Robust Dynamic Transmission and Renewable Generation Expansion Planning: Walking Towards Sustainable Systems

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by  Cristina Roldán

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Robust Dynamic Transmission and Renewable
Generation Expansion Planning: Walking Towards
Sustainable Systems
Cristina Rolda´n, Agust´ın A. Sa´nchez de la Nieta, Member, IEEE, Roberto M´ınguez, and Raquel
Garc´ıa-Bertrand, Senior Member, IEEE
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1
0
2 Abstract—Nowadays, the transition from a conventional gen- G+ Set of all prospective new generators which can be
g
n eration system to a renewable generation system is one of the installed at differentphases associated with groupg.
a most difficult challenges for system operators and companies. g Index for groups of generators built per phases.
J Thereareseveralreasons:thelong-standingimpactofinvestment G− Set of all generators to be uninstalled or dismantled
decisions, the proper integration of renewable sources into the
0
system,thepresentandfutureuncertainties,andtheconvenience during the study period.
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to consider an integrated year-by-year representation of both i Index related to generators.

] uncertainties and investment decisions. However, recent break- j Index associated with loads.
C throughsinDynamicTransmissionNetworkExpansionPlanning k Index referring to lines.
(DTNEP)have demonstrated that theuseof robustoptimization
O l Counter index for each iteration.
might render this problem computationally tractable for real
. systems. This paper intends to consider not only the capacity L Setofallexistingtransmissionlinesatthebeginning
h
expansion of lines, but the construction and/or dismantling of time horizon considered.
t
a of renewable and conventional generation facilities as well. L+ Set of all prospective transmission lines.
m TheDynamicTransmission Network andRenewableGeneration
N Set of all networks buses.
Expansion Planning (DTNRGEP) problem is formulated as an
[ n Index related to buses.
adaptive robust optimization problem with three levels. First
1 level minimizes the investment costs of transmission network n(i) Bus index where the i-th generating unit is located.
v and generation expansion planning, the second level maximizes n(j) Bus index where the j-th demand is located.
9 system operational costs with respect to uncertain parameters, T Set of indices of years.
4 while the third level minimizes those operational costs with ΨD Set of indices of the demand located at bus n.
6 respect to operational decisions. The method is tested for two n
ΨG Set of indices of the generating units located at bus
8 cases: i) an illustrative example based on Garver IEEE system n
0 and ii) a case study using the IEEE 118-bus system. Numerical n.
. resultsfromtheseexamplesdemonstratethattheproposedmodel U(t) Set of indices of the uncertain variables for time
1
allows making optimal decisions towards reaching a sustainable period t.
0
power system, while overcoming problem size limitations and
7 computational intractability for realistic cases. Constants:
1
: IndexTerms—powersystems,renewablegenerationexpansion bk Susceptance of line k (S).
v planning, robust optimization, transmission network expansion cG Generator i operational cost (e/MWh).
i i
X planning. cGI Generator i investment cost (e).
i
r cLkI Line k investment cost (e).
a NOMENCLATURE cS Consumer j load-shedding cost (e/MWh).
j
This section states the main notation used in this paper for e(t) Percentageofload shedbythe j-th demandforyear
j
quick reference. t.
Indices and Sets: h(t) Nominal value evolution factor for demand j and
µ,j
D Set of indices of demand. period t.
G Set of indices of all generation units installed at the h(t) Dispersion value evolution factor for demand j and
σ,j
beginningof time horizon considered which can not period t.
be removed from the system. fmax Line k capacity (MW).
k
G+ Set of all prospective and independent new possible I Discount rate.
generators. Ng Total number of generators built per phases in the
group g.
N Number of study periods.
y
C. Rolda´n, A. A. Sa´nchez de la Nieta, and R. Garc´ıa-Bertrand
o(k) Line k sending-end bus.
are with the Department of Electrical Engineering, Universidad de
Castilla-La Mancha, Ciudad Real, Spain (e-mail: [email protected], r(k) Line k receiving-end bus.
[email protected],[email protected]) Π Transmission expansion investment budget (e).
L
R.M´ıngueziswithHidralabIngenierayDesarrollo,S.L.,Spin-OffUCLM, Π Generation expansion investment budget (e).
Hydraulics Laboratory, Universidad de Castilla-La Mancha, Ciudad Real E- G
13071,Spain(e-mail: [email protected]). σ Annual weighting factor (h).
