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IEEE TRANSACTIONS ON SMART GRID, VOL. X, NO. X, XX 2016 1
Optimal price-energy demand bids for aggregate
price-responsive loads
Javier Saez-Gallego, Mahdi Kohansal, Student Member, IEEE, Ashkan Sadeghi-Mobarakeh, Student
Member, IEEE, and Juan M. Morales, Senior Member, IEEE
Abstract—In this paper we seek to optimally operate a retailer
that, on one side, aggregates a group of price-responsive loads and
on the other, submits block-wise demand bids to the day-ahead
and real-time markets. Such a retailer/aggregator needs to tackle
uncertainty both in customer behavior and wholesale electricity
markets. The goal in our design is to maximize the profit
for the retailer/aggregator. We derive closed-form solutions for
the risk-neutral case and also provide a stochastic optimization
framework to efficiently analyze the risk-averse case. In the latter,
the price-responsiveness of the load is modeled by means of a non-
parametric analysis of experimental random scenarios, allowing
for the response model to be non-linear. The price-responsive load
models are derived based on the Olympic Peninsula experiment
load elasticity data. We benchmark the proposed method using
data from the California ISO wholesale electricity market.
Index Terms—Price-energy bidding, demand response, elec-
tricity market, smart grid, data-driven.
NOTATIO N
The main notation used throughout the paper is stated below
for quick reference. Other symbols are defined as required.
A. Indexes and sets
tTime period t∈ {1,2, . . . 24}.
bBidding block b∈ {1,2,...B}.
wRealization of the stochastic variables, represented as
scenarios w={1,2,...N}.
B. Input stochastic processes
XLoad.
ΛDDay-ahead price.
ΛRReal-time price.
ΠRetail price.
C. Decision variables
XDStochastic process representing scheduled energy in
the day-ahead market.
xD
t,w Scheduled energy in the day-ahead market for time
tand scenario w.
J. Saez-Gallego, J. M. Morales (corresponding author) are with the
Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark (email
addresses: {jsga, jmmgo}@dtu.dk), and their work is partly funded by
DSF (Det Strategiske Forskningsr˚
ad) through the CITIES research center
(no. 1035-00027B) and the iPower platform project (no. 10-095378). M.
Kohansal and A. Sadeghi are with the Department of Electrical and Computer
Engineering, University of California, Riverside, USA. E-mail: {mkohansal,
asadeh004}@ece.ucr.edu. Their work is supported in part by the United States
National Science Foundation grants 1253516, 1307756, and 1319798.
ut,b Price bid for time tand block b.
Remark: a subscript tunder the stochastic processes indicate
the associated random variable for time t.
D. Parameters
φwProbability of each scenario w.
xt,w Load at time tand scenario w.
λD
t,w Day-ahead price at time tand scenario w.
λR
t,w Real-time price at time tand scenario w.
πt,w Retail price at time tand scenario w.
CbWidth of energy block b.
LFraction of the load that must be purchased in the
day-ahead market.
βProbability of occurrence of chance constraint.
I. INTRODUCTION
With the increasing deployment of smart grid technologies
and demand response programs, more markets around the
world are fostering demand bids that reflect the response of
the consumers to changing electricity prices [1], [2]. In this
paper, we consider the case of a retailer who procures energy
to a pool of consumers in a typical two-settlement electricity
market, as for example, the California wholesale electricity
market CAISO [3]. The retailer submits price-energy demand
bids to the day-ahead market, and only energy quantity bids to
the real-time market in order to counterbalance the deviations
from the scheduled day-ahead energy market to the actual load.
The possibility of arbitrage is indirectly allowed depending on
the submitted bid to the day-ahead market and the realization
of the stochastic processes affecting the problem.
We assume that the load is price-responsive, in the sense
that it may change depending on the price of electricity
during the considered period. The retailer passes the retail
price onto her consumers, who react accordingly. We do not
make any assumption about the means that the consumers
use to adjust their consumption based on the retail price,
because the proposed methodology relies on historical data
of aggregate load and retail price to estimate the relationship
between them. Also because of this, we do not need to make
any assumption on the nature of the load that the retailer
aggregates. Furthermore, we consider that the retail price is
directly linked to the market price and prefixed beforehand, for
example, as a prearranged percentage of the day-ahead price.
Therefore, the retail price is out of the retailer’s control (in the
short term at least). Finally, note that the communication flow
between the retailer and her consumers is one-directional: the
IEEE TRANSACTIONS ON SMART GRID, VOL. X, NO. X, XX 2016 2
price is communicated to the consumers by the retailer, who,
in turn, observes the aggregate load.
The contributions of the paper are summarized as follows:
•An analytic solution to the problem of finding opti-
mal block-wise price-energy demand bids in the day-
ahead market when risk is not considered. Moreover,
we propose a mixed-integer linear programming solution
approach to the risk-averse case.
