Value of forecasts in planning under uncertainty

Value of forecasts in planning under uncertainty
2015 American Control Conference
Palmer House Hilton
July 1-3, 2015. Chicago, IL, USA
Value of forecasts in planning under uncertainty
Konstantinos Gatsis, Ufuk Topcu, and George J. Pappas
Abstract— In environments with increasing uncertainty, such
as smart grid applications based on renewable energy, planning
can benefit from incorporating forecasts about the uncertainty
and from systematically evaluating the utility of the forecast
information. We consider these issues in a planning framework
in which forecasts are interpreted as constraints on the possible
probability distributions that the uncertain quantity of interest
may have. The planning goal is to robustly maximize the expected value of a given utility function, integrated with respect
to the worst-case distribution consistent with the forecasts.
Under mild technical assumptions we show that the problem
can be reformulated into convex optimization. We exploit this
reformulation to evaluate how informative the forecasts are in
determining the optimal planning decision, as well as to guide
how forecasts can be appropriately refined to obtain higher
utility values. A numerical example of wind energy trading in
electricity markets illustrates our results.
I. I NTRODUCTION
The intrinsically uncertain nature of renewable energy
sources, for example their dependence on local weather conditions [1], impedes the reliable operation of the electricity
grid [2]. Forecasts about the renewable energy generation
provide a means to cope with this uncertainty. Hence the
successful incorporation of forecasts into planning grid operation emerges as an important challenge, as well as the
problem of obtaining forecasts that provide the most valuable
information for planning.
Uncertainty is usually considered in a stochastic framework during the grid planning stage. Generation, for example, is modeled as a random variable, and planning decisions
are made a certain length of time before the generation is
realized at operation time. Planning in this context can provide probabilistic guarantees, for example, that undesirable
events will happen with low probability, or that expected
operation costs/utilities are optimized. Examples can be
found in, e.g., dispatch problems [3], unit commitment [4],
and participation in electricity markets [5], [6].
Solving such stochastic optimization problems relies on
the availability of the underlying probability distribution of
the uncertain quantities. The renewable generation forecasts
however do not provide complete descriptions of the underlying uncertainty, in the sense that they do not completely
describe the probability distribution of the generation. Certain questions arise in this context:
1) How can forecasts of the quantities of interest be
incorporated in a stochastic planning framework?
This work was supported in part by NSF CNS award number 1312390,
and by TerraSwarm, one of six centers of STARnet, a Semiconductor
Research Corporation program sponsored by MARCO and DARPA. The
authors are with the Department of Electrical and Systems Engineering,
University of Pennsylvania, 200 South 33rd Street, Philadelphia, PA 19104.
Email: {kgatsis, utopcu, pappasg}@seas.upenn.edu.
978-1-4799-8686-6/$31.00 ©2015 AACC
2) How informative is a set of given forecasts to a specific
planning problem?
The first question is crucial to reliably integrate renewable
generation in the grid. The second question becomes practically relevant if obtaining forecasts during the planning
stage incurs significant cost. This could be, for example, the
computational cost of running complex renewable generation forecasting algorithms, typically combining numerical
weather prediction models, historical data, local measurements, and simulations [7]. Alternatively these costs could
be monetary, as is the case when grid operators and power
producers do not create their own forecasts but instead
purchase them in the form of products from companies specializing in forecasting [8], [9]. Understanding how valuable
are the forecasts in determining planning decisions could
help reduce such associated costs.
In this paper we take a step towards mathematically
formalizing and answering the above questions. We consider
forecasts in a probabilistic form. This type of forecasts has
recently attracted attention as a more sophisticated alternative
to the traditional point forecasts for the highly variable wind
power [6], [7], [10]. In particular, probabilistic forecasts
provide information about the probability distribution of
some unknown random quantity of interest, in contrast to the
less descriptive point forecast which is a single value thought
as the mean value of the random quantity. Motivated by [6],
[7], one example of probabilistic forecasts are prediction
intervals, i.e., bounds on the probability that the quantity
takes values in certain intervals.
We consider a general stochastic planning framework
under probabilistic forecasts (Section II). We look for a
planning decision that maximizes the expected value of a
given utility function, but the probability distribution of the
uncertain quantity, with respect to which the expectation is
computed, is unknown and only described via the forecasts.