2
tG− Timeperiodwhengeneratori∈G− isuninstalledor years.Keepingthefulldynamiccomplexityoftheprob-
i
dismantled. lem has been considered to be highly complex, mostly
Primal variables: resulting in computationally intractable problems.
u(t) Vectorof randomoruncertainparametersforyeart,
including generation capacities and loads (MW). B. Literature Review
f(t) Line k power flow for year t (MW). Transmission and generationexpansionplanninghave been
k
g(t) Power production of generating unit i for year t extensively studied areas from the beginning of power sys-
i
(MW). tems [5]. These problems have been carried out by means
p(t) Power consumption of demand j for year t (MW). of several mathematical programming techniques, such as
j
r(t) Load shed of demand j for year t (MW). stochastic programming [6], genetic algorithms [7], or Adap-
j tive Robust Optimization (ARO) [8], among others.
x(t) Binary variable representingnew line k construction
k Transmission expansion planning was first described in
at the beginning of year t.
1970 by [9]. A new strategy for transmission expansion in
x˜(t) Line k status (existing vs no existing) at the begin-
k a competitive electricity market is proposed in [10], while
ning of year t.
[11] studies the minimum load cutting problem existing in
y(t) Binary variable representing new generator i con-
k theprocessof transmissionnetworkexpansionplanningwhen
struction at the beginning of year t.
loadisuncertainwithingivenintervals.Abi-levelapproachfor
y˜(t) Generator i status (existing vs no existing) at the
k transmissionexpansionplanningwithin a marketenvironment
beginning of year t.
is proposed in [12]. In an attempt to improve computational
θn(t) Bus n voltage angle for year t (radians). tractability, several robust approaches are presented in [13],
[14],[15],[16]byusingARO.Theyprovedthatcomputational
I. INTRODUCTION tractabilityispossibleforreal-sizesystems.However,allthese
A. Motivation approaches take a wide spectrum of simplifying assumptions
byconsideringstaticandsequentialstaticmodelsinanattempt
The new objective of Kyoto Protocol for reducing Green-
to consider the year-by-year representation of investment de-
houseGases(GHG)encouragesthedevelopmentofrenewable
cisions.Incontrast,analternativeadaptiverobusttransmission
energy sources within electric systems [1]. The main reason
network expansion planning formulation is proposed by [17],
is to fight against the growing trend of worldwide average
which keeps the full dynamic complexity of the problem and
temperature and climate change, and thus, it is expected that
reaches the global optimal solution of the problem.
vastamountsofnewgenerationfacilities,speciallyrenewable,
Regarding generation expansion planning, it was first eval-
will be built in the medium-term future.
uated in 1981 by [18]. Reference [19] attempted to solve
Transmission network and renewable generation expansion
the problem by using Lagrangian relaxation and probabilistic
planning analyze the issue of how to expand or reinforce
productionsimulation.Anheuristicapproachforpowergener-
an existing power transmission network, incorporate new
ation expansion planning with emission control is introduced
renewable generation facilities and dismantling the old ones
in[20],whilea modifiedversionbasedonchance-constrained
to adequately service system loads over a given time horizon
optimization is implemented by [21]. In contrast, in [22] a
whiledecreasingGHGemissions.Thisproblemischallenging
two-levelgame-theoreticmodel is used. There are also robust
for several reasons [2]:
approaches dealing with capacity expansion planning, a two-
1) Transmission and generation investment decisions have
stagerobustoptimizationmodelconsideringtheuncertaintyin
along-standingimpactonthepowersystem asa whole.
investment costs is developed in [23].
2) Transmission and generationinvestments,specially new
Finally, joint consideration of transmission and generation
generationsources,mustbeintegratedappropriatelyinto
expansion planning under uncertainty is checked in [24] and
the existing system.
a comprehensive review is presented in [25]. Transmission
3) Consumption and renewable energy generation uncer-
andgenerationexpansionplanningunderrisk usingstochastic
tainties, such as with wind and solar power plants,
programming is provided by [26], and a market-based model
complicatetheproblemresolution.Notethatwindpower
with uncertainties is presented by [27]. A static (one period)
has been the most developed renewable technology
robust approach is modelled in [28], where electric demand
in the last decade, while the next renewable technol-
andwindpowergenerationarecontinuousuncertainvariables,
ogy, in constant evolution, is the photovoltaic power.
modelled through bounded intervals. In this case, availability
The introduction of this type of renewable sources in
ofunitsandlinesisrepresentedbymeansofuncertaindiscrete
the generation mix increases the uncertainty about the
variables, whose behaviour is modeled thorough probability
feasibility of generation. Some references attempt to
distributions.
