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IEEE TRANSACTIONS ON SMART GRID, VOL. 4, NO. 4, DECEMBER 2013
Decomposition Algorithms for Market Clearing With
Large-Scale Demand Response
Nikolaos Gatsis, Member, IEEE, and Georgios B. Giannakis, Fellow, IEEE
Abstract—This paper is concerned with large-scale integration
of demand response (DR) from small loads such as residential
smart appliances into modern electricity systems. These appliances
have intertemporal consumption constraints, while the satisfaction of the end-user from operating them is captured through
utility functions. Incorporation of the appliance scheduling flexibility and user satisfaction in the system optimization points to
maximizing the social welfare. In order to solve the resultant very
large optimization problem with manageable complexity, dual
decomposition is pursued. The problem decouples into separate
subproblems for the market operator and each aggregator. Each
aggregator addresses its local optimization aided by the end-users’
smart meters. The subproblems are coordinated with carefully
designed information exchanges between the market operator and
the aggregators so that user privacy is preserved. Numerical tests
illustrate the benefits of large-scale DR incorporation.
Index Terms—Aggregators, decomposition algorithms, demand
response, demand-side management, pricing.
Convex and compact feasible set for
power consumption of smart appliance
corresponding to end-user of
aggregator over the entire scheduling
, and
denotes
horizon (
the nonnegative reals).
NOMENCLATURE
Parameters of end-user utility functions.
Total energy requirement of smart
appliance .
Start and end times of smart appliance .
Minimum and maximum power
consumption of smart appliance .
Maximum power provided by aggregator
.
Vector
at .
A. Constants, Sets, and Indices
Number of scheduling periods, period
index.
of base load on all buses
Reactance of line
.
Bus admittance matrix for DC power
flow.
Number of buses, bus indices.
Number of lines.
Matrix
bus angles.
Number of generators, generator index.
relating power flows to
Bus/generator indicator matrix
if generator is
on bus .
Number of aggregators, aggregator
index.
Number of end-users corresponding to
aggregator , end-user index.
Bus/aggregator indicator-matrix
if aggregator
is on bus .
Set of smart appliances of residential
end-user of aggregator , and smart
appliance index.
Vectors
of line flow limits.
Generator ’s minimum and maximum
output.
Lower and upper bounds for multipliers
.
Generator ’s ramp-up and ramp-down
limits.
Algorithm iteration index and auxiliary
index.
Generator ’s initial power output.
Optimal value of market clearing.
Manuscript received October 02, 2012; revised February 08, 2013; accepted
April 01, 2013. Date of publication July 12, 2013; date of current version
November 25, 2013. This work was supported by NSF grant 1202135; and by
the Institute of Renewable Energy and the Environment (IREE) under Grant
RL-0010-13, University of Minnesota. Paper no. TSG-00684-2012.
The authors are with the Digital Technology Center and the Department of
Electrical and Computer Engineering, University of Minnesota, Minneapolis,
MN (email: [email protected]; [email protected]).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSG.2013.2258179
Upper bound on optimal value at
iteration .
Tolerance for termination criterion.
B. Variables
1949-3053 © 2013 IEEE
Output of generator at period .
Consumption of smart appliance at .
GATSIS AND GIANNAKIS: DECOMPOSITION ALGORITHMS FOR MARKET CLEARING WITH LARGE-SCALE DEMAND RESPONSE
Total power provided by aggregator to
its end-users at .
Bus angle at period
Vector
collecting
Vector
collecting
Vector
collecting
Vector
collecting
for all .
for all .
.
for all
Vector collecting
for all .
Vector collecting
for all and ,
collects the power consumptions
i.e.,
of all smart appliances corresponding to
aggregator .
C. Functions
Convex cost function of generator .
Concave utility function of smart
appliance and end-user .
Concave utility function of smart
appliance and end-user at period .
Partial Lagrangian function of market
clearing formulation.
Summands of the previous Lagrangian
.
function
Dual function of market clearing
formulation.
Summands of the previous function.
Full Lagrangian function.
D. Lagrange Multipliers
Multiplier corresponding to the balance
equation for bus at period .
Multiplier corresponding to the balance
equation for aggregator at period .
Vector collecting
for all .
Vector collecting
for all .
Vector collecting
for all .
Variables of master programs
.
Auxiliary variables of master programs.
Vectors collecting the previous variables.
D
I. INTRODUCTION
EMAND response (DR) is an important resource management task promising to enable interaction of end-users
with the grid of the future. DR amounts to adaptation of the
end-user power consumption in response to time-varying energy pricing. In the context of modern power markets, DR has
been proposed as an alternative form of generation or load bid
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[1], [2]. DR aggregators will sign up groups of individual residential and commercial loads to offer large enough DR bids into
the market. Bidirectional communication between aggregators
and end-users will be provided by the advanced metering infrastructure (AMI) [3], with the smart meters installed at end-users’
premises being the AMI terminals.
