discrete limits of the transformers withautomatic control of taps. Equations (9)–(14) constrain thecapacity of continuous and discrete reactive sources that arepresent or that are to be allocated to the EPS.

The solution provided by (1)–(15) must enable feasibleoperation of the power system in a particular load scenarioor for a particular contingency. The final reactive powerexpansion plan is obtained by combining the reactive powersource requirements of the individual situations. Someaspects of this problem are (a) existing reactive sources orallocated reactive sources obtained from the simulation of aprevious scenario must be considered; (b) the candidate busset for installation of new sources must be defined. Thedefinition of this set is critical and must be made carefully,since a poor definition may lead to impractical solutionsand (c) the reactive source type to be considered must bedefined. All these sources can be classified as fixed ordiscrete (capacitor/reactor banks) or controlled (synchronousmachines or static controlled VAr devices).

In Table 1, a variable summary of the MINLP problem isshown. Note that if new reactive power sources are notnecessary in the EPS, an optimal reactive power dispatch issolved considering continuous and discrete variables.

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3 Solution techniqueThe B&B is an enumerative algorithm that uses the relaxationand separation strategy to solve a complex optimisationproblem. It has been used to analyse and study MILPproblems where it is capable of arriving at a global optimum.The solution structure is based on building an enumeratedtree, where the nodes represent the current sub-problemsand the branches represent the new restrictions that must beconsidered in the next sub-problems. By means of this tree,it is possible to test if a given feasible solution is optimal,comparing it implicitly or explicitly to each one of thecurrent feasible solutions of the MILP problem. A sub-problem can be fathomed using the elimination criteriapresented later. The B&B process stops when all the sub-problems have been fathomed, that is, when all the searchspace solutions have been analysed. The optimum globalsolution is the stored incumbent at the end of the process.

The B&B algorithm for MILP problems is very simple.However, it presents some difficulties when applied to realproblems such as (a) long processing time for large-scaleproblems; (b) memory problems when storing theinformation that identifies each non-fathomed sub-problem;(c) high programming level needed to implement an efficientsoftware and (d) in the proposed approach, NLP problemsshould be solved at each node of the tree, generating anadditional complicating factor. Considering the MINLP (1)–(15) and relaxing the conditions (11)–(15), the NLP problemto be solved for each planning scenario is shown in (16)

min w =∑

i[I

[CFXiri + Cci qci + Cri qri] + g∑

j[J

[y1j + y2j]

(16)

such that

Pk(u, V , t) − I GPGg + I LPCl = 0, k [ NB, g [ G, l [ L

Qk(u, V , t) − I GQGg + I LQCl + I q(qci − qri)

+ I u(qc0u − qr0

u ) + I D(y1j − y2j) = 0, k [ NB, g [ G,

l [ L, i [ I , u [ U , j [ J

Table 1 Variables summary for the MINLP problem

Variables operation Investment

continuous Vk, ∀k [ NB qcm, ∀m [ M

uk, ∀k [ NB qrm, ∀m [ M

QGg, ∀g [ G

discrete tl, ∀l [ NT rm, ∀m [ M

qc0u, ∀u [ U qcn, ∀n [ N

qr0u , ∀u [ U qrn, ∀n [ N


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QGlowerg ≤ QGg ≤ QGupper

g , g [ G

V lowerk ≤ Vk ≤ V

upperk , k [ NB

y1j ≥ 0, j [ J

y2j ≥ 0, j [ J

min{T } ≤ t lowerl ≤ tl ≤ t

upperl ≤ max{T }, l [ NT

0 ≤ qcm ≤ qcupperm rm, m [ M

0 ≤ qrm ≤ qrupperm rm, m [ M

min{Sd}rn ≤ qclowern ≤ qcn ≤ qcupper

n ≤ max{Sd}rn, n [ N

min{Sd}rn ≤ qr lowern ≤ qrn ≤ qrupper

n ≤ max{Sd}rn, n [ N

min{Sd} ≤ qc0,lowern ≤ qc0

n ≤ qc0,uppern ≤ max{Sd}, u [ U

min{Sd} ≤ qr0,lowern ≤ qr0

n ≤ qr0,uppern ≤ max{Sd}, u [ U

0 ≤ r loweri ≤ ri ≤ r

upperi ≤ 1, i [ I

where min{T } and max{T } are, respectively, the minimumand maximum values for the set NT and min {Sd } andmax{Sd } are, respectively, the minimum and maximum valuesfor the set N. t lower