We interpret the probabilistic forecasts as constraints defining
a set of possible probability distributions for which the
decision needs to account. Consequently a robust (max-min)
planning problem is formulated to determine the decision
that maximizes the worst-case expected utility, where the
worst-case expectation is selected from the set of probability
distributions consistent with the forecasts.
In Section III we utilize Lagrange duality theory [11] to
show that the robust planning problem admits an equivalent
reformulation to a single maximization problem with the
introduction of auxiliary (dual) variables. This resulting
problem is convex under certain relatively mild assumptions
on the planning utility function, but it includes an infinite
number of constraints. For the special case of forecasts in
the form of prediction intervals we show that the constraints
can be reduced to a finite number (Section III-A), resulting in
5043
II. P ROBLEM DESCRIPTION
We consider a decision-making framework under uncertainty about some quantity of interest x, for example the
unknown renewable power generation. We assume that x
takes real values in a subset X ⊆ R of the reals and is
modeled as a random variable following some probability
distribution on X . As we will make clear, this distribution is
not known completely. Hence there is uncertainty about the
distribution of the random quantity of interest.
Suppose a decision-maker needs to select the value of
a controlled parameter b before the random quantity x is
realized, i.e., drawn from its probability distribution. We
assume b takes values in a convex subset B ⊆ R of the
reals. To rank the different choices for b we assume that a
function J(x, b) models the utility to the user if decision b
is taken a priori and the unknown quantity takes value x.
Technically we assume the utility function is continuous in
both variables x and b, and also concave in b for every value
of x ∈ X , and concave in x for every value of b ∈ B.
Example 1. Consider the case of a wind power plant
participating in a day-ahead electricity market [5], [6]. The
producer provides a capacity bid in the market, which we can
denote by a decision variable b, representing the estimated
renewable power generation that will be supplied to the grid
Profit J(x,b)
0.5
Profit J(x,b)
a standard (finite-dimensional) convex optimization problem
that can be solved readily [12].
The problem of selecting the worst-case probability distribution given probabilistic descriptions has been discussed
previously in the context of optimal uncertainty quantification [13]. The difference in our formulation however is
that an additional planning optimization level is considered. Related planning formulations under uncertainty about
probability distributions have been considered in, e.g., [14],
[15] and references therein, where the optimal planning
is solved in a data-driven fashion based on samples from
the unknown underlying distribution. Conceptually similar
approaches have been considered in the context of power
systems, where expected values in stochastic optimization
are approximated using samples obtained from wind power
forecasting algorithms [4], [16]. In contrast, planning in our
work is decoupled from the data collection and does not
rely on sample-based approximations, but instead utilizes the
output of the forecasting procedures, i.e., the probabilistic
forecasts themselves.
A further novelty of our approach is that it allows to
characterize how valuable is the forecast information in
determining the optimal planning decision. This characterization follows from a sensitivity analysis of the optimal planning objective value with respect to the forecast constraints
(Section IV). Moreover, under the hypothesis that forecasts
can be refined by, e.g., a forecasting oracle, we develop a
sensitivity-driven procedure that sequentially selects forecast
refinements to yield a higher robust planning utility value.
Numerical examples of the proposed approach for trading
wind energy in electricity markets [5], [6] are presented in
Section V. Finally, we conclude with a discussion on our
contributions and future work in Section VI.
0.4
0.3
0.2
0.1
0
0.2
0.4
0.6
Bidding b
0.2
0
−0.2
x
0
0.4
0.8
1
b
0
0.2
0.4
0.6
0.8
1
Power generation x
Fig. 1. Profit J(x, b) from bidding in electricity markets for p = 1,
q = 1.6. On the left generation is kept constant x = 0.5 while bid b varies.
On the right bid is kept constant b = 0.5 while generation x varies.
at a future time interval. We let b take values in the unit
interval, b ∈ [0, 1], which can be interpreted as a normalized
percentage with respect to the maximum capacity of the
generator. If we denote by x the actual realized power, also
normalized in x ∈ [0, 1], a simple model [5] of the producer’s
monetary profit from a realization x after bidding b is
J(x, b) = p b − q[b − x]+ ,
(1)
where [θ]+ = max{θ, 0}. The constant p > 0 rewards
high bids, while the constant q > 0 penalizes a shortfall
of generation compared to the bid, i.e., when b > x. It is
assumed that q > p, implying that the profit decreases for
higher shortfalls. The utility function satisfies our general
convexity assumptions as can be seen in Fig. 1.