•The dynamic price-responsive behavior of consumers is
modeled based on scenarios. The conditional probability
of the load given a certain retail price trajectory is
estimated using a non-parametric approach.
•We assess the practicality of the proposed methodology
by using data from a real-world experiment.
The estimation of demand bids has been extensively studied
in the past years [4, ch. 7]. Several papers share the common
goal of estimating price-energy bids relative to specific types
of load, for example, time-shiftable loads [5], electric vehicles
[6] and thermostatically-controlled loads [7]. Our methodology
differs with those in the fact that we do not make any
assumption on the nature of the load. Methodologies based on
forecasting tools [8], [9] generally do not make assumptions
on the type of price-responsive load either, but, on the other
hand, do not tackle the bidding problem.
Besides the works on forecasting, another group of papers
focus on finding the optimal bid for generic loads. The
work in [10] elaborates on a robust bidding strategy against
procurement costs higher than the expected one, considering
uncertainty in the prices only. Uncertain prices and demand
are taken into account in [11] but minimizing imbalances and
disregarding the economic side of the bidding. Our approach
resembles that of [12] with the main differences being that
we use data to estimate the price-response of the dynamic
load, and that we consider energy-block bidding in a one-
price balancing market as the US CAISO [3]. Authors of
[13] consider, from the theoretical point of view, the problem
of allocating a deterministic load by deciding which fraction
should be purchased in the day-ahead market and which in
the real-time market. Finally, authors of [14] study demand
curves in an arbitrage- and risk-free situation by using a game
theory.
Regarding the generation of scenarios of the stochastic
processes, our methodology is inspired from [15]–[17]. From
the application point of view, our approach differs in the
final goal, as they deal with wind energy production. To our
knowledge, there is no previous work that characterizes the
dynamic price-responsive load with a set of scenarios. From
the methodological point of view, our approach differs with
the existing literature in the estimation of the conditional
distribution of the price-responsive load, taking into account
the full trajectory of the day-ahead price. This enables us to
capture the full dynamics of the load across the hours of the
next operational day. The real-time price is modeled in an
analogous manner. In both cases, we model their distributions
using a non-parametric approach that allows for non-linear
responses to a given day-ahead price trajectory.
The paper is structured as follows. In Section II we intro-
duce the retailer’s bidding problem. Section III provides the
analytic solution to the risk-neutral case. In Section IV we
formulate the stochastic optimization model for solving the
bidding problem with risk constraints. Section V elaborates
on the scenario-generation technique. Next, in Section VI we
analyze results from the bidding problem under the generated
scenarios. Finally, in Section VII we draw conclusions and
implications.
II. PRO BL EM FORMULATION
Consider a utility retailer/aggregator that seeks to maximize
its profit based on the revenue that it collects from its loads, the
payments it makes to the day-ahead market, and the payments
it makes or receives in the real-time market. Mathematically
speaking, we need to solve the following optimization pro-
blem:
Maximize
XD
t,ut,b
E24
X
t=1 ΠtXt−ΛD
tXD
t−
ΛR
tXt−XD
t(1a)
subject to
XD
t=
B
X
b=1
CbI(ut,b ≥ΛD
t)∀t, b (1b)
ut,b+1 ≤ut,b ∀t, b = 1 . . . B −1(1c)
PXD
t∈[(1 −L)Xt,(1 + L)Xt]≥β∀t(1d)
λ≤ut,b ≤λ∀t, b (1e)
where I(·)is the 0-1 indicator function.
The objective function (1a) is the expected total daily profit,
composed of three terms. The first term represents the revenue
that the retailer makes form selling energy to the consumers
at the retail price. The second term represents the cost of
purchasing energy from the day-ahead market. The third term
accounts for the cost/revenue of purchasing/selling energy
from/to the real-time market. The energy purchased or sold
in the real-time market is equal to the difference between the
purchased quantity at the day-ahead market and the realized
load, i.e., Xt−XD
t.
Constraint (1b) defines the scheduled energy in the day-
ahead market to be equal to the sum of the width of the blocks
of energy which have a price-bid higher than the market price.
In other words, blocks of energy will be purchased if their
price-bid is higher or equal to the day-ahead price. Note that
ut,b is the decision variable which determines the shape of the
submitted bidding curve to the day-ahead market.
Constraint (1d) models the risk-aversion of the retailer
through two parameters. Parameter Lrepresents the maximum
fraction of the load that can be procured in the real-time
market. This parameter could be defined by the retailer, but
could also be constrained by the ISO as a way to avoid
putting too much pressure on the real-time market, this way
safeguarding and prioritizing the security and stability of the
power system. Values of Lclose to 1 indicate that the full
amount of the load can potentially be bought in the real-time
market. On the other hand, as Ldecreases, we give priority
to purchasing energy in the day-ahead market. Parameter β
IEEE TRANSACTIONS ON SMART GRID, VOL. X, NO. X, XX 2016 3
indicates the minimum probability with which the constraint
(1d) must be fulfilled. Values of βclose to 1 indicate a hard
constraint, while lower values of βindicate that the constraint
is loose. The parameter βcan be interpreted as the aversion of
the retailer towards purchasing a certain fraction of the load in
the day-ahead market. Low values of βcan be interpreted as a
sign that the retailer seeks to profit from arbitrage rather than
from serving the load. As we show in the case study, higher
values of βyield lower expected profit but also lower risk.