l , qclowern , qrlower

n , qc0,lowern , qr0,lower

n , rloweri and

tupperl , qcupper

n , qruppern , qc0,upper

n , qr0,uppern , r

upperi are the lower

and upper bounds in the tl , qcn, qrn, qc0n , qr0

n , ri variables thatcharacterise or identify the node in the B&B tree. Decisionvariables of problem (16) are the voltage magnitude, phaseangle, transformer tap ratio, reactive power generation,reactive power source that can be installed, reactive powercompensation installed, auxiliary reactive power injection andbinary decision variable to allocate reactive power sources. Itshould be observed that the NLP problems at each node ofthe B&B tree are only different in t lower

l , qclowern , qrlower

n ,qc0,lower

n , qr0,lowern , r lower

i and tupperl , qcupper

n , qruppern , qc0,upper

n ,

qr0,uppern , r

upperi vector variables. The auxiliary reactive power

injections (y1j and y2j) penalised in the objective function areused to identify a feasible solution for the reactive powerdispatch and planning problem [13]. g is the weight factorand it is made very large in order to validate the unfeasiblecriterion (criterion (a) in step (v)) of the B&B algorithm.Hence, by solving the unrestricted problem (16), we obtain asolution in which some variables violate the requirement thatthey are discrete. At each node of the B&B algorithm, theneighbouring smaller and larger integer values of the non-integer solution value of the variable are selected.

Four features related to the B&B algorithms that should beanalysed separately are (a) the choice of a variable for sub-problem separation, (b) the choice of the next sub-problemto be solved, (c) fathoming tests and (d) how to work withNLP problems.

The choice of a variable for separation and the next sub-problem to be solved is important as they determine thesize of the B&B tree and the computational efficiency ofthe algorithm. There is no optimum way to determine the

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variable for separation, and the next sub-problem to beused for analysis generates a small-sized B&B tree. In thiswork, the LIFO strategy [17] is chosen to determine thenext sub-problem. In order to determine the variable forseparation, a priority list using the difference between thenon-integer value (of integer variables) and its nearestinteger is used.

Fathoming tests verify that, in the feasible region of a sub-problem, there are no better solutions than those alreadyknown (the current incumbent) and, therefore, they shouldbe eliminated from the search. In B&B algorithms for theMILP problem, there are three fathoming tests. Thus, asub-problem is fathomed in the following cases: (i) if thesolution for the sub-problem is integer, indicating that thisinteger solution is the best one for this feasible region, (ii)if the solution for the sub-problem is bigger than thecurrent incumbent solution, indicating that if there areinteger solutions in the feasible region, the best integersolution should be greater than the current incumbent and(iii) if the solution for the sub-problem is unfeasible,indicating that there are no integer solutions in the feasibleregion of the sub-problem.

The proposed B&B algorithm, for each scenario planning,consists of the following steps:

1. Make k ¼ 0 (number of sub-problems generated), to solvethe problem (16) that generates the optimal solution of theinitial sub-problem and define an initial incumbentsolution (w∗ = 1) and the current sub-problem (PCk).

2. Choose an integer variable tl , qcn, qrn, qc0n , qr0

n or ri , whichhas a non-integer current value t∗l , qc∗n , qr∗n , qc0,∗

n , qr0,∗n or r∗i ,

for separation.

3. Generate two new descendent sub-problems from thecurrent sub-problem (PCk), adding the following restrictions

First descendent (PCk+1): tl ≤ ⌊t∗l ⌋ or qcn ≤ ⌊qc∗n⌋ orqrn ≤ ⌊qr∗n⌋ or qc0

n ≤ ⌊qc0,∗n ⌋ or qr0

n ≤ ⌊qr0,∗n ⌋ or ri ≤ ⌊r∗i ⌋

Second descendent (PCk+2): tl ≥⌊t∗l ⌋+ Dl or qcn ≥ ⌊qc∗n⌋+ Dn or qrn ≥ ⌊qr∗n⌋ + Dn or qc0

n ≥ ⌊qc0,∗n ⌋ + Dn or qr0

n ≥⌊qr0,∗

n ⌋ + Dn or qr0n ≥ ⌊qr0,∗

n ⌋ + Dn or ri ≥ ⌊r∗i ⌋ + 1

where⌊x⌋ is the nearest lower integer of x.

4. Store PCk+1 in the list of candidates sub-problems forfurther analysis, and to solve the problem PCk+2, makek ¼ k + 2.