reduce the uncertainty by means of synergies between
renewable generators, i.e. combined wind and hydro-
C. Aims and Contributions
pumpgenerators[3],orusingphysicalbilateralcontracts
[4]. The purpose of this paper is three-fold:
4) The expansion planning problem is by nature a multi- 1) To propose an effective model, which allows making
stage problem that entails planning a horizon of several optimaldecisionsin an integratedway to transformany
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power system into a sustainable system. It would allow subject to
to make decisions about where and when new trans-
1
mission lines and/or new generation facilities, such as, Π ≥ cLIx(t) (2)
L (1+I)t−1 k k
wind and photovoltaicpower plants, should be installed tX∈T kX∈L+
during a long-term time frame of several years. x˜(t) =1; ∀k ∈L,∀t∈T (3)
k
2) To extend the ARO formulation associated with the
p=t
dynamic expansion planning problem proposed by [17] x˜(t) = x(p); ∀k ∈L+,∀t∈T (4)
k k
for consideringalso the constructionand/ordismantling
p=1
of renewable and conventional generation facilities. X
x(t) ≤1; ∀k ∈L+ (5)
3) To show that computational tractability for a year-by- k
year, multi-year or multi-stage representation of in- tX∈T
vestment decisions (dynamic approach) associated with x(t) ∈{0,1}; ∀k ∈L+,∀t∈T (6)
k
transmission and generation capacities is possible for Π ≥ 1 cGIy(t) (7)
real-size systems. In addition,globaloptimality is guar- G (1+I)t−1 i i
anteed. tX∈T i∈G+X∪Gg+;∀g
y˜(t) =1; ∀i∈G,∀t∈T (8)
i
It is worth stressing that the proposed model is highly
flexible in what regards to generation capacity expansion
p=t
possibilities.Itispossibletodismantleconventionalgeneration y˜(t) = y(p); ∀i∈G+∪G+;∀g;∀t∈T; (9)
i i g
facilities reaching their lifetimes during the time period con-
p=1
X
sidered,itispossibletoconsidertheconstructionofrenewable
y(t) ≤1; ∀i∈G+∪G+;∀g (10)
generation facilities in different phases, and the inclusion of i g
t∈T
conventional facilities without uncertainties. In summary, the X
model allows to take into consideration all practical aspects yi(t) ∈{0,1}; ∀i∈G+∪Gg+;∀g;∀t∈T (11)
required to effectively transform any power system into a y˜(t) =1; ∀i∈G−,∀t=1,...,tG− (12)
i i
sustainable power system. y˜(t) =0; ∀i∈G−,∀t=tG− +1,...,N (13)
i i y
y(t) ≤y˜(t);∀i∈G+;∀g;∀t∈T (14)
i+1 i g
y(t) +y(t) ≤1;∀i∈G+;∀g;∀t∈T (15)
i+1 i g
D. Paper Structure where the objective function (1) is the sum of the present
values of cost over the time horizon, i.e. net present cost
(NPC). Equations (2)-(6) are the constraints related to the
The rest of the paper is structured as follows. Section II
construction of lines as presented in [17], which: i) limit the
describes the robust formulation of the DTNRGEP problem.
maximum expansion investment (2), ii) force the line status
The proposed decomposition method to solve the problem is
to 1 for all existing transmission lines at the beginning (3),
describedinSectionIII.SectionIVprovidesnumericalresults
and iii) once the line has been constructed(4), iv) ensure that
for two examples. Finally, in Section V relevant conclusions
no line is constructed more than once throughout the time
are drawn.