A common assumption in planning problems under uncertainty is that the underlying probability distribution F
of the unknown quantity x is known, and one searches
for the optimal decision b that maximizes the expected
utility Ex∼F J(x, b). In this paper however the underlying
distribution F is not completely known but only forecasts
about the random quantity x are available. We consider
probabilistic forecasts [6], [7], which provide information
about the underlying probability distribution F of x. More
specifically we consider given (measurable) functions gi (x),
for i = 1, . . . , n, of the random quantity x, and forecasts
stating that the expected value of these functions is upper
bounded by some parameters εi , i.e.,
Ex∼F gi (x) ≤ εi , i = 1, . . . , n.
(2)
Both the functions gi (x) and the parameters εi in these
inequalities are given to the decision-maker, e.g., provided by
some forecasting algorithm. We interpret the forecasts (2) as
a set of constraints on the unknown underlying distribution
F , narrowing the uncertainty of the decision-maker regarding
the distribution of x. Based on this interpretation, we will
interchangeably use the terms forecasts and constraints when
referring to (2).
The constraint-based characterization of the uncertainty in
(2) matches many common types of forecasts. For example
bounds on mean value, second moment, or higher-order
moments can be expressed in the form of (2) by appropriately
selecting the function gi (x). In the following example we
detail another special case, the prediction intervals. We will
revisit this case later.
5044
distributions consistent with forecasts (2) is
True probability density
Prediction intervals
F(ε) = {F ∈ DX : Ex∼F gi (x) ≤ εi , i = 1, . . . , n}, (6)
0
0.1667
0.3333
0.5
0.6667
0.8333
1
Renewable Generation Power x
Fig. 2. Pictorial representation of prediction intervals (Example 2). The
true but unknown probability density function of the renewable generation is
plotted. Forecasts are available in the form of prediction intervals, which are
upper and lower bounds on the true probability mass placed on m = 6 equal
intervals of generation amounts. For illustration purposes the prediction
intervals are shown by blocks, with the areas under the blocks equal to
the upper and lower bounds, δ i and δ i respectively, according to (3).
Example 2. An example motivated by recent forecast methods for renewable generation [6] are prediction intervals.
Suppose the renewable generation x takes values in a normalized interval X = [0, 1], partitioned into m sub-intervals
[xi−1 , xi ) for i = 1, . . . , m, where x0 = 0, xm = 1, and the
forecasts take the form
δ i ≤ P(xi−1 ≤ x < xi ) ≤ δ i ,
i = 1, . . . , m.
(3)
In other words, this forecast provides upper and lower bounds
on the probability that x takes value in each of the intervals.
These forecasts can be reformulated into the form of (2) by
defining the indicator functions
1 if xi−1 ≤ x < xi ,
gi (x) = 1{[xi−1 , xi )} =
(4)
0 otherwise,
for i = 1, . . . , m. Then the constraints (3) are equivalently
written in the form of (2) as
Ex∼F gi (x) ≤ δ i , and
Ex∼F − gi (x) ≤ −δ i .
(5)
A pictorial representation is given in Fig. 2.
We make the following technical assumption about the
forecasts provided in (2).
Assumption 1. There exists a probability distribution F
on X such that the forecasts (2) are satisfied with strict
inequality, i.e., Ex∼F gi (x) < εi for all i = 1, . . . , n.
This assumption states that the constraints (2) are feasible,
and furthermore that strict feasibility holds. To motivate the
feasibility, we can think of the actual underlying distribution
of the random quantity, which is not known to the decisionmaker, as satisfying the constraints (2). The strict feasibility
is assumed for certain technical reasons in order to establish
the results in the sequel of this paper, but is not practically
restrictive. The parameters εi can always be perturbed to
make the strict inequality assumption hold.
Given the constraint-based interpretation of the forecasts
in (2), we can alternatively describe the planner’s uncertainty
concerning the true distribution F of the unknown quantity
x by a set F containing the (possibly uncountably many)
distributions that satisfy the forecasts (2). The set of all
where we denote the set of all probability distributions on X
by DX . Here we parametrize the set with the forecast bounds
εi , i = 1, . . . , n, grouped in a vector ε. For the current
problem development, ε is fixed, but we introduce this
parametrization for later use in Section IV. By Assumption 1
the set F(ε) has a nontrivial interior, e.g., the true underlying
distribution belongs in the set.
