Unit Commitment with N-1 Security and Wind Uncertainty

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by  Kaarthik Sundar

Unit Commitment with N-1 Security and Wind Uncertainty
Kaarthik Sundar‡, Harsha Nagarajan†,MilesLubin¶, LineRoald§, Sidhant Misra†, Russell Bent†, DanielBienstock$
† LosAlamos NationalLaboratory, LosAlamos, NM,UnitedStates.
‡ Department ofMechanical Engineering, Texas A&MUniversity, CollegeStation,TX, UnitedStates.
¶ Operations Research Center, Massachusetts InstituteofTechnology, Cambridge, MA,UnitedStates.
§ Power Systems Laboratory, Department of Electrical Engineering, ETH Zurich, Switzerland.
$ Department ofIndustrial Engineering and Operations Research and
Department of AppliedPhysics andApplied Mathematics,Columbia University, NY, UnitedStates.
Abstract—As renewable wind energy penetration rates continue to proposed. In [6], a stochastic UC variant was considered where
increase,oneofthemajorchallengesfacinggridoperatorsisthequestion the authors developed a two-stage dual-decomposition algorithm,
6
of how to control transmission grids in a reliable and a cost-efficient
1 accounting for wind via a scenario-based approach. The approach
manner.Thestochasticnatureofwindforcesanalterationoftraditional
0 methods for solving day-ahead and look-ahead unit commitment and was validated on a 225 bus model of the California power system;
2 dispatch.Inparticular,uncontrollablewindgenerationincreasestherisk the average computation time for a solving a 42-scenario model
of random component failures. To address these questions, we present approximately 6 hours. A variation of this problem was studied in
n
an N-1 Security and Chance-Constrained Unit Commitment (SCCUC) [7], [8], where the authors developed a sampling-based approach to
a
that includes the modeling of generation reserves that respond to wind
J account for wind uncertainty via a chance-constrained formulation
fluctuationsandtertiaryreservestoaccountforsinglecomponentoutages.
0 The basic formulation is reformulated as a mixed-integer second-order (without [7] and with [8] generation reserves modeling) and further
3 coneproblemtolimittheprobabilityoffailure.Wedevelopthreedifferent corroborated theiralgorithms using Monte Carlosimulations against
algorithmstosolvetheproblemtooptimalityandpresentadetailedcase their deterministic variant on a modified IEEE 118-bus network and
study on the IEEE RTS-96 single area system. The case study assesses
] the IEEE 30-bus network respectively. Reference [9] developed a
Y the economic impacts due to contingencies and various degrees of wind
transmission-constrained UC formulation where the uncertainty is
powerpenetrationintothesystemandalsocorroboratestheeffectiveness
S ofthe algorithms. modeled using an interval formulation. They test their improved
. interval formulation against existing stochastic, interval and robust
s Index Terms—Unit commitment, wind uncertainty, security con-
c straints, mixed-integer second-order conic programs UCtechniques intermsof solutionrobustness and cost ontheIEEE
[ RTS-96 test system. In contrast to this paper, none of these papers

I. INTRODUCTION consider N-1securityconstraints andtheassociated tertiaryreserves
1
Transmission grids play a vital role in the supply and the deliv- modeling.
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9 ery of electric power. As renewable wind energy penetration rates Within the literature, the number of papers that do consider
7 continuetogrow,reliableandcost-efficientoperationoftransmission contingency modelingislimited.Forinstance, in[5],authors solved
0 gridsbecomesincreasinglyimportant.However,thestochasticnature a simpler OPF problem (without UC modeling) using a standard
0 of wind power necessitates an alteration of traditional methods sampling-methodtoaccountforwindandN-1securityconstraints.In
0 for solving day-ahead and look-ahead Unit Commitment (UC) and termsofmodelcomprehensiveness,themodelmostsimilartooursis
2. generation dispatch. The fluctuations caused by uncontrolled wind that of [10]. They developed a two-stage adaptive robust UC model
0 generation can bring system components closer to their physical with security constraints and nodal net injection uncertainty. The
6 limits, making generator and line outages more likely to occur. As modelincludesadeterministicuncertaintyset,unliketheprobability
1 a result, power system operators are interested in securing the grid distribution of this paper, to model the wind. They proposed a
: against component failures in the presence of these resources. Benders-baseddecomposition algorithmtohandlelinecontingencies
v
i More formally, the security of a power grid refers to its ability over a real-world large scale system, however they do not consider
X to survive contingencies, while avoiding disruption of service [1]. generator contingencies. It is the combination of both types of
r The failure to secure a power system could potentially result in contingencies,especiallythegeneratorcontingencies, thatmakesour
a
cascading events [2]. The concept of N-1 security assessment was problemmoredifficulttosolve.Giventhecomplexityofourproblem
developed to quantify this notion of security (see [3] and references resulting from the size of the network (24 buses) and N-1 security
therein). A power system is N-1 secure if it can survive all single constraints on lines and generators for each one-hour time step
component outages. While N-1 is an important security criteria, an over 24 hours, our algorithm compares favorably from a scalability
N-1 secured power system still faces the risk of cascading events perspective.
