52 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 1, FEBRUARY 2013
Adaptive Robust Optimization for the Security
Constrained Unit Commitment Problem
Dimitris Bertsimas, Member, IEEE, Eugene Litvinov, Senior Member, IEEE,Xu
Andy Sun, Member, IEEE,
Jinye Zhao, Member, IEEE, and Tongxin Zheng, Senior Member, IEEE
Abstract—Unit commitment, one of the most critical tas
ks in
electric power system operations, faces new challenges as the
supply and demand uncertainty increases dramatically due to
the integration of variable generation resour
cessuchaswind
power and price responsive demand. To meet these challenges, we
propose a two-stage adaptive robust unit commitment m od el for
the security constrained u nit commitment p
roblem in the presence
of nodal net injection uncertainty. Compared to the conventional
stochastic programming approach, the proposed m odel is more
practical in that it only requires a deter
ministic uncertainty
set, rather than a hard-to-obtain probability distribution on the
uncertain data. The unit commitment solutions of the proposed
model are robust against all possib
le realizations of the modeled
uncertainty. We develop a practical solution methodology based
on a combination of Benders decomposition type algorithm and
the outer approximation techniq
ue. We present an extensive nu-
merical study on the real-world large scale power system operated
by the ISO New England. Computational results demonstrate
the economic and operational a
dvantages of our model over the
traditional reserve adjustment approach.
Index Terms—Bilevel mixed-integer optimization, power system
control and reliability, robust and adaptive optimization, security
constrained unit commitm
ent.
I. INTRODUCTION
U
NIT commitm ent (UC) is one of the most critical deci-
sion processes performed by system operators in deregu-
lated electricity markets as w ell as in vertically integrated util-
ities. The objective of the UC problem is to find a unit com -
mitment schedule that minimizes the com m itment and dispatch
costs of meeting t he forecast system load, taking into account
various physical, inter-temporal constraints for generating re-
sources, transm ission , and system reliabili ty requirem ents.
During the normal real-time operation, system operator dis-
patches the committed generation resources to satisfy the actual
demand and reliability r equirem ents. In the event that the actual
system condition significantly deviates from the expected con-
dition, system operator n eeds to t ake corrective actions such as
Manuscript recei ved March 16, 2011; revised September 18, 2011 and April
02, 2012; accepted May 25, 2012. Date of publication July 24, 2012; date of
current version January 17, 2013. Paper no. TPWRS-00231-2011.
D. Bertsimas is with the Sloan School of Managem en t an d the O p erations
Research Center, Massachusetts Institute of Technology, Cambridge, MA 02139
USA (e-mail: dbertsim@mit.edu) .
E. Litvinov, J. Zhao, and T. Zheng are with the Department of Business Ar-
chitecture & Technology, ISO New England, Holyoke, MA 01040 USA (e-mail:
elitvinov@iso-ne.com; jzhao@iso-ne.com; tzheng@iso-ne.com).
X. A. Sun is with the H. Milton Stewart Schoo l of Indu strial and Systems
Engineering, Georgia Institu te of Technology, Atlanta, GA 3 0332 USA (e-mail:
sunx@mit.ed u).
Dig
ital Object Identifier 10.1109/TPWRS.2012.2205021
committing expensive fast-start generators, voltage
reduction,
or load shedding in emergency situation to maintain sy
stem se-
curity. The main causes of the unexpected events come
from
the uncertainties associated with the load f o
recast error, changes
of system interchange schedules, generator
’s failure to follow
dispatch signals, and unexpected transmis
sion and generation
outages.
In recent y ears, higher penetration of vari
able generation re-
sources (such as wind power, solar power, an
d distributed gen-
erators) and more price-responsive d
emand participation have
posed new challenges to the unit comm
itment process, espe-
cially in the independent system ope
rator (ISO) managed elec-
tricity markets. It becom es import
ant for the ISOs t o have an
effective methodology that produ
ces robust unit commitment
decisions and ensures the system
reliability in the p resence of
increasing real-time uncertai
nty.