Having defined the utility J(x, b) of different choices b
to the decision-maker, as well as the possible distributions
F ∈ F(ε) of the random quantity x that the decision-maker
can anticipate according to the probabilistic forecasts, we
now pose a planning problem. The decision-maker needs to
determine the value b that robustly maximizes the worst-case
expected utility anticipated from the forecasts, mathematically captured as a robust optimization problem
P ∗ (ε) = maximize
b∈B
inf
F ∈F (ε)
Ex∼F J(x, b).
(7)
In other words, the planner looks for a decision that leads
to the most favorable objective value assuming the worstcase distribution among the ones consistent with the forecast
F(ε). We denote the robustly optimal objective value and
decision by P ∗ (ε), b∗ (ε) respectively, assuming they exist,
again parametrized by the forecast bounds ε for later use.
The difficulty in solving the robust planning problem (7)
lies in the max-min formulation. For every choice b, one
needs to solve a minimization problem with respect to the
distribution F to evaluate how good the choice is, and then
infer how to improve on b to maximize the objective function.
To overcome this complexity, in Section III, we examine how
the robust planning problem (7) can be equivalently written as a single-layer optimization problem. For the special
case of forecasts given by prediction intervals (Example 2),
we obtain an equivalent convex optimization problem from
which b∗ (ε) can be determined efficiently.
Then in Section IV we characterize how informative the
n forecasts given by (2) are in determining the optimal
planning decisions by examining the sensitivity of the robust
planning value P ∗ (ε) to changes in the forecast parameters
ε. Based on this analysis, we also develop a methodology
for improving the planning utility by appropriately refining
forecasts.
III. ROBUSTLY OPTIMAL PLANNING
The robust planning problem described in (7) involves
a max-min structure that makes it hard to determine the
robustly optimal decision b∗ (ε). In this section we follow
an alternative path based on Lagrange duality theory [11].
Under certain technical conditions the inner minimization
in (7) over probability distributions F of the unknown
quantity x is equivalent to its Lagrange dual (maximization)
problem. Replacing then the inner minimization in (7) with
the equivalent maximization yields a maximization problem
(with a single optimization layer) over the planning decision
5045
variable b and additional (dual) variables. This result is stated
in the following theorem, and the proof relies on [11].1
robust planning follows from the solution to the problem
Theorem 1. If Assumption 1 holds, the robust planning problem defined in (7) is equivalent to the following optimization
problem
maximize
b,λ,η
−
n
X
m
X
maximize
b∈B, λ≥0, η
subject to
λi δ i − λ̄i δ̄i − η
(11)
i=1
J(x, b) +
m
X
(λ̄i − λi ) 1{[xi−1 , xi )}
i=1
λ i εi − η
i=1
subject to J(x, b) +
n
X
+ η ≥ 0,
(8)
λi gi (x) + η ≥ 0, ∀x ∈ X , (9)
b ∈ B, λ ∈
J(x, b) + (λ̄i − λi ) + η ≥ 0,
η ∈ R.
(10)
According to the theorem, the optimal choice b∗ (ε) of the
robust planning problem (7) can be equivalently found by the
optimization problem (8)-(10), and the optimal values of the
two problems are equal. The advantage of the representation
in (8)-(10) is that it bypasses the max-min structure of (7).
There is a finite number of optimization variables in (8)-(10),
i.e., the decision b, as well as the dual variables η ∈ R and
the vector λ ∈ Rn+ . The caveat, although, is that the number
of constraints is infinite and uncountable. Such optimization
problems are called semi-infinite – see [18] for an overview.
Nevertheless it is a convex optimization problem since the
objective is linear and the constraint (9) for each value of x
defines a convex set due to the concavity of function J(x, b)
in variable b for any value x.
General approaches for solving semi-infinite programs can
be found in [18]. We examine however next the special
case in which the forecasts (2) are given in the form of
prediction intervals, described in Example 2. In that case
it turns out that the number of constraints in the equivalent
problem (8)-(10) can be reduced to a finite number. Hence
we can pose the robust planning decision under prediction
interval forecasts as a standard finite-dimensional convex
optimization problem, for which efficient algorithms exist.
After this special case, we show in the following section
that the optimal values of the additional variables λ in
the optimization problem (8)-(10) can be interpreted as
indicators of how valuable are the given forecasts in determining the optimal planning. Based on this interpretation,
we also examine in the following section how updates on
the forecasts can yield a higher planning objective value.