if one does not take into account the uncertain deviations of wind In this paper, we formulate the SCCUC as a large Mixed-Integer
from its forecast value. One approach for controlling risk is chance Second-Order ConeProgram(MISOCP),leveraging therecent work
constraints, where an upper bound on the probability of constraint in [4], [11]. This formulation models two kinds of reserves, namely
violation is included in the OPF [4], [5]. Here, we adopt the latter thegeneratorreservesandthetertiary/spinningreservestoaccountfor
approach and develop a comprehensive model that incorporates all thewindfluctuationsandsinglelineorgeneratoroutage,respectively.
aspects of day-ahead planning and security discussed above - UC We develop three approaches to solve the MISOCP with each ap-
for generators, N-1 security constraints for a line or a generator proachbuildingonitspredecessor.Thefirstalgorithmisatraditional
outage, chance constraints to ensure reliability with respect to wind sequential linear outer approximation for the Second-Order Cone
inastochasticsense,reservesfromgeneratorsandtertiary(spinning) (SOC)constraints [4]. The second algorithm, the scenario-based de-
reserves. composition, inaddition tousingthesequential outer approximation
Literature review: We review briefly some of the variations of technique,exploitsablockdiagonalstructureoftheconstraintmatrix
the model explored in the literature and the algorithms that were to decompose the formulation. The third algorithm is a Benders-
like algorithm incorporating the classical Benders feasibility cuts to pup,init - time i has been on before t=0, hrs
i
invalidate solutions tosmall SOC feasibilitysubproblems. Extensive pdown,init - time i has been on before t=0, hrs
i
computationalexperimentationbasedonthesingleareaIEEERTS-96 pon-off - on-off status of i at t=0 (1 if pup,init>0, 0 otherwise)
i i
systemisusedtocomparetheeffectivenessofeachoftheapproaches. pi(0) - power output of generator i at t=0, MW
We also make a detailed comparison of the SCCUC formulation to RU - ramp-up limit of i, MW/hr
i
the deterministic variant of the problem to illustrate the advantages RD - ramp-down limit of i, MW/hr
i
of solving the SCCUC over its deterministic counterpart. c - cost of block s of stepwise start-up cost function of i, $
is
T - upper limit of block s of the stepwise start-up cost of i, hrs
is
II. SCCUCFORMULATION T - upper limit of block s of the stepwise start-up cost of i, hrs
is
Throughout this paper we utilize the linearized DC power flow µb(t) - constant forecast output of wind farm at bus b for hour t,
model. While this model has its limitations, it is the model most MW
generallyusedintheunitcommitmentliterature.Inordertoquantify ωb(t) - actual wind deviations from forecast µb(t), at hour t
theprimarycomplexitiesandbenefitsofthispaper’scorecontribution r - index of the reference bus
(chance constrained response to wind fluctuations) we isolate this R - bounds on the amount of reserves that can be purchased, MW
modification to the traditional unit commitment model. Future work B - bus admittance matrix for the network
will consider more realistic power flow models. Bc - bus admittance matrix for the network under contingency c
Intherestofthearticle,boldsymbolsdenoterandomvariables.In
A. Nomenclature
particular,ωb(t)istherandomvariablethatmodelsωb(t)forhourt.
(i) Sets: IntheSCCUC,weassumethatthedeviationsωb(t)areindependent
B - set of buses, indexed by b and normally distributed with zero mean and variance σb(t)2 (see
L - set of lines, indexed by ℓ [4]).Weassumethatωb(t)arenotcorrelatedintimeoracrossspace
G - set of all generators, indexed by i geographically, however, thisassumption can be relaxed to a certain
Gb - subset of generators located at bus b degreew.l.o.g.=Thesewinddeviationsdrivetherandomfluctuations
S - set of generating units’ start-up cost blocks, indexed by s in the controllable generator injectors p (t), and line flows f (t).