5. Verify the fathoming tests: a sub-problem is fathomed ifone of the following conditions is satisfied:

(a) if the sub-problem is unfeasible (y1j = 0 or y2j = 0, forat least one j [ J );

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(b) if wfound . w∗ + 1;

(c) if the optimal solution of the sub-problem satisfiesintegrality constraints. If so, if the optimum is lower thanthe incumbent, then make w∗ = wfound and apply test (b)for all candidates not fathomed yet.

If the current sub-problem does not satisfy any of thefathoming criteria (a), (b) or (c), go to step 2. Otherwise,go to step 4;

6. If non-fathomed sub-problems exist, solve the last sub-problem on the list of candidates and go to step 2. If allthe sub-problems have been fathomed, stop. The process isfinished and the current incumbent solution is the bestsolution.

The need for an efficient algorithm to solve the non-linearproblems at each node of the B&B tree is emphasised and,for that, a non-linear IPM is used. Another relevant aspectof the proposed B&B algorithm is the fathoming test (b)adding a safety factor (1) to find local optimum in solvingthe NLP problems. Thus, for a sub-problem to befathomed, there should be a lower bound of the objectivefunction that is greater than the incumbent solution withan additional safety value added. It should be observedthat, in the B&B algorithm for MILP problems, theobjective function value of a successor sub-problem isgreater than or equal to the objective function value of itsantecessor. However, for non-linear models, this property isnot guaranteed and the safety factor tries to avoid thisproblem.

Fig. 1 shows in a geometrical manner the necessity ofconsidering this safety factor in the fathoming test of theB&B algorithm to solve MINLP problems. From Fig. 1, it

Figure 1 Illustration of search space and MINLP non-convexity problem

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can be observed that, owing to the non-linear characteristicof the problem, if there is no security factor 1, the localminimum (l node) can do the fathoming of all nodes thatcan be the route to find the global optimum. The 1-valueguarantees that the k node (wk , w∗

l + 1) is not fathomedand helps the B&B algorithm to converge to the globaloptimum ( g node) or to another local minimum withbetter quality. Thus, the proposal presented increases theprobability of finding a global optimum solution but mightneed to solve an additional number of NLP problems. Inthis work, the value of 1 is set to a percentage of w∗ thatdepends on the size of the test system. Note that choosinga big value for 1 just increases the number of NLPs thatthe B&B algorithm must solve, whereas a small value for 1can make the algorithm stop at a local optimum.

It is possible that the NLP solution will not converge topractical applications. In that case, the strategy must be tofathom the sub-problem in order to continue the process ofsolving the B&B algorithm. This type of problem clearlyrepresents an additional limitation to the process. However,in the tests that were carried out, this type of problem didnot occur.

4 Test and resultsThe B&B algorithm and the IPM were implemented inFORTRAN 90 programming language. The bifactorisationmethod proposed by Zollenkopf [18] was used to solve thelinear equation systems in the IPM. Three systems wereused for the tests [19]: the IEEE 30-bus, IEEE 118-busand the IEEE 300-bus systems, operating under differentcontingencies and load levels scenarios. All cases were runon a Pentium IV PC with 3.4 GHz and 1 GB RAM.

4.1 Test with the IEEE 30-bus system

This system consists of 30 buses, 5 generators and 41branches. Buses 2, 5, 8, 11, 13 and 30 were adopted ascandidate buses for new discrete reactive sources expansion.In Table 2, the limits and costs of candidate reactive powersources are shown. The step sizes for reactive power sourcesare Dn = 2 MVAr and the safety factor for fathoming testis 1 = 3%. Nominal and high (103%) load levels and a

Table 2 IEEE-30 bus system data

Buses CFXi,US$ Cci andCri,US$/MVAr

qcupperm and

qrupperm ,MVAr

2 30 1 30

5 30 1 45

8 15 1 40

11 30 1 40

13 35 1 30

30 30 1 30


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contingency (#1: branches 1–2) were considered. The lowerand upper limits of voltage magnitude allowed for each busare 0.95 and 1.05 p.u., respectively. In Table 3, the resultsobtained are presented, where NLP is the total NLPproblem numbers used by the B&B algorithm.