A. Robust planning under prediction intervals
Consider forecasts given in the form of prediction intervals
described in (3). By their reformulation into the generic form
of forecasts presented in (5), we can pose the robust planning decision problem (7) into its equivalent form provided
by Theorem 1 in (8)-(10). In particular, introducing dual
variables λ̄1 , . . . , λ̄m for the upper bound constraints in (5),
and λ1 , . . . , λm similarly for the lower bounds, we have that
1 The proofs of the results in this paper are omitted due to space
limitations, but can be found in [17].
(12)
We note that the continuum of constraints (12) can be
equivalently separated into the given intervals
i=1
Rn+ ,
for all x ∈ [0, 1].
for all x ∈ [xi−1 , xi ), (13)
for all i = 1, . . . , m. Note also that by the assumption that
the function J(x, b) is concave in x for all b, it follows that
J(x, b) ≥ min{J(xi−1 , b), J(xi , b)}, where the inequality
is tight by continuity of function J(x, b). Hence instead of
checking (13) for a continuum of values x ∈ [xi−1 , xi )
for each i = 1, . . . , m, we only need to check at the two
endpoints xi−1 and xi . In other words, the problem (11)(12) can be equivalently written as
maximize
b∈B, λ≥0, η
subject to
m
X
λi δ i − λ̄i δ̄i − η
(14)
i=1
J(xi−1 , b) + λ̄i − λi + η ≥ 0,
(15)
J(xi , b) + λ̄i − λi + η ≥ 0, i = 1, . . . , m.
This optimization problem is convex and has a finite number
of constraints. The number of constraints is twice the number
of initial forecast intervals because essentially each forecast
of the robust optimization is converted into two constraints,
one for each of the two interval endpoints. The solution
of (14) can be determined efficiently by standard convex
optimization algorithms.
IV. R ELATIVE VALUE OF FORECASTS
In this section we aim to quantify the value of the given
forecasts (2) to the planner who needs to solve the robust
planning problem (7). In particular, (2) defines a set of n
given forecasts and we are interested in determining how
valuable information does each one of them provide to the
planner. To this end, we examine how sensitive the robustly
optimal objective P ∗ (ε), defined in (7), is to the given
forecast values ε, i.e., what would the objective value become
for deviations from the given parameters ε.
The forecast parameters ε determine the objective P ∗ (ε)
defined in (7) via the set of distributions F(ε) appearing
in the constraint of the inner minimization. To examine the
sensitivity of P ∗ (ε) when the given forecast parameters ε
change to some new values ε − ∆ε for some ∆ε ∈ Rn ,
with the negative sign chosen for convention, we leverage the
problem reformulation provided by Theorem 1. We exploit
the well-known convex optimization fact that in general the
optimal values of the Lagrange dual variables λ express the
sensitivity of the objective of a convex optimization problem
with respect to changes in the constraints - see [12, Ch. 5.6].
In our case, the robust planning (7) involves a two-layer
(max-min) optimization objective, but a similar sensitivity
result can be obtained, as we state next.
5046
Theorem 2. If Assumption 1 holds, then
P ∗ (ε − ∆ε) ≥ P ∗ (ε) +
n
X
λ∗i (ε) ∆εi ,
(16)
i=1
holds for any ∆ε ∈ Rn , where λ∗i (ε), for i = 1, . . . , n, is
the optimal solution to problem (8).
The theorem provides a lower bound on the optimal
planning objective value of (7) after changes in the forecast
bounds ε provided in (2). The change in the lower bound
compared to P ∗ (ε) is proportional to the change from ε to
ε − ∆ε. The rate of change in each direction i = 1, . . . , n
is given by the optimal value of the (dual) variable λ∗i (ε).
We can thus interpret the values λ∗i (ε) as indicators of how
valuable each one of the forecasts (2) is to the planner.
A forecast i with a high value λ∗i (ε) is more informative
compared to other forecasts j 6= i because a small change in
the given forecast bound εi gives a significant change in the
(lower bound of the) objective. Moreover the values λ∗i (ε)
are already obtained during solving problem (8), hence no
further computation is required to determine them.