i ℓ
K - set of generating units’ production cost blocks, indexed by k Finally, we let Ω(t) = ω(t) denote the total deviation
W - set of buses with wind farms in the wind from the forecPastb.∈WFor notional convenience, we use
T - set of discretized times (hours), indexed by t p¯(t), µ¯(t), d¯(t), ω¯(t), δ¯c(t), α¯(t), α¯c(t), r¯+(t), r¯−(t), r¯up(t) to
C - set of contingencies, indexed by c denote the vector of power generation, constant wind forecast, wind
(ii) Binary decision variables: deviations, additional generation during contingencies, participation
xi(t) - generator on-off status at time t factorsofthecontrollablegenerators,theparticipationfactorsduring
yi(t) - generator 0-1 start-up status contingencies, generationupanddownreservesandtertiaryreserves
zi(t) - generator 0-1 shut-down status respectively.
wis(t)-generator start-upblockidentification;1ifiisstartedupat
the beginning of hour t after being down for s hours, 0 otherwise
B. Generation control
(iii) Continuous decision variables:
sci(t) - start-up cost of i at hour t, $ As mentioned previously, we assume that the random wind de-
pi(t) - power output of i for hour t, MW viations drive the controllable generator injections during each time
r+(t) - generation up reserve power output of i at hour t, MW period. Thus, the controllable generators respond proportionally to
i
r−(t) - generation down reserve power output of i at hour t, MW the wind fluctuations [4], [11] as
i
rup(t) - tertiary reserve power output of i at hour t, MW
giik(t) - power output on segment k of cost curve of i at t, MW pi(t)=pi(t)−αi(t)Ω(t). (1)
αi(t) - participation factor of i at hour t
fℓ(t) - real power flow over line ℓ at hour t, MW Here, αi(t) ≥ 0 is the participation factor for the controllable
δc(t) - power provided by i for contingency c at hour t, MW generator i. It was shown in [4] that when Piαi(t) = 1, Eq. (1)
αic(t) - participation factor of i for contingency c at hour t guarantees balance of generation and load for every time period t.
i
fc(t)- realpower flow over lineℓfor contingency cathour t,MW
ℓ
(iv) Parameters: C. Line flows
β - susceptance of line ℓ
ℓ The random fluctuations in line flows f (t) for the line ℓ depend
pmin - minimum output of generator i, MW ℓ
i on the wind fluctuations implicitly, through the random bus angles
pmax - maximum output of generator i, MW
pmi ax -maximumpoweroutputofiinproductioncostblockk,MW θb(t) which satisfy
i,k
db(t) - demand at bus b for hour t, MW p¯(t)+µ¯(t)−d¯(t)+ω¯(t)−Ω(t)α¯(t)=Bθ(t). (2)
a0 - no-load cost of the generator i, $
i
a1 - linear cost coefficient for r+ and r− for i, $/MW The bus admittance matrix B is invertible after removing the row
i i i
a2 - linear cost coefficient for rup for i, $/MW andcolumncorrespondingtothereferencebusr∈B.Followingthe
i i
Kk - slope of the kth segment of the cost curve for i, $/MW DC power flow model [12], the flows f (t) are a linear function of
i ℓ
fmax - capacity of line ℓ, MW thebusangles,hencewedenotethe|L|×|B|matrixM asthelinear
ℓ
L - min. time i has to run from the start of planning horizon, hrs mapfrompowerinjectionstolineflows.Thentherandomlineflows
i
L -min.timeihastobeofffromthestartofplanninghorizon, hrs for each line ℓ are computed as:
i
UT - minimum up-time of i, hrs
i f (t)=M p¯(t)+µ¯(t)−d¯(t)+ω¯(t)−Ω(t)α¯(t) . (3)
DTi - minimum down-time of i, hrs ℓ (ℓ,·)(cid:0) (cid:1)
D. Reserve generation 2) Generation limits:
resIenrvethse(rS+C(tC)U,rC−,(tw))eanmdotdheeltetwrtioarytyrpeesservoefsr(ersuepr(vte)s):.Tgheenegreantieorn- pmi in·xi(t)≤pi(t)≤pmi ax·xi(t) ∀i∈G,t∈T, (10)
i i i
ationreservesareusedtorespondtowindfluctuationsandthetertiary 0≤ri−(t),ri+(t),riup(t)≤R·xi(t) ∀i∈G,t∈T, (11)
reserves are used to respond to generator outages. The linear cost
coefficients for purchasing generation reserves and tertiary reserves δc(t)≤rup(t) ∀i∈G,t∈T,c∈C, (12)
i i
from a generator i at time is given by a1 and a2, respectively.