Note that, for large Land small β, constraint (1d) becomes
irrelevant, indicating the neutrality of the retailer towards risk.
Constraint (1c) ensures that the estimated bidding curve is
monotonically decreasing which is a typical requirement in
electricity markets. Finally, constraint (1e) set lower and upper
bounds to the price bids, which are given by the market rules
[18]. All in all, the expected profit depends on the decision
variable “price-bid” and also on the realization of the input
stochastic variables.
The maximum number of blocks that is allowed depend on
the market rules [18] as well. The width of each block Cb
must be set by the retailer depending on the magnitude of the
load.
As in practice, here we assume that the retail price is given
exogenously, in other words, it is not a decision variable of the
retailer. The main driver for this consideration is the fact that
the retail price must, to a certain extent, represent the true cost
of electricity. This might not always be the case if the retail
price is subject to the will of the retailer. As a consequence, the
retailer’s bidding strategy does not directly affect the behavior
of the load, since the behavior of the load depends on the retail
price and other factors such as the weather conditions. Another
implication is that only the profit of the retailer is affected by
her bidding strategy and the realized market prices.
III. CLO SE D-FORM ANALYTICAL SOLUTION IN ABSENCE
OF RI SK CONSTRAINTS
In this subsection we elaborate on the closed-form analytic
solution to problem (1), when the risk constraint (1d) is
disregarded, or equivalently, when L→ ∞ and/or β= 0.
The retailer’s bidding problem (1) can be decomposed by
time period, so that 24 smaller optimizations problems can be
solved instead, one for each time t.
In the risk-neutral case, each of these smaller optimization
problems writes as follows:
Max.
XD
t,ut,b EXtΠt−ΛR
t−EXD
tΛD
t−ΛR
t(2)
subject to (1b), (1c) and (1e). The advantage of reformulation
(2) is that we can perform simpler optimization problems in
parallel. Note that the first term of (2) is constant with respect
to the decision variables ut,b and XD
t, whereas the last term
is not. Hence, both the stochastic load Xtand the retail price
Πtcan be dropped out of the optimization problem (1) in this
case. Interestingly, this implies that, in the risk-neutral case,
the retailer’s optimal bidding strategy is not affected by the
price-responsive nature of the load.
Next we analyze the case when ΛDand ΛRare statistically
independent. Results are presented in Theorem 1. For ease of
reading, and given that the maximization problem (2) can be
decomposed per time period, we drop the time index tin the
remaining of this section.
Theorem 1: The optimal price bid u∗
bin problem (2), when
the day-ahead and real-time prices are independent, is equal
to the expected value of the real-time price.
The proof of Theorem 1 is given in Appendix A. Theorem 1
also shows that, given the risk-neutral setup and independent
prices, we do not obtain extra benefit from bidding a curve
instead of a single price-quantity bid.
The assumption of statistically independent prices is not
necessarily fulfilled in practice (see, for example, [19, Fig. 1]).
For this reason, in Theorem 2 below, we provide the analytic
solution to problem (2) when ΛDand ΛRare statistically
dependent.
Theorem 2: A global optimum solution to problem (2)
satisfies that the price bids for all blocks is equal to u∗.
Moreover, u∗is equal to either λ,λ, or EΛR|ΛD=u∗
with d
du EΛR|ΛD=u∗<1in the latter case.
The proof of Theorem 2 is given in Appendix B. One could
interpret the result of Theorem 2 in the following way: the
optimal price bid will be the one for which price consistency
holds, namely, for which the expected real-time price is equal
to the day-ahead price. A second conclusion drawn from
Theorem 2 is that the maximum profit is achieved with the
same price-bid for each block. If there is more than one price
bid that maximizes the expected profit (i.e., several global
maxima), then the price bid for each block can be chosen
indistinctly between them. Similarly as with Theorem 1, we
do not obtain extra benefit from bidding a curve when prices
are dependent.
From a practical point of view, Theorem 1 and 2 allow us
to simplify the demand curve to a simple price-quantity bid.
By taking into account this implication, we can obtain the
optimal price bid in the case when the distributions of prices
are discrete, which allow us to compute the optimal price bid
when the uncertainty is modeled by scenarios. The optimal
price bid can be chosen by evaluating the profit in the local
maxima, which are characterized according to the following
remark:
Remark 2: Given a discrete set of scenarios for the
day-ahead and real-time prices, let us consider the re-
ordered pair of terms {λD
w,EΛR|ΛD=λD
w}such that
λD
w≤λD
w+1. Local maxima1are achieved at the stationary
points u∗=λD
wsuch that λD
w≤EΛR|ΛD=λD
wand
λD
w+1 >EΛR|ΛD=λD
w+1.
