∎

11institutetext:  Corresponding author: Wei Xie<EMAIL_ADDRESS>22institutetext: Northeastern University, Boston, MA 02115
33institutetext: Yuan Yi 44institutetext: Rensselaer Polytechnic Institute,
Troy, NY 12180
55institutetext: Zhi Zhou 66institutetext: Argonne National Laboratory,
Argonne, IL 60439
77institutetext: Keqi Wang 88institutetext: Northeastern University, Boston,
MA 02115

# Data-Driven Stochastic Optimization for Power Grids Scheduling under High
Wind Penetration

Wei Xie  Yuan Yi  Zhi Zhou  Keqi Wang

(Received: date / Accepted: date)

###### Abstract

To address the environmental concern and improve the economic efficiency, the
wind power is rapidly integrated into smart grids. However, the inherent
uncertainty of wind energy raises operational challenges. To ensure the cost-
efficient, reliable and robust operation, it is critically important to find
the optimal decision that can correctly and rigorously hedge against all
sources of uncertainty. In this paper, we propose data-driven stochastic unit
commitment (SUC) to guide the power grids scheduling. Specifically, given the
finite historical data, the posterior predictive distribution is developed to
quantify the wind power prediction uncertainty accounting for both inherent
stochastic uncertainty of wind power generation and input model estimation
error. For complex power grid systems, a finite number of scenarios is used to
estimate the expected cost in the planning horizon. To further control the
impact of finite sampling error induced by using the sample average
approximation (SAA), we propose a parallel computing based optimization
solution methodology, which can quickly find the reliable optimal unit
commitment decision hedging against various sources of uncertainty. The
empirical study over six-bus and 118-bus systems demonstrates that our
approach can provide more efficient and robust performance than the existing
deterministic and stochastic unit commitment approaches.

###### Keywords:

Stochastic Programming, Unit Commitment, Parallel Computing, Wind Power, Power
Grids Scheduling, Renewable Energy

##  1 INTRODUCTION

Wind power is rapidly incorporated into power grids in an effort to combat the
climate change and improve power system resilience Hargreaves_Hobbs_2012 ;
Zhou_etal_2013 ; Zhao_Wu_2014 . In the past few years, the wind energy
capacity expanded explosively Jiang_Wang_Guan_2012 ; Hargreaves_Hobbs_2012 ;
Zhao_Wu_2014 . It is also projected that the wind power penetration will
continue to grow in the near future Zhou_etal_2013 . However, the inherent
volatility of wind energy has a significant impact on the system operation
Jiang_Wang_Guan_2012 ; Hargreaves_Hobbs_2012 ; Papavasiliou_Oren_2013 . To
ensure a cost-efficient and reliable power grid scheduling, the stochastic
unit commitment (SUC) model is widely used in the literature, especially under
the situations with high wind penetration Wang_Shahidehpour_Li_2008 ;
Ruiz_etal_2009 ; Tuohy_etal_2009 ; Wang_Guan_Wang_2012 . Decision makers seek
the unit commitment decision minimizing the total expected cost of the power
production to meet the demand, which explicitly accounts for the inherent
stochastic uncertainty of wind power generation Wang_Shahidehpour_Li_2008 ;
Ruiz_etal_2009 ; Tuohy_etal_2009 ; Zhou_etal_2013 .

However, the existing SUC approaches tend to ignore two sources of
uncertainty, which can lead to inferior and unreliable unit commitment
decisions. First, the underlying true statistical input model characterizing
the wind power generation uncertainty is unknown and estimated by finite
historical data, which can induce the input model estimation uncertainty,
called model risk. The existing SUC approaches, called the empirical approach,
tend to take the estimated statistical model as the true one Ruiz_etal_2009 ;
Tuohy_etal_2009 ; Wang_Guan_Wang_2012  and ignore the model estimation error
Xie_etal_2018 . Second, to solve SUC, the sample average approximation (SAA),
using finite number of scenarios to approximate the expected cost in the
planning horizon, can introduce the finite sampling error, which is also
typically ignored in the existing SUC approaches Ruiz_etal_2009 ;
Tuohy_etal_2009 ; Wang_Guan_Wang_2012 .

In this paper, we introduce a data-driven SUC model and further develop a
parallel computing based optimization solution methodology. Our study can lead
to the optimal unit commitment decision which can appropriately hedge against
all sources of uncertainty. Basically, both parametric and nonparametric
statistical model can be used to characterize the inherent stochastic
uncertainty of wind power generation. Since the underlying input model is
unknown and estimated by finite historical data, this induces the model
estimation uncertainty and we quantify it with the posterior distribution.
Then, we utilize the posterior predictive distribution to quantify the wind
power generation prediction uncertainty, which accounts for both stochastic
uncertainty and model estimation error. Thus, driven by the scenarios
generated by the posterior predictive distribution, we propose the data-driven
SUC, which leads to the optimal decision simultaneously hedging against both
wind power generation stochastic uncertainty and input model estimation error.

Built on data-driven SUC, we further introduce a parallel computing based
optimization approach, called the optimization and selection (OPSEL), which
can efficiently control the impact from finite sampling error induced by SAA.
Specifically, to solve SUC problems, we observe that the computational effort
is heavily invested in searching for the optimal unit commitment decision.
Compared with the optimization search, given a candidate unit commitment
decision, it takes much less time to assess its performance. Thus, the OPSEL
approach includes the parallel optimization search and the best candidate
decision selection. In the optimization step, we utilize the parallel
computing to simultaneously solve a sequence of finite sample approximated
data-drive SUC problems and obtain candidate solutions. Then, in the selection
step, we use the rank and selection approach to efficiently evaluate these
candidate decisions and select the best decision. The proposed approach can be
used by general SUC to control the impact of finite sampling error and quickly
find the optimal unit commitment decision, which is critically important for
guiding the dynamic scheduling decision for complex power grids in the intra-
day market.

The main contributions of this paper are listed as follows.

  1. 1.

To the authors’ best knowledge, there is no existing approach explicitly
accounting for all three sources of uncertainty: (1) inherent stochastic
uncertainty of wind power generation, (2) SUC input model estimation
uncertainty, and (3) finite sampling error induced by using the sample average
approximation on the expected cost in the planning horizon of SUC. We propose
a data-driven stochastic optimization framework that can appropriately hedge
against all sources of uncertainties and quickly deliver a reliable cost-
efficient optimal dynamic unit commitment decision.

  2. 2.

The proposed data-driven SUC leads to the optimal unit commitment decision
simultaneously hedging against both the inherent stochastic uncertainty of
wind power generation and the SUC input model estimation uncertainty. It can
be applied to cases with either parametric or nonparametric wind power
forecast models.

  3. 3.

Our OPSEL approach can utilize the parallel computing and quickly solve for
the optimal unit commitment decisions hedging against the impact of finite
sampling error induced by SAA, which is large especially for complex power
grids with high wind power penetration.

The organization of this paper is as follows. We review the related literature
in the next section. We formally state our problem and introduce the proposed
data-driven SUC modeling in Section 3. In Section 4, we introduce the OPSEL
approach that can quickly solve SUC and control the impact of SAA finite
sampling error. Both six-bus and 118-bus test cases are used to study the
performance of our approach in Section 5 and the results demonstrate that the
proposed data-driven SUC framework has the clear advantages. We conclude this
paper in Section 6. The code of proposed data-driven SUC framework is
available on GitHub at https://github.com/kw48792/data-driven-suc.

##  2 LITERATURE REVIEW

In the power system literature, several streams of optimization approaches
have been developed for the stochastic unit commitment (SUC) problem. The
first one and the most commonly used one is called the empirical SUC. Given
the historical data, it first estimates the underlying statistical input model
characterizing the wind power generation variation, and then takes the
estimate as the true one. Stochastic programming was first introduced to solve
the unit commitment problem with uncertainty from load Takriti_etal_1996 .
This approach has frequently been applied in recent research, as renewables
are integrated into the power system on a large scale; see the examples in
Tuohy_etal_2009 ; Wang_Guan_Wang_2012 ; madaeni2013impacts ;
Huang_Zheng_Wang_2014 . While the empirical SUC accounts for the inherent
stochastic uncertainty of renewable energies, it fails to account for
statistical input model estimation uncertainty and finite sampling error
induced by SAA, which could lead to inferior decisions.

The second stream is the robust optimization (RO). Without assuming the
distribution modeling wind power generation variability, this approach focuses
on the worst-case scenario, with the objective minimizing the worst-case cost.
The studies in Jiang_Wang_Guan_2012 ; Bertsimas_etal_2013  employed RO to
smart grids with high wind power penetration, Zhao_etal_2013  further extended
RO to multi-stage cases, and Lee_etal_2014  included the transmission line
constraints in RO. However, RO is too conservative; see Xiong_etal_2017 .
While some efforts have been made to reduce the conservativeness
Jiang_Wang_Guan_2012 ; Zhao_Zeng_2012 ; Lorca_Sun_2015 ; kazemzadeh2019robust
, since RO only considers the worst-case without taking into consideration of
the likelihood of all scenarios, the conservativeness issue persists.

The third stream, called the distributionally robust optimization (DRO), is
proposed to overcome the limitation of RO; see for example Bian_etal_2015 ;
Wang_etal_2016J ; Xiong_etal_2017 . The distributional robust unit commitment
model minimizes the worst-case expected cost over a set of probability
distributions, called an ambiguity set, accounting for the input model
estimation uncertainty. DRO is a special case of the composite measure
approach where separate measures are used to quantify the input model
estimation uncertainty and stochastic uncertainty, and then the composite of
these measures is used in the objective Xiong_etal_2017 . Even although this
approach could produce less conservative decisions than RO, it fails to take
into account the possibility of distribution candidates being the true one.
Hence, the resulting scheduling decision is still too conservative and costly.

The fourth stream is called the minimax regret optimization. The regret is
defined by the objective difference between the current solution without
knowing the uncertain parameters and the perfect-information solution. Jiang
et al. Jiang_etal_2013  introduced an innovative minimax regret unit
commitment model aiming to minimize the maximum regret of the day-ahead
decision over all possible realizations of the uncertain wind power
generation. While the minimax regret optimization can deliver less
conservative results than RO, like DRO, it also fails to take into account the
possibility of distribution candidates being the true one. Hence, it suffers
the similar drawback as DRO.