This paper is concerned with large-scale incorporation of DR
into an energy market. The context includes DR from end-users
with small-scale electricity consumption, such as residential
end-users. Residential loads participating in DR programs are
air conditioning (A/C) units, heaters, pool pumps, or plug-in
hybrid electric vehicle (PHEV) charging. The major research
challenges are the high computational complexity associated
with coordination of a very large number of end-users, and also
the incorporation of various intertemporal constraints of their
loads needed to obtain system-wide benefits. To this end, the
present work advocates optimal decomposition methodologies
of manageable complexity to develop algorithms for DR coordination across the grid that preserve end-user preferences,
constraints, and privacy, while relying upon decentralized
communication protocols between the market operator, the
aggregators, and the end-users.
Approaches to incorporate DR bidding have focused either on
large customers able to shed load, or, on DR bidding from aggregators [4], [5]. However, the intertemporal load shifting capabilities have not been leveraged in [4] and [5]. Recently, aggregator
bidding strategies for charging a PHEV fleet have been also proposed [6]–[8]. These works model a single aggregator as a price
taker, and offer bidding strategies to maximize the aggregator’s
profit. The present work on the other hand, investigates the effect of multiple aggregators on achieving system-wide benefits such as social welfare maximization, and reduction of the
system marginal prices.
The intertemporal constraints of the demand side and their
impact on system performance indices have been investigated
for demand-side bidding by introducing parameters indicating
the load shifting and recovery possibilities [9]–[13]. In a related approach, a market where the demand is allowed to bid
for total energy across a horizon is studied in [14]. Price elasticity matrices (PEMs) representing the willingness of loads to
shift their consumption depending on the prices have been utilized for market clearing in [15] and [16]. The process iterates
between price determination from the market clearing formulation, demand adjustment using the PEMs, and feeding back the
adjusted demand into the market clearing algorithm.
The aforementioned works [9]–[16] focus on large-scale
customers. However, the ideas of aggregating many individual
small-scale user preferences and constraints, and decomposing
the resultant large-scale optimization problem (e.g., involving
generation and end-user coordination) have not been explored.
Recently, a method to aggregate multiple loads with intertemporal constraints based on polytope addition techniques was
advocated in [17], where the aggregated load control capability
was utilized to accommodate fluctuations of the generation.
The difference with [17] is that in addition to the intertemporal scheduling constraints, utility functions capturing the
individual user satisfaction are adopted here, which makes the
overall optimization more involved. Incorporation of aggregated end-user preferences into power system scheduling has
been pursued through dual decomposition in [18]. The main
limitations of [18] are: a) algorithm convergence is not fully
addressed; b) end-users must announce entire demand-price
functions to the aggregators, which may raise privacy concerns;
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and c) the transmission network is not accounted for, while
only strictly convex costs are considered.
There is also a large body of literature dealing with distributed
demand response, see e.g., [19]–[24]. These works focus on a set
of end-users served by a single load-serving-entity, and consider
only the demand-supply power balance, without accounting for
the power network. The distribution network is introduced in
[25]. The decomposition techniques developed in all aforementioned works however are not tailored to power systems with
multiple generators and aggregators.
The present work postulates a set of DR aggregators per bus
of the transmission network. Each aggregator controls loads of
several end-users. Each end-user has preferences about its controllable load operation, captured by utility functions and constraint sets. The objective is social welfare maximization for
day-ahead system scheduling, while transmission network constraints are included in the form of DC power flows. The difficulty here is that small end-users cannot participate directly in
the market because a) individual end-users may not be willing
to reveal their utility functions as well as individual scheduling
constraints, and b) their power consumption is small (in the
order of kW), and it would be a burden for the market operator
to solve an optimization problem directly coordinating a significantly large number of customers.
Seasoned to cope with these issues, the approach here applies dual decomposition to the optimization problem after introducing carefully selected auxiliary variables representing the
total DR-controllable power consumption at each bus, and also
additional coupling constraints. Leveraging Lagrangian relaxation of the coupling constraints, the large-scale optimization
decomposes into manageable optimization problems of favorable structure. The market operator and each aggregator is assigned one of these problems. The aggregator solves its problem
in coordination with the users’ smart meters. The problems that
the market operator and aggregators solve need to be coordinated, and this is accomplished through properly designed information exchanges, which do not need to reveal the end-user
preferences.
For any algorithm based on dual decomposition, part of the
design is a) to bring up the constraints that will be considered
coupling, and therefore associated with Lagrange multipliers,
and b) to select a suitable algorithm for multiplier update.