Previous studies of uncertain
ty management in the UC
problem can be divided into tw
ogroups.Thefirst group
commits and dispatches gene
rating resources to meet a deter-
ministic forecast load, an
d h andles uncertainty by im posing
conservative reserve requ
irements. The second group relies
on stochastic optimizati
on techniques. The first group, the
so-called reserve adju
stment method, is widely used in today’s
power industry. Much o
f research along this vein, including
[1]–[4], has focused
on analyzing the levels of reserve require-
ments based on determ
inistic criteria, such as loss of the largest
generatororsystem
import change. Such an approach is easy to
implement in pract
ice. However, committing extra generation
resources as reser
ves could be an economically inefficient way
to handle uncert
ainty, especially when the reserve requirement
is determined l
argely by some ad-hoc rules, rather than by a
systematic ana
lysis. Also, since the UC decision only considers
the expected o
perating condition, even w ith eno ugh reserve
available, t
he power system m ay still suffer capacity inade-
quacy when th
e real-time condition , such as load, dev iates
significan
tly from the expected value. This is confirmed b y
the I SO’s o
perational experience as well as by the numerical
simulati
on shown later.
The stocha
stic opti mi zation approach explicitly incorporates
a probabi
lity distribution of the uncertainty, and often relies o n
pre-samp
ling discrete scenarios of the uncertainty realizations
[5]–[9 ]
. This approach has some practical limitations in the ap-
plicat
ion to large scale power systems. First, it may be diffi-
cult to
identify an accurate probability distribution of the un-
certa
inty. Second, stochastic UC solutions only provide proba-
bilis
tic guarantees to the system reliability. To obtain a reason-
ably
high guarantee requires a large number of scenario sam-
ple
s, which results i n a problem that is computat ion a lly inten-
0885-8950/$31.00 © 2012 IEEE
BERTSIMAS et al.: ADAPTIVE ROBUST OPTIMIZATION FOR THE SECURITY CONSTRAINED UNIT COMMITMENT PROBL
EM 53
sive. To i mprove the robustness of stoch astic UC solutions, Ru iz
et al. [10] proposed a hybrid approach combining the reserve re-
quirement and stochastic optimization methods. A recent work
[11] proposed a framework that combines uncertainty quantifi-
cation with stochastic optimization. This framew ork could also
be integr ated into the robu st optimization formulation proposed
below.
Robust optimization has recently gained substantial pop-
ularity as a modeling framework for optimization und er
parameter uncertainty, led by the w ork in [12]–[18]. Th e ap-
proach is attractive in several aspects. First, it only requires
moderate info rmation about the underlying u ncertainty, such as
the m ean and the range of the uncertain data; and the framework
is flexible enough that the modeler can always incorporate more
probabilistic information such a s correlati on to the uncerta int y
model, when such inform ation is available. For instance, the
method of uncer tai nty quanti fication (UQ) proposed in [11] can
be integrated into th e ro bust optimization UC m o d el, where the
UQ module updates the uncer tain ty m odel as more information
is obtained in time. Second, the robust model constructs an
optimal solutio n that immuni zes against all realizations of
the uncer tain data within a determin istic uncertainty set. This
robustness is a d esirable feature, especially when the penalty
associated with infeasible solutions is very high, as the case
in the power system operations. Hence, the concept of robust
optimization is con sistent with the risk-averse fashion in which
the power systems are operated. Robust optimization has been
broadly applied in engineering and managemen t sciences,
such as structural design, integrated cir cuit design, statistics,
inventory management, to name a few. See [19] and references
therein.
In this paper, w e propose a two -stage adaptiv e robust opti-
mization model for the security constrained unit commitmen t
(SCUC) problem, where the first-stage UC decision and the
second-stage dispatch decision are robust against all uncertain
nodal net injection realizations. F urthermo re, the second-stage
dispatch solu tion has full adaptability to the uncertainty. T he
critical constraints such as network co nstraints, ramp rate con-
straints and transmission security constraints are incorporated
into the proposed model as w ell. It is key to design a proper
uncertainty set to contr ol the conservatism of th e robust solu-
tion. We use a special techniq ue p roposed in [17] and [18] for
this purpose. We develop a practical solution method, and ex-
tensively test the method on a real-world power system. Papers
[20] and [21] proposed similar robust optim ization U C models.