Additionally, there is another stream called the chance-constrained
programming. It was first introduced to model the UC problem with random wind
power generation in ozturk2003stochastic ; ozturk2004solution . The objective
is to satisfy the net load (load minus wind) with a specified high probability
level over the entire time horizon while minimizing the operating cost. The
original SUC problem is decomposed to a sequence of deterministic versions of
the UC problem that converge to the solution of the chance-constrained
program. In wang2011chance , the SUC problem with uncertain wind power
generation is formulated as a chance-constrained two-stage stochastic program.
A combined sample average approximation (SAA) algorithm is further developed
to solve the problem. Similar to the empirical SUC, this method relies on the
assumption that underlying input model of uncertain variables can be
accurately estimated with historical data.

In this paper, we propose a data-driven SUC with the prediction risk
accounting for both wind power generation stochastic uncertainty and the input
model estimation error. To assess the performance of any candidate decision,
the sample average approximation (SAA) is used to approximate the expected
future cost shapiro_2009 ; Wang_Guan_Wang_2012 . To hedge against the impact
of finite sampling error, we propose the OPSEL approach. This approach is
built based on ranking-and-selection techniques, which are originally
introduced for time-consuming stochastic simulation optimization
Chen_etal_2000 ; Boesel_Nelson_Kim_2003 ; Kim_Nelson_2007 ; Powell_Ryzhov_2012
. Since each simulation run could be time-consuming, given a fixed number of
candidate system designs and a limited computational budget, they are
statistical comparison approaches developed to efficiently find the best
design by sequentially allocating more simulation resources to the promising
candidates. Existing ranking and selection approaches include the indifference
zone method Kim_Nelson_2001 ; Boesel_Nelson_Kim_2003 , the Expected
Improvement (EI) methods Chick_Inoue_2001 ; Frazier_Powell_Dayanik_2008 ;
Chick_Branke_Schmidt_2010 , and Optimal Computing Budget Allocation (OCBA)
Chen_etal_2000 ; Chen_etal_2008 ; xiao2013optimal ; xu2015simulation . The
objective is to maximize simulation efficiency, expressed as the probability
of correct selection (PCS) of the underlying best candidate under limited
computing budget. We consider OCBA in the our proposed OPSEL procedure because
it has several advantages. First, it guarantees the asymptotically correct
selection Glynn_Juneja_2004 . Second, the convergence rate provided by OCBA is
at least as good as other ranking-and-selection methods Ryzhov_2016 . Third,
it demonstrates good finite sampling performance in many studies
Quan_etal_2013 .

##  3 PROBLEM STATEMENT AND DATA-DRIVEN SUC

In this section, we first describe the two-stage stochastic unit commitment
(SUC) problem in Section 3.1. Since the underlying input model, characterizing
the inherent stochastic uncertainty of wind power generation, is unknown, it
is estimated by using finite historical wind power data. This introduces the
input model estimation uncertainty. In Section 3.2, we review the existing
empirical SUC approach which takes the estimated model as the true one and
ignores the model estimation uncertainty. We also discuss the existing
deterministic unit commitment (UC) model, which ignores the prediction risk
induced from both wind power inherent stochastic uncertainty and model risk.
Then, in Section 3.3, we propose the data-driven SUC accounting for both
sources of uncertainties. Our proposed stochastic unit commitment modeling
approach can be applied to situations with both parametric and nonparametric
forecast model for wind power.

###  3.1 Stochastic Unit Commitment Model

Let $\boldsymbol{\xi}$ denote the random wind power generation, and let
$F^{c}$ represent the underlying “correct” statistical input model for SUC
with $\boldsymbol{\xi}\sim F^{c}$. Here, we consider a general formulation of
the two-stage stochastic unit commitment problem Zheng_Wang_Liu_2015 ;
Wang_etal_2016

$\displaystyle\small\min_{\mathbf{u}}G(\mathbf{u})$ $\displaystyle\equiv$
$\displaystyle\mathbf{C}^{su}\mathbf{u}+\mbox{E}_{\boldsymbol{\xi}\sim
F^{c}}\left[\min_{\mathbf{y}}\mathbf{C}^{fuel}\mathbf{y}(\mathbf{u},\boldsymbol{\xi})\right]$
(1) s.t. $\displaystyle A\mathbf{u}\leq B$ (3) $\displaystyle
H\mathbf{u}+Q\mathbf{y}(\mathbf{u},\boldsymbol{\xi})\leq M(\boldsymbol{\xi})$

where $\mathbf{C}^{su}$ is the first-stage cost coefficient, consisting of
various startup and shutdown costs, and the coefficient $\mathbf{C}^{fuel}$
represents the fuel cost; see Zheng_etal_2013 ; Zheng_Wang_Liu_2015 ;
Hobbs_Rothkopf_O'Neill__Chao_2001 . The first-stage unit commitment decision
for thermal generators, denoted by $\mathbf{u}$, is made prior to the
realization of $\boldsymbol{\xi}$. The second-stage economic dispatch
decision, denoted by $\mathbf{y}(\mathbf{u},\boldsymbol{\xi})$, is made after
the unveiling of $\boldsymbol{\xi}$ and it depends on $\mathbf{u}$.

Objective (1) includes the cost incurred in the first stage and the expected
dispatch cost incurred in the planning horizon. The general constraints for
the first- and second-stage decisions can be expressed in (3) and (3). Denote
the optimal unit commitment decision by $\mathbf{u}^{\star}$ and the optimal
objective by
$G(\mathbf{u}^{\star})\equiv\mathbf{C}^{su}\mathbf{u}^{\star}+\mbox{E}_{\boldsymbol{\xi}\sim
F^{c}}[\min_{\mathbf{y}}\mathbf{C}^{fuel}\mathbf{y}(\mathbf{u}^{\star},\boldsymbol{\xi})]$.

###  3.2 Review of Existing Empirical Stochastic Unit Commitment (SUC) and
Deterministic Unit Commitment (UC) Approaches

In this section, we briefly summarize the deterministic UC and SUC modeling
approaches and then discuss their limitations. Traditionally, scheduling and
dispatch in power system operations have been done by using deterministic
methods, and this is still the industry practice in most
regionszhou2016stochastic . Some recent studies also use the deterministic UC
to analyze the impact of renewable resources on power system operations
ummels2007impacts ; delarue2008adaptive .

Basically, given the historical data, various methods can be used for wind
power forecasting, such as persistence-based forecasting method
Nielsen_etal_1998 ; Gneiting_etal_2007 ; Kavasseri_2009  and ARMA
modelummels2007impacts . The deterministic UC makes the optimal decision,
denoted by $\mathbf{u}^{\star,d}$, based on the point predictor of future wind
power generation in the planning horizon, denoted by $\boldsymbol{\hat{\xi}}$,
by solving the deterministic optimization,

$\displaystyle\small\min_{\mathbf{u}}G^{p}(\mathbf{u})$ $\displaystyle\equiv$
$\displaystyle\mathbf{C}^{su}\mathbf{u}+\min_{\mathbf{y}}\mathbf{C}^{fuel}\mathbf{y}(\mathbf{u},\boldsymbol{\hat{\xi}})$
(4) s.t. $\displaystyle A\mathbf{u}\leq B$ $\displaystyle
H\mathbf{u}+Q\mathbf{y}(\mathbf{u},\boldsymbol{\hat{\xi}})\leq
M(\boldsymbol{\hat{\xi}}).$

Thus, deterministic UC does not consider the prediction risk induced by both
wind power generation inherent stochastic uncertainty and input model
estimation uncertainty. It will deliver an inferior and unreliable decision,
especially when the penetration of renewable energy increases.

Differing with the deterministic UC, the empirical SUC considers the impacts
from stochastic uncertainty of wind power generation on electric power systems
Tuohy_etal_2009 ; Wang_Guan_Wang_2012 ; Huang_Zheng_Wang_2014 . Basically,
given the historical data $\mathcal{D}$, to find the optimal decision, the
empirical SUC takes the input model estimate, denoted by $F^{e}$, of unknown
$F^{c}$ as the true one, and then solves the stochastic optimization,

$\displaystyle\small\min_{\mathbf{u}}G^{p}(\mathbf{u})$ $\displaystyle\equiv$
$\displaystyle\mathbf{C}^{su}\mathbf{u}+\mbox{E}_{\boldsymbol{\xi}\sim
F^{e}}\left[\min_{\mathbf{y}}\mathbf{C}^{fuel}\mathbf{y}(\mathbf{u},\boldsymbol{\xi})\right]$
(5) s.t. $\displaystyle A\mathbf{u}\leq B$ $\displaystyle
H\mathbf{u}+Q\mathbf{y}(\mathbf{u},\boldsymbol{\xi})\leq M(\boldsymbol{\xi}).$

Then, a Monte Carlo approach can be used to generate a finite number of
scenarios from $F^{e}$, use the sample mean to approximate the expected future
cost in the planning horizon, and obtain the optimal decision, denoted by
$\mathbf{u}^{\star,e}$. Thus, the empirical UC ignores the input model
estimation uncertainty and finite sampling error from SAA. It could lead to an
inferior and unreliable unit commitment decision, especially under the
situations when the wind power penetration is high and the amount of
representative real-world wind power historical data is limited; see as shown
in the case studies in Section 5.

###  3.3 Data-Driven Stochastic Unit Commitment Model

In this paper, we propose a data-driven SUC accounting for both stochastic
uncertainty of wind power generation and input model estimation uncertainty.
Given the historical data $\mathcal{D}$, the posterior distribution of
underlying input model $F^{c}$ is used to quantify the model estimation
uncertainty. Then, the posterior predictive distribution, denoted by $F^{p}$,
can quantify the prediction risk induced from both sources of uncertainties.
Thus, the proposed data-driven SUC model becomes,

$\displaystyle\small\min_{\mathbf{u}}G^{p}(\mathbf{u})$ $\displaystyle\equiv$
$\displaystyle\mathbf{C}^{su}\mathbf{u}+\mbox{E}_{\boldsymbol{\xi}\sim
F^{p}}\left[\min_{\mathbf{y}}\mathbf{C}^{fuel}\mathbf{y}(\mathbf{u},\boldsymbol{\xi})\right]$
(6) s.t. $\displaystyle A\mathbf{u}\leq B$ $\displaystyle
H\mathbf{u}+Q\mathbf{y}(\mathbf{u},\boldsymbol{\xi})\leq M(\boldsymbol{\xi}).$

It can lead to cost-efficient, reliable and robust optimal unit commitment
decision, denoted by $\mathbf{u}^{\star,p}$, which can hedge against the
prediction risk induced by stochastic uncertainty of wind power generation and
input model estimation uncertainty.