Note that, due to market rules, the price bid have a maxi-
mum and minimum allowed values. In practice, one needs to
check also if the maximum profit is achieved when the price
bid is equal to one of its bounds. Using Remark 2 one can
find the optimal price-bid by just performing a finite set of
simple calculations.
As a final remark, it is noteworthy to say that the results
from Theorem 1 and 2 show that the solution to (2) does not
depend on the retail price, neither on the load. From a practical
point of view this means that the risk-neutral retailer acts as
1The proof is available upon request.
IEEE TRANSACTIONS ON SMART GRID, VOL. X, NO. X, XX 2016 4
a financial trader, making profit by selling and buying energy
in both markets.
IV. SCENARIO-BAS ED SOLUTION IN PR ES EN CE O F RIS K
CONSTRAINTS
In this section we present a solution to problem (1) using
a scenario-based approach. The input for every time tis a set
of Nscenarios, each one characterized by a realization of the
retail price πt,w, the day-ahead price λD
t,w, the real-time price
λR
t,w, and the load xt,w . Each scenario has a probability of
occurrence of φw.
We reformulate constraint (1b) by adding a binary variable
yt,w,b. Then, constraint (1b) is replaced by:
xD
t,w =X
b
yt,w,bCb∀t, w
ut,b −λDA
t,w ≤Myt,w,b ∀t, w, b
−ut,b +λDA
t,w ≤M(1 −yt,w,b)∀t, w , b
yt,w,b ∈ {0,1} ∀t, w, b
(3)
where M is a large enough constant. The equations above
imply that yt,w,b = 1 if ut,b ≥λD
t,w and 0 otherwise.
Next, we reformulate constraint (1d) by adding two extra
binary variables. We first define zt,w = 1 if xD
t,w ≤(1 −
L)xt,w, and zt,w = 0 otherwise. Secondly, we define zt,w = 1
if xD
t,w ≥(1 + L)xt,w, and zt,w = 0 otherwise. Consequently,
the chance constraint (1d) can be replaced by the following
set of equations:
xD
t,w −(1 −L)xt,w ≤M(1 −zt,w)∀w
−xD
t,w + (1 −L)xt,w ≤Mzt,w ∀w
xD
t,w −(1 + L)xt,w ≤Mzt,w ∀w
−xD
t,w + (1 + L)xt,w ≤M(1 −zt,w)∀w
1
NX
wzt,w +zt,w≤1−β .
(4)
All in all, taking into consideration the reformulations
presented above, the optimal price-bid is found by maximizing,
for every time t,
Maximize
xD
t,w,ubX
w
φwπt,wxt,w −λD
t,wxD
t,w−
λR
t,w(xt,w −xD
t,w)(5)
subject to (1c), (1e), (3), and (4).
V. SCENARIO GENERATI ON
In this section we elaborate on the modeling of the stochas-
tic variables by scenarios. The proposed approach to generate
scenarios has several advantages. First, we do not need to make
any assumption on the type of price-responsive load we model.
The response of the load to the price is directly observed in
the data and modeled by a non-parametric distribution. For
this very same reason, the response of the load to the price is
allowed to be non-linear. Second, it is a fast approach, hence,
big datasets can be quickly processed. Finally, the proposed
approach is adequate for bidding purposes, since forecasting
the load is not the main goal of the paper but rather account
for its uncertainty in order to make an informed decision.
Each scenario is characterized by a 24-long sequence of
day-ahead prices, real-time prices, retail prices and observed
load. The proposed method to approximate their joint distri-
bution is summarized as follows. First of all, we model the
marginal distribution of the day-ahead price. Note that the day-
ahead price is not dependent on the real-time price, neither on
the bid of a small price-taker consumer. Second, we model the
distribution of the load conditioned on the retail price using
a non-parametric approach. Lastly, we model the distribution
of the real-time price conditioned on the day-ahead price. The
real-time price depends on the day-ahead price, but not on the
load of a price-taker retailer.
The rest of this section is organized as follows. First, in
Section V-A, we briefly elaborate on the technique to generate
scenarios of day-ahead price. Then, for each scenario of day-
ahead price, we generate conditional scenarios of real-time
price and load in Section V-B.
A. Day-ahead Price Scenarios
The first step in the scenario generation procedure is to
model the day-ahead price using an Autoregressive Integrated
Moving Average model (ARIMA). We choose the most ad-
equate model according to the AICc criteria [22]. Using the
estimated model, we draw scenarios using the methodology
explained in [15]. Because the scenarios are used in day-
ahead trading, they are generated in a rolling horizon manner
everyday at 12:00 with a lead time of 13 to 36 hours.
B. Load and Real-time Price Scenarios
In this section, we elaborate on the proposed methodology
to draw scenarios from the distribution of load conditioned
on the retail price. The methodology to generate conditional
real-time price scenarios is analogous, hence, we omit it for
brevity.