i i
E. Post-contingency generation outputs Xrnup(t)≥pi(t) ∀i∈G,t∈T, (13)
n∈G
The generation output of the generator i after the outage of a
generator c during hour t is modeled as pi(t)−ri−(t)≥pmi in·xi(t) ∀i∈G,t∈T, (14)
pci(t)=pi(t)−αci(t)Ω(t)+δic(t) (4) pi(t)+ri+(t)+riup(t)≤pmi ax·xi(t) ∀i∈G,t∈T, (15)
gwehneerrea,toαrci(itc)o≥rre0spisontdhiengnetwopthaertiocuiptaagtieonoffagcetnoerrfaotorrthceacnodntprocl(lta)blies Pr(ri−(t)≥Ω(t)αi(t))≥1−εi ∀i∈G,t∈T, (16)
i
tbhaelarnacnedowme egnefnoerrcaetor injection during the outage. To ensure power Pr(ri+(t)≥−Ω(t)αi(t))≥1−εi ∀i∈G,t∈T, (17)
Xαci(t)=1, Xδic(t)=0, and δcc(t)=−pc(t) (5) Pr(ri−(t)≥Ω(t)αci(t))≥1−εi ∀i∈G,t∈T,c∈C, (18)
i i
For an outage of line c during time t, the participation factors do Pr(ri+(t)≥−Ω(t)αci(t))≥1−εi ∀i∈G,t∈T,c∈C, (19)
not change i.e.,αci(t)=αi(t) and δic(t)=0for all thecontrollable 0≤αi(t),αci(t)≤xi(t) ∀i∈G,t∈T,c∈C. (20)
generators i.
The constraints in (10) – (15) enforce the generation limits and
F. Post-contingency line flows the reserve capacity limits for the generator i at every hour t. In
The effect of a line or generator outage c changes the topology particular,constraint(13)ensuresthatthetotaltertiaryreservesfrom
ofthesystem, whichisrepresentedbythematrixM definedinSec. allthegeneratorsduringanhourtmustbegreaterthanthemaximum
II-C. Let Mc denote the matrix M corresponding to the topology power generated by any generator during that hour. This guarantees
after an outage c. For generator outages, we have M =Mc. Using thatenough tertiaryreservesarepurchased ateachhour tocover for
these notations, we model the line flow during a contingency c as any generator outage. The constraints in (16) – (19) are the chance
follows: constraints on the generation limits. They ensure that generation
reserves respond to wind fluctuations feasibly with high probability
fcℓ(t)=M(cℓ,·)(cid:0)p¯(t)+δ¯c(t)+µ¯(t)−d¯(t)+ω¯(t)−Ω(t)α¯c(t)(cid:1) bothduringnormaloperationandcontingencies.Theconstraints(20)
(6) impose the bounds on the participation factors.
where,fc(t)istherandomlineflowonthelineℓduringcontingency 3) Piecewise linear production cost of the generators:
ℓ
c at time t. pi(t)= Xgik(t) ∀i∈G,t∈T, (21)
G. Optimization problem k∈K
WiththenotationsandmodelingconsiderationsinSec.II-A–II-F, 0≤gik(t)≤pmi,kax·xi(t) ∀i∈G,t∈T,k∈K. (22)
we present a formulation of the SCCUC. The objective function of
Constraint (21) definesthe power generated by each generator iand
the SCCUC minimizes the operating cost of the generators which
at each hour t as the sum of power generated on each block of the
includes the no-load cost, start-up cost, the running cost of all the
production cost curve and the constraint (22) enforces the limits on
generators and the cost of the generation and tertiary reserves, i.e.
the power generated on each block.
minXX(cid:8)a0i ·xi(t)+XKik·gik(t)+sci(t)+ 4) Stepwise start-up cost of the generators:
i∈Gt∈T k∈K (7) Xwis(t)=yi(t) ∀i∈G,t∈T, (23)
(cid:2)a1i ·(ri+(t)+ri−(t))+a2i ·riup(t)(cid:3)(cid:9) s∈S
The choice of the objective function is motivated by the models in Tis
[9], [13], [14]. The optimization issubject toconstraints (1)-(6) and wis(t)≤Xzi(t−s) ∀i∈G,t∈T,s∈S, (24)
the following constraints: Tis
1) Binary variable logic:
sci(t)=Xwis(t)·cis ∀i∈G,t∈T. (25)
yi(t)−zi(t)=xi(t)−xi(t−1) ∀t∈T,i∈G, (8) s∈S
The start-up cost for a generator i varies with the number of
yi(t)+zi(t)≤1 ∀t∈T,i∈G. (9)
consecutive time periods i has been off before it is started up.