Given the historical data $\mathcal{D}$, the posterior distribution
characterizing the input model estimation uncertainty can be obtained by the
Bayes’ rule, $p(F|\mathcal{D})\propto p(F)p(\mathcal{D}|F),$ where $p(F)$
denotes the prior and $p(\mathcal{D}|F)$ denotes the likelihood of historical
data. Then, the density of posterior predictive distribution $F^{p}$,

$f^{p}(\boldsymbol{\xi})\equiv\int p(\boldsymbol{\xi}|F)p(F|\mathcal{D})dF,$
(7)

can quantify the overall prediction uncertainty of wind power generation with
$p(F|\mathcal{D})$ characterizing the input model estimation uncertainty and
$p(\boldsymbol{\xi}|F)$ characterizing the prediction uncertainty induced by
wind power generation inherent volatility or stochastic uncertainty.

The proposed data-driven SUC can be applied to situations where the parametric
family of underlying input model $F^{c}$ is known, e.g., Wang_Guan_Wang_2012 ;
Zhang_Wang_Wang_2014 ; Pandzic_etal_2016 . In Sections 5.1.1 and 5.2.1, when
we study the six- and 118-bus power grid systems, we use the normal
distribution assumption for illustration. The proposed data-driven SUC in (6)
can also apply to the situations where there is no strong prior information on
the distribution family for $F^{c}$. In Sections 5.1.2 and 5.2.2, we suppose
that there is no strong parametric assumption on the underlying input model
$F^{c}$ and use the Bayesian nonparametric probabilistic forecast introduced
in our previous study Xie_etal_2018  as an illustration.

##  4 OPTIMIZATION AND SELECTION

When we solve the stochastic unit commitment optimization, a finite number of
scenarios is typically used to approximate the expected cost in the planning
horizon. It induces the finite sampling error, which can be large especially
for complex power grids with high wind power penetration. In this section, we
develop an optimization and selection (OPSEL) approach which can utilize the
parallel computing to quickly solve the data-driven SUC and find the optimal
decision accounting for the impact of finite sampling error induced by SAA.
This study can be applied to general stochastic UC problems and it can
accelerate the real-time reliable decision making.

Since the time used to assess the performance of any given unit commitment
decision is much less than the optimization search, we propose an optimization
solution methodology including two parts: optimization search and best
candidate decision selection. For the optimization search part as presented in
Section 4.1, we simultaneously solve $L$ independent SAA approximated data-
driven SUC problems through parallel computing. It returns a set of optimal
candidate solutions quantifying the impact of finite sampling error induced by
SAA. For the best candidate decision selection part as shown in Section 4.2,
we apply the ranking-and-selection methodology to efficiently allocate more
computational budget to the most promising candidate decisions, improve the
assessment of their performance, and quickly select the best decision. Thus,
the combination of data-driven SUC and OPSEL can lead to the optimal and
reliable unit commitment decision, which can hedge against: (1) inherent
stochastic uncertainty of wind power generation, (2) input model estimation
uncertainty, and (3) finite sampling error induced by SAA.

###  4.1 Parallel Optimization Search Accounting for SAA Finite Sampling Error

The second-stage expected economic dispatch cost
$\mbox{E}_{\boldsymbol{\xi}\sim
F^{p}}\left[\min_{\mathbf{y}}\mathbf{C}^{fuel}\mathbf{y}(\mathbf{u},\boldsymbol{\xi})\right]$
does not have a closed-form expression, and it is typically approximated by
using sample average approximation (SAA) (see the introduction of SAA in
shapiro_2009 ), which introduces the finite sample error. Specifically, we
generate $S$ scenarios, $\boldsymbol{\xi}^{s}\stackrel{{\scriptstyle
i.i.d.}}{{\sim}}F^{p}$ for $s=1,2,\ldots,S$, and use them to estimate the
expected cost in the planning horizon. The SAA approximated data-driven SUC in
(6) becomes

$\displaystyle\small\min_{\mathbf{u}}$
$\displaystyle\bar{G}^{p}(\mathbf{u})=\mathbf{C}^{su}\mathbf{u}+\frac{1}{S}\sum_{s=1}^{S}\left[\min_{\mathbf{y}}\mathbf{C}^{fuel}\mathbf{y}(\mathbf{u},\boldsymbol{\xi}^{s})\right]$
(8) s.t. $\displaystyle A\mathbf{u}\leq B$ $\displaystyle
H\mathbf{u}+Q\mathbf{y}(\mathbf{u},\boldsymbol{\xi}^{s})\leq
M(\boldsymbol{\xi}^{s})\quad\mbox{for }s=1,2,\ldots,S.$

To accurately approximate the expected second-stage cost, the number of
scenarios $S$ needs to be sufficiently large. It can be computationally
infeasible to solve for complex power grids with high wind power penetration,
especially when the quick decision making is required for intra-day unit
commitment problems. Thus, SAA in (8) typically introduces the unignorable
finite sampling error.

To efficiently employ the computational resource and quickly deliver the
optimal UC decision hedging against various sources of uncertainty, we exploit
the parallel computing with $L$ available CPUs. With each $\ell$-th CPU, we
first generate an independent set consisting of $S$ scenarios,
$\boldsymbol{\xi}^{1},\ldots,\boldsymbol{\xi}^{S}$, drawn from $F^{p}$, and
then we solve the corresponding SAA approximated data-driven SUC problems in
(8). In the case study, we use the L-Shaped algorithm for optimization
Zheng_etal_2013 ; Zheng_Wang_Liu_2015 . Thus, based on the parallel
optimization search, we obtain the optimal unit commitment candidate
decisions, denoted by $\widehat{\mathbf{u}}^{\star}_{\ell}$ with
$\ell=1,\ldots,L$, quantifying the impact of finite sampling error.

###  4.2 Ranking and Selection based Best Decision Selection

To eliminate the impact of finite sampling error quantified by candidates
$\widehat{\mathbf{u}}^{\star}_{1},\ldots,\widehat{\mathbf{u}}^{\star}_{L}$, we
efficiently utilize the computational resource to assess the performance of
candidate decisions,
$G^{p}(\widehat{\mathbf{u}}^{\star}_{1}),\ldots,G^{p}(\widehat{\mathbf{u}}^{\star}_{L})$,
and select the best one,

${\mathbf{u}}^{\star}_{b}\equiv\mbox{argmin}_{\widehat{\mathbf{u}}^{\star}_{\ell}\in\\{\widehat{\mathbf{u}}^{\star}_{1},\ldots,\widehat{\mathbf{u}}^{\star}_{L}\\}}G^{p}(\widehat{\mathbf{u}}^{\star}_{\ell}).$

with the subscript $b$ denoting the best unit commitment decision. The number
of CPUs, $L$, could be large. For each candidate, the performance
$G^{p}(\widehat{\mathbf{u}}^{\star}_{\ell})$ with $\ell=1,2,\ldots,L$ can be
assessed by using Monte Carlo approach. We sequentially allocate the
computational budget to the most promising candidates
$\widehat{\mathbf{u}}^{\star}_{\ell}$ so that we can provide more accurate
estimation of their expected cost $G^{p}(\widehat{\mathbf{u}}^{\star}_{\ell})$
and efficiently select out the best solution.

Basically, built on the results from optimal search described in Section 4.1,
we further use the OCBA approach Chen_etal_2000  to guide the sequential
allocation of computational resource to promising candidates. Here, the each
unit of computational budget is measured by the cost of solving one second-
stage economic dispatch problem for each scenario. Let $N_{k,\ell}$ be the
accumulated number of scenarios assigned to the candidate solution
$\widehat{\mathbf{u}}^{\star}_{\ell}$ for $\ell=1,\ldots,L$ until the $k$-th
iteration of sequential candidate selection, and the objective estimate is

$\bar{G}^{p}_{k}(\widehat{\mathbf{u}}_{\ell}^{\star})=\mathbf{C}^{su}\widehat{\mathbf{u}}_{\ell}^{\star}+\frac{1}{N_{k,\ell}}\sum_{s=1}^{N_{k,\ell}}\left[\min_{\mathbf{y}}\mathbf{C}^{fuel}\mathbf{y}(\widehat{\mathbf{u}}_{\ell}^{\star},\boldsymbol{\xi}^{s})\right].$
(9)

With more scenarios, we can estimate the performance
${G}^{p}_{k}(\widehat{\mathbf{u}}_{\ell}^{\star})$ more accurate. The number
of initial scenarios is $N_{0,\ell}=S$ since the data-driven SUC approximated
with $S$ samples is solved by the $\ell$-th CPU to obtain
$\widehat{\mathbf{u}}^{\star}_{\ell}$; see Eq. (8). At the $k$-th iteration,
we allocate $\Delta T$ additional scenarios to the candidates
$\widehat{\mathbf{u}}^{\star}_{\ell}$ with $\ell=1,\ldots,L$. Then, we solve
the corresponding economic dispatch problems for the new generated scenarios
and update the objective estimate for
$G^{p}(\widehat{\mathbf{u}}^{\star}_{\ell})$ by using Eq. (9).