From Table 3, it is observed that it is not necessary toallocate the reactive power source to the base case in thenominal load or in the high load, indicating that theavailable reactive power sources in the EPS are sufficient toobtain a feasible steady-state operation and the B&Balgorithm found this state operation using an NLPproblem. However, for the contingency case, it is necessaryto allocate a total reactive power of 88 and 112 MVAr forthe nominal and high load, respectively. The B&Balgorithm selected buses 2, 5, 8 and 13 to allocate reactivepower sources with a capacity of 30, 28, 40 and 14 MVAr,respectively, to obtain a feasible operation for all cases andload levels. In buses 2 and 8, it is necessary to allocate thereactive power capacity at its maximum. Note that fewerNLP problems and seconds were used by the B&Balgorithm to find the best solutions.


This system consists of 118 buses, 54 generators and 186branches. For this test system, 13 buses were considered toallocate new discrete reactive sources. Table 4 shows limitsand costs of the candidate reactive power sources. Threeload levels (nominal, low (80%) and high (120%)) and four

Table 3 Results obtained for the IEEE-30 bus system

Load level Buses Base case Contingency

#1: 1–2

qci,MVAr qci,MVAr

nominal load 2 – 30

5 – 18

8 – 40

w – US$ 163

NLP 1 71

CPU time 1 s 34 s

high load 2 – 30

5 – 28

8 – 40

13 – 14

w – US$ 222

NLP 1 33

CPU time 1 s 30 s


contingencies (#1: branches 5–8, #2: branches 26–30, #3:branches 65–68 and #4: branches 89–92) were considered.The lower and upper limits of voltage magnitude allowedfor each bus were 0.97 and 1.03 p.u., respectively. The stepsizes for reactive power sources were Dn = 12.5 MVAr andthe safety factor for fathoming test is 1 = 3%. Table 5shows the results obtained.

From Table 5, it is observed that, for all cases (base andcontingencies), it is necessary to allocate reactive powersources. The B&B algorithm selected buses 10, 12, 21, 76,83, 85 and 94 to allocate reactive power sources with212.50, 75, 37.50, 62.50, 75, 75 and 50 MVAr capacities,respectively, to obtain a feasible operation for all cases andload levels. In bus 10, it was necessary to allocate aninductive reactive power source. As in the previous case,few NLP problems and seconds were used by the B&Balgorithm to find the best solutions. Note that the resultsare discrete based on Dn.


This system consists of 300 buses and 411 branches. For thistest system, 81 and 2 buses were considered to allocate newdiscrete reactive and continuous sources, respectively.Table 6 shows limits and costs of the candidate reactivepower sources. Three load levels (nominal, low (80%) andhigh (108%)) and two contingencies (#1: branches 3–1and #2: branches 156–157) were considered. The lowerand upper limits allowed for voltage magnitude of each buswere 0.95 and 1.05 p.u., respectively. The step sizes forreactive power sources were Dn = 12.5 MVAr and the


Buses CFXi(US$) Cci andCri,US$/MVAr

qcupperm and

qrupperm ,MVAr

10 12 1 100

12 10 1 100

21 10 1 100

31 10 1 100

35 5 1 100

49 5 1 100

56 10 1 100

60 5 1 100

71 10 1 100

76 5 1 100

83 10 1 100

85 5 1 100

94 10 1 100

969


97

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Load level Buses Base case Contingency

#1: 5–8 #2: 26–30 #3: 65–68 #4: 89–92

qci,

MVArqri,

MVArqci,

MVArqri,

MVArqci,

MVArqri,

MVArqci,

MVArqri,

MVArqci,

MVArqri,

MVAr

nominalload

10 – – – 12.50 – – – – – –

21 – – – – 12.50 – – – – –

76 25 – 25 – 25 – 25 – 25 –

83 – – – – – – – – 62.50 –

w US$ 30 US$ 54.50 US$ 52.50 US$ 30 US$ 102.50

NLP 3 5 5 3 9

CPUtime

2 s 3 s 3 s 2 s 12 s

low load 10 – – – 12.50 – – – – – –

w – US$24.50

– – –

NLP 1 3 1 1 1

CPUtime

1 s 2 s 1 s 1 s 1 s

high load 10 – – – 12.50 – – – – – –

12 – – 75 – – – – – – –

21 – – 12.50 – 37.50 – – – – –

76 62.50 – 62.50 – 62.50 – 50 – 62.50 –

83 37.50 – 37.50 – 37.50 – 50 – 75 –

85 – – – – – – – – 75 –

94 37.50 – 37.50 – 37.50 – 37.50 – 50 –

w US$ 162.50 US$ 294.50 US$ 210 US$ 162.50 US$ 292.50

NLP 33 45 21 27 15

CPUtime

40 s 45 s 30 s 28 s 19 s

0

safety factor for fathoming test is 1 = 3%. Table 7 shows theresults obtained.