Remark 1. The sensitivity analysis provided by Theorem 2
is performed with respect to the given forecast parameters ε,
i.e., they are obtained locally for ε. At some other forecast
parameter ε0 , the sensitivities, i.e., the values λ∗i (ε0 ) will
differ. This difference matches the intuition that different
forecasts provide different information. On the other hand
the theorem provides a lower bound on the objective value
at any deviation ε − ∆ε ∈ Rn , not just locally around ε. It
is also worth noting that the theorem does not provide any
guarantee on how far the lower bound is with respect to the
actual objective for such deviations.
Given the sensitivity-based characterization of how informative some given forecasts are, we next propose a method
for refining/updating the forecasts in a way that increases the
utility of the decision-making problem.
A. Forecast refinements
Suppose the decision-maker has access to a forecasting
oracle, which upon request can refine one of the n given
forecasts in (2) – see Remark 2. Inspecting the form of the
forecasts in (2) written as inequalities, a refined forecast of
some inequality i can be represented by decreasing the bound
εi to a lower value εi − ∆εi . Equivalently, such decrease has
the effect of shrinking the set F(ε) of possible probability
distributions that the decision-maker has to consider, as
defined in (6), to a subset F(ε − ∆ε) ⊆ F(ε).
A question that arises in this context is how the decisionmaker should select which of the n forecasts of (7) to
refine. This question is particularly important if the cost of
refining forecasts is high, e.g., the computational cost of
running extra simulations of a forecasting algorithm. The
characterization of Theorem 2 suggests that the forecast that
provides the most valuable information should be selected,
i.e., the forecast i with the highest value λ∗i (ε). Based on this
intuition we propose the iterative forecast refinement procedure shown in Algorithm 1. On every iteration sensitivity
values are obtained by solving the planning problem, and a
Algorithm 1 Sensitivity-driven forecast refinement
Input: Utility function J(x, b), forecast functions gi (x) for
i = 1, . . . , n, initial forecast parameters ε(0) ∈ Rn
Output: Robustly optimal planning decision b
1: k ← 0
2: repeat
3:
Solve robust planning (8)-(10) with forecast parameters ε(k), and determine optimal planning b∗ (ε(k)) and
sensitivity values λ∗ (ε(k)) ∈ Rn
4:
Select j = argmax1≤i≤n λ∗i (ε(k))
5:
Request refinement in jth forecast εj (k + 1) ≤ εj (k)
6:
Keep εi (k + 1) = εi (k) for all i 6= j
7:
k ←k+1
8: until Termination condition
9: return Final planning decision b∗ (ε(k))
refinement for the forecast bound with the highest sensitivity
is requested (e.g. from the oracle) while all other bounds
are kept constant. The procedure terminates, for example, if
no further refinements are possible, or if the objective value
improvements become insignificant.
Even though this procedure is motivated mainly by intuition, without theoretical claims in terms of optimality,
convergence, etc., we note the following fact. On each
iteration, according to (16) of Theorem 2, the robustly
optimal utility value increases by at least an amount equal
to λ∗j (ε(k)) (εj (k + 1) − εj (k)). This minimum increase is
proportional to the amount of change in the selected forecast
parameter. If the selected forecast j cannot be refined, i.e.,
εj (k + 1) = εj (k), then no improvement in decision-making
is achieved. In general, however, the result matches the
intuition that with a better forecast, i.e., imposing a more
constraining set F(ε − ∆ε) in the minimization of (7), the
objective value of the planning can only improve, i.e., the
optimal value of (7) cannot decrease.
In the following section we present a numerical example
for a wind power plant participating in an electricity market
(Example 1) under prediction interval forecasts (Example 2).
We demonstrate the methodology derived by Theorem 1 for
determining the robustly optimal bidding decision. Additionally we perform the sensitivity analysis developed in this
section and we implement the sensitivity-driven procedure
for refining forecasts.
Remark 2. The methodology for refining forecasts in this
section is developed without considering a specific forecasting procedure. For example, if an oracle obtains the forecasts
by sampling from the true distribution of the unknown
quantity through simulations of an underlying stochastic
model, forecasts can be refined by drawing extra samples.