Constraint(8)determinesifthegeneratorisstarteduporshutdownat Constraint (23) ensures exactly one start-up cost from the set of
hourtbasedofitson-offstatusbetweenhourtandt−1.Constraint start-up cost blocks is chosen for the generator i. Constraint (24)
(9) ensures that a generator i is not started up and shut down in the identifies the appropriate start-up block by implicitly counting the
same hour t. number of consecutive time periods the generator has been in the
off state. Finally, constraint (25) selects the actual start-up cost that theproblem.Eachalgorithmpresentedinthissectiondiffersinsteps
shows up in the objective function. (i) and (iii). All algorithms are implemented within a single branch-
5) Minimum up and down time, ramping: and-bound tree by using solver callbacks.
xi(t)=poin-off ∀i∈G,t≤Li+Li, (26) A. Outer approximation
t The formulation in Sec. II contains chance constraints (Eq. (35)–
Xyi(n)≤xi(t) ∀i∈G,t≥Li,t=t−UTi+1, (27) (38)) that can be reformulated as Second-Order Cone (SOC) con-
n=t straints. The chance constraints in Eq. (16)–(19) are a special case
wherethereformulationislinear(see[11]).Whilethisreformulation
t
Xzi(n)≤1−xi(t) ∀i∈G,t≥Li,t=t−DTi+1, (28) ofchance constraintsasSOCconstraintsisuseful,[4]alsoobserved
that off-the-shelf commercial solvers were not able to handle large
n=t
scaleCCOPFinstancesi.e.,continuousSOCproblems.So,toaddress
RDi ≥pi(t−1)−pi(t) ∀i∈G,t∈T, (29) thisissue, weuse the following approach: weomit the reformulated
SOCconstraintscorrespondingtoEq.(35)–(38)fromtheformulation
RUi ≥pi(t)−pi(t−1) ∀i∈G,t∈T. (30) we provide to the solver. Whenever the solver obtains an integer
feasible solution to this relaxed problem, we check if it satisfies
The constraint in Eq. (26) sets the on-off status of the generator
all the SOC constraints that were ignored. If not, we add a linear
i based on the initial conditions. Notice that both L and L will
i i outer approximation of the infeasible SOC constraint and continue
not take positive values simultaneously. The constraints inEqs. (27)
solving the original problem. This process of adding linear outer
and (28) enforce the minimum up time and minimum down time
approximations of violated SOC constraints sequentially has been
constraints for generator i for the remaining time intervals of the
observed to be computationally efficient for the CCOPF and robust
planning horizon. Theconstraints(29) and(30)enforce theramping
CCOPF problems [4], [11].
limitson consecutive periods on every generator i.
Weprovidedtheformulation,asstatedinSec.II-G,tothemodeling
6) Power flow:
tool JuMPChance [15], which enables the user to select between
Xαci(t)=1,αcr(t)=0 ∀t∈T,c∈C, (31) solutions via sequential outer approximation or via reformulation to
i∈G an SOC problem. Despite using sequential outer approximations to
address the issue of the SOC constraints, solving the full problem,
Xαi(t)=1,αr(t)=0 ∀t∈T, (32) as is, can be time consuming even for small SCCUC instances. We
i∈G also observe that the constraints of the SCCUC formulation have
an inherent block-diagonal structure witha few coupling constraints
X(pb(t)+µb(t)−db(t))=0 ∀t∈T, (33) that can be exploited to develop more efficient exact algorithms. In
b∈B the following section, we discuss a modified version of a traditional
scenario-baseddecompositionalgorithmthatexploitsthisblockdiag-
Xδic(t)=0 and δcc(t)=−pc(t) ∀t∈T,c∈C, (34) onal structure of the constraint matrix to compute optimal solutions
i∈G to the SCCUC.
Pr(fℓ(t)≤fmax)>1−εℓ ∀ℓ∈L,t∈T, (35)
B. Scenario-based decomposition
Pr(fℓ(t)≥−fmax)>1−εℓ ∀ℓ∈L,t∈T, (36) Inthissection, wepresent ascenario-based decomposition (SBD)
approach tosolve theSCCUC.Thisalgorithm isan improvement to
Pr(fc(t)≤fmax)>1−εc ∀ℓ∈L,t∈T,c∈C, (37)
ℓ ℓ theouterapproximationalgorithm.WehandletheSOCconstraintsin
exactlythesamewayasfortheouterapproximation.Inaddition,we
Pr(fc(t)≥−fmax)>1−εc ∀ℓ∈L,t∈T,c∈C. (38)
ℓ ℓ alsoleaveout constraintscorresponding toasubset ofcontingencies
Thepowerflowequationsareadaptedfrom[4],[11]formultipletime C1 and solve the relaxed problem. Whenever the solver obtains a
periods.Constraints(31)–(32)aretheconstraintsontheparticipation feasiblesolutiontotherelaxedproblem,inadditiontocheckingifall
factors that guarantees balance of generation and load (Sec. II-B theSOCconstraintsaresatisfiedbythecurrentfeasibledispatch,we
and II-E). Constraints (33) and (34) impose the demand-generation alsocheckifthedispatchviolatesanyofthecontingenciesinC1.The