Specifically, based on Chen_etal_2000 , the optimal budget allocation
$N_{k,\ell}$ is obtained by solving

$\displaystyle\frac{N_{k,\ell}}{N_{k,\ell^{\prime}}}$ $\displaystyle=$
$\displaystyle\left(\frac{\delta_{k-1,\ell^{\prime}}}{\delta_{k-1,\ell}}\right)^{2}\mbox{
for }\ell,\ell^{\prime}\neq b$ (10) $\displaystyle N_{k,b}$ $\displaystyle=$
$\displaystyle\widehat{\sigma}_{k-1,b}\sqrt{\sum_{\ell=1,\ell\neq
b}^{L}\frac{N^{2}_{k,\ell}}{\widehat{\sigma}^{2}_{k-1,\ell}}}$
$\displaystyle\sum_{\ell=1}^{L}N_{k,\ell}$ $\displaystyle=$ $\displaystyle
L\times S+k\times\Delta T$

where $N_{k,b}$ denotes the number of scenarios assigned to the current best
candidate selected from the $(k-1)$-th iteration

$\widehat{\mathbf{u}}^{\star}_{k-1,b}\equiv\mbox{argmin}_{\widehat{\mathbf{u}}^{\star}_{\ell}\in\\{\widehat{\mathbf{u}}^{\star}_{1},\ldots,\widehat{\mathbf{u}}^{\star}_{L}\\}}\bar{G}^{p}_{k-1}(\widehat{\mathbf{u}}^{\star}_{\ell}).$
(11)

The estimate of variance,
${\sigma}_{\ell}^{2}=\mbox{Var}\left[\min_{\mathbf{y}}\mathbf{C}^{fuel}\mathbf{y}(\mathbf{u},\boldsymbol{\xi})\right]$,
is the sample variance obtained from the $(k-1)$-th iteration,

$\displaystyle\widehat{\sigma}_{k-1,\ell}^{2}=\frac{1}{N_{k-1,\ell}-1}\sum_{s=1}^{N_{k-1,\ell}}\Big{[}\mathbf{C}^{su}\widehat{\mathbf{u}}^{\star}_{\ell}\Big{.}\Big{.}+\min_{\mathbf{y}}\mathbf{C}^{fuel}\mathbf{y}(\widehat{\mathbf{u}}^{\star}_{\ell},\boldsymbol{\xi}^{s})-\bar{G}^{p}_{k-1}(\widehat{\mathbf{u}}^{\star}_{\ell})\Big{]}^{2}$

and $\delta_{k-1,\ell}$ denotes the standardized distance between
$\widehat{\mathbf{u}}^{\star}_{\ell}$ with the current estimated best
candidate $\widehat{\mathbf{u}}^{\star}_{k-1,b}$,

$\delta_{k-1,\ell}\equiv\frac{\bar{G}^{p}_{k-1}(\widehat{\mathbf{u}}^{\star}_{\ell})-\bar{G}^{p}_{k-1}(\widehat{\mathbf{u}}^{\star}_{k-1,b})}{\widehat{\sigma}_{k-1,\ell}}.$
(12)

By applying Eq. (10) and (12), the budget allocation to any candidate
$\widehat{\mathbf{u}}^{\star}_{\ell}$ is directly related to its standardized
distance with the current best estimate, which can allocate more computational
resource to the promising candidates and efficiently select out the best
decision. Thus, in the $k$-th iteration, the number of new scenarios allocated
to the candidate $\widehat{\mathbf{u}}^{\star}_{\ell}$ for $\ell=1,2,\ldots,L$
is,

$\Delta N_{k,\ell}=\max(0,N_{k,\ell}-N_{k-1,\ell}).$ (13)

Then, we solve the additional $\Delta N_{k,\ell}$ second-stage economic
dispatch problems and update the objective estimate
$\bar{G}_{k}^{p}(\widehat{\mathbf{u}}^{\star}_{\ell})$.

The OPSEL procedure is provided in Algorithm 1. We denote the overall
computational budget in terms of number of scenarios allocated for the
candidate selection by $T$. In Step (1), we utilize $L$ CPUs to simultaneously
solve the sample average approximated SUC problems with form (8) and obtain
the optimal candidate decisions $\widehat{\mathbf{u}}^{\star}_{\ell}$ with
$\ell=1,2,\ldots,L$. Then, in Steps (2) and (3), we sequentially allocate the
computational resource to
$\widehat{\mathbf{u}}^{\star}_{1},\ldots,\widehat{\mathbf{u}}^{\star}_{L}$ and
select the best candidate decision hedging against the finite sampling error.

Step (0) Specify $S$ and the total budget $T$, the total number of scenarios
used for the candidate solution selection.

Step (1) Utilize $L$ CPUs to simultaneously solve the SUC problems (8)
approximated by $S$ scenarios, and obtain the optimal candidate decisions
$\widehat{\mathbf{u}}^{\star}_{1},\ldots,\widehat{\mathbf{u}}^{\star}_{L}$.

Step (2) At the $k$-th iteration of selection, allocate $\Delta T$ new
scenarios to
$\widehat{\mathbf{u}}^{\star}_{1},\ldots,\widehat{\mathbf{u}}^{\star}_{L}$ by
using (13) and solve the additional second-stage dispatch problems. Update
$\bar{G}^{p}_{k}(\widehat{\mathbf{u}}^{\star}_{\ell})$ for $\ell=1,\ldots,L$
and $\widehat{\mathbf{u}}^{\star}_{k,b}$ by applying (9) and (11).

Step (3) Repeat Step (2) until reaching to the budget $T$. Return
$\widehat{\mathbf{u}}^{\star}_{k,b}=\mbox{argmin}_{\widehat{\mathbf{u}}^{\star}_{\ell}\in\\{\widehat{\mathbf{u}}^{\star}_{1},\ldots,\widehat{\mathbf{u}}^{\star}_{L}\\}}\bar{G}^{p}_{k}(\widehat{\mathbf{u}}^{\star}_{\ell}).$

ALGORITHM 1 The Optimization and Selection Procedure

##  5 Empirical Studies

We use the six-bus system from Wang_etal_2017  in Section 5.1 and the
derivative 118-bus system from pena2017extended  in Section 5.2 to compare the
performance of proposed data-driven SUC framework with the deterministic unit
commitment (UC) and the empirical SUC under the situations when the parametric
input model for wind power generation is known or not. In the empirical
studies, we consider the two-stage SUC problem,

$\displaystyle\min$ $\displaystyle\quad
G(u_{i,t})=\sum_{t=1}^{T}\sum_{i=1}^{I}(C^{i}F_{mi}u_{i,t}+SU_{i,t}+SD_{i,t})$
(14)
$\displaystyle+\mbox{E}\left[\min_{P_{i,t},P_{b,t}^{ens},P_{w,t}^{wc}}\sum_{t=1}^{T}\sum_{i=1}^{I}C^{i}F_{ai}P_{i,t}+\sum_{t=1}^{T}\sum_{b=1}^{B}C^{ens}P_{b,t}^{ens}+\sum_{t=1}^{T}\sum_{w=1}^{W}C^{wc}P_{w,t}^{wc}\right]$

$\displaystyle\textrm{s.t.}\quad\quad\quad\quad\quad\quad\quad\quad\quad
u_{i,t}-u_{i,t-1}\leq u_{i,k}\quad\forall k=t,\ldots,\min(T,t+T_{i}^{on}-1)$
(15) $\displaystyle u_{i,k}\leq 1+u_{i,t}-u_{i,t-1}\quad\forall
k=t,\ldots,\min(T,t+T_{i}^{off}-1)$ (16)
$\displaystyle\sum_{i=1}^{I}P_{i,t}+\sum_{w=1}^{W}(P_{w,t}^{c}-P_{w,t}^{wc})=\sum_{b=1}^{B}(P^{D}_{b,t}-P_{b,t}^{ens})$
(17) $\displaystyle-
PL_{\ell,\max}\leq\sum_{b=1}^{B}G_{\ell-b}P^{D}_{b,t}+\sum_{i=1}^{I}G_{\ell-i}P_{i,t}+\sum_{w=1}^{W}G_{\ell-w}(P_{w,t}^{c}-P_{w,t}^{wc})\leq
PL_{\ell,\max}$ (18) $\displaystyle u_{i,t}P_{i,\min}\leq u_{i,t}P_{i,t}\leq
u_{i,t}P_{i,\max}\quad\forall i,\quad\forall t$ (19) $\displaystyle\quad\quad
u_{i,t}\quad\mbox{binary}$ (20)

The objective in (14) is to minimize the total expected cost occurring in the
planning horizon $T$, including the start-up cost $SU_{i,t}$, the turn-off
cost $SD_{i,t}$ and minimal thermal operation cost $C^{i}F_{mi}u_{i,t}$
incurring in the fist stage, where $C^{i}$ is the fuel price and $F_{mi}$ is
the amount of fuel consumption for the minimal output for generator $i$. Since
the thermal generator consumes the extra fuel to produce $P_{i,t}$, the
additional production cost in the second-stage is $C^{i}F_{ai}P_{i,t}$, where
$F_{ai}$ is the fuel consumption needed to generate $P_{i,t}$ power
production. The penalty cost of non-satisfied demand for bus $b$ at time
period $t$ is $C^{ens}P_{b,t}^{ens}$, where $C^{ens}$ is the unit load
shedding price and $P_{b,t}^{ens}$ is the amount of unmet load at bus $b$ in
time period $t$. Lastly, for the wind power, we may not use up all its
capacity $P_{w,t}^{c}$ and there exists a wind farm curtailment
$P_{w,t}^{wc}$. The wind curtailment cost at the $t$-th hour for wind farm $w$
is $C^{wc}P_{w,t}^{wc}$, where $C^{wc}$ is the per unit monetary reward for
the wind production and $P_{w,t}^{wc}$ is the amount of wind curtailment.
Constraints (15)–(16) formulate the minimum up and down time requirement,
Constraints (17)–(18) are for the nodal power balance and the DC power flow
constraint, and Constraint (19) enforces the minimum and maximum generator
output limits.

###  5.1 Empirical Study with Six-Bus System

For the six-bus system, it consists of three thermal units, a wind farm and
seven transmission lines as depicted in Fig. 1. The three thermal units are
located in No.1, No.2 and No.6 buses, while No.4 Bus hosts a wind farm; see
Table 1 for the description of the six-bus system. Tables 2 and 3 describe the
characteristics of the thermal generators, while the characteristics of the
transmission lines are provided in Table 4. Since the uncertainty in wind
supply typically dominates, in this study, we assume deterministic loads and
stochastic wind supply Zhou_etal_2013 . Following Wang_etal_2017 , we use the
2006 data of the U.S. Illinois power system for the load $P_{b,t}^{D}$ and the
wind supply $P_{w,t}^{c}$. Then, the wind penetration level measured by the
ratio of wind power generation to actual demand,
$R=\sum_{w=1}^{W}\sum_{t=1}^{T}{P_{w,t}^{c}}/\sum_{b=1}^{B}\sum_{t=1}^{T}{P_{b,t}^{D}}$,
is 37.8%. Additionally, the wind curtailment price is fixed at $50\$/MWh$ and
the load shedding price is set at $3500\$/MWh$ Wang_etal_2017 .