The former design consideration dictates how different tasks
will be assigned to network entities, while the latter determines messages exchange and the convergence speed. The
contributions of the present paper can then be summarized
as follows: 1) The scheduling problem is formulated in a
way that dual decomposition yields separate problems for the
market operator and each aggregator. 2) As a result, an enticing
feature of the proposed method is that the market operator
can account for the transmission network and simultaneously
rely on current state-of-the-art algorithms for e.g., DC optimal
power flow (OPF), without modification. 3) The operation of
the aggregator preserves user privacy, and does not require
revealing price-demand functions or other user preferences. 4)
The cutting plane method with disaggregated cuts (see e.g.,
[26, Ch. 7]) is advocated as an attractive means of updating
the multipliers. This choice is tailored to the structure of the
problem, and yields faster convergence than standard cutting
plane or subgradient methods.
The remainder of this paper is organized as follows. Section II
presents the optimization problem for market clearing with DR
IEEE TRANSACTIONS ON SMART GRID, VOL. 4, NO. 4, DECEMBER 2013
from a large number of residential end-users. The decomposition algorithm is developed in Section III, alongside system
pricing and economic considerations. Numerical tests are in
Section IV, and Section V concludes the paper. The Appendix
presents a short review of cuttinng plane methods.
II. MARKET CLEARING FORMULATION
Before the optimization formulation is presented, it is instructive to detail the end-user preferences, and their modeling
through utility functions and constraint sets. Specifically, it is
postulated that user corresponding to aggregator has a set
of smart appliances. The power consumption of smart appliance across the horizon
is constrained to be in a set
. This set models the possible intertemporal operational
constraints, and is assumed convex, closed, and bounded. Moreover, a user may draw satisfaction from using the appliance at
different power levels. This is captured by postulating a utility
, assumed to be concave in
. Note that
function
in general, the utility function depends jointly on the power consumption across different periods; thus, it is a function of the
.
entire vector
This smart appliance consumption model is very general, and
can accommodate various cases of interest. To make the exposiand
are given
tion concrete, three examples for
next.
Example 1: Consider appliances that need to consume a specific amount of energy over a horizon in order to complete the
desired tasks, but their actual consumption is allowed to vary
from period to period. Specific cases include charging a PHEV
must
or operating a pool pump. The prescribed total energy
and an end time
,
be consumed between a start time
while the consumption must remain within bounds
and
per period. Set
then takes the form
(1)
Moreover, the utility function can be simply selected to be zero
here.
Example 2: The effect of charging profile to battery lifetime
has been the theme of several studies; see e.g., [27] and [28]
concerning lithium-ion batteries, which are popular choices for
PHEVs. To minimize impacts on the battery lifetime, avoiding
charging at full power and postponing the start of charging have
been advocated. Such charging profile characteristics can be in, in addicorporated here by judicious selection of
tion to imposing constraint (1).
Specifically, to avoid charging at full power,
can
be selected as
(2)
GATSIS AND GIANNAKIS: DECOMPOSITION ALGORITHMS FOR MARKET CLEARING WITH LARGE-SCALE DEMAND RESPONSE
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so as to encourage larger deviations from
. To postpone the
start of charging, the utility can be chosen as
(3)
increasing in to encourage smaller
at
with weights
at later slots.
the beginning of the horizon, and larger
Example 3: Appliances with desired set points of operation
can also be considered. Here too, the appliance is to be operated
and
, and the utility function is written as the
between
superposition of per-period utilities; that is,
Fig. 1. Power system example featuring 6 buses, 3 generators, 4 aggregators,
and base loads at three of the buses.
(4)
(6g)
Each of these functions attains its maximum at the given desired
operating point, which yields maximum satisfaction or comfort
to the end-user, and can be variable across the horizon. The set
simply takes the form
(6h)
(6i)
(5)
Note that contrary to Example 1, the particular form of
in
(5) does not entail coupling across periods.
is a polyhedron,
It is interesting that in all examples,
meaning that it is described by a set of linear equalities and
inequalities. This turns out to be true for many cases of interest,
including e.g., thermostatically controlled loads; see [19] for
details and additional examples. Further, it is worth mentioning
that user may also have a base load which is not schedulable
but constant; this can be included in the model by simply taking
to be a singleton for a particular .
The aim is to formulate an optimization problem for system
scheduling and dispatch, which takes into account the end-users.
allow for load
In a nutshell, the intertemporal constraints in
allow for load adjustment
shifting, while the utilities
(increase or decrease). It is exactly these features that model
the DR capabilities of residential loads, and these need to be
integrated in the system scheduling.