From Table 7, it is observed that, for all cases, it isnecessary to allocate reactive power sources. For the highload case, considering contingency #2, the problem isunfeasible. This unfeasibility occurs due to the fact that, forthis load level, the maximum magnitudes of reactivesources and the candidate bus set are inadequate and donot meet the operation restrictions regarding the specifiedminimum voltage profile. To obtain a feasible operation forall cases and load levels, the B&B algorithm selected buses55, 97, 103, 104, 149, 150, 152, 157, 158, 171, 252, 259and 260 to allocate reactive power sources with 21.41, 50,


50, 25, 50, 37.50, 50, 100, 75, 1.72, 50, 25 and12.50 MVAr capacities, respectively. As in the previouscases, few NLP problems and seconds were used by theB&B algorithm to find the best solutions and results arediscrete based on Dn.

Fig. 2 shows the B&B tree for contingency #1. It can beverified that ten nodes (1, 7, 9, 11, 13, 15, 21, 23, 28 and29) were fathomed for the security factor and another fournodes (4, 6, 26 and 30) because the PNLIM problem isunfeasible. The B&B algorithm found two discretesolutions at nodes 22 and 25. The best solution is to install11 compensators shown at node 25. Non-linearity of theproblem is shown by the values of w19, w24 and w25,


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Buses CFXi(US$) Cci andCri,US$\MVAr

qcupperm and

qrupperm ,MVAr

131, 155, 164, 165, 166, 169, 177, 192, 199, 200, 201, 206, 209, 212,215, 217, 218, 220, 221, 222, 248, 249, 251, 252, 254, 255, 256, 259,260, 261, 262

10 1 100

8, 10, 19, 63, 69, 76, 77, 80, 120, 122, 125, 126, 128, 295, 296 12 1 100

88 5 1 50

99, 100, 135, 156, 170, 247, 264, 268, 269, 280, 282, 287, 292, 294 7 1 100

97 25 1 50

103 30 1 50

133, 145, 152, 157, 158, 265, 267 5 1 100

55a 40 1 30

104 20 1 30

117 15 1 40

171a 45 1 50

98 10 1 20

150 10 1 40

132 20 1 80

258 20 1 40

253 20 1 100

149 15 1 50

245 15 1 60

250, 263 15 1 100

aContinued reactive power source.

T

where reduction of the objective function is shown when itgoes on the B&B tree.

5 ConclusionsA fast and efficient approach that integrates a non-linearB&B algorithm with the IPM is proposed to solve thereactive power dispatch and planning problem. The B&Balgorithm used few NLP problems and seconds to find thebest solutions for all cases. An important aspect of thismethodology is its efficiency and robustness in solvingMINLP problems with continuous and discrete variables.This methodology makes it possible to compute thesettings of reactive power dispatch corresponding to apredetermined number of load levels and contingencies thatdo not significantly interfere with the needs of changinggenerations of active and reactive power generation plants.This is a small database to permit the choice of settingswithin the large range of operating conditions that can betypically encountered during the running of the system, and


other dispatch algorithms are needed to facilitate systemoperation.

The results obtained show that the model and solutiontechnique can solve the optimal reactive power dispatchproblem. In addition, this methodology enables minimalinvestments in new reactive sources to satisfactorily meetoperational conditions of power systems under differentoperational scenarios to be determined whenever necessary.The safety factor for fathoming test (1) should beempirically calibrated according to the physicalcharacteristics of the system studied. In the test systemssimulated in this work, when this factor is adjustedbetween 3 and 5%, it does not affect the B&B tree size orthe quality of the solutions obtained. If 1 values are higherthan 5%, the result will be B&B trees too big for a systemthat operates under heavy load conditions and under sometest contingencies. In order to avoid problems of IPMconvergence that may occur at some B&B tree nodes, thenumber of iterations and parameters that supply the IPM

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convergence diagnostic should be controlled. If the NLPproblem does not converge, this node of the B&B treeshould be considered unfeasible.