horizon considered (5), and v) establish the binary nature of
line investment decisions (6). Constraints (7)-(15) are novel
and associated with generation facilities. Constraint (7) keeps
the maximum amount of generation investment within the
available budget. Constraints (8) and (9) make the generation
II. ROBUST DYNAMIC TRANSMISSION NETWORK AND
status equal to 1 for all existing generation facilities at the
RENEWABLEGENERATION EXPANSION PLANNING
beginning of the time horizon considered which will not
FORMULATION
be dismantled, and once the generation facility has been
constructed,respectively,whileconstraint(10)ensuresthatno
generation facility is constructed more than once. Constraint
A detailed formulationofthe robustDTNRGEPproblemis (11) establishes the binary nature of generation investment
as follows: decisions. For generators to be dismantled during the study
period(∀i∈G−),constraint(12) makethe generationi status
equal to 1 until the facility is dismantled, i.e. t ≤ tG−,
i
Minimize 1 cLIx(t) while constraint (13) makes the status equal to 0 once it is
x(kt),x˜(kt),yi(t)y˜i(t) tX∈T (1+I)t−1 kX∈L+ k k dfoisrmeaancthledg,eni.eer.atti>ontGig−ro.uCpontostrbaeinctson(1s4tr)uacntedd(1in5)ceonnssuerceutthivaet
c(t) phases,theorderofconstructionissequentialaccordingtothe
+ cGIy(t)+ op ; (1)
i i (1+I) generatorsetGg+ order,thusphasei+1cannotbeconstructed
i∈G+X∪Gg+;∀g before phase i. Note that it would be straightforward to

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consider the possibility to dismantle old lines analogously to Equation(16) representsthe worst operationalcosts, which
the generation case by adapting constraints (12) and (13). maximizes generation and load-shedding costs. Constraints
Given the values of the first-stage decision variables (17)-(25) represent operational constraints such as setting
x˜(t),y˜(t), operational costs c(t) in (1) for each period t; ∀t∈ power balance, line flows, reference bus, flow and voltage
k i op
T areobtainedusingthefollowingoptimizationproblem.Note angle limits, etc. Check reference [17] for more details about
thatthedualvariablesassociatedwithconstraintsareprovided these constraints. Restriction (26) makes the level of demand
separated by a colon. match the uncertaindemandvariable.Constraint(27) is novel
and sets the power generation to be lower than the uncertain
generation capacity variable multiplied by the binary variable
c(otp) =Maximum Minimumσ cGi gi(t)+ y˜(t),whichestablishesifthecorrespondinggeneratorisactive
u(t) ∈U(t) gi(t),p(jt),rj(t), i∈G∪G+X∪G−∪Gg+;∀g foir period t. In case it is not active, i.e. y˜(t) = 0, the power
θn(t),fk(t)  generation is set to zero. Constraint (28) limi its load-shedding
to a percentage of the uncertain demand variable.
+σ cSr(t) ; (16) Constraints (29)-(34) define the polyhedral uncertainty set
j j 
j∈D analogouslyas itis donein [17].Randomgenerationcapacity
X
 uG(t) depends on binary variable zG(t), if the binary variable
subject to i i
zG(t) is1,maximumgenerationcapacityisset tothe nominal
i
g(t)− f(t)+ f(t)+ r(t) value u¯G(t) minus the maximum deviation allowed from the
i k k j i
iX∈ΨGn k|oX(k)=n k|rX(k)=n jX∈ΨDn nominalvalueuˆiG(t), otherwise maximumgenerationcapacity
= p(t) :λ(t); ∀n∈N;∀t∈T (17) is set to the nominal value u¯G(t). Analogously with uncer-
j n i
jX∈ΨDn tain demands uDj(t), although in this particular case, demand
f(t) =b x˜(t)(θ(t) −θ(t) ):φ(t); ∀k ∈L∪L+;∀t∈T nominalvaluesanddispersionareallowedtoevolveduringthe
k k k o(k) r(k) k timehorizonusingparametersh(t) andh(t).Theseparameters
(18) µ,j σ,j
allow to introduce the possible evolution of demands and
θn(t) =0:χ(nt); n:slack;∀t∈T (19) their uncertainties (see reference [17] for more details). The
f(t) ≤fmax :φˆ(t); ∀k ∈L∪L+;∀t∈T (20) level of uncertainty is controlled throughout the uncertainty
k k k budgets ΓG and ΓD, which sets the maximum number of
f(t) ≥−fmax :φˇ(t); ∀k ∈L∪L+;∀t∈T (21)
k k k generators whose maximum capacity might be different from
θ(t) ≤π :ξˆ(t); ∀n∈N\n:slack,∀t∈T (22) their nominalvalues and the maximum load levels that might
n n
θ(t) ≥−π :ξˇ(t); ∀n∈N\n:slack,∀t∈T (23) changewith respectto nominalvalues,respectively.However,
n n
unlike in reference [17] where the generation uncertainty
gi(t) ≥0; budget was constant, in this case the uncertainty budget
∀i∈G∪G+∪G+∪G−;∀g;∀t∈T (24) for each time period ΓG is a function of the number of
g
r(t) ≥0; ∀j ∈D,∀t∈T (25) active generators for each period, i.e. y˜i(t),∀i. Let remind the
j reader that the uncertaintybudget is the maximum number of
p(t) =uD(t) :αD(t); ∀j ∈D,∀t∈T (26) generatorswhosemaximumcapacityisallowedtodepartfrom
j j j
their nominal values, if the number of generator increases,
g(t) ≤uG(t)y˜(t) :ϕG(t); the uncertainty budget should increase to keep an analogous
i i i i level of protection against uncertainty. Note that what is the
∀i∈G∪G+∪G−∪G+;∀g;∀t∈T (27)
g appropriate selection of this function ΓG y˜(t),∀i is out of
rj(t) ≤e(jt)uDj(t) :ϕDj(t); ∀j ∈D,∀t∈T (28) thescopeofthepaper.Finally,constraint(3(cid:16)5)iisals(cid:17)onovelfor
uG(t) =u¯G−uˆGzG(t); this work and sets the binary variables related to generators
i i i i
to zero in case generatorsare not active at time period t, thus
∀i∈G∪G+∪G−∪G+;∀g;∀t∈T (29)
g they can not account for uncertainty budget in (31).