For this subsection, we consider a scenario of day-ahead
prices ˜
λD={λD
1, . . . , λD
24}that is generated using the metho-
dology explained in Section V-A. Under the considered setup,
as explained in the sections above, the retail price is given
exogenously. In the case study, we assume the retail price to
be proportional to the day-ahead price, that is, Π = kλD.
Therefore a scenario of retail price is directly specified from
a scenario of day-ahead price.
The procedure outlined next allows us to weigh the his-
torical trajectories, such that trajectories with a retail price
“closer” to the given retail price ˜πweigh more. These weights
are used later in this section to compute the conditional density
function of the load, given ˜π. To begin with, we define π(j)as
the 24-long vectors of retail price, with each element referring
to an hour of the day, and with jreferring to the index of the
historical day considered. Then, we compute the Euclidean
distance d(j)=||π(j)−˜π||. In this way, we “summarize”
each historical price trajectory π(j)with a single value, so that
trajectories “closer” to the given retail price ˜πhave a lower
distance. Next, we use a Gaussian kernel to weight trajectories,
IEEE TRANSACTIONS ON SMART GRID, VOL. X, NO. X, XX 2016 5
90 100 110 120 130 140 150
0.0 0.1 0.2 0.3 0.4
Distance (kWh)
Weight
●Load
K = 0.5
K = 1
K = 10
Fig. 1. The weights of the historical retail price trajectories are shown against
their distance to the price reference.
such that the weights are equal to w(j)0=f(d(j)), where fis
the probability density function of a normal distribution with
mean 0 and standard deviation σf. For the case study, we used
σf=Kσd, meaning that the standard deviation for fis equal
to the standard deviation of the distances σd, multiplied by
abandwidth parameter K. Finally, we normalize the weight
w(j)=w(j)0
Pw(j)0so that their sum is equal to 1.
The effect of the bandwidth parameter Kover the weights
can be seen in Fig. 1. On its x-axis, we represent d(j)and on
the y-axis the weights w(j). A smaller bandwidth penalizes
price references further away. This is the reason why, when
K= 0.5, there are few scenarios with a weight significantly
greater than zero. On the other hand, when K= 10, all
scenarios weigh similarly.
The procedure to generate each scenario is inspired from
[15] and [16]. In short, we first transform the load data to
a normal distribution using a non-parametric transformation.
Then, we compute its covariance, and finally, generate random
correlated Gaussian errors that are transformed back to the
original distribution. The procedure consists of the following
seven steps:
1) For each hour of the day, we compute a non-parametric
estimation of the density of the price-responsive load [23]
conditional on a retail price trajectory ˜π. We do this by
computing the kernel density estimator at hour twith the
weights w(j)in the following way:
ˆ
ft(x|˜π) = 1
J
J
X
j=1
w(j)Gh(x−x(j)
t),(6)
where Gh(x)is a kernel (non-negative function that
integrates to one and has zero mean), his its bandwidth,
and x(j)
tis the observed load at time tand day j. An
example of a estimated density using a Gaussian kernel
is shown in Fig. 2, for different values of Kand same
h. For Kclose to zero (K= 0.5in the case study),
the weighting gives relatively high importance to few
observations, therefore, the estimated density is more
localized around them.
2) Using ˆ
ft(x|˜π)from Step 1, we compute the cumulative
density function, called ˆ
Ft(x|˜π).
80 100 120 140 160
0.00 0.02 0.04 0.06
Load (kWh)
Density
●● ●●●● ● ● ●●●● ●●● ●●●●●●● ●●● ● ● ●● ●● ●●●●●●● ●●● ●●● ●● ● ●●● ● ●●●● ●● ●●●● ●●
●Load
K = 0.5
K = 1
K = 10
Fig. 2. The estimated conditional distribution of the load given the retail
price is shown for different values of the bandwidth parameter K.
3) The transformed load values y(j)
t=ˆ
Ft(x(j)
t|˜π)for
every hour tfollow a uniform distribution U(0,1). Then,
we normalize the load data through the transformation
z(j)
t= Φ−1(y(j)
t), where Φ−1(Y)is the probit function.
Consequently, (z(1)
t, . . . , z(J)
t)≡Zt∼N(0,1).
4) We estimate the variance-covariance matrix Σof the
transformed load Z, relative to the 24 hours of the day.
One could do it recursively as in [15].
5) Using a multivariate Gaussian random number generator,
we generate a realization of the Gaussian distribution
˜
Z∼N(0,Σ).
6) We use the inverse probit function to transform ˜
Zto a
uniform distribution, that is, ˜
Y= Φ( ˜
Z).
7) Finally, we obtain a scenario of load by transforming back
˜
Yusing the inverse cumulative density function from step
2, that is, ˜xt=ˆ
F−1
t(˜
Yt|˜π),∀t. Numerically, we use a
smoothing spline to interpolate ˆ
F−1
t(˜
Yt|˜π).