However, a large number of extra samples might be required
to improve upon the estimates of all the integrals in (2), e.g.,
in a Monte Carlo fashion. By focusing on only one of the n
integrals during refinement, as the oracle in our hypothesis
does, the number of required samples can be reduced. This
can be performed by variance reduction techniques, such as
importance sampling [19], which aim at sampling more often
5047
0.5
True Expected Value
Worst−Case Value
Optimal Point
Profit
0.4
Iter
Iter
Iter
Iter
0.3
1
2
3
4
λ̄1
λ̄2
λ̄3
λ̄4
λ̄5
λ̄6
0.66
0.80
0.80
1.07
0.39
0.53
0.53
0.80
0.12
0.27
0.27
0.53
0.00
0.00
0.00
0.27
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TABLE I
S ENSITIVITIES TO UPPER FORECAST BOUNDS
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Iter
Iter
Iter
Iter
Bid value b
Fig. 3. Expected profit and worst-case expected profit as a function of
the chosen bidding for the case considered in the numerical example. The
worst-case value is always an under-approximation of the true one. The
robustly optimal bidding balances between the shortfall penalties and gains
from a high bid.
from values important in estimating the selected integral.
V. A PPLICATION : B IDDING IN ELECTRICITY MARKETS
UNDER PREDICTION INTERVALS
In this section we apply our theoretical results in a
numerical example of bidding renewable energy in electricity
markets, introduced in Example 1, with the utility function
in (1) representing the profit from a bid b ∈ [0, 1] and
a generation x ∈ [0, 1]. Suppose the generator obtains
forecasts about the future generation value x in the form of
prediction intervals, introduced in Example 2. We consider
the problem of determining the value of bidding that robustly
maximizes the expected profit to the generator subject to
the given forecasts about the renewable generation. This
problem is mathematically described by (7). We adopt the
convex optimization reformulation presented in Section IIIA, in particular the form (14)-(15).
We consider forecasts for the probability that the random
generation x takes values in each of m equal intervals that
partition the total range of values [0, 1], i.e., xi = i/m for
i = 0, . . . , m – see also Example 2. To derive the lower and
upper bounds, δ i and δ̄i respectively according to (3), we
adopt the following procedure. We fix some true underlying
probability distribution of the generation x and we produce
bounds δ i , δ̄i by perturbing the true value P(xi−1 ≤ x < xi )
computed with the underlying distribution. Specifically we
consider perturbation by a constant amount below and above
P(xi−1 ≤ x < xi ) for each i = 1, . . . , m. Then only the
resulting lower and upper bounds, δ i and δ̄i respectively,
are available to the generator, not the underlying probability
distribution used to create them.
Given m = 6 prediction intervals shown in Fig. 2, we
first solve the convex optimization (14)-(15) for reward and
penalty values p = 1, q = 1.6 in (1) and get an optimal
bidding value b∗ ≈ 0.67. In Fig. 3 we illustrate the value
of the worst-case expected profit, i.e., the objective in the
planning problem (7), for all bid values b as well as the
optimal point. For comparison we also illustrate the expected
profit computed using the underlying true probability distribution of x for all bid values b. The worst-case expected
profit is an under-approximation of the true expected value,
since the forecasts provide incomplete information about
the underlying distribution. The optimal choice b∗ balances
between the two extremes 0 and 1, as expected by intuition,
1
2
3
4
λ1
λ2
λ3
λ4
λ5
λ6
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.14
0.00
0.00
0.00
0.41
0.27
0.27
0.00
0.41
0.27
0.27
0.00
TABLE II
S ENSITIVITIES TO LOWER FORECAST BOUNDS
with b = 0 being the most ”secure” bid but yielding a zero
expected profit, and b = 1 being the most ”risky” bid since
it is certain that a shortfall x < b = 1 will occur.
Moreover, we examine how the information provided by
the given forecasts of Fig. 2 is valued by the decision maker.
According to Theorem 2 of Section IV the sensitivity of the
robust bidding profit to the set of given forecasts is captured
by the optimal values of the dual variables. In particular the
sensitivity to upper and lower bounds of prediction intervals,
δ̄i and δ i respectively, for i = 1, . . . , m, is captured by the
dual variables λ̄1 , . . . , λ̄m and λ1 , . . . , λm respectively, as
introduced in Section III-A. The sensitivity values in our
example are shown in the first row of Tables I, II.
We observe that at each interval i = 1, . . . , m, at most one
of the upper or lower bound sensitivities λ̄i , λi is non-zero.