uD(t) =u¯Dh(t) +uˆDh(t)zD(t);
j j µ,j j σ,j j
∀j ∈D;∀t∈T (30) III. PROPOSED DECOMPOSITION METHOD
zG(t) ≤ΓG y˜(t),∀i ; ∀t∈T (31) The aim of this section is to extend the solution procedure
i i presented in [17] to solve the robust DTNRGEP problem
i∈G∪G+X∪G−∪Gg+;∀g (cid:16) (cid:17) described in Section II. Since the decomposition method has
zD(t) ≤ΓD; ∀t∈T (32) a bi-level structure, the first step is to merge the initial three-
j
j∈D level formulation (1)-(35) into a two-level problem.
X
zG(t) ∈{0,1};∀i∈G∪G+∪G−∪G+;∀g;∀t∈T (33)
i g
zjD(t) ∈{0,1};∀j ∈D;∀t∈T (34) A. Second-level formulation: subproblem
For given values for the first-stage variables x˜(t) and y˜(t)
zG(t) ≤y˜(t);∀i∈G∪G+∪G−∪G+;∀g;∀t∈T (35) k i
i i g for each time period, the problem set out by (16)-(35) might
5
be decomposed into the following single-level maximization theuncertainparametervaluesu(t) withintheuncertaintysets
problem for each time period t; ∀t∈T: to give the least desirable operational costs for each year.
The resulting formulation associated with subproblems is
c(t) = Maximize
op a mixed-integer linear programming problem, which can be
u,λ(nt),φ(kt),χ(nt),φˆ(kt), solvedefficientlybyusingstate-of-the-artmixed-integermath-
φˇ(kt),ξˆn(t),ξˇn(t),αDj(t),ϕGi (t),ϕDj(t) ematical programming solvers such as CPLEX or Gurobi.
φˆ(t)−φˇ(t) fmax+ π ξˆ(t)−ξˇ(t)
k k k n n B. First-level formulation: master problem
 kX∈L(cid:16) (cid:17) n∈NX\n:slack (cid:16) (cid:17)  First level formulation corresponds to the master problem.
 +i∈G∪G+X∪G−∪Gg+;∀g(cid:16)uGi (t)y˜i(t)ϕGi (t)(cid:17)  UTh(tu)s,foforregaicvhenyveaalrueosbotafintheedufnrocemrtasiencopnarda-mleevteelrsvuabluperosbul(etm) ∈s,
+ uD(t)αD(t)+e(t)uD(t)ϕD(t) the master problem at iteration ν corresponds to:
j j j j j
 jX∈D(cid:16) (cid:17) (36) Minimize (1+1I)t−1 cLkIx(kt)
subject to: x(kt),x˜(kt),yk(t),y˜k(t), tX∈T kX∈L+
g(t),p(t),r(t),θ(t),f(t)
λ(nt()i)+ϕGi (t) ≤ (1+σ I)cGi ; i,ll =j,1l,.j.,.l,νn−,l1k,l +i∈G+X∪Gg+;∀gcGi Iyi(t)+γ(t);
∀i∈G∪G+∪G−∪Gg+;∀g;∀t∈T (37) (54)
−λ(t) +αD(t) ≤0; ∀j ∈D;∀t∈T (38)
n(j) j subject to
σ
λ(nt()j)+ϕDj(t) ≤ (1+I)cSj; ∀j ∈D;∀t∈T (39) γ(t) ≥ (1+σ I) cGi gi(,tl)+ (1+σ I) cSjrj(,tl);
−λ(t) +λ(t) +φ(t)+φˆ(t)+φˇ(t) =0; i∈GX∪G+ jX∈D
o(k) r(k) k k k ∪G−∪G+;∀g
∀k ∈L∪L+;∀t∈T (40) g
∀t∈T,l =1,...,ν−1 (55)
− b x˜(t)φ(t)+ b x˜(t)φ(t)
k k k k k k γ(t) ≥0; ∀t∈T (56)
k|oX(k)=n k|rX(k)=n
+ξˆ(t)+ξˇ(t) =0; ∀n∈N\n:slack;∀t∈T (41) Constraints (2)−(15) (57)
n n
Equations (17)−(28); l=1,...,ν−1. (58)
− b x˜(t)φ(t)+ b x˜(t)φ(t)
k k k k k k
k|oX(k)=n k|rX(k)=n Constraint (55) are primal decomposition cuts, while vari-
+χ(t) =0; n:slack;∀t∈T (42) ablesγ(t) relatestoyearonyearoperationalcosts.Themaster
n problemincludesonevariableg(t),p(t), r(t), θ(t) andf(t) for
−∞≤λ(t) ≤∞; ∀n∈N;∀t∈T (43) i,l j,l j,l n,l k,l
n each year and for each realizationof the uncertainparameters
−∞≤φ(t) ≤∞; ∀k ∈L∪L+;∀t∈T (44) obtained from the subproblem (36)-(53) at every iteration.