Following the standard approach in power systems, the focus
here is to maximize the system social welfare for day-ahead
system scheduling. Therefore, the optimization is based on the
DC optimal power flow (OPF), and stands as follows:
The objective in (6a) is to minimize the negative social welfare. Equality (6b) amounts to the per bus balance. Inequalities
(6c) and (6d) are the standard generator output and ramp limit
constraints. Network line flow constraints are accounted for in
(6e). Taking bus 1 as reference without loss of generality, its
bus angle is constrained to zero in (6f). Constraint (6g) gives
lower and upper bounds on the energy provided from aggregators. Equality (6h) amounts to the aggregator balance equation; that is, energy allocated to the aggregator is consumed by
its end-users. Finally, (6i) is the smart appliance constraint. The
, and are
following example clarifies how matrices
constructed.
Example 4: Consider the power system in Fig. 1, whose
topology is an adaptation of the Western System Coordinating
, and
Council system [29]. Noting that
, matrices
and
take the following form:
(7)
With
denoting the reactance of line
has elements [30]
, matrix
(8)
(6a)
(6b)
if there is no line between
where by convention,
, so
buses and . Finally, matrix has dimensions
that if line
connects buses and , the entries
of
are
(6c)
(9)
(6d)
(6e)
(6f)
are expressed in p.u. (per unit), then the rightIf reactances
hand sides of (8) and (9) need to be multiplied with the p.u. base
to have their
(e.g., 100 MVA), in order for matrices and
correct values when used in (6b) and (6e).
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IEEE TRANSACTIONS ON SMART GRID, VOL. 4, NO. 4, DECEMBER 2013
Problem (6) maximizing the system social welfare presents
a principal method for incorporation and coordination of DR
from small-scale users. However, one faces two chief challenges
when it comes to solving (6):
and sets
are private, and
i) Functions
cannot be revealed to the market operator, who would
otherwise be typically responsible for solving (6).
and
were to be made known,
ii) Even if
as variables would render the overall
including all
problem intractable, due to the sheer number of variables.
(Recall that residential consumptions are in the order of
kWh, while generation in the order of MWh.)
The aggregator holds a critical role in successfully coping
with these two challenges through an appropriate optimization
decomposition and an iterative market clearing process, as detailed in the ensuing section.
The dual function is defined as
–
(14)
The dual decomposition method iterates between two steps:
S1) Lagrangian function minimization given the current multipliers, and S2) multiplier update, using the results of the Lagrangian minimization. It is clear from (11) that the Lagrangian
minimizations,
minimization can be decoupled into
where one is performed by the market operator, and each of the
ones by the aggregators.
remaining
index iterations, and
Specifically, letting
, the market operator at iteration
given the multipliers
minimizes
[cf. (12)] subject to the constraints
(6b)–(6g); that is,
III. DECOMPOSITION ALGORITHM
A. Dual Decomposition
Dual decomposition is used here in order to decouple the
problem into simpler ones that will be solved by the market
operator and the aggregators relying on the AMI. Dual decomposition is a general method which finds several applications
in power systems, among others [31]. However, there are two
design choices that must be adapted to the problem at hand: i)
selection of the coupling constraints, with which Lagrange multipliers need to be associated; and ii) an efficient method to update the multipliers.
The only coupling constraints considered will be (6h); the remaining constraints will be kept implicit. Let be the Lagrange
multiplier corresponding to (6h). Then, the (partial) Lagrangian
function is
(10)
Upon straightforward re-arrangements, the Lagrangian function can be written as
(15a)
(15b)
(15c)
(15d)
(15e)
(15f)
(15g)
The last line in (15) emphasizes that the previous optimization
returns the solution denoted as
for
all . It is worth stressing that (15) is a standard DC OPF
problem which includes generator costs, and also “supply
offers” from the aggregators through the objective term
. Therefore, this problem is tractable
using today’s methods.
Turning attention to the remaining Lagrangian minimizaan optimizations, each aggregator must solve per iteration
, and constraints (6i)
tion problem with objective
for all and . It is easily seen from (13) that this problem
decouples per residential end-user, yielding the following
minimization per
(11)
(16a)
where
(12)
(13)
(16b)
for
The last line stresses that the optimization returns
all smart appliances. Optimization problem (16a-16b) is easy
GATSIS AND GIANNAKIS: DECOMPOSITION ALGORITHMS FOR MARKET CLEARING WITH LARGE-SCALE DEMAND RESPONSE
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because it is convex, and can be handled efficiently by the smart
meters installed at the end-user premises. In fact, for most examples of Section II, the problem can be solved in closed form.
needed to this end can be transmitted to
The multipliers
the user’s smart meter using the AMI.