There is no guarantee that the algorithm will obtain aglobal minimum cost solution to satisfy the constraints forall the contingency-load combinations. While the


Loadlevel

Buses Base case Contingency

#1: 3–1 #2:156–157

qci,MVAr qci,MVAr qci,MVAr

nominalload

149 25 37.50 25

152 50 50 25

157 – – 100

158 – – 75

w US$ 95.00 US$107.50

US$ 255

NLP 5 5 9

CPUtime

26 s 24 s 41 s

low load 157 – – 50

w – – US$ 55

NLP 1 1 9

CPUtime

1 s 1 s 57 s

high load 55 21.41 10 unfeasibleproblem

97 50 50

103 50 50

104 25 25

149 50 50

150 37.50 37.50

157 75 75

171 1.36 1.72

252 – 50

259 – 25

260 – 12.50

w US$500.27

US$606.72

–

NLP 21 31 –

CPUtime

70 s 104 s –


technique of taking the maximum setting of each reactivesource over all combinations will obviously obtain asolution that satisfies all of them, there is no guarantee thatit is optimal. A cheaper solution may exist which differsfrom the optimal of the individual cases, and is costlierthan some of them, but will also satisfy all the constraints.That does not detract from the validity of the technique,but one must be careful not to make unsubstantiated claimsof optimality.

Defining the candidate bus set determines the quality ofthe solution obtained for the problem according to testsconducted with IEEE-30, IEEE-118 and IEEE-300 bussystems operating under different load and contingencyconditions. A misestimated bus set leads to unfeasibleplanning problems or to non-economic solutions. Thus,definitions of a candidate bus set must be done verycritically and carefully, using simulations to identify busesthat present physical problems as to reactive needs support.

The main contribution of this work is that the proposedalgorithm deals directly with MINLP problems, solving, ateach node of the B&B tree, an NLP problem which is theoptimal dispatch of reactive sources, where discretisationrestrictions of the adjustment of capacitors and inductor arerelaxed.

6 AcknowledgmentsWe acknowledge the financial support received from theBrazilian institutions CAPES and CNPq (grant no.301060/2006–1).

Figure 2 Branch-and-bound tree for the IEEE 300-bussystem


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7 References

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[2] GRANVILLE S.: ‘Optimal reactive dispatch through interiorpoints methods’, IEEE Trans. Power Syst., 1994, 9, (1),pp. 136–146

[3] WU Y.C., DEBS A.S., MARSTEN R.E.: ‘A direct nonlinearpredictor corrector primal-dual interior point algorithmfor optimal power flows’, IEEE Trans. Power Syst., 1994, 9,(2), pp. 876–883

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[5] LEBOW W.M., MEHRA R.X., NADIRA R., ET AL.: ‘Optimization ofreactive volt– ampere (VAr) sources in system planning’.EPRI Report EL-3729, 1984, vol. 1

[6] LEBOW W.M., ROUHANI R., NADIRA R., ET AL.: ‘A hierarchicalapproach to reactive volt ampere (VAr) optimization insystem planning’, IEEE Trans. Power Appar. Syst., 1985,PAS-104, (8), pp. 2051–2057

[7] GRANVILLE S., PEREIRA M.V.F., MONTICELLI A.: ‘An integratedmethodology for VAr sources planning’, IEEE Trans. PowerSyst., 1988, 3, (2), pp. 549–557

[8] IBA K., SUZUKI H., SUZUKI K., SUZUKI K.: ‘Practical reactivepower allocation/operation planning using successivelinear programming’, IEEE Trans. Power Syst., 1988, 3, (2),pp. 558–566

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[10] HONG Y., LIU C.: ‘A heuristic and algorithmic approach toVAr planning’, IEEE Trans. Power Syst., 1992, 7, (2),pp. 505–512

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[12] MANTOVANI J.R.S., GARCIA A.V.: ‘A heuristic method forreactive power planning’, IEEE Trans. Power Syst., 1996,11, (1), pp. 68–74

[13] MANTOVANI J.R.S., MODESTO S.A.G., GARCIA A.V.: ‘VArplanning using genetic algorithm and linearprogramming’, IEE Proc. Gener. Transm. Distrib., 2001,148, (3), pp. 257–262

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[16] HAFFNER S., MONTICELLI A., GARCIA A., ROMERO R.: ‘Specialisedbranch-and-bound algorithm for transmission networkexpansion planning’, IEE Proc. Gener. Transm. Distrib.,2001, 148, (5), pp. 482–488

[17] TAHA H.A.: ‘Integer programming: theoryapplications and computations’ (Academic Press,New York, 1975)

[18] ZOLLENKOPF K.: ‘Bi-factorisation – basic computationalalgorithm and programming techniques’, in REID J.K. (ED.):‘Large sparse set of linear equations’ (Academic Press,1971)

[19] University of Washington. Available at: www.ee.washington.edu/research/pstca (accessed on 11/19/2008)

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Reactive power dispatch and planning using a non-linear branch-and-bound algorithm

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