The reason is that at most one of two bounds (cf. (3)) is tight
for the worst-case distribution at the robustly optimal point
of problem (7). Moreover, we see that the optimal profit is
sensitive to the probability of both having a low generation
(captured by λ̄1 , λ̄2 , λ̄3 ) as well as having a high generation
(captured by λ4 , λ5 , λ6 ). In other words information about
these events is the most informative. The sensitivity is higher
in the low generation case because the profit is significantly
affected by shortfalls in this case.
Next we adopt the sensitivity-driven methodology of Section IV to refine the given forecasts so that they become
more informative for robust bidding. Following Algorithm 1
we iteratively select to refine the forecast with the highest
sensitivity value, i.e., one of the upper or lower bounds δ̄i
and δ i for an interval i in (3). If an upper bound δ̄i is
selected it gets decreased, and respectively if a lower bound
δ i is selected it gets increased. In our example, the resulting
new bound needs to be consistent with the true underlying
distribution. For simplicity here we model a forecasting
oracle who can upon request refine each upper/lower bound
by a constant decrease/increase, and the new bound is close
enough to the true value so that the oracle cannot refine it
any further. In general, hovever, Algorithm 1 can be applied
regardless of how the forecasting oracle operates.
The sensitivity values after each iteration of Algorithm 1
are shown in Tables I, II. As already mentioned, λ̄1 has
the higher value initially, so the upper bound on the first
interval is refined (i.e., reduced). On the second iteration,
after solving again the optimal planning (14)-(15), the new
5048
Normalized Profit
0
−0.05
−0.1
Iteration 1
Iteration 2
Iteration 3
Iteration 4
−0.15
−0.2
0
0.2
0.4
0.6
0.8
1
Bid value b
Fig. 4. Normalized worst-case profit as a function of bid value after iterations of the forecast refinement algorithm. The plotted values are normalized
by subtracting the true underlying expected profit. The refinements increase
the worst-case profit and bring it closer to the true value. The robustly
optimal biding values are also indicated.
sensitivity values are obtained and again λ̄1 is the largest.
Since the forecasting oracle described in our example cannot
reduce the corresponding bound any further, the bound with
the second largest sensitivity is selected to be refined, which
is λ̄2 . On the third iteration, again λ̄1 , λ̄2 are the largest, but
due to the oracle, the third largest λ̄3 is selected, and so on.
We note that after some iterations the sensitivities of all lower
bounds λi become zero, meaning that they do not provide
any valuable information, and refining/increasing them does
not necessarily offer an improvement in profit.
Finally, to examine the improvements in the bidding profit
after these iterations of the refinement algorithm, we plot in
Fig. 4 the worst-case expected profit, i.e., the objective in the
planning problem (7), for all bid values b. For visualization
reasons the values are normalized by subtracting the true
expected profit (the latter is larger, as shown in Fig. 3, so
the plotted values are negative). As the forecasts are refined,
the worst-case expected profit increases and gets closer to the
actual expected profit (the baseline zero). In other words, the
refined forecasts provide a more accurate model of the true
objective. The largest discrepancy between worst-case and
true expected value happens at the high bid region, since the
worst-case scenario assumes that generation takes the lowest
possible values with the highest possible probability, making
the associated shortfall costs large. In this example even
though forecast refinements increase the optimal profit value,
the optimal bidding decision in Fig. 4 is the same. This is
a consequence of the piecewise linear utility function in (1).
For more general utility functions the robust decision may
change as well during refinements, along with the optimal
objective values.
VI. C ONCLUDING REMARKS AND FUTURE WORK
We examine how planning problems under uncertainty can
utilize forecasts about uncertain quantities. By equivalently
expressing forecasts as constraints on the possible probability distributions that the uncertain quantity can follow,
the problem is cast as a robust optimization. The optimal
planning seeks to maximize the expected value of a given
utility function, integrated with respect to the worst-case
distribution consistent with the forecasts. A reformulation
into a convex optimization problem is presented, from which
we can extract information about how valuable are the
forecasts in determining the optimal planning decision.
Supposing a forecasting oracle can refine the given forecasts upon request, a sensitivity-driven approach to iterative
forecast refinement is presented. Coupling the theoretical approach with practical forecasting algorithms requires further
exploration to account for, e.g., the ability of such algorithms
to provide new forecasts instead of strict refinements, or the
limited ability for further refining in other cases. Future work
also includes generalizations of the approach to uncertain
quantities correlated over time, as in the scenario-based
forecasts of [20], or over space, e.g., renewable generation
at different physical locations in the grid.
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