Notice that summing the optimal values of (15) and (16)
, that is,
yields the dual value at
(17)
Having described the Lagrangian minimization subproblems,
the next step is to design a method for updating the multipliers
using the solutions of (15) and (16).
B. Multiplier Update
The choice of the multiplier update method is crucial, because fewer update steps imply less communication between
the market operator and the aggregators. A popular method of
choice in the context of dual decomposition is the subgradient
method, which is typically slow. Possible alternatives exhibiting
faster convergence are methods using multiple subgradients,
such as cutting plane or bundle methods [26, Ch. 7]. For concreteness, two versions of the cutting plane method are adopted
and compared here; one is the standard cutting plane method
(CPM) [26, Sec. 7.2] and the other uses the concept of disaggregated cuts [26, p. 409]. The two methods are reviewed in
the Appendix for completeness, while the present subsection focuses on giving the multiplier update rules, and describing the
implementation of the algorithm. It is worth pointing out that
the CPM with disaggregated cuts is better suited to the problem
at hand yielding faster convergence, because it exploits the fact
that the dual function can be written as a sum of separate terms
[cf. (14)]. Numerical tests in Section IV illustrate differences in
terms of convergence speed.
and
The two methods utilize lower and upper bounds
so that the optimal multipliers lie strictly between these
bounds. In practice, sufficiently small or large numbers can be
selected for the lower and upper bounds, respectively. At iteration , the CPM with disaggregated cuts amounts to solving
the following problem:
(18a)
(18b)
(18c)
(18d)
Fig. 2. Information exchanges between aggregator, market operator, and smart
meter.
where
in (18a) denotes transpose.
The previous problem is typically called the master program,
, and , where
and its variables are the coefficients
the latter are collected in vectors and , respectively, for
and
. One recognizes readily that
(18) is a linear program with special structure due to its constraints; therefore, it can be efficiently solved. Moreover, note
that the solutions of (15) and (16) enter (18) through the terms
and
in (18a), and likewise for
and
in (18d). The main pur, which will be
pose of (18) is to yield the multipliers
used in the next iteration for (15) and (16). These multipliers are
obtained as the optimal multipliers corresponding to (18d). To
obtain these multipliers, problem (18) must be solved by an algorithm that yields not only the optimal solution, but also the
optimal Lagrange multipliers (e.g., primal-dual interior point
,
methods). The algorithm is initialized with arbitrary
which are used in the minimizations (15) and (16).
Problem (18) that yields the updated multipliers
can be solved at the market operator. To this end, the following quantities are needed from each aggregator per iteration
and
. To obtain the
latter, the user’s smart meter must transmit to the aggregator
and
using
the sums
the AMI. The latter among these is the scheduled total power
consumption at period , and the former is a single scalar
number. Then, the total consumption of all users is formed
at the aggregator level as
, and along with the
scalar quantity
, they are both transmitted
to the market operator, in order to solve (18). This information
exchange is depicted in Fig. 2. The upshot here is that the
proposed decomposition and solution method respects user
and
are never revealed.
privacy, as
When applied to (6), the method is guaranteed to converge to
as
. In fact, if (6) is a linear
the optimal multipliers
program, the method terminates in a finite number of steps. This
and
are piecewise linear
is the case when
are polyhedral constraint sets (cf. the
functions, and all
discussion after Example 3).
It is worth noting that the optimal value of (18) is an approximation—in fact an upper bound—of the optimal value
of (6), as explained in the Appendix. Therefore, termination of
the algorithm should in practice be based on the proximity beand the dual value at
tween the current upper bound
, which is an estithe latest Lagrange multipliers
mate of the duality gap. For instance, for a prescribed , termination can be declared when
(19)
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IEEE TRANSACTIONS ON SMART GRID, VOL. 4, NO. 4, DECEMBER 2013
To obtain power generation and consumption schedules, the
following linear combinations need to be formed:
TABLE I
ITERATIVE ALGORITHM. THE ABBREVIATIONS MO, AGG, SM ARE USED FOR
THE MARKET OPERATOR, AGGREGATOR, AND SMART METER
(20a)
(20b)
(20c)
(20d)
These quantities will converge to the optimal solution of (6).
The overall algorithm is summarized in Table I.
The standard CPM operates in a fashion similar to the CPM
with disaggregated cuts. The main difference is the master program, which takes the following form:
(21a)
denoting the optimal multiplier vectors for constraints (6h) and
(6b), it holds that
(21b)
(21c)
(21d)
(22)
Proof: First note that condition
implies that constraint (6g) can be dropped from problem (6)
without altering the optimal solution or the optimal value of the
problem. Next, consider the Lagrangian function for (6) where
constraint (6b) is dualized with multipliers , in addition to (6h)
(cf. (10)). The Lagrangian function takes the following form:
Compared to (18), problem (21) has fewer equations (
versus
) and fewer variables (
versus
). The overall algorithm proceeds as summarized in Table I, where (21) is used in Step 27. In order to
are used
obtain the primal variables, the coefficients
in all linear combinations of (20).