However, their proposals ignored reserve constraints and did
not study critical issues such as th e impact of robust solutions
on system efficiency, operational stability, and ro bust ness of the
UC solutions. Our research was cond ucted independently of the
work in [20] and [21 ]. The main contributions of our paper are
summarized below.
1) We formulate a tw o-stage adaptive robust optimizatio n
model for the SCUC problem. Given a pre-specified nodal
net injection uncertainty set, the two-stage adaptive robust
UC model obtains an “immunized against uncertainty”
first-stage commitment decision and a second-stage
adaptive dispatch actions by minimizing the sum of the
unit commitment cost and the dispatch cost under the
worst-case realization of uncertain nodal net injection.
The nodal net injection un certainty set models variable
resources such as non-dispatchable wind generation,
real-time demand variation, and interchan ge uncertainty.
The parameters in the uncertainty set provide control over
the conservatism of the robust solution.
2) We develop a practical solution methodo log y to solve the
adaptive robust model. In particular, w e design a two-level
decomposition a pproach. A B enders decomposition type
algorithm is employed to decompose the ov erall pro blem
into a master problem involving the first- stage commit-
ment decisions at the outer level and a bilinear subproblem
associated with the second-stage dispatch actions at the
inner level, which is solved by an outer approx im a tio n ap -
proach [22], [23]. The proposed so lution m etho d applies to
general polyhedral uncertainty sets. Computation al study
shows the efficiency of the method.
3) We condu ct extensive numerical experim ents on the real-
world large scale power system operated by the ISO New
England. We study the performance of the adaptive robust
model and provide detailed comparison w ith the current
practice, the reserve adjustment approach. Specifically, we
analyze the merit of the adaptive robust model from three
aspects: economic efficiency, contribution to real-time op-
eration reliability, and robustness to probability distribu-
tions o f the uncertainty.
The paper is organized as follows. Section II d escribes the de-
terministic SCU C formulation. Secti on III introduces the two-
stage adaptive robust SCUC formulatio n. Section IV discusses
the solution m ethodology. Section V presents computational re-
sults, including a discussion on the proper way to choose the
level of conservatism in the robust model. Section VI concludes
with discussions.
II. D
ETERMINISTIC SCUC PROBLEM
The deterministic SCUC problem is extensively studied in
the power system literature (e.g.,[24],[25]).PleaseseetheAp-
pendix for a detailed m odel. Here w e present a compact matrix
formulation:
(1)
(2)
(3)
(4)
The binary v ariable
is a vector of commitment related deci-
sions including the on/off and start-up/shut-down status of each
generation unit for each time interval of the comm itment pe-
riod, usually 24 h in an ISO settin g. The continuous variable
is a vector of dispatch related decision including the generation
output, load consumption levels, resource reserve levels, and
power flows in the transmission network for each time interval.
By conven tio n, generation, reserve, and flow tak e positive sign,
whereas load takes n egative sign.
54 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 1, FEBRUARY 2013
The objective function is to minimize the sum o f comm itment
cost
(including start-up, no-load, and shut-d own costs)
anddispatchcost
over th e planning horizon. Constrain t
(1), involving only com mitment v ar iables, contain s minim um
up and down, and start-up/shut-dow n co nstraints. Constrain t
(2) includes dispatch related constraints such as energy balance
(equality can always be w ritten as two opposite inequalities),
reserve requirement, reserve capacity, transmission limit, and
ramping constraints. Constraint ( 3) couples the commitment
and dispatch decisions, including minimum and m aximum
generation capacity constraints. Constraint (4) emphasizes the
fact that the uncertain nodal ne t injections are fixed at expected
values in the deterministic model (
selects t he components
from
that correspond to uncertain resources).
III. T
WO-STAGE ADAPTIVE ROBUST SC UC FORMULATION
In this section, we first discuss the uncertainty set, which is
a key building blo ck of the robust model. Then, we introduce
the two-stage adaptive robust SCUC f ormulation and provide a
detailed explanation.