balance during normal operations and contingencies for all time violatedSOCconstraintsareaddedaslinearouterapproximationcuts
periods.Finally,constraints(35)–(38)arethechanceconstraintsfor and the constraints corresponding to the infeasible contingencies in
the line flows during normal operations and during contingencies. setC1aredirectlyaddedtotherelaxedproblem.Wenotethatonceall
theconstraintscorrespondingtoaninfeasiblecontingencyinc ∈C
1 1
III. ALGORITHMS areaddedtotherelaxedproblem,thiscontingencywillnotbeviolated
In this section, we present three different algorithms to solve by any subsequent feasible solutions and hence, c can be removed
1
the SCCUC problem. The common underlying ideas for all the from the list of contingencies C . Checking if the contingencies in
1
three algorithms are: (i) we relax the formulation by ignoring a C are feasible for the current dispatch involves solving an SOC
1
few constraints and provide the relaxed formulation to a branch- feasibility problem for every hour t and every contingency c ∈C ,
1 1
and-bound solver, (ii) whenever a feasible solution is obtained to given by the constraints (12), (18), (19), (20), (31), (34), (37), and
this relaxed problem, we check if the solution is feasible for the (38).ThesearesmallSOCfeasibilityproblemsandcanpotentiallybe
constraints that were ignored, (iii) if one or more constraints are solvedinparallel.Thissmallimprovementtotheouterapproximation
violated for the current feasible solution, then we add a cut to the algorithmresultsinaspeedupofafactoroftwoaswewillobserve
original problem that invalidates the solution and continue solving in Sec. IV.
C. Benders decomposition TABLEI. COMPUTATIONTIMESINSECONDS.
TheSBDapproach iseffectivewhenonlyafewofthecontingen-
W%=10% W%=20% W%=30%
cies are “active” in the final solution. However, we found that in a
L% OA SBD Benders OA SBD Benders OA SBD Benders
numberofcases,especiallywhenwedecreasetheallowedprobability
70 1483 586 58 507 467 110 1230 467 54
violationsǫcℓ,alargefractionofthecontingenciesneededtobeadded 80 1467 743 63 1262 580 130 1641 571 115
before convergence is achieved. As an alternative, we developed 90 1046 691 106 951 803 39 1736 552 121
100 1117 713 94 1353 720 85 984 716 112
a Benders-like decomposition. The relaxed problem we provide to
the solver is the entire formulation defined in Section II except for
the constraints (37)-(38). Instead of treating these constraints as is,
A. Test system and wind data
we use an extended formulation which avoids forming (and storing
in memory) the entire dense matrix Mc defined in Section II-F We use the IEEE single-area RTS-96 system with modifications
performed on the base system similar to the ones described in [9].
for each contingency. Benders decomposition (generalized to SOCP
The system comprises of 24 buses including 17 load buses, 38
subproblems)isthenappliedtothisextendedformulation.Whenever
transmissionlinesand32conventionalgenerators.Thetotalinstalled
an integer feasible solution is found by the solver, we invoke the
capacity of the generators is 3405 MW. Among the 32 generators,
Benders cut generation procedure briefly described below.
ofF(oθrc,aγcco,nftcin)g∈enRcy2|Bc|+∈|LC|,sdaetfiisnfeyiLngFcγ(cp¯=(t)θ,cδ¯c=(t)0,,α¯c(t))astheset 2daataresentu[2c0le]aprraonvdide1sitsheawhyinddrodagtean.eWraitnodr.faTrhmesNloRcEatLioWnseasrteermnaWppinedd
r r totheIEEERTS-96respectingthelengthsofthelines(see[9]).The
test system contains a total of 9 wind farms with a total generation
XBbcnθnc = X[pi(t)+δic(t)]+µb(t)−db(t) ∀b∈B, (39) capacityof3900MW.Thelocationsofthewindfarms,theindividual
generationcapacityofeachwindfarm;thestepwisegenerationcost,
n∈B i∈Gb
start-upcost, ramping restrictions, upand down-time restrictionsfor
X Bbnγnc = Xαci(t) ∀b∈B\{r}, (40) eachgenerator;andtheloadprofiledatafora24-hourperiodusedfor
thecasestudyaremadeavailablebyauthorsin[9]and[21].Usingthe
n∈B,n6=r i∈Gb
1000 wind generation scenarios for each wind farmgenerated in[9]
fmcn =βmn(θmc −θnc) ∀(m,n)∈Lc, (41) viavariousstatisticalmethods,themeanandstandarddeviationofthe
wind power injection for each time period was estimated assuming
Pr(fmcn+βmnΩ(t)(γnc −γmc )+βmnω¯T(t)(πmc −πnc) thatthewindpowerinjectionsforeachhourareindependentnormally
(42) distributedrandomvariables.Thecostcoefficientsa0i, a1i,anda2i for
≤fmax)≥1−ǫc ∀(m,n)∈Lc, thegenerationandtertiaryreserves,respectively,areadopteddirectly
mn mn
fromtheIEEERTS-96generationcost coefficientdataandthewind
Pr(fmcn+βmnΩ(t)(γnc −γmc )+βmnω¯T(t)(πmc −πnc) power is assumed to have zero marginal cost.