The procedure outline above generates a scenario of load
conditioned on the retail price. Steps 5 to 7 are repeated as
many times as needed if more scenarios of load per retail price
are desired.
VI. CA SE ST UDY
In this section we first introduce the datasets and the
generated scenarios using the methodology from Section V.
Then, in Section VI-B, we analyze in detail the solution
of the bidding model with and without considering risk.
Afterwards, in Section VI-C, we benchmark the performance
of the proposed models and present the final conclusions.
A. The Data and Practical Considerations
The scenarios of day-ahead and real-time prices are gen-
erated using historical hourly values from CAISO [3]. We
use three months of training data, from August to October
2014. The test period spans over November 2014. For the retail
price and price-responsive load, we use data from the Olympic
Peninsula experiment [21]. In this experiment, the electricity
price was sent out every fifteen minutes to 27 households that
participated in the experiment. The price-sensitive controllers
and thermostats installed in each house decided when to turn
IEEE TRANSACTIONS ON SMART GRID, VOL. X, NO. X, XX 2016 6
30 40 50 60 70
Hour
Day−ahead price ($)
0 4 8 12 16 20
●●●●●●●●●●●●●●●
●●●
●●●●●●
●Predicted price
Actual price
Scenario
Fig. 3. Actual price, point forecast and generated scenarios for the day-ahead
price.
on and off the appliances, based on the price and on the house
owner’s preferences. The training and test months are the same
as for the CAISO data, but relative to year 2006.
Some practical considerations need to be addressed. Firstly,
that the day-ahead price and the retail price come from two
different datasets. For this reason, prices are normalized. The
second practical consideration is that we assume Π = kΛD
with k= 1 even though it is not fulfilled in practice. However,
this does not affect the comparison of the proposed models,
due to the fact that we use the same set of scenarios for the
benchmark and for all the models. This issue could be solved
in future work when data from new experiments becomes
available.
Throughout the case study, we set a total of 20 blocks, where
the width is equality distributed between a maximum and a
minimum bidding quantities, set to be equal to the historical
range of the scenarios at every hour. They are represented by
the dotted lines in Fig. 6(b).
For the case study we use a total of 150 scenarios. For
the estimation of the densities, we use a Gaussian kernel
with a bandwidth hgiven by Silverman’s rule of thumb [23].
Also, the bandwidth parameter is set to K= 0.5. For the
model of the day-ahead price we use an ARIMA(3,1,2)(1,1,1)
with a seasonal period of 24 hours. The Root Mean Square
Error (RMSE) for the model of the day-ahead price (13 to
36 lead hours) is, on average, $3.22, which is in line with
the forecasting performance that other authors have achieved
using similar methods [24].
A subset of the generated scenarios of day-ahead price is
given in Fig. 3. By graphical inspection we conclude the
scenarios of day-ahead price are a plausible representation of
the actual day-ahead price and its uncertainty.
B. Model Analysis
To begin with, we discuss the results from the risk-neutral
model (2). The solution to this model, for a given set of
scenarios, is calculated either using Remark 2 or by solving
(5) with β= 0. In Fig. 4, we show the scenarios of retail
price and load in dots for hour 20 of November 1st, and hour
2 of November 2nd. The estimated bidding curve is displayed
as a dashed green line. In accordance with Theorem 2, the
resulting bidding curve is flat.
16 18 20 22 24
25 35 45
Hour 2
Load
Price ($)
35 40 45 50 55
30 50 70 90
Hour 20
Load
Price ($)
●
LRisk UncRisk Scenarios
Fig. 4. The left figure is relative to the 2nd of November, while the right
figure is relative to the 1st of November.
100 200 300 400
β
Exected profit ($)
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4
L
Exp. profit
Frontier line
Fig. 5. On the left axis, in dashed red, the expected profit for every β, with L
being in the feasibility frontier line. On the right axis, the feasibility frontier
line is shown for combinations of βand L.
Next, we discuss the results from the risk-averse model (5).
We start by analyzing the effect of the risk parameters Land
βon the expected profit and the feasibility of the problem. In
Fig. 5 we show, on the right axis, the feasibility frontier plot
for Land βand its standard deviation in a shadowed area.
We calculated it empirically, using data relative to the 1st of
November. The combinations of Land βshown below the
displayed dark line result, on average, in an infeasible solution.
The frontier line is dependent on the scenarios of load: higher
variability in the scenarios of load will require a greater value
of Lfor the problem to be feasible. On the left axis of Fig.
5, we show the expected profit for the risk-averse problem,
with the combination of βand Lthat lay on the frontier line.
Naturally, the highest profits are achieved for low values of
β, that is, when the retailer is less risk averse. From now on,
we set the risk parameter βto 0.8. The value of Lis chosen
from the frontier plot, to be as small as possible.