The first step to build a ro bust model is to construct an un -
certainty set. Unlike the stochastic optimization approach, the
uncertainty mode l in a robust optim izat ion f ormulation is not
a probability distrib ution, but rather a deterministic set. In this
paper, the uncertain parameter is the no dal net injection. We
consider the fol lowing uncertain ty set of n odal net injection at
each time period
in the planning horizon :
(5)
where
is the set of nodes that have uncertain inj ections,
is the number of such nodes, is the vector
of uncertain net injections at time
, is the nominal va lue of
the net injection o f node
at time , is the deviation from
the nominal n et injection value of node
at time ,theinterval
is the range of the uncertain ,andthein-
equality in (5) controls the total deviation of all injections from
their nominal values at time
. The parameter is the “budget
of uncertainty”, taking values between 0 and
.When ,
the uncertainty set
is a singleton, correspon ding to
the nominal determinis tic case. As
increases, the size of the
uncertainty set
enlarges. This means that larger total devia-
tion from the expected net injection is considered, s o that the
resulting robust UC solutio ns are more conservative and the
system is protected against a higher degree of uncertainty. Wh en
, equals to the entire hypercube definedbythein-
tervals for each
for .
Now we formu late the two-stage adaptive robust SC UC
model as follows:
(6)
where
and . The objective
function has two parts, reflecting the two-stage nature of the
decision. The first part is the commitment cost. T he second part
is the worst case second-stage dispatch cost.
From this formulation, we can see that the commitm ent deci-
sion
takes into account all possible future net injection repre-
sented in the uncertainty set. Such a U C solution remains fea-
sible, thus robust, for any realization of the uncertain net in-
jection. In compariso n, the traditional UC solution only guar-
antees feasibility for a single nominal net injection, whereas
the s tochastic o pti mization UC solution only c onsi ders a finite
set of preselected scenarios of the uncertain net injection. Fur-
thermore, in our formulation t he optimal second-stage decision
is a function of th e uncertai n net injectio n , theref ore,fully
adaptive to any realization of the uncertainty. Notice that
is
also a function of the first-stage decision
. However, we write
it as
to emphasize the adaptability of the second-stage de-
cision to the uncertai nty
. The functional form of is de-
termined im p lici tly by the opt imization prob lem , as opposed to
beingpresumedasinthecaseofaffinely adaptive policies (see
[26] and discussion in the conclusion part of this paper).
The full
adaptability properly models the economic d ispatch procedure
in the real-time opera tion.
The above formulation can be recast in the following
equivalent form, which is suitable for developing numerical
algorithms:
(7)
where
is the
set of feasible dispatch solutions for a fixed commitment deci-
sion
and nodal net injection realization . Notice that the worst
case dispatch cost has a max-min form, w here
determines the economic dispatch cost for a fixed commitment
and net injection , which is then maximized over the un-
certainty set
. A slack variable is included in the energy bal-
ance constraints in the inner max-min problem to ensure its
feasibility.
It is useful to write out the dual of the disp atch problem
. Denote its cost by :
(8)
where
, ,and are the multipliers of the constraints (2)–(4),
respectively.
Now, the secon d-stage problem
is
equivalent to a bilinear op tim ization problem given as follows:
(9)
where the constraints involving variable
de-
fine a polyhedral set, deno ted as
. Due to the bilinear struc-
ture of the objective function, the optimal solution o f pro blem
BERTSIMAS et al.: ADAPTIVE ROBUST OPTIMIZATION FOR THE SECURITY CONSTRAINE D UNIT COM MITM ENT PROBL
EM 55
(9) is an extreme point of the polyhedron
, and similarly the optimal s olution is an extreme point
of
. T herefore, if we denote all the extreme points of as
, and all the extrem e points of as ,
can be written equivalently as
(10)
which shows that
is the maximum of a finitely many affine
functions in
, hence is a convex piecewise linear func-
tion in
. However, in general we h ave no knowledge of the
extreme points of
and , and computing is non-trivial.