Figure 1: Six-Bus System Table 1: Bus Data Bus ID | Type | Thermal Unit | Wind Farm | Load Share
---|---|---|---|---
NO 1 | Thermal | G1 |  |
NO 2 | Thermal | G2 |  |
NO 3 |  |  |  | 20%
NO 4 | Wind |  | W1 | 40%
NO 5 |  |  |  | 40%
NO 6 | Thermal | G3 |  |
Table 2: Thermal Generator Data

Unit | Pmax(MW) | Pmin(MW) | Ini.State (h) | Min Off(h) | Min On (h)
---|---|---|---|---|---
G1 | 220 | 90 | 4 | -4 | 4
G2 | 100 | 20 | 2 | -3 | 2
G3 | 30 | 10 | -1 | -1 | 1

Table 3: Thermal Generator Data

Unit | Fuel consumption Function |  Start up Fuel (MBtu) |  Shut down Fuel (MBtu) |  Fuel Price ($)
---|---|---|---|---
| a
---
(MBtu)
| b
---
(MBtu/MWh)
| c
---
(MBtu/MW2h)
G1 | 176.9 | 13.5 | 0.0004 | 180 | 50 | 1.2469
G2 | 129.9 | 32.6 | 0.001 | 360 | 40 | 1.2461
G3 | 137.4 | 17.6 | 0.005 | 60 | 0 | 1.2462

Table 4: Transmission Line Data

Line No. | From Bus | To Bus | X (p.u) | Flow Limit (MW)
---|---|---|---|---
1 | 1 | 2 | 0.17 | 200
2 | 1 | 4 | 0.258 | 100
3 | 2 | 4 | 0.197 | 100
4 | 5 | 6 | 0.14 | 100
5 | 2 | 3 | 0.037 | 200
6 | 4 | 5 | 0.037 | 200
7 | 3 | 6 | 0.018 | 200

####  5.1.1 Case Study with Parametric Forecast Model

In this section, suppose the input model of wind power generation $F^{c}$
follows the normal distribution Wang_Shahidehpour_Li_2008 ;
Wang_Guan_Wang_2012 ; Pandzic_etal_2016  with unknown parameters estimated by
valid historical data. Here, we consider the day-ahead unit commitment with
the planning horizon equal to 24 hours. In each day $d$, suppose that the wind
power generation at the $t$-th hour follows a normal distribution,
$\xi_{d,t}\sim N(\mu_{d,t}^{c},\phi_{d,t}^{2})$, where $\mu_{d,t}^{c}$ and
$\phi_{d,t}^{2}$ are mean and variance. Thus, the underlying true input model
$F^{c}$ for $\xi_{d,t}$ in SUC (1) is $N(\mu_{d,t}^{c},\phi_{d,t}^{2})$. To
evaluate the performance, we pretend that the true parameter $\mu_{d,t}^{c}$
is unknown. Suppose that wind power at the $t$-th hour in the past $m$ days
follows the same distribution. To predict $\xi_{d,t}$, we use the valid
historical observation
$\mathcal{D}_{d,t}\equiv\\{\xi^{r}_{(d-m),t},\ldots,\xi^{r}_{(d-1),t}\\}$ with
$t=1,\ldots,24$ hour, where $\xi^{r}_{(d-i),t}$ denotes the real wind power
observation at the $t$-th hour in day $(d-i)$ with $i=1,\ldots,m$. Here, we
set $\mu_{d,t}^{c}$ equal to the 2006 wind power generation.

The empirical SUC takes the estimated input model coefficient, sample mean, as
the true one. Thus, the input model $F^{e}_{d,t}$ for SUC is
$N(\bar{\xi}^{r}_{d,t},\phi_{d,t}^{2})$, where the sample mean of the
historical data
$\bar{\xi}^{r}_{d,t}=\frac{1}{m}\sum_{i=1}^{m}\xi^{r}_{(d-i),t}$ is the plug-
in estimate of unknown parameter $\mu^{c}_{d,t}$ For the proposed data-driven
SUC, the model estimation uncertainty is characterized by the posterior
distribution. Without strong information about the mean $\mu_{d,t}^{c}$, we
use the non-informative prior, a normal distributed with mean zero and
infinite variance Yang_Berger_1997 . The posterior distribution is
$p(\mu_{d,t}|\mathcal{D}_{d,t})=N(\bar{\xi}^{r}_{d,t},{\phi_{d,t}^{2}}/{m})$.
Then, the resulting posterior predictive distribution $F^{p}_{d,t}$ is
$N(\bar{\xi}^{r}_{d,t},(1+\frac{1}{m})\phi_{d,t}^{2})$, which quantifies the
prediction risk accounting for both input model estimation uncertainty and
wind power generation inherent stochastic uncertainty,

We compare the performance of unit commitment decisions obtained from the
data-driven SUC with the empirical SUC under various settings with standard
deviation $\phi_{d,t}=5\%\mu_{d,t}^{c},10\%\mu_{d,t}^{c},20\%\mu_{d,t}^{c}$.
Let $n_{d}$ denote the total number of days used for the evaluation. Let
$\widehat{\mathbf{u}}_{d}^{\star,p}$ and $\widehat{\mathbf{u}}^{\star,e}_{d}$
denote the 24-hour optimal unit commitment decisions obtained by data-driven
and empirical SUCs with $d=1,\ldots,n_{d}$. Then, the total expected costs
obtained by these approaches are

$\sum G^{p}=\sum_{d=1}^{n_{d}}G(\widehat{\mathbf{u}}_{d}^{\star,p})\mbox{ and
}\sum G^{e}=\sum_{d=1}^{n_{d}}G(\widehat{\mathbf{u}}_{d}^{\star,e}).$ (21)

Since there is no closed form, the sample average approximations,
$\bar{G}(\widehat{\mathbf{u}}^{\star,p}_{d})$ and
$\bar{G}(\widehat{\mathbf{u}}^{\star,e}_{d})$, are used to estimate the true
objectives. To determine a proper scenario size $S_{e}$ so that we can
estimate $G(\widehat{\mathbf{u}}^{\star,p}_{d})$ and
$G(\widehat{\mathbf{u}}^{\star,e}_{d})$ accurately, we conduct a side
experiment and consider the high uncertainty case with
$\phi_{d,t}=20\%\mu_{d,t}^{c}$. In addition, since the empirical approach
ignores the input model parameter estimation uncertainty, the unit commitment
decisions highly fluctuate with the random wind power observations and its
solution quality is more volatile. Thus, to decide the proper sample size that
can ensure the accurate estimation of the expected total cost, we consider the
empirical approach. Specifically, we first apply the L-shaped algorithm to
solve the empirical SUC with the expected cost approximated by SAA having
$S=50$ and obtain a unit commitment decision
$\widehat{\mathbf{u}}_{d}^{\star,e}$. Then, we estimate the expected cost by
using $S_{e}$ scenarios to obtain
$\bar{G}(\widehat{\mathbf{u}}_{d}^{\star,e})$ and calculate the relative
error,
$\mbox{relativeError}\equiv{|\bar{G}(\widehat{\mathbf{u}}_{d}^{\star,e})-G(\widehat{\mathbf{u}}_{d}^{\star,e})|}/{G(\widehat{\mathbf{u}}_{d}^{\star,e})}$,
where $G(\widehat{\mathbf{u}}_{d}^{\star,e})$ denotes the objective value
obtained by using $10^{5}$ scenarios. Suppose that $10^{5}$ is large enough so
that the estimation error of $G(\widehat{\mathbf{u}}_{d}^{\star,e})$ is
negligible. The maximum relative error obtained from $10$ day-period is
recorded in Table 5. We observe that $S_{e}=1000$ achieves accuracy with the
maximum relative error not exceeding $1.0\%$. Balancing the computational cost
and the accuracy, we use $S_{e}=1000$ to evaluate the true expected cost.

Table 5: The Maximum Absolute Relative Difference for
$G(\widehat{\mathbf{u}}_{d}^{\star e})$ Estimation

$S_{e}$ | $10^{2}$ | $5\times 10^{2}$ | $10^{3}$ | $5\times 10^{3}$ | $10^{4}$ | $5\times 10^{4}$
---|---|---|---|---|---|---
relativeError | 3.0% | 1.5% | 0.9% | 0.9% | 0.7% | 0.1%

The wind power data in 2006 October are used for evaluation. Let $m=1$. In the
study, we set the scenario size $S=50$ and get the optimal decision estimates
$\widehat{\mathbf{u}}_{d}^{\star,e}$ and $\widehat{\mathbf{u}}_{d}^{\star,p}$
by solving the sample average approximated empirical and data-driven SUCs. For
cases with $\phi_{d,t}=5\%\mu_{d,t}^{c},10\%\mu_{d,t}^{c},20\%\mu_{d,t}^{c}$,
the results of $\sum G^{p}$ and $\sum G^{e}$ in (21) for the one-month period
are recorded in Table 6. We also record the relative expected saving obtained
by our method, denoted by $r\Delta G$,

$r\Delta G=\frac{\sum G^{e}-\sum G^{p}}{\sum G^{p}}.$

The results in Table 6 demonstrate that the proposed data-driven SUC
significantly outperforms the empirical SUC. When
$\phi_{d,t}=5\%\mu_{d,t}^{c}$, the total expected cost-saving by our approach
is $208,730$, which represents a $8.8\%$ lower cost than the empirical SUC. As
$\phi_{d,t}$ increases, the advantages of data-driven SUC tend to be larger.
When $\phi_{d,t}=20\%\mu_{d,t}^{c}$, our approach outperforms the empirical
approach by $15.4\%$ savings. Thus, the proposed data-driven SUC can lead to
better and more robust unit commitment decision and the advantage becomes
larger when the wind penetration is higher.

Table 6: Total costs obtained by data-driven SUC, empirical SUC, and
deterministic UC approaches under the cases with different levels of wind
power penetration

| $\sum G^{p}$ | $\sum G^{e}$ | $r\Delta G$
---|---|---|---
$\phi_{d,t}=5\%\mu_{d,t}^{c}$ | 2,152,146 | 2,360,877 | 8.8%
$\phi_{d,t}=10\%\mu_{d,t}^{c}$ | 2,443,784 | 2,844,019 | 14.1%
$\phi_{d,t}=20\%\mu_{d,t}^{c}$ | 2,671,475 | 3,156,688 | 15.4%

####  5.1.2 Case Study with Nonparametric Forecast Model

In the real application, we often do not have any strong prior knowledge about
the underlying input model $F^{c}$ and its distribution family is typically
unknown. Thus, we consider a unit commitment problem with nonparametric
forecast models. Here, we compare the performance of various approaches,
including (1) the deterministic unit commitment (UC); (2) the empirical SUC
accounting for wind power stochastic uncertainty; and (3) the data-driven SUC
accounting for both wind power stochastic uncertainty and input model
estimation uncertainty. Specifically, for the data-driven SUC, we use the
Bayesian nonparametric wind power forecast model proposed in our previous
study Xie_etal_2018 . For the empirical SUC and deterministic UC, we use
probabilistic and deterministic persistence models Nielsen_etal_1998 ;
Gneiting_etal_2007 ; Kavasseri_2009 . At the time period $h_{i}$, the
probabilistic persistence model is based on the empirical predictive
distribution for $\boldsymbol{\xi}_{h_{t}+i}$ specified by

$\\{\boldsymbol{\xi}^{r}_{h_{t}}-\boldsymbol{\xi}^{r}_{h_{t}-j}+\boldsymbol{\xi}^{r}_{h_{t}-j-i}:j=0,\ldots,h_{t}-i-1\\}$

where $\boldsymbol{\xi}^{r}_{h_{t}}$, $\boldsymbol{\xi}^{r}_{h_{t}-j}$ and
$\boldsymbol{\xi}^{r}_{h_{t}-j-i}$ are wind power observations at time periods
$h_{t}$, $h_{t}-j$ and $h_{t}-j-i$, respectively. The deterministic
persistence model simply takes the previous historical observation as the
point estimator, i.e.,
$\boldsymbol{\xi}_{h_{t}+i}=\boldsymbol{\xi}^{r}_{h_{t}}$.