(43)
B. Performance of proposed algorithms
≥−fmax)≥1−ǫc ∀(m,n)∈Lc,
mn mn
TheSCCUCformulationhasfouruser-definedparametersnamely
where,πbc isthebthrowoftheinverseof theadmittancematrixBc, ǫi, ǫci, ǫℓ, and ǫcℓ. For the rest of the computational experiments
after excluding the row and column corresponding to the reference in the paper, we set their values to 1%, 2%, 10%, and 20%
busr.ThesetLc isdefinedasL\{c}ifcisalinecontingencyand respectively. We then sequentially vary the loading levels and the
L for generator contingencies. wind penetration levels and solve the resulting SCCUC instances
By [4], [11], the constraints (37)-(38) are satisfied if and only usingeachofthealgorithmsproposedinSec.III.Thefirstalgorithm
if the set LFc(p¯(t),δ¯c(t),α¯c(t)) is not empty. We therefore test is the sequential linear outer approximation algorithm (OA), the
feasibility of the solution which the solver finds by solving the second is the SBD presented in Sec. III-B, and the last one is the
SOCP feasibility problem corresponding to LFc(p¯(t),δ¯c(t),α¯c(t)) Benders decomposition inSec. III-C. For the SBD,the set C1 is the
for each contingency and time period. If the problem is infeasible, setofallgeneratorcontingencies.Thecomputationswerecarriedout
we compute, via SOCP duality, a cut analogous to the Benders on aDellPrecision T5500 workstation (Intel Xeon E5630 processor
feasibilitycut whichinvalidatesthesolution(p¯(t),δ¯c(t),α¯c(t))[16, 2.53GHz, 12GB RAM). For all the runs, the optimality tolerance
Prop.2.4.2].Thesecutsareaddedbyusingsolvercallbackswithinthe was set to 1%. The algorithms were implemented using Julia and
branch-and-boundtreeaspreviouslydiscussed.Atechnicalchallenge JuMPChance [22] with CPLEX and Mosek [17], [18] as the LP
weencounteredwasobtainingvaliddualraystoinfeasibleSOCPs,a and conic solvers respectively. Table I shows the computation time
featurenotsupportedbyCPLEX[17].Instead,weusedMosek[18], of the three algorithms for varying wind penetration (W%) and
which has this functionality, to solve the SOCP subproblems. load levels (L%). We observe from the results in Table I that the
Benders decomposition outperforms the other two algorithms for
IV. CASESTUDY all the test instances. We also note that this trend was observed
consistently for different choices of subsets of of contingencies, C ,
1
Inthissection,wedemonstratethebenefitsofsolvingtheSCCUC in the SBD. Hence, throughout the rest of the article, we use the
relativetothedeterministicversionoftheunitcommitment problem Benders decomposition algorithm for all the runs.
using the single area IEEE RTS-96 system [19]. The comparison
is based on a variety of factors including nominal operational cost, C. Comparison of the SCCUC to its deterministic counterpart
number of line and generator violations, amount of generation and We now compare the performance of the SCCUC with its de-
tertiaryreservesallotted,etc.Wealsoinvestigatetheperformance of terministic counterpart, which assumes ωb(t) = 0. Both the deter-
proposedalgorithmswithregardstocomputationtimeandscalability. ministic and chance constrained unit commitment with N-1 security
Operational Cost [$]110202468x 105 Absolute Cost Difference [$]−246820000000000000000 TPNRRSortoeeaot ssardLeetluo−rrvvcaUeetdpioss nrru+p, r− Empirical Violation Probability per Constraint000...0246 Generators Lines Generators Lines
DeterministicChance−Constrained Change in cost Deterministic Chance Constraints
Figure1. Totalcostandvalueofthedifferentcostcomponentsforthedeter- Figure 2. Empirical violation probability of each generation and line con-
ministic and chance constrained unit commitment (left). The cost difference straintinthedeterministic (left)andthechance constrained (right)solution.
(right) shows the change in costwhen movingfrom thedeterministic tothe
chance constrained unitcommitment.