To see this, notice that the objective fun ctio n of (9 ) contains
a non-concave bilinear term
, and b ilinear programs are in
general NP-hard to solve. Another way to see this is from the
formulation (7), where
can be written equivalently as
Notice that the objective value of the inner problem
is a convex fu nction i n . Therefore, e val -
uating
needs to maxim ize a convex function, which
is generally NP-h a rd. Throughout the paper, we assume
for all feasible . This can be ensured by adding
penalty terms in the dispatch constraints. We omit the penalty
terms here for a clear presentation.
IV. S
OLUTION METHOD TO SOLVE
THE
ADAPTIVE ROBUST MODE L
As analyzed in the previous section, the adaptive robust for-
mulation ( 7) is a two-stag e problem. The first-stage is to find
an optimal comm itment decision
. The second-stage is to find
the worst -case dispatch cost under a fixed co mm itment solution.
Naturally, we w ill have a two-level algorithm. T he outer level
employs a Benders decomposition (BD) type cutting plane al-
gorithm to obtain
using the information (i.e., cuts) generated
from the inner level, which appr oxi mat e ly solves the bilinear
optimization pro blem (9 ).
A. Outer Level: Benders D ecomposition Algorithm
The Benders decomposition algorithm is described below.
Initialization: Let
beafeasiblefirst-stage solution.
Solve
defined by (9) to get an initial solu-
tion
. Set the outer level lower bound
, upper bound and the number
of iteration
. Choose an outer level convergence
tolerance level
.
Iteration
:
Step 1) Solve BD master problem.
The m aster problem of BD is the following mixed
integer program (MIP):
(11)
Let
be the optimum. S et
.
Step 2) Solve B D subproblem
.
We will discuss the m e tho dology to
solve
in the next subsection. Let
be the optimal solu-
tion. Set
.
Step 3) Check the outer level convergence.
If
, stop and return .Other-
wise, let
, and go to step 1.
To speed up the convergence of the ab ove BD alg orithm, we
find it helpful to add dispatch constraints
to the BD
master problem (11) at certain iteration
when or
has im proved slo wly .
B. Inner Level: Solve
An outer approximation (OA) algorithm [22], [23] is used to
solve the bilinear program (9), where the bilinear term in the
objective is linearized around intermediate solutio n poi nts and
linear approximations are added to t he OA formulation. Since
the problem (9) is nonconcave, only a lo cal optim um is guar-
anteed by the OA algorith m. To verify the quality of the solu-
tion, we test on different initial conditions and observe fast con-
vergence and consistent results. The OA algorithm is described
below.
Initialization: Fix the unit com mitment decisio n
passed
from the
th iteration of the outer level BD algorithm.
Find an initial n et inj ection
. S et the inner level
lower bound
, upper bo und and
number of iter a tion
. Choose an inner l evel conver-
gence tolerance level
.
Iteration
:
Step 1) Solve OA su bprob lem
.
Solve
, the dual dispatch problem de-
fined by (8). Let
be the optimal solu-
tion. Set
.Define ,the
linearization of the bilinear term
at ,
as follows:
Step 2) Check the inner level convergence. If
, then s top and o utp ut the current solu-
tion.Otherwise,set
,gotoStep3ofthe
OA algorith m.
Step 3) Solve OA master problem.
Solve the linearized version of
,defined in
the following:
Since the uncertainty set isassumedtobe
polyhedral,
is a linear program. De-
note
as the op-
56 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 1, FEBRUARY 2013
Fig. 1. Flow chart of the proposed two-level algorithm. SFT stands for simul-
taneous feasibility test, where security constraints, such as
constraints,
are tested at current solutions and violated constrain ts are added seq uentially.
timal solution. Set the inner level upper bound as
.
We want to remark th at the above inner level algorithm works
for general polyhedral uncertainty set
, not restricted to the
budgeted uncertainty set of (5). T he nex t theorem shows that
the Benders cuts generated b y the inner level are valid cuts.