Since the Bayesian nonparametric forecast proposed in Xie_etal_2018  and
persistent models focus on the short term prediction, we consider the unit
commitment problem for the intra-day market; see Analui_Scaglione_2017 . The
one-hour ahead intra-day unit commitment problem has the planning horizon with
$n_{h}=4$ hours and we make the unit commitment decision of the $t$-th hour at
hour $h_{t}$, where $h_{t}\equiv(\lceil\frac{t}{n_{h}}\rceil-1)\cdot n_{h}.$
It means that we make the one-hour ahead intra-day unit commitment decisions
every four hours. For example, we make $t=1,\ldots,4$ hour intra-day unit
commitment decisions at $h_{t}=0$. Suppose that all three generators in the
six-bus system are fast start generators that can be committed/de-committed at
the intra-day market.

At any time period $h_{t}+1$, we can use $m$ latest historical data for the
wind power prediction,
$\mathcal{D}=\\{\xi^{r}_{h_{t}-m+1},\ldots,\xi^{r}_{h_{t}}\\}$, where
$\xi^{r}_{h_{t}-i}$ is the hourly wind power observation happened $i$ hours
prior to $h_{t}$. Following Xie_etal_2018 , we set $m=100$. For the Bayesian
nonparametric forecast approach, we apply the sampling procedure developed in
Xie_etal_2018 . For the probabilistic persistence model, we apply the sampling
procedure developed in Gneiting_etal_2007 .

Denote the optimal unit commitment decisions between hours $h_{t}+1$ and
$h_{t}+n_{h}$ on day $d$ obtained from data-driven SUC, empirical SUC and
deterministic UC by $\widehat{\mathbf{u}}^{\star,p}_{d,h_{t}}$,
$\widehat{\mathbf{u}}^{\star e}_{d,h_{t}}$ and $\widehat{\mathbf{u}}^{\star
d}_{d,h_{t}}$ for $d=1,\ldots,n_{d}$ and $h_{t}=0,4,\ldots,20$. Let the set
$\mathcal{H}\equiv\\{0,4,\ldots,20\\}$. Since $F^{c}$ is unknown, we evaluate
the performance of these solutions by the actual incurred cost; see the
details in Zhou_etal_2013 . Denote
$\boldsymbol{\xi}_{d,h_{t}}^{r}\equiv({\xi}^{r}_{d,(h_{t}+1)},\ldots,{\xi}^{r}_{d,(h_{t}+n_{h})})$
as the real wind power realizations between hours $h_{t}+1$ and $h_{t}+n_{h}$
on day $d$. Then, the real cost $G^{r}_{d,h_{t}}$ including both commitment
and economic dispatch costs is,

$G^{r}_{d,h_{t}}(\mathbf{u}_{d,h_{t}})\equiv\mathbf{C}^{su}\mathbf{u}_{d,h_{t}}+\min_{\mathbf{y}}\mathbf{C}^{fuel}\mathbf{y}(\mathbf{u}_{d,h_{t}},\boldsymbol{\xi}_{d,h_{t}}^{r}).$

Thus, the total cost occurring in the $n_{d}$ days is

$\sum
G^{\gamma}_{r}\equiv\sum_{d=1}^{n_{d}}\sum_{h_{t}\in\mathcal{H}}G^{r}_{d,h_{t}}(\widehat{\mathbf{u}}^{\star,\gamma}_{d,h_{t}})$
(22)

where $\gamma=p$ is for the data-driven SUC with the Bayesian nonparametric
forecast model Xie_etal_2018 , $\gamma=e$ is for the empirical SUC with the
probabilistic persistent model, and $\gamma=d$ is for the deterministic UC
with the persistent model.

The wind power data of 2006 October are used to evaluate the performance of
these approaches. The results are recorded in Table 7. We also report the
relative savings with respect to deterministic UC,

$r\Delta G=\frac{\sum G^{d}_{r}-\sum G^{p}_{r}}{\sum
G^{d}_{r}}~{}~{}~{}\mbox{and}~{}~{}~{}r\Delta G=\frac{\sum G^{d}_{r}-\sum
G^{e}_{r}}{\sum G^{d}_{r}},$

obtained by the data-driven and empirical SUC approaches. The data-driven SUC
leads to the total aggregated cost $2,822,705$, while the empirical SUC has a
total incurred cost $2,969,178$. Thus, our proposed method has a total saving
$146,473$, which represents a $5\%$ savings. Compared with the deterministic
UC, the advantage of our data-driven SUC is even more substantial and it leads
to a total saving $638,756$. That means our data-driven SUC could provide
better and more reliable performance.

Table 7: Total costs obtained by data-driven SUC, empirical SUC, and
deterministic UC approaches when there is no strong prior knowledge on wind
power generation input model

| Total Cost | $r\Delta G$
---|---|---
Data-Driven SUC: $\sum G^{p}_{r}$ | 2,822,705 | 18.4%
Empirical SUC: $\sum G^{e}_{r}$ | 2,969,178 | 14.2%
Deterministic UC: $\sum G^{d}_{r}$ | 3,461,462 | 0

Therefore, since the data-driven SUC rigorously considers the wind power
generation inherent stochastic uncertainty and input model estimation
uncertainty, it outperforms the empirical SUC and deterministic UC.

####  5.1.3 Performance of OPSEL Approach

In this section, we use the intra-day unit commitment problem described in
Section 5.1.2 to study the performance of the OPSEL approach developed to
hedge against the finite sampling uncertainty induced by using SAA. We
consider the data-driven SUC with the nonparametric Bayesian forecast model
Xie_etal_2018 . To illustrate the impact of finite sampling uncertainty, we
use a representative day, October 2nd, for demonstration. Fig. 2 plots the
incurred cost
$\sum_{h_{t}\in\mathcal{H}}G^{r}_{d,h_{t}}(\widehat{\mathbf{u}}^{\star,p}_{d,h_{t},\ell})$
for $\ell=1,\ldots,L$ obtained by utilizing $L=20$ CPUs to independently solve
the SAA approximated data-driven SUC optimization problem in (8), where
$\widehat{\mathbf{u}}^{\star,p}_{d,h_{t},\ell}$ is obtained from the $\ell$-th
CPU. Each decision is obtained by using SAA with the number of scenarios,
$S=50$. In the plot, each dot represents one optimal decision, the horizontal
axis provides the CPU index $\ell$, and the vertical axis provides the
incurred cost. We can observe that the cost is widespread, ranging from about
$55,000$ to as much as around $120,000$. Thus, Fig. 2 demonstrates the obvious
impact of finite sampling error induced by SAA.

Figure 2: The scatter plot of incurred costs by using the optimal UC decisions
obtained from $L=20$ CPUs to illustrate the impact of finite sampling
uncertainty from SAA.

Then, we study the performance of OPSEL approach introduced in Section 4.2.
Suppose there are $L$ CPUs available for parallel computing. Specifically, for
each intra-day unit commitment problem at the $h_{t}$-th hour on day $d$, we
consider the sample average approximated data-driven SUC in (8) with $S=50$.
By solving $L$ intra-day unit commitment problems in parallel, we obtain the
optimal unit commitment decisions for the next $n_{h}$ hours, denoted by
$\widehat{\mathbf{u}}^{\star,p}_{d,h_{t},1},\ldots,\widehat{\mathbf{u}}^{\star,p}_{d,h_{t},L}$.
After that, the OPSEL is used to estimate the objective values
$G^{p}(\widehat{\mathbf{u}}^{\star,p}_{d,h_{t},1}),\ldots,G^{p}(\widehat{\mathbf{u}}^{\star,p}_{d,h_{t},L})$
and efficiently select the best unit commitment decision, denoted by
$\widehat{\mathbf{u}}^{\star,p}_{d,h_{t},b}$.

We compare the proposed OPSEL with the sample average approximated data-driven
SUC approach that ignores the finite sampling error by using the wind power
data occurring in October $1$st–$31$st. We evaluate these two approaches with
the actual incurred cost. If we ignore the finite sampling error, the total
cost is $\sum
G^{p}_{r}=\sum_{d=1}^{n_{d}}\sum_{h_{t}\in\mathcal{H}}G^{r}_{d,h_{t}}(\widehat{\mathbf{u}}^{\star,p}_{d,h_{t}})$.
For the OPSEL approach, the total incurred cost is

$\sum
G^{p,b}_{r}\equiv\sum_{d=1}^{n_{d}}\sum_{h_{t}\in\mathcal{H}}G^{r}_{d,h_{t}}(\widehat{\mathbf{u}}^{\star,p}_{d,h_{t},b}),$

where
$G^{r}_{d,h_{t}}(\mathbf{u}_{d,h_{t},b})\equiv\mathbf{C}^{su}\widehat{\mathbf{u}}^{\star,p}_{d,h_{t},b}+\min_{\mathbf{y}}\mathbf{C}^{fuel}\mathbf{y}(\widehat{\mathbf{u}}^{\star,p}_{d,h_{t},b},\boldsymbol{\xi}_{d,h_{t}}^{r})$
and it is the cost occurring between hours $h_{t}+1$ and $h_{t}+n_{h}$ on day
$d$.