Fig. 6(a) shows the scenarios of day-ahead price (continuous
lines), together with the estimated optimal price bids (horizon-
tal segments), for each hour. We observe that the magnitude
of the price bid depends on the scenarios of day-ahead price.
In Fig. 6(b), we show the amount of energy bought in the day-
ahead market for each scenario and the span of the bidding
blocks in dashed lines. In Fig. 6(c), we show the scenarios of
load. On average, in day-ahead market we buy approximately
the expected value of the load.
IEEE TRANSACTIONS ON SMART GRID, VOL. X, NO. X, XX 2016 7
20 40 60 80
1:24
Price ($)
(a)
10 30 50 70
Hour of the day
Load (Kwh)
(b)
10 30 50 70
Hour of the day
Load (Kwh)
0 4 8 12 16 20
(c)
Fig. 6. In (a), the day-ahead scenarios (lines) are shown together with the
estimated price-bids (horizontal segments). The day-ahead purchase for each
scenario, and the load in each scenario, are shown in (b) and (c), respectively.
The estimated price bid by the risk-averse model is repre-
sented by the continuous red line in Fig. 4. Note that, at hour 2,
the estimated price-responsiveness is much smaller than during
hour 20. The reason is that, according to the scenarios of load,
the load shows a lower variation during the early morning than
during the early night.
C. Benchmark: Results in November and December
In this subsection, we benchmark the following models:
ExpBid: Single block model, where E1is equal to the
expected value of the load, and the price-bid of the single
block is equal to infinity. In other words, we always buy the
expected load in the day-ahead market. No optimization is
needed as the solution is trivial.
LRisk: Risk-averse model (5) with 20 bid blocks. The
price-bid for each block is optimized.
UncRisk: Unconstrained risk model (2). The solution can
be obtained by using Remark 2 or by solving (5) with β= 0.
In order to reproduce the real-time functioning of the
markets, we validate the models using a rolling horizon
procedure. Everyday at 12:00, we generate scenarios for the
next operational day, and afterwards obtain the optimal bidding
curve for all the benchmark models. The data from the last
two months is used in the scenario-generation procedure, and
the process is repeated daily all over the months of November
and December.
In Table I we show the mean (1st column) and the standard
deviation (2nd) of the profit for the three benchmark models,
during November and December. We observe that the simple
model ExpBid under-performs the rest of the models and,
indeed, delivers a negative expected profit. The risk-optimized
TABLE I
MEA N AND S TANDA RD D EVI ATIO N OF TH E PRO FIT F OR TH E
BENCHMARKED MODELS DURING NOVE MBE R AN D DECEMBER.
Mean Std. dev.
ExpBid -1.78 34.52
LRisk 22.26 45.22
UncRisk 188.82 259.62
problem LRisk yields positive expected profit, with a variance
greater than the ExpBid model but substantially lower than for
the UncRisk problem. The risk-unconstrained model UncRisk,
as anticipated, provides the highest mean returns.
VII. CONCLUSION
In this paper we consider the bidding problem of a retailer
that buys energy in the day-ahead market for a pool of
price-responsive consumers. Under the considered setup, the
deviations from the purchased day-ahead energy are traded
at the real-time market. We provide an analytic solution in
the case that the retailer is not risk averse. Additionally, we
formulate a stochastic programming model for optimal bidding
under risk aversion. The price-responsiveness of the consumers
is derived from a real-life dataset, and the modeling approach
is non-parametric where non-linear relationships are allowed.
The analytic results show that, in the risk-unconstrained
case, the optimal bid is a single price, meaning that there is
no extra benefit from bidding a curve. On the other hand,
the computational results from the risk-averse case show
that a block-wise bidding curve successfully mitigates the
risk in terms of profit volatility. Altogether, the proposed
methodology allows the retailer to optimally bid in the day-
ahead market, whether it is for expected-profit maximization
(by leveraging arbitrage opportunities), or for the purpose of
safely procuring energy.
Future work can follow several directions. From the view-
point of the modeling of the retailer’s trading problem, it could
be of interest to compare our chance-constraint approach for
risk control with alternative ones. For example, one could
consider, instead of the chance constraint (1d), a weighted
sum of the expected value of the retailer’s profit and its
Conditional Value at Risk (CVaR) in the objective function
of problem (1), possibly establishing a link between the risk-
related parameters that define both approaches. Likewise, it
would be interesting to extend the retailer’s trading problem
(1) to allow for complex market bids. On a data-modeling
level, one could explore how to extend the proposed method
to generate scenarios when weather variables are considered.
Finally, on a practical level, a possible avenue for future
research would be to compare the results compiled in the case
study of this paper with those that would be obtained for a
control group of price-irresponsive loads. This way, we could
properly assess how much the retailer benefits from supplying
price-sensitive loads and accounting for their price-responsive
nature.
IEEE TRANSACTIONS ON SMART GRID, VOL. X, NO. X, XX 2016 8
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APPENDIX A
PROO F OF TH EO RE M 1
We start by computing the expected profit, conditional on
the day-ahead price (i.e., we treat ΛD=λDas a parameter).