Theorem 1: The dual solutions,
,ofthe th
inner level problem generate valid in equal iti es for the second-
stage value function
, i.e.,
Proof: The dual solutions generated by
the OA subproblem
are extreme points o f the dual
polytope
, because is a linear optimizatio n problem
over
. Therefore, using (10), we have
which shows that the Benders cuts added to the BD master
problem are valid cuts, i.e., the epigraph of
is above the
Benders cuts.
The overall algorithmic framework is summarized in Fig. 1.
In our implementation, we include all co ntingen cy tran smissio n
constraints in the BD subproblem.
V. C
OMPUTATIONAL EXPERIMENTS
In this section, we present a computat ional study to evalu ate
the perform ance of the ad a ptive robust (AdaptRob) approach
and the reserve adjustment (ResAdj) approach. We test on the
power system operated by the ISO New Eng land Inc. (ISO-NE).
We have the following data and uncertainty model.
The system and market data: The system has 312 generating
units, 174 loads, and 2816 nodes. We select 4 representative
transmission constraints that interconnect four major load zones
in the ISO -NE’s system (we also demonstrate the performance
of our algorithms with 1876 transmission constraints). The
market dat a is taken from a normal winter day of the ISO- N E ’s
day-ahead energy market. In particular, we have 24-h data
of generators’ offer curves, reserve offers, expected nodal
load, system r eserve requirement (10-min spinning reserve,
10-min n on-spin ning reserve, and 30-min reserve), an d various
network param eters. The average total generation capacity per
hour is 31 929 MW and the average forecast load per hour is
16 232 MW.
The uncertainty model: We use the b udgeted uncertainty set
definedin(5),inwhich
is the nominal load given in the data.
We set the range of load variation to be
for each
load
at time . The budget of uncertainty takes values in
the entire range of 0 to
. We will discuss the proper way to
choose
within this range th at results in the best performance
of the r obust solution in t he follo wing subsection.
For the ResAdj approach, we solve the deterministic UC
problems presented in Section II at the expected load level
with adjusted reserve requirement. In particular, w e model the
reserve adjustment as follows:
where is the system reserve req uir ement of type at time ,
composed of the basic reserve level
and an adjustment part
proportional to the total variation of lo ad. Here, the uncertainty
budget
also controls the level of conservatism of the ResAdj
solution.
The com putational experiments proceed as follows.
1) Obtain UC solutions: Solve the ResAdj and AdaptRob UC
models, respectively, for different uncertainty budg ets:
.
2) Dispatch simulation: For each UC solution, solve the dis-
patch problem repeatedly for two sets of 1000 rando mly
generated loads.
One set of randomly sampled load follows a un ifo rm distri-
bution in the interval
for each load at tim e
. The other set follow s a n ormal distribution with mean and
standard dev iation
, which results in an 85 percentile
of load falling between
and the load is trun-
cated for nonnegative values. Notice that samples from both sets
may fall o utsid e of the budgeted uncertainty set, especially when
the budget
is close to 0. I n this way, we can test the perfor-
mance of the robu st formulation when the uncertainty mode l is
inaccurate.
To mimic the high cost of dispatching fast-start units or load
shedding in the real-time operation , we introduce a slack vari-
able
to energy balance, reserve requirement, and transm issio n
constraints in the dispatch simulation. If the real-time dispatch
incurs any energy deficiency, reserve shortage, or transmission
violation, at least one component of
will be po sitiv e. The dis-
patch cost is the sum of production cost and penalty cost, i.e.,
,where is set to $5000/MWh for each component.
The proposed tw o-level algorithm for the two-stage adap-
tive robust model is imp lemented in GAMS. The mixed integer
program and linear program in the alg orithm are s olved with
CPLEX 12 .1 .0 on a PC laptop w ith an Intel Core(TM) 2Duo
2.50-GHz CPU an d 3 GB of memory . We set the convergence
tolerance for the outer level BD algorithm to be
,and
the convergence tolerance for the inner level algorithm to be
. The MIP gap for the BD master problem is set to
. The average computation time to solve the robust UC
problem is 6.14 h. The average computation time for the re-
serve adjustm ent approach is 1.65 h. The compu tational results