Having described how to optimally solve (6), the next subsection deals with pricing issues.
C. Pricing Considerations
The algorithm of Section III-B returns the optimal Lagrange
for constraint (6h). In addition, the optimal mulmultipliers
tiplier
corresponding to the nodal balance equation (6b) represents the system marginal prices. The two multiplier vectors
are related as explained in the next proposition.
holds
Proposition 1: Suppose that
for the optimal
and for all and . With
and
(23)
After straightforward rearrangements, the Lagrangian becomes
GATSIS AND GIANNAKIS: DECOMPOSITION ALGORITHMS FOR MARKET CLEARING WITH LARGE-SCALE DEMAND RESPONSE
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TABLE II
ARE IN $/MWH; THE REST ARE
GENERATOR PARAMETERS. PARAMETERS
IN MW
(24)
To obtain the dual function, the Lagrangian function must
be minimized with respect to all primal variables, that is,
. The only term in (24) that involves
is the last summand, where
enters linearly, while
is unconstrained. It is well known that the minimum of a linear
unless the function is identically 0. Therefore,
function is
holds, which
the dual function is finite only if
leads to (22)
.
Equation (22) provides an easy way to obtain system prices
. In fact, using the properties of matrix
, (22) imfrom
for all aggregators sitting on bus . When
plies that
applied to the power system of Example 4, Proposition 1 im, and
. It is
plies that
is
also worth mentioning that condition
not restrictive, because the energy provided from the aggregator
will always be nonzero, and a sufficiently large upper bound can
be selected from historical records related to that bus.
The system prices are used to obtain the profits of generators
and payments of the aggregators in the standard way. Specifi,
cally, the profit of a generator will be
where here indicates the bus where generator is situated.
Similarly, aggregator’s payment to the market operator will
be
, as per the price relation (22). Note that
without DR, it is possible to think of aggregators as conventional load-serving entities that need to purchase energy for their
customers, and their demand is inelastic. In this case, they need
, which is not a variable. From an opto purchase fixed
to be
timization-theoretic point of view, allowing for
an optimization variable through DR, enlarges the feasible set,
which in turn leads to higher social welfare. Also the system
prices are expected to be smaller. This implies that the aggregators incur smaller payments, and can afford passing part of
the associated savings on to the end-users. The positive effect
of DR is demonstrated numerically in Section IV.
IV. NUMERICAL TESTS
The effectiveness of formulation (6) and the decomposition
algorithm are illustrated on the system of Fig. 1, where each of
the 4 aggregators serves 1000 residential end-users.
The scheduling horizon consists of twenty-four 1-hour periods, starting from the hour ending at 1 A.M. until the hour
ending at 12 A.M.. The generator cost function has the form
for all and . All generator parameters are
listed in Table II.
The network reactances have values
p.u., at a base of 100
Fig. 3. Upper and lower limits for random residential non-schedulable load;
and system base load. The former limits are scaled to 5 kWh, while the latter to
50 MWh.
TABLE III
PARAMETERS OF RESIDENTIAL APPLIANCES. ALL LISTED HOURS ARE THE
ENDING ONES; W.P. MEANS WITH PROBABILITY
MVA. Flow limits were not imposed, so that congestion effects
are not prevalent.
Each end-user has a PHEV to be charged overnight, and a
pool pump to be operated during the day. Because the night interval (say, 6 P.M. to 6 A.M. of the next day) is split into two
parts at the beginning and at the end of the scheduling horizon,
55% of the PHEV capacity will be charged during the first part,
and the remaining 45% during the second part. Effectively, the
end-user has 3 appliances as in Example 1, while the utility
is selected to be zero. The PHEV parameters are
function
randomly chosen using values from [32], and the pool pump
parameters from [33]; all details are given in Table III. Moreover, each residential end-user has a non-schedulable base load.
It is chosen randomly between upper and lower limits with daily
variation depicted in Fig. 3 and scaled to 5 kWh, following [34,
Sec. 2.2]. The upper bound on each aggregator’s consumption
MW.
was set to
Finally, there is a system total base load depicted also in
Fig. 3, which follows variation of the total MISO actual load
for September 25, 2012, [35], scaled to a peak 50 MW. This
load is equally split among buses 4, 5, and 6, as shown in Fig. 1.
The algorithms of Section III-A are used for the solution, with
, and
for the termination criterion (19). Note that the problem here involves 4,000 end-users,
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IEEE TRANSACTIONS ON SMART GRID, VOL. 4, NO. 4, DECEMBER 2013
Fig. 4. Convergence of the master program objective value and dual value (dein the caption).
noted as
Fig. 6. Total system load with and without DR.