Here, we use $L=4$ CPUs. The results of total operation cost in October
obtained by using the data-driven SUC with and without OPSEL are recorded in
Table 8. The OPSEL approach can control the impact of finite sampling error
induced by SAA and leads to a cheaper total cost, $\sum G^{p,b}_{r}\leq\sum
G^{p}_{r}.$ The relative cost saving caused by using the proposed OPSEL
approach, $r\Delta G\equiv\frac{\sum G^{p}_{r}-\sum G^{p,b}_{r}}{\sum
G^{p}_{r}},$ is about 20%. To further check the robustness of our approach, we
study the performance of OPSEL under various settings of $\Delta T$ and $T$.
It is obvious that our procedure produces substantially better $\sum
G^{p,b}_{r}$ solutions than $\sum G^{p}_{r}$ under all settings of $\Delta T$
and $T$. Therefore, our OPSEL approach can effectively control the impact of
finite sampling error and significantly improve the reliability and cost-
efficiency of unit commitment decisions.

Table 8: The results of total incurred costs obtained by data-driven SUC with
and without the proposed OPSEL

| Cost | $r\Delta G$
---|---|---
Data-driven SUC without OPSEL $\sum G^{p}_{r}$ | 2,822,705 | 0
SUC + OPSEL $\sum G^{p,b}_{r}$ $(T=500,\Delta T=100)$ | 2,323,507 | 17.7%
SUC + OPSEL $\sum G^{p,b}_{r}$ $(T=1000,\Delta T=50)$ | 2,186,869 | 22.5%
SUC + OPSEL $\sum G^{p,b}_{r}$ $(T=1000,\Delta T=100)$ | 2,282,531 | 19.1%
SUC + OPSEL $\sum G^{p,b}_{r}$ $(T=1000,\Delta T=200)$ | 2,134,574 | 24.4%

We also record the overhead computational burden induced by the OPSEL
candidate selection. We observe that the total running time of first-stage
optimization search by using the L-Shaped optimization for the one-month
period is about $40,000$ seconds, while the total time spent on the candidate
selection is around $1,500$ seconds. Thus, the overhead burden is negligible.

###  5.2 Empirical Study with 118-Bus System

To evaluate the scalability and robustness of our approach, in this section,
we consider the derivative 118-bus system including 54 thermal units, three
wind farms and 186 transmission lines; see the description in pena2017extended
. Similar to the 6-bus system case, we assume deterministic loads and
stochastic wind supplyZhou_etal_2013  and use the 2006 data of the U.S.
Illinois power system for the load and the wind power generation. The whole
system’s wind penetration $R$ is 9.9%. In addition, we set the wind
curtailment at 30$/MWh and load shedding price at 3500$/MWh. We just consider
constraints (4)-(9) in Wang_etal_2017  as in section 5.1.

####  5.2.1 Case Study with Parametric Forecast Model

One month wind power data are used to study the performance of the proposed
data-driven SUC and the empirical SUC. We use the same settings with those
used in Section 5.1.1, and consider one day-ahead unit commitment problem
here. When we solve the SUC problems for the optimal decision estimates
$\widehat{\mathbf{u}}_{d}^{\star,e}$ and $\widehat{\mathbf{u}}_{d}^{\star,p}$,
we set the scenario size to be $S=50$. Then, we use $S_{e}=1000$ to evaluate
the true expected cost $\sum G^{p}$ and $\sum G^{e}$. The cases with standard
deviation $\phi_{d,t}=5\%\mu_{d,t}^{c},10\%\mu_{d,t}^{c},20\%\mu_{d,t}^{c}$
and $m=1,5,10$ are analyzed and the results are recorded in Table 9. As $m$
goes big, the input model estimation uncertainty decreases. According to Table
9, the advantage of proposed data-driven SUC increases as the wind penetration
increases, the wind power generation variation becomes larger, and the amount
of valid historical data decreases.

Table 9: Total expected costs when $\phi_{d,t}=5\%\mu_{d,t}^{c},10\%\mu_{d,t}^{c},20\%\mu_{d,t}^{c}$ and $m=1,5,10$ Standard Deviation | $m=1$ | $m=5$ | $m=10$
---|---|---|---
$\phi_{d,t}=5\%\mu_{d,t}^{c}$ | 8.22% | 1.26% | 0.11%
$\phi_{d,t}=10\%\mu_{d,t}^{c}$ | 13.80% | 5.51% | 0.14%
$\phi_{d,t}=20\%\mu_{d,t}^{c}$ | 20.59% | 12.36% | 2.60%

####  5.2.2 Case study with Nonparametric Forecast Model

Here we consider the case when there is no strong prior information on the
underlying wind power generation distribution and the nonparametric
distributions are used for wind power forecast, which is similar with that
used in Section 5.1.2. We use the 118-bus test case to study the performance
of: (1) deterministic unit commitment, (2) empirical SUC and (3) data-driven
SUC. We utilize the deterministic persistence as prediction model for the
deterministic unit commitment, use the probabilistic persistence model for the
empirical SUC, and implement Bayesian nonparametric wind power forecast model
Xie_etal_2018  for data-driven SUC with prediction risk accounting for both
wind generation stochastic uncertainty and model estimation error. We consider
the unit commitment problem for the intra-day market with 4 hours planning
horizon and set the amount of historical data used for the forecast to be
$m=100$. One month wind power data are used to evaluate the performance and
the results are recorded in Table 10. The empirical SUC, accounting for wind
power generation stochastic uncertainty only, leads to 1.12% savings compared
with the deterministic UC method. The proposed data-driven SUC, accounting for
both wind power stochastic uncertainty and input model estimation uncertainty,
leads to 2.58% savings. Thus, the proposed data-driven SUC, accounting for all
sources of uncertainties, can lead to more cost-efficient and robust
decisions, and the advantage is larger as the wind power penetration
increases.

Table 10: Total costs of operation obtained from various approaches when $R=9.9\%$ Approach | Penalty Cost | Penalty Ratio | Total Cost | $r\Delta G$
---|---|---|---|---
Deterministic UC: $\sum G^{d}_{r}$ | 1,156,456.31 | 2.21% | 52,327,089.47 | 0.00%
Empirical SUC: $\sum G^{e}_{r}$ | 537,661.31 | 1.04% | 51,743,500.01 | 1.12%
Data-Driven SUC: $\sum G^{p}_{r}$ | 0.00 | 0.00% | 50,974,697.80 | 2.58%

We also record the penalty cost induced when the energy production does not
meet the demand; see Eq. (14). The proportions of penalty cost obtained by the
deterministic unit commitment, the empirical SUC, and the proposed data-driven
SUC are shown in Table 10. The decision made by the deterministic UC method
leads to the total penalty cost $ 1,156,456.31. The proportion of the penalty
cost to the total cost $\sum G^{d}_{r}$ is 2.21%. The empirical SUC method has
the total penalty cost $ 537,661.31 contributing 1.04% percentage of the total
cost $\sum G^{e}_{r}$. Thus, the results indicate that the load unmet risk can
be hedged by utilizing the proposed data-driven SUC, and it can lead to cost
saving, more reliable and robust power grids with high renewable energy
penetration.

####  5.2.3 Performance of OPSEL Approach

Similar to Section 5.1.3, we investigate the performance of the proposed OPSEL
approach by using the 118-bus system. Let $L=3$, $\Delta T=50$ and $T=500$.
The results of total operation cost obtained by using the data-driven SUC with
and without OPSEL are recorded in Table 11. The relative cost saving by
implementing our OPSEL approach is about 2.36%. Thus, the proposed OPSEL
approach demonstrates the promising performance on the large-scale system by
hedging against the impact of finite sampling error induced by SAA.

Table 11: Total incurred costs of operation on 118-bus system

| Cost | $r\Delta G$
---|---|---
Data-driven SUC without OPSEL $\sum G^{p}_{r}$ | 50,974,697.80 | 0.00%
SUC + OPSEL $\sum G^{p,b}_{r}$ $(T=50,\Delta T=500)$ | 49,771,552.75 | 2.36%

##  6 CONCLUSION

In this paper, we first propose a data-driven stochastic optimization to guide
the power system unit commitment decision, which can simultaneously hedge
against the underlying stochastic uncertainty or volatility of wind power
generation and statistical input model estimation error. Then, we introduce an
optimization and selection approach that can efficiently utilize the parallel
computing to quickly find the optimal unit commitment decision, while
controlling the impact of finite sampling error induced by SAA. Various case
studies on a six-bus system and a 118-bus system verify the advantages of our
proposed data-drive SUC for both parametric and nonparametric forecast models.
They also demonstrate that our OPSEL procedure can further deliver the optimal
unit commitment decision hedging against the impact of finite sampling error.