k
−∞≤χ(t) ≤∞; n:slack;∀t∈T (45)
n C. Solution method
φˆ(t) ≤0; ∀k ∈L∪L+;∀t∈T (46)
k The solution method consists of iteratively solving the
φˇ(t) ≥0; ∀k ∈L∪L+;∀t∈T (47) following master and subproblems at each iteration ν:
k
ξˆ(t) ≤0; ∀n∈N\n:slack;∀t∈T (48) Master problem: For given realizations of the uncer-
n
tain parameters obtained from subproblems at the pre-
ξˇn(t) ≥0; ∀n∈N\n:slack;∀t∈T (49) vious iterations, new first-stage variables x˜(t), y˜(t) val-
−∞≤αD(t) ≤∞; ∀j ∈D;∀t∈T (50) ues are calculated by means of (54)-(58k).,νThie,νopti-
j
ϕG(t) ≤0; ∀i∈G∪G+∪G−∪G+;∀g;∀t∈T (51) mal objective function lower bound is updated z(lo) =
i g 1 ( cLIx(t) + cGIy(t)+
ϕDj(t) ≤0; ∀j ∈D;∀t∈T (52) γ(tt)∈)T. (1+I)t−1 k∈L+ k k,ν i∈G+∪Gg+;∀g i i,ν
P P P
Constraints (29)-(35) (53) Subproblems, oneforeachyear:Forgivenvaluesofthe
first-stage decision variables x˜(t), y˜(t), uncertain parame-
Subproblems (36)-(53) result from substituting in problem k,ν i,ν
ters within the uncertainty set giving the least desirable
(16)-(35) for each time period t, the third-level problem
operational costs (16), i.e. u(t) and c(t) , respectively,
by its dual. An important aspect for the resolution of sub- ν op,ν
are calculated by solving subproblems in (36)-(53). The
problems is the linearization of bilinear terms in (36), i.e.,
optimalobjectivefunctionupperboundisupdatedz(up) =
Pe(jti)∈uGDj∪(tG)+ϕ∪DjG(−t)∪G.g+T;∀hge(cid:16)ulGiin(et)ay˜ri(izt)aϕtiGio(nt)(cid:17)pr+ocPesjs∈Dis(cid:16)udeDjs(ct)rαibDje(dt)i+n Pc(otpt),∈νT).(1+I1)t−1(Pk∈L+cLkIx(kt,)ν+Pi∈G+∪Gg+;∀gcGi Iyi(,tν)+
more detail in(cid:17)[16]. Note that variable y˜(t) is considered a This iterative process is repeated until the absolute value
i
parameterwithinoursubproblem.These subproblemsprovide of the relative difference between upper and lower bounds is
6
belowaselectedthreshold.Formoredetailsaboutthestructure Existing lines (cid:9262) Existing generators
of the algorithm, applied just for the capacity expansion Prospective lines
Prospective generators
problem, check reference [17].
The advantage of this bi-level formulation given by (36)- 240 MW 100 MW 80 M W 15 0 M(cid:9262)W 50 MW 40 MW
(53) and (54)-(58) is that it has the same problem structure bus 5 bus 1
than that defined by [29], and thereforethe proposedcolumn-
70 MW
and-constraintgenerationmethodguaranteesconvergencetoa
global optimum.
350 MW
(cid:36382)
bus 3
IV. EXAMPLES
40 MW
240 MW
In this section, the numerical results for an illustrative
bus 2
example based on Garver system [9] and a case study using
theIEEE118-bustestsystem[30]areportrayedtoanalyzethe
150 MW
joint study of transmission network and generation expansion
planning.