Fig. 7. Marginal prices (Lagrange multipliers).
Fig. 5. Convergence of the Lagrange multipliers.
corresponding to roughly
variables for the
demand side (with the setup of Table III). Fig. 4 depicts the evoand the dual
lution of the master program objective
for the standard CPM and the CPM with
value
disaggregated cuts. Recall that the algorithm terminates when
these values are close, as per (19). It is clearly seen that the standard version requires at least three times more iterations than the
one with disaggregated cuts. The punchline here is that the selection of the multiplier update method is important in order to
ensure faster convergence. Practically, faster convergence implies fewer communication between the market operator and
the aggregators (cf. Table I). Fig. 5 depicts the Lagrange mul, also illustrating differences in convertiplier sequence
gence speed.
The results of the algorithm are compared to a case where
there is no DR. To obtain the results for this case, the power
consumption from all adjustable appliances is derived as
(25)
which shows the consumption is distributed uniformly over the
allowable operation interval. The total power consumption is
then computed from (6h), and the resultant values of
are
used to solve (6a)–(6f).
Fig. 6 depicts the resultant total system load, where it is
shown that accounting for residential DR offers the benefit of
smoothing out the total load, and in particular it reduces the
peak load by over 10 MWh. The effect on prices is illustrated
in Fig. 7. Because there are no congestion effects, all buses
have the same ’s, which are depicted in Fig. 7. There are two
periods where the system marginal price is higher without DR
than with DR, namely the periods 5 A.M. to 6 A.M. and 8 P.M.
to 9 P.M.. The reason is that the higher system load at those
two periods requires additional generation. To better illustrate
this situation, the power production per generator is depicted
in Figs. 8 and 9. For instance, for the period 8 P.M.–9 P.M.,
Generators 1 and 2 already operate at their capacity in Fig. 9.
Generator 3 also contributes to support the load, setting the
price at 50 $/MWh, in contrast to the situation in Fig. 8.
The previous remarks are further substantiated by Tables IV
and V, where the various costs and payments are listed. Specifically, the generation costs and profits with and without DR are
given in Table IV. It is immediately seen that the generator costs
are smaller when DR is accounted for. On the other hand, the
generator profits are smaller. This is explained by the fact that
the system marginal prices are smaller with DR. Table V lists
the aggregator payments to the market. It is clearly seen that the
total payments are over $3000 less with DR than without DR.
Part of these savings can be passed on to the end-users through
appropriate pricing and rebate schemes.
Finally, the load factor for the two scenarios is evaluated (see
[34, Sec. 2.2] for definition). With DR, the load factor is 0.7039,
while without DR it drops to 0.6435. A load factor closer to 1
means smoother total load (smaller peak and higher valleys).
GATSIS AND GIANNAKIS: DECOMPOSITION ALGORITHMS FOR MARKET CLEARING WITH LARGE-SCALE DEMAND RESPONSE
1985
method relies on dualizing only the aggregator balance equality
constraint in order to separate problems for the market operator and each aggregator. In a nutshell, the approach has the following desirable characteristics: a) it allows the market operator
to integrate DR resources in a large-scale fashion; b) it admits a
scalable distributed solution tapping into the two-way communication network in smart grids; and c) it captures end-user preferences while respecting privacy concerns. It is interesting to
pursue extensions where residential DR is incorporated in joint
energy and reserve markets, and also in market clearing formulations that include deterministic or stochastic security constraints.
Fig. 8. Generator output with DR.
APPENDIX
REVIEW OF CUTTING PLANE METHODS
The purpose of this Appendix is to provide a short review
on cutting plane methods, in order to motivate the ideas of
Section III; see [26, Ch. 7] or [36, Ch. 7] for detailed discussions.
Consider the following prototype convex optimization with
linear constraints:
(26a)
(26b)
(26c)
Fig. 9. Generator output without DR.
TABLE IV
COSTS AND LOAD FACTOR WITH AND WITHOUT DR
Problem (26) is separable, in the sense that the objective and
constraints are sums of terms, and each of these terms depends
on different optimization variables. Constraint (26b) correcaptures constraints (6b)–(6g), while
sponds to (6h). Set
, corresponds to (6i).
Dualizing constraint (26b) with multiplier vector , the dual
where
is
function is written as
defined as the minimum of the partial Lagrangian function
(27)
TABLE V
AGGREGATOR PAYMENTS WITH AND WITHOUT DR
The dual problem is to maximize the dual function with respect
to the Lagrange multipliers:
(28)
This desirable effect is therefore enabled here by the incorporation of DR, and in particular, by the user load intertemporal
shifting availability.