We disregard the first term in (2) since it is constant with re-
spect to the decision variables uband XD, and therefore, does
not affect the solution. The expected profit (2) conditioned on
the day-ahead price λDis thus given by
EXDΛR−λD|ΛD=λD=(7)
B
X
b=1
Iut,b ≥λD
tCbEΛR−λD.(8)
Note that, since λD
tis given, XDcan be computed as
PB
b=1 Iut,b ≥λD
tCb. We distinguish three cases:
(a) When EΛR> λD, the second term in (8) is positive,
hence the expected profit is maximized when ub≥
ub+1 ≥λD,∀b. This implies that uB≥λD.
(b) When EΛR< λD, the second term in (8) is negative,
hence the profit is maximized when ub+1 ≤ub<
λD,∀b. This implies that u1≤λD.
(c) When EΛR=λD, any solution that satisfies ub+1 ≤
ubis optimal.
Finally, we conclude that the expected value of the real-time
price is an optimal price bid, since u∗
b=EΛRmaximizes
the retailer’s expected income in the three cases above.
APPENDIX B
PROO F OF TH EO RE M 2
Analogously as in Appendix A, from Equation (8), the
expected profit conditioned on ΛD
t=λDis proportional to
B
X
b=1
Iut,b ≥λD
tCbEΛR|ΛD=λD−λD.(9)
Next, recall that, from the basic properties of the expected
value, EX{g(X)}=R∞
−∞ g(x)fX(x)dx. We compute the
expected value of (9) with respect to ΛD, which is equal to:
ZΛD
g(λD)fΛD(ΛD=λD)dλD(10)
with g(λD)equal to (9). Arranging terms, we obtain that (10)
is equal to
B
X
b=1
Cb Zub
−∞ Z∞
−∞
λRfΛR(λR|ΛD=λD)dλR×
fΛD(λD)dλD−Zub
−∞
λDfΛD(λD)dλD!.(11)
Now we relax problem (2) by dropping constraint (1c).
Then, the problem becomes decomposable by block, since (11)
is a sum of Belements. For notational purposes, let us rename
each of the Bterms in the summation in (11) by hb(ub).
Note the functions hb(ub)are continuous, since the integral
of a continuous function is continuous. Then, for each block,
the relaxed problem consists in maximizing the continuous
function hb(ub)subject to λ≤ub≤λ. By the intermediate
IEEE TRANSACTIONS ON SMART GRID, VOL. X, NO. X, XX 2016 9
value theorem, we know that the maximum of each term in
the summation will be achieved either at u∗
b=λ, at u∗
b=λ,
or otherwise inside the interval λ, λ.
Considering the case when u∗
bis inside the interval, we
proceed to find the ubsuch that it maximizes hb(ub). In
order to achieve this, we calculate d
dubhb(ub)=0. Note that
d
du Ru
−∞ φ(x)dx =φ(u). With this in mind, the derivative of
hb(ub)is equal to
Z∞
−∞
λRfΛR(λR|ΛD=ub)dλRfΛD(λD=ub)
−ubfΛD(λD=ub).(12)
Assuming that fΛD(λD=ub)is different than zero, and
solving d
dubhb(ub)=0, we obtain the stationary point:
nu∗
b|u∗
b=EΛR|ΛD=u∗
bo.(13)
Next we calculate the second derivative2of d2
du2
b
hb(u∗
b). Its
sign depends on the value of (d
dub
EΛR|ΛD=u∗
b−1),
which can be interpreted as the sensitivity of the expected
real-time price to the day-ahead price at the stationary point.
Depending on the sign of the second derivative, we distinguish
three cases:
(a) When d
dub
EΛR|ΛD=u∗
b<1,u∗
bis a local maximum.
From a practical point of view, it means that at day-
ahead price λD=u∗
b, any marginal increase of this price
will imply a comparatively lower marginal increase in the
expected real-time price, hence, it becomes not profitable
to buy energy from the day-ahead market at price levels
greater than u∗
b.
(b) When d
dub
EΛR|ΛD
t=u∗
b>1,u∗
bis a local minimum.
(c) When d
dub
EΛR|ΛD=u∗
b= 1, the solution u∗
bis an
inflection point that delivers an expected profit equal to
zero.
After having identified the possible candidates ubthat might
maximize hb(ub), it is easy to see that at least one global
optimum to problem (2) satisfies that all ubare all equal to
each other, i.e., ub=u∗,∀b. This is so because functions
hbare all identical for all blocks, hence, the solution u∗
bthat
yields the highest expected profit for one block bwill also
deliver the highest expected profit for the remaining blocks.
Finally, we should point out that this global solution to
the relaxed problem (2)—without constraint (1c)— naturally
satisfies constraint (1c), hence, it must also be a global solution
to the original problem (2).
2The calculation of d2
du2
b
hb(u∗
b), where u∗
bis given by (13), is available
upon request.