Allnumericaltestshavebeenimplementedandsolvedusing
bus 6 bus 4
CPLEX within GAMS [31] on a Windows DELL PowerEdge
(cid:36382) 600 MW 160 MW 200 MW
R920serverwith two IntelXeonE7−4820processorsclock-
ing at 2 GHz and 768 Gb of RAM. The stopping tolerance
Fig.1. Garver’s6-bustestsystem.
for all cases is equal to ε=10−6.
TABLEI
A. Illustrative Example. Garver’s 6-bus System
CANDIDATERENEWABLEGENERATORSRELATEDTOGARVER’S6-BUS
TESTSYSTEM.
The model is initially tested in the Garver 6-bus system
PowerCapacity O&MCosts Investment cost
depicted in Figure 1. This system is composed of 6 buses, Gen. Bus u¯Gi (MW) (e/MWh) (Me)
3 generators, 5 levels of inelastic demand and 6 lines. Data 4 1 50 17.8 50
forgenerationanddemandcapacities,andsupplyandbidding 5 1 70 17.5 80
prices are given in [16]. The load-shedding cost is equal to 6 1 40 17.5 40
7 2 150 16.5 200
hundred times the bidding price for each level of demand. 8 4 200 15.0 198
LinedataareobtainedfromTableIofreference[12],including 9 5 100 17.0 110
construction costs.
Regarding the uncertainty sets, the maximum capacity of
Regarding expansion possibilities, 6 generation units and
conventional generators can decrease a maximum of 50%
3 transmission lines between each pair of buses could be
installed. The characteristics of possible generation units, with respect to their nominal values, i.e. uˆGi = 0.5u¯Gi ; i =
which are assumed to be wind units, are shown in Table I. 1,2,3,whileforrenewablegeneratorstheirmaximumcapacity
can decrease a 100% with respect to their nominal values,
The first three generation units belong to one group (g = 1)
i.e. uˆG = u¯G; i = 4,...,9. Load levels may change a
tobeconstructedandinstalledinphases(atdifferenttimes)at i i
maximum of 20% with respect to their nominal values, i.e.
the same bus 1, so in case the model decides to include them
in the generation expansion planning they must be installed uˆDj = 0.2u¯Dj ; j = 1,...,5. Finally, annual growth rates for
sequentially, i.e. G+ ≡{4,5,6}. In addition, it is known that load nominal values and dispersion are equal to 1.2%, i.e.
the existing genera1tor at node 1 is going to be dismantled at h(µt,)j =h(σt,)j =1.012(t−1).
time period8 becauseitreachesitsusefullife, i.e. G− ≡{1}. Four case studies are analyzed, considering two different
The rest of sets associated with generation are G ≡ {2,3} combinations of uncertainty budgets associated with gener-
and G+ ≡ {7,8,9}. Operational and investment costs for ation capacities and demands and two different generation
generatorspresentedinTableIcomefrom[3].Inwhatregards investment budgets. It is worth stressing that the inclusion
to the characteristics of the new possible transmission lines, of new generators implies updating the uncertainty budget
they are also attained from Table I of reference [12]. The associatedwithgeneration,weusethefollowingstepfunction
maximum available investment budget for transmission lines to define ΓG y˜(t),∀i : if 1 or 2 generators are built, the
i
is 40 million euros. generation unc(cid:16)ertainty(cid:17)budget increases in one unit; if 3 or
The time horizon considered is 25 years and the discount 4 generatorsare built, the uncertaintybudgetincreases in two
rate is 10%. The weighted factor σ is equal to the number units;andfinally,if4or5generatorsarebuilt,theuncertainty
of hours in one year, i.e. 8760, so that the load-shedding budget increases in three units.
and power generation costs are related to years, which can Results about investment cost, and lines and generators
be comparable with the annualized investment cost. built for each case study using the proposed model are given
7
in Table II. From this table the following observations are TABLEII
pertinent: RESULTSFORGARVER’S6-BUSTESTSYSTEMILLUSTRATIVEEXAMPLE.
