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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.