## References

  * [1] J. J. Hargreaves and B. F. Hobbs.  Commitment and dispatch with uncertain wind generation by dynamic programming.  IEEE Transactions on Sustainable Energy, 3(4):724 – 734, 2012.
  * [2] Z. Zhou, A. Botterud, J. Wang, R.J. Bessa, H. Keko, J. Sumaili, and V. Miranda.  Application of probabilistic wind power forecasting in electricity markets.  Wind Energy, 16(3):321–338, 2013.
  * [3] Z. Zhao and L. Wu.  Impacts of high penetration wind generation and demand response on lmps in day-ahead market.  IEEE Transaction On Smart Grid, 5(1):220–229, 2014.
  * [4] R. Jiang, J. Wang, and Y. Guan.  Robust unit commitment with wind power and pumped storage hydro.  IEEE Transactions on Power Systems, 27(2):800–810, 2012.
  * [5] A. Papavasiliou and S. S. Oren.  Multiarea stochastic unit commitment for high wind penetration in a transmission constrained network.  Operations Research, 61(3):578–592, 2013.
  * [6] J. Wang, M. Shahidehpour, and Z. Li.  Security-constrained unit commitment with volatile wind power generation.  IEEE Transactions on Power Systems, 23(3):1319 – 1327, 2008.
  * [7] P. A. Ruiz, C. R. Philbrick, and P. W. Sauer.  Wind power day-ahead uncertainty management through stochastic unit commitment policies.  In 2009 IEEE/PES Power Systems Conference and Exposition, pages 1–9, 2009.
  * [8] A. Tuohy, P. Meibom, E. Denny, and M. O’Malley.  Unit commitment for systems with significant wind penetration.  IEEE Transactions on Power Systems, 24(2):592 – 601, 2009.
  * [9] Q. Wang, Y. Guan, and J. Wang.  A chance-constrained two-stage stochastic program for unit commitment with uncertain wind power output.  IEEE Transactions on Power Systems, 27(1):592 – 601, 2012.
  * [10] W. Xie, P. Zhang, R. Chen, and Z. Zhou.  A nonparametric bayesian framework for short-term wind power probabilistic forecast.  Accepted, 2018.
  * [11] S. Takriti, J. Birge, and E. Long.  A stochastic model for the unit commitment problem.  IEEE Transactions on Power Systems, 11(3):1497–1508, 1996.
  * [12] Seyed Hossein Madaeni and Ramteen Sioshansi.  The impacts of stochastic programming and demand response on wind integration.  Energy Systems, 4(2):109–124, 2013.
  * [13] Y. Huang, Q. P. Zheng, and J. Wang.  Two-stage stochastic unit commitment model including non-generation resources with conditional value-at-risk constraints.  Electric Power Systems Research, 116:427–438, 2014.
  * [14] D. Bertsimas, E. Litvinov, X. A. Sun, J. Zhao, and T. Zheng.  Adaptive robust optimization for the security constrained unit commitment problem.  IEEE Transactions on Power Systems, 28(1):52 – 63, 2013.
  * [15] C. Zhao, J. Wang, and J.-P. Watson.  Multi-stage robust unit commitment considering wind and demand response uncertainties.  IEEE Transactions on Power Systems, 28(3):2708 – 2718, 2013.
  * [16] C. Lee, C. Liu, S. Mehrotra, and M. Shahidehpour.  Modeling transmission line constraints in two-stage robust unit commitment problem.  IEEE Transactions on Power Systems, 29(3):1221 – 1231, 2014.
  * [17] P. Xiong, P. Jirutitijaroen, and C. Singh.  A distributionally robust optimization model for unit commitment considering uncertain wind power generation.  IEEE Transactions on Power Systems, 32(1):39 – 49, 2017.
  * [18] L. Zhao and B. Zeng.  Robust unit commitment problem with demand response and wind energy.  In Power and Energy Society General Meeting, pages 1–8. IEEE, 2012\.
  * [19] Á. Lorca and X. A. Sun.  Adaptive robust optimization with dynamic uncertainty sets for multi-period economic dispatch under significant wind.  IEEE Transactions on Power Systems, 30(4):1702 – 1713, 2015.
  * [20] Narges Kazemzadeh, Sarah M Ryan, and Mahdi Hamzeei.  Robust optimization vs. stochastic programming incorporating risk measures for unit commitment with uncertain variable renewable generation.  Energy Systems, 10(3):517–541, 2019.
  * [21] Q. Bian, H. Xin, Z. Wang, and K. P. Wong.  Distributionally robust solution to the reserve scheduling problem with partial information of wind power.  IEEE Transactions on Power Systems, 30(5):2822–2823, 2015.
  * [22] Z. Wang, Q. Bian, H. Xin, and D. Gan.  A distributionally robust co-ordinated reserve scheduling model considering cvar-based wind power reserve requirements.  IEEE Transactions on Sustainable Energy, 7(2):625–636, 2016.
  * [23] R. Jiang, J. Wang, M. Zhang, and Y. Guan.  Two-stage minimax regret robust unit commitment.  IEEE Transactions on Power Systems, 28(3):2271 – 2282, 2013.
  * [24] Ugur Aytun Ozturk.  The stochastic unit commitment problem: a chance constrained programming approach considering extreme multivariate tail probabilities.  PhD thesis, University of Pittsburgh, 2003.
  * [25] U Aytun Ozturk, Mainak Mazumdar, and Bryan A Norman.  A solution to the stochastic unit commitment problem using chance constrained programming.  IEEE Transactions on Power Systems, 19(3):1589–1598, 2004.
  * [26] Qianfan Wang, Yongpei Guan, and Jianhui Wang.  A chance-constrained two-stage stochastic program for unit commitment with uncertain wind power output.  IEEE Transactions on Power Systems, 27(1):206–215, 2011.
  * [27] A. Shapiro, D. Dentcheva, and A. Ruszczynski.  Lectures on Stochastic Programming: Modeling and Theory.  SIAM, Philadelphia, 2009.
  * [28] C. H. Chen, J. Lin, E. Yucesan, and S. E. Chick.  Simulation budget allocation for further enhancing the efficiency of ordinal optimization.  Journal of Discrete Event Dynamic Systems: Theory and Applications, 10:251–270, 2000.
  * [29] J. Boesel, B. L. Nelson, and S.-H. Kim.  Using ranking and selection to “clean up” after simulation optimization.  Operations Research, 51(5):814 – 825, 2003.
  * [30] Seong-Hee Kim and B. L. Nelson.  Recent advances in ranking and selection.  In 2007 Winter Simulation Conference, pages 162–172, Dec 2007.
  * [31] Warren B. Powell and Ilya O. Ryzhov.  Ranking and Selection, chapter 4, pages 71–88.  Wiley-Blackwell, 2012.
  * [32] Seong-Hee Kim and Barry L. Nelson.  A fully sequential procedure for indifference-zone selection in simulation.  ACM Transactions on Modeling and Computer Simulation, 11(3):251–273, 2001.
  * [33] Stephen E. Chick and Koichiro Inoue.  New two-stage and sequential procedures for selecting the best simulated system.  Operations Research, 49(5):732–743, 2001.
  * [34] Peter I. Frazier, Warren B. Powell, and Savas Dayanik.  A knowledge-gradient policy for sequential information collection.  SIAM Journal on Control and Optimization, 47(5):2410–2439, 2008\.
  * [35] S. E. Chick, J. Branke, and C. Schmidt.  Sequential sampling to myopically maximize the expected value of information.  INFORMS Journal on Computing, 22(1):71–80, 2010.
  * [36] Chun-Hung Chen, Donghai He, Michael Fu, and Loo Hay Lee.  Efficient simulation budget allocation for selecting an optimal subset.  INFORMS Journal on Computing, 20(4):579–595, 2008.
  * [37] Hui Xiao, Loo Hay Lee, and Kien Ming Ng.  Optimal computing budget allocation for complete ranking.  IEEE Transactions on Automation Science and Engineering, 11(2):516–524, 2013.
  * [38] Jie Xu, Edward Huang, Chun-Hung Chen, and Loo Hay Lee.  Simulation optimization: A review and exploration in the new era of cloud computing and big data.  Asia-Pacific Journal of Operational Research, 32(03):1550019, 2015\.
  * [39] P. Glynn and S. Juneja.  A large deviations perspective on ordinal optimization.  In R. G. Ingalls, M. D. Rossetti, J. S. Smith, and B. A. Peters, editors, Proceedings of the 2004 Winter Simulation Conference, pages 101–112. IEEE Computer Society, Washington, DC, 2004.
  * [40] I. O. Ryzhov.  On the convergence rates of expected improvement methods.  Operations Research, 64(6):1515 – 1528, 2016.
  * [41] N. Quan, J. Yin, S. H. Ng, and L. H. Lee.  Simulation optimization via kriging: a sequential search using expected improvement with computing budget constraints.  IIE Transactions, 45(7):763–780, 2013.
  * [42] Q. P. Zheng, J. Wang, and A. L. Liu.  Stochastic optimization for unit commitment - a review.  IEEE Transactions on Power Systems, 30(4):1913 – 1924, 2015.
  * [43] Yishen Wang, Zhi Zhou, Cong Liu, and A. Botterud.  Evaluating stochastic methods in power system operations with wind power.  In 2016 IEEE International Energy Conference (ENERGYCON), pages 1–6, 2016.
  * [44] Qipeng P. Zheng, Jianhui Wang, Panos M. Pardalos, and Yongpei Guan.  A decomposition approach to the two-stage stochastic unit commitment problem.  Annals of Operations Research, 210(1):387–410, Nov 2013.
  * [45] B.F. Hobbs, M.H. Rothkopf, R.P. O’Neill, and Hung po Chao.  The Next Generation of Electric Power Unit Commitment Models.  Kluwer Academic, Norwell, MA, USA, 2001.
  * [46] Z Zhou, C Liu, and A Botterud.  Stochastic methods applied to power system operations with renewable energy: A review.  Technical report, Argonne National Lab.(ANL), Argonne, IL (United States), 2016.
  * [47] Bart C Ummels, Madeleine Gibescu, Engbert Pelgrum, Wil L Kling, and Arno J Brand.  Impacts of wind power on thermal generation unit commitment and dispatch.  IEEE Transactions on energy conversion, 22(1):44–51, 2007.
  * [48] Erik Delarue and William D’haeseleer.  Adaptive mixed-integer programming unit commitment strategy for determining the value of forecasting.  Applied Energy, 85(4):171–181, 2008.
  * [49] Nielsen Torben Skov, Joensen Alfred, Madsen Henrik, Landberg Lars, and Giebel Gregor.  A new reference for wind power forecasting.  Wind Energy, 1(1):29–34, 1998.
  * [50] Gneiting Tilmann, Balabdaoui Fadoua, and Raftery Adrian E.  Probabilistic forecasts, calibration and sharpness.  Journal of the Royal Statistical Society: Series B (Statistical Methodology), 69(2):243–268, 2007.
  * [51] Rajesh G. Kavasseri and Krithika Seetharaman.  Day-ahead wind speed forecasting using f-arima models.  Renewable Energy, 34(5):1388 – 1393, 2009.
  * [52] Yao Zhang, Jianxue Wang, and Xifan Wang.  Review on probabilistic forecasting of wind power generation.  Renewable and Sustainable Energy Reviews, 32:255 – 270, 2014.
  * [53] H. Pandžić, Y. Dvorkin, T. Qiu, Y. Wang, and D. S. Kirschen.  Toward cost-efficient and reliable unit commitment under uncertainty.  IEEE Transactions on Power Systems, 31(2):970–982, 2016.
  * [54] Y. Wang, Z. Zhou, A. Botterud, and K. Zhang.  Optimal wind power uncertainty intervals for electricity market operation.  IEEE Transactions on Sustainable Energy, 9(1):199 – 210, 2018.
  * [55] Ivonne Pena, Carlo Brancucci Martinez-Anido, and Bri-Mathias Hodge.  An extended ieee 118-bus test system with high renewable penetration.  IEEE Transactions on Power Systems, 33(1):281–289, 2017.
  * [56] R. Yang and J. Berger.  A catalog of noninformative priors.  ISDS Discussion Paper, pages 97–42, 1997.
  * [57] B. Analui and A. Scaglione.  A dynamic multistage stochastic unit commitment formulation for intraday markets.  IEEE Transactions on Power Systems, 33(4):3653–3663, 2018.