V. SUMMARIZING REMARKS AND FUTURE DIRECTIONS
This work is motivated by the vision to incorporate residential DR in a large scale. Accounting for the intertemporal constraints, as well as user scheduling preferences and satisfaction,
leads to social welfare maximization. The dual decomposition
where strong duality is supposed to hold here. Let
and
be lower and upper bounds so that an optimal solution
.
of (28) is included in the region
Cutting plane methods in general tackle (28) by solving a
sequence of problems, where each problem is to maximize a
piecewise linear overestimator of
.
Specifically, suppose that the method has so far generated the
after steps. Let
be the primal miniiterates
mizer in (27) corresponding to
. Observe that the vector deis a subgradient of function
fined as
at point
, and it therefore holds that
(29)
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IEEE TRANSACTIONS ON SMART GRID, VOL. 4, NO. 4, DECEMBER 2013
Clearly, the right-hand side of (29) is a linear overestimator
. The minimum of the right-hand side of (29) over
is a concave and piecewise linear overestimator
. The CPM with disaggregated cuts maximizes the sum
of
of these overestimators. Mathematically, the problem can be expressed as
of
(30a)
(30b)
(30c)
(30d)
The solution of this problem yields the next iterate
,
for reasons that will
while its optimal value is denoted as
be clear shortly.
The purpose of adding constraint (30c) is to ensure that (30)
has an optimal solution. Notice that (30) is a linear program,
for which one can solve equivalently its dual problem. To this
denote Lagrange multipliers corresponding to
end, let
(30b), and
corresponding to (30c). The dual function can
be written after rearrangements as
(31)
The dual problem is to minimize (31) over the Lagrange multipliers
, and .
It holds by definition that
. Moreover, because the maximization over and
is unconstrained, the dual function
is finite
only when all terms in parentheses in (31) are zero. Taking into
account the previous considerations, the dual problem of (30)
takes the form
(32a)
(32b)
(32c)
(32d)
Problem (32) corresponds exactly to the master program (18).
Assigning Lagrange multipliers to (32c), it is not hard to verify
that the dual of (32) is (30). This fact confirms that the next
can be obtained either as solution of (30), or
iterate
as optimal Lagrange multipliers corresponding to (32c).
Since (30) is obtained as an overestimator of
cause of (28), it is deduced that
, and be(33)
which forms the basis for the termination criterion in (19), and
also reveals that (30) and (32) yield an upper bound on .
Finally, the standard CPM does not consider overestimators
, but rather for the entire
. Confor every summand
straint (30b) is replaced by
(34)
and the objective is to maximize . Assigning multipliers
to (34), one is led to (21). The punchline is that the CPM with
disaggregated cuts takes advantage of the separability of (26).
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Nikolaos Gatsis (S’04) received the Diploma degree
in electrical and computer engineering from the
University of Patras, Patras, Greece, in 2005 with
honors. He received the Ph.D. degree in electrical
engineering with minor in mathematics from the
University of Minnesota, Minneapolis, MN, USA,
in 2012. His research interests are in the areas of
smart power grids, renewable energy management,
wireless communications, and networking, with an
emphasis on optimization methods and resource
management.
Georgios B. Giannakis (F’97) received his Diploma
in electrical engineering from the National Technical
University of Athens, Greece, in 1981. From 1982
to 1986 he was with the University of Southern
California (USC), Los Angeles, CA, USA, where he
received his M.Sc. degree in electrical engineering
in 1983, M.Sc. degree in mathematics in 1986, and
Ph.D. degree in electrical engineering in 1986.
Since 1999 he has been a professor with the University of Minnesota, Minneapolis, MN, USA, where
he now holds an ADC Chair in Wireless Telecommunications in the ECE Department, and serves as director of the Digital Technology Center. His general interests span the areas of communications, networking, and statistical signal processing—subjects on which he has published
more than 350 journal papers, 580 conference papers, 20 book chapters, two
edited books, and two research monographs (h-index 104). Current research focuses on compressive sensing, cognitive radios, cross-layer designs, wireless
sensors, social and power grid networks. He is the (co-) inventor of 21 patents
issued, and the (co-) recipient of 8 best paper awards from the IEEE Signal Processing (SP) and Communications Societies, including the G. Marconi Prize
Paper Award in Wireless Communications. He also received Technical Achievement Awards from the SP Society (2000), from EURASIP (2005), a Young Faculty Teaching Award, and the G. W. Taylor Award for Distinguished Research
from the University of Minnesota. He is a Fellow of EURASIP, and has served
the IEEE in a number of posts, including that of a Distinguished Lecturer for
the IEEE-SP Society.