• For the first combination of uncertainty budgets, ΓG and Case Input Inv.Cost Newlines Newgenerators
ΓD equal to 1 and 2, lines and generators built are the study ΓG,ΓD,ΠG(Me) (Me) From To Period Bus Period
1 5 5 4 8
same.Notethatthecandidategeneratorthatremainstobe
2 3 1 5 3
builthasaninvestmentcostof200millioneuros,thusthe 2 6 1 1 1
a) 1,2,350 384.802
rise considered in the generation investment budget does 2 6 8 1 2
2 6 1 1 3
not allow the constructionof the remaining generatorand
3 5 1 - -
no more lines are needed. 4 6 1 - -
• Totalcostfor casesa) and c) is 14627.918and64927.727 4 6 1 - -
1 5 5 4 8
million euros, respectively. Therefore, the higher uncer-
2 3 1 5 3
tainty is, the higher total cost is. Note that lines are built 2 6 1 1 1
b) 1,2,450 384.802
in earlier periods when the total cost is higher. 2 6 8 1 2
2 6 1 1 3
• Total cost for case studies b) and d) is 14627.918 and
3 5 1 - -
63081.771 million euros, respectively. Note that the con- 4 6 1 - -
figuration of built generators changes and lines are built 4 6 1 - -
1 5 1 4 8
early in d) due to the higher cost.
2 6 2 5 1
• Incasestudiesc)andd),theincreaseininvestmentbudget 2 6 1 1 1
c) 2,3,350 385.190
for generators allows to build one more line or more 2 6 1 1 4
3 5 1 1 5
generators.
4 6 1 - -
• It is worth stressing the complexity of this problem. For 4 6 1 - -
thesameuncertaintybudgets,thesolutionsassociatedwith 1 5 1 2 2
different investment budgets are not incremental. Thus 2 3 2 5 1
2 6 6 1 1
the necessity to consider the year-by-year dynamic of the d) 2,3,450 475.124 2 6 1 1 2
problem. 2 6 1 1 7
3 5 1 - -
Regarding computational tractability, the number of itera- 4 6 1 - -
tion required are 16, 15, 10 and 16, respectively, for study 4 6 1 - -
casesa), b),c) andd).Themaximumcomputingtime iseight
hours for the case a).
TABLEIII
CANDIDATEGENERATORSRELATEDTOIEEE118-BUSTESTSYSTEM
B. IEEE 118-bus test system example EXAMPLE.
Bus PowerCapacity O&MCost Investmentcost
We also apply the proposed model on a bigger and more Number
(MW) (e/MWh) (Me)
realistic case using the IEEE 118-bus test system [30]. The 55 1 90 15.2 120
system is composed of 118 buses, 186 existing lines, 54 56 4 50 17.8 50
generatorsand91loads.We assumethatthegeneratorlocated 57 4 70 17.5 80
58 4 40 17.5 40
at bus 4 stops working at period 8 because it reaches its
59 6 100 17.1 110
usefullife,,i.e.G− ≡{4}.Generationcapacitiesanddemand 60 10 180 15.3 184
loads can be found in [16]. The load-shedding cost is ten 61 14 100 17.0 110
62 14 90 15.2 120
times the bidding price of each level of demand. In addition,
63 18 150 16.0 145
the same 61 existing lines given in [16] can be duplicated 64 20 50 17.6 50
to build additional lines. Data for all lines are taken from 65 20 50 17.6 50
[30].Thecharacteristicsofnewpossiblegeneratorsareshown 66 20 60 15.4 55
67 21 130 16.5 135
in Table III and there are two groups of generators to be
68 22 200 15.0 198
installedsequentiallyatbuses4and20,i.e.G1+ ≡{56,57,58} 69 27 80 15.9 90
and G+ ≡ {64,65,66}. The investment budgets for the 70 38 110 16.7 123
genera2tors and transmission lines are 1500 and 100 million 71 39 200 15.1 200
72 50 90 17.0 118
euros, respectively. The discount rate is 10% and the time 73 51 150 16.6 153
horizon is 25 years. 74 62 110 16.8 103
Due to uncertainty, conventional generators can decrease 75 75 110 16.9 147
76 80 170 16.0 164
a 50% with respect to their nominal values, i.e. uˆG =
i 77 88 200 14.9 198
0.5u¯Gi ; i = 1,...,54, while for renewable generators their 78 93 100 17.0 110
maximum capacity can decrease a 100% with respect to their 79 94 200 15.0 198
nominalvalues,i.e. uˆG =u¯G; i=55,...,84. Demandlevels 80 96 140 16.3 158
i i 81 101 170 15.2 180
maychangeamaximumof50%withrespecttotheirnominal 82 114 190 15.5 191
values. Annual growth rates for load nominal values and 83 116 110 16.6 112
dispersion are equal to 1.2%, i.e. h(t) =h(t) =1.012(t−1). 84 118 90 17.3 102
µ,j σ,j
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