TABLEII. NUMBEROFCOMMITTEDUNITSANDALLOCATEDRESERVES mplesns 1000
ComUnmitistted ReseTrevretsiar[yMW] ReGseernveersat[iMonW] umber of Sawith Violatio 500 CCDhoeantensrtcmreai−ninisetdic
N 0
5 10 15 20
Deterministic 417 5383.7 482.1 Hour
Chanceconstrained 421 5357.5 954.3
Figure3. Thenumberofwindfluctuationsamples(outof1000)thatleadto
oneormoreconstraint violations.
constraints are solved for a case where the forecast wind power
productionaccountsfor20%ofthetotalload.Sincethedeterministic Theempiricalviolationprobabilities,evaluatedforeachconstraint
unit commitment withN-1securityconstraints assumes thereareno separately,areshowninFig.2.Weobservethatthemaximumvalue
windfluctuations,itwillnotrequireanygenerationreservestocover of the empirical violation probabilities for the optimal dispatch of
for thewindpower fluctuations. Inorder tomake afair comparison, the deterministic problem is 50% for both the generator and line
weassumethatthesystemoperatormaintainsaminimumgeneration limit constraints. Interestingly, the optimal solution of the chance-
reserve requirement equal to 0.5% of the load and impose this constrainedproblemisabletoeffectivelylimittheempiricalviolation
constraint as a part of the optimization problem. Furthermore, to probabilities to 2% and 20% for generators and lines, respectively.
obtain a more interesting case, the transmission capacities of the Wenotethattheseviolationsincludetheonesthatoccurduringsingle
IEEE RTS-96 system were decreased to 90% of their original base component outages.
case value in [9]. For a system operator, it is tantamount not only to limit the
The total cost of the unit commitment and the different cost violation probability for a specific line or generator limit constraint,
components are shown in Fig. 1, with the deterministic and chance but also to reduce the frequency at which any any line or generator
constrained cost on left and the cost differences to the right. The limit is violated during a 24-hour period. Hence, Fig. 3 compares
number of committedunitsand theamount of allocated reservesare the number of wind realizations out of 1000 that actually lead to
shown in Table II. We observe that the total cost of the SCCUC at least one constraint violation during each hour of the day. It is
solution is only slightly greater than the cost of its deterministic clear from the Fig. 3 that for the optimal dispatch obtained from
counterpart, with a major difference showing up in the cost of the deterministic problem, the empirical probability that at least one
generation reserves and aminor one inthe no-load cost. Thereason constraint is violated during each hour varies from 20 to 100%,
fortheincreasedcostofgenerationreservesinthechance-constrained while for the optimal solution to the chance constrained version of
version is that it allocates twice as much generation reserves as the problem, the risk is much lower, with a meager 10-20 samples
the deterministic case in order to accommodate the wind power creatingviolationsformosthoursofthedayandwith225beingthe
fluctuations; as mentioned previously the deterministic counterpart maximumnumber ofsampleswithviolationsduringanyhour ofthe
is immune to wind fluctuations and it allocates the bare minimum day.
amount of generation reserves as required by the system operator
(0.5% of the load). The same reason is valid for the higher number
D. Influence of wind penetration and wind variability
ofcommittedunitsforthechance-constrainedcaseandtheincreased
no-load cost (see Table II). Toassesstheinfluenceofwindpowerpenetrationandwindpower
Next, we compare the two variants in their abilityto handle wind variabilityonthecostofunitcommitmentandamountofreserves,we
fluctuations during normal operations and contingencies. To do so, generate two classes of test instances. In the first class of instances,
we solve both the variants and obtain the corresponding optimal we fix the wind penetration level and vary the normalized standard
solution.Toassesstheperformanceofboththedeterministicandthe deviation of the wind fluctuation linearly and in the second class,
chanceconstrainedsolutionsundersimulatedwindpowerconditions, we do the vice-versa. The Table III shows the variation of the
we evaluate the line flows and generator outputs using the optimal number of committed units, the amount of generation reserves and
dispatches for 1000 wind realizations and compute the number of the total SCCUC cost for different levels of wind penetration. We
generatorandlineviolationsthatoccurforeachoftherealization.The observe that the increase in the wind penetration levels decrease the
samples are drawn from the multivariate normal distribution whose overall SCCUC cost. The amount of generation reserves is more or
means and variances were known a priori and were used as input less constant because the normalized standard deviation of the wind
to the SCCUC i.e., we assume perfect knowledge of the probability fluctuations is a constant for the first class of test instances. The
distribution of the wind fluctuations. decrease inthenumber of unitscommitted canalsobeexplained by
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ACKNOWLEDGEMENTS
ThisworkwassupportedbytheAdvancedGridModelingProgram
oftheOfficeofElectricitywithintheU.SDepartmentofEnergyand
theCenterforNonlinearStudiesatLosAlamosNationalLaboratory.