# Towards Improving Operation Economics: A Bilevel MIP-Based Closed-
Loop Predict-and-Optimize Framework for Prescribing Unit Commitment

Xianbang Chen, Yikui Liu, Lei Wu This work was supported in part by the Army
Combat Capabilities Development Command (CCDC) project on Resiliency of Energy
Resources and Supply Chain for the Industrial Base and the PSEG Foundation
gift. The authors are with the ECE Department, Stevens Institute of
Technology, Hoboken, NJ, 07030 USA. (Email: xchen130, yliu262,
<EMAIL_ADDRESS>
###### Abstract

Generally, system operators conduct the economic operation of power systems in
an open-loop predict-then-optimize process: the renewable energy source (RES)
availability and system reserve requirements are first predicted; given the
predictions, system operators solve optimization models such as unit
commitment (UC) to determine the economical operation plans accordingly.
However, such an open-loop process could essentially compromise the operation
economics because its predictors myopically seek to improve the immediate
statistical prediction errors instead of the ultimate operation cost. To this
end, this paper presents a closed-loop predict-and-optimize framework,
offering a prescriptive UC to improve the operation economics. First, a
bilevel mixed-integer programming model is leveraged to train cost-oriented
predictors tailored for optimal system operations: the upper level trains the
RES and reserve predictors based on their induced operation cost; the lower
level, with given predictions, mimics the system operation process and feeds
the induced operation cost back to the upper level. Furthermore, the
embeddability of the trained predictors grants a prescriptive UC model, which
simultaneously provides RES-reserve predictions and UC decisions with enhanced
operation economics. Finally, numerical case studies using real-world data
illustrate the potential economic and practical advantages of prescriptive UC
over deterministic, robust, and stochastic UC models.

###### Index Terms:

Unit commitment, prescriptive analytics, predict-and-optimize, bilevel mixed-
integer programming.

## Nomenclature

### Sets and Indexes

$\mathcal{B}/b$

Set/index of branches.

$\mathcal{E}/e$

Set/index of iterations.

$\mathcal{I}/i$

Set/index of non-RES units, i.e.,
$\mathcal{I}=\mathcal{I}^{ns}\cup\mathcal{I}^{qs}$, where $\mathcal{I}^{ns}$
and $\mathcal{I}^{qs}$ are sets of non-quick-start and quick-start non-RES
units.

$\mathcal{J}/j$

Set/index of RES units.

$\mathcal{K}/k$

Set/index of piecewise linear segments of a generation curve.

$\mathcal{Q}/q$

Set/index of load buses.

$\mathcal{S}/s$

Set/index of training scenarios.

$\mathcal{T}/t,t^{\prime}$

Set/indexes of hours. The total number of hours is $T$.

$\mathcal{T}^{su/sd}_{i}$

Set defined as $\\{{T}^{su}_{i},\cdots,T\\}/\\{{T}^{sd}_{i},\cdots,T\\}$.

$|\cdot|$

The cardinality of a set.

### Decision Variables

$c^{ant/act}$

Anticipated/actual system cost.

$D_{it}/U_{it}$

Shutdown/startup status of unit $i$ at hour $t$.

$I_{it},P_{it}$

Unit commitment and generation schedule of unit $i$ at hour $t$.

$O_{it}$

Non-spinning reserve (NR) commitment of unit $i$ at hour $t$.

$P_{itk}^{sg}$

Generation schedule of unit $i$ in segment $k$ at hour $t$.

$R_{it}^{sr/nr}$

Spinning reserve (SR)/NR schedule of unit $i$ at hour $t$.

$S^{1/2/3/4}_{t}$

Slack variables at hour $t$.

$\mathcal{W}(\cdot)/\mathcal{R}(\cdot)$

Predictor of RES power/reserve requirements.

$W_{jt}$

Generation schedule of RES $j$ at hour $t$.

$\boldsymbol{x},\boldsymbol{y}$

Vectors of decision variables in the UC model, including binaries
$\boldsymbol{x}=$ $\\{\boldsymbol{I},$ $\boldsymbol{U},$ $\boldsymbol{D},$
$\boldsymbol{O}\\}$ and continuous
$\boldsymbol{y}=\\{\boldsymbol{P},\boldsymbol{P}^{sg},\boldsymbol{W},\boldsymbol{R}^{sr},\boldsymbol{R}^{nr}\\}$.

$\boldsymbol{z}$

Vector of decision variables in the ED model, including binaries
$\\{\boldsymbol{I}^{ed},\boldsymbol{I}^{ed,qs},\boldsymbol{U}^{ed},\boldsymbol{D}^{ed}\\}$
and continuous
$\\{\boldsymbol{P}^{ed},\boldsymbol{P}^{ed,sg},\boldsymbol{W}^{ed},\boldsymbol{S}^{1/2/3/4}\\}$.

${\cdot}^{dv/ev/gv}$

Duplicated/enumerated/generated variables.

${\cdot}^{\star}$

Indicating the optimal solution to a variable.

### Parameters

$B_{b}$

Transmission capacity of branch $b$.

$C_{ik}^{sg}$

Generation cost of unit $i$ in segment $k$.

$C_{i}^{su/nl}$

Startup/no-load cost of unit $i$.

$C^{1/2/bs}$

Penalty cost of slack variables.

$\boldsymbol{f}^{w/r}$

Feature vector of predictor $\mathcal{W}(\cdot)/\mathcal{R}(\cdot)$.

$\mathcal{F}_{b}(\cdot)$

Power flow function of branch $b$.

$\hat{L}_{qt}/\tilde{L}_{qt}$

Predicted/actual demand of load $q$ at hour $t$, forming the vector
$\hat{\boldsymbol{l}}/\tilde{\boldsymbol{l}}$.

$\mathcal{P}^{w/r}$

Feasible region of predictor $\mathcal{W}(\cdot)/\mathcal{R}(\cdot)$.

$\underline{P\mkern-6.0mu}\mkern 4.0mu_{i}/\bar{P}_{i}$

Minimum/maximum generation limit of unit $i$.

$\bar{P}_{ik}^{sg}$

Power limit of unit $i$ at segment $k$.

$R_{i}^{su/sd}$

Startup/shutdown ramping capacity of unit $i$.

$R_{i}^{\uparrow/\downarrow}$

Upward/downward ramping capacity of unit $i$.

$\bar{R}_{i}^{sr/nr}$

SR/NR limit of unit $i$.

$\hat{R}_{t}^{sr/nr}$

Predicted SR/NR requirement at hour $t$, which together form the vector
$\hat{\boldsymbol{r}}$.

$T_{i}^{su/sd}$

Minimum on/off time requirement of unit $i$.

$\hat{\boldsymbol{w}}^{\diamond}/\hat{\boldsymbol{r}}^{\diamond}$

Tailored counterpart of raw prediction
$\hat{\boldsymbol{w}}/\hat{\boldsymbol{r}}$.

$\hat{W}_{jt}/\tilde{W}_{jt}$

Predicted/actual available power of RES $j$ at hour $t$, forming the vector
$\hat{\boldsymbol{w}}/\tilde{\boldsymbol{w}}$.

##  I Introduction

###  I-A Problem Statement

This paper focuses on the economic operation of power systems administered by
system operators[1]. Generally, the system economic operation is conducted in
an open-loop predict-then-optimize (O-PO) process, as shown in Fig. 1(a):

  * •

The available power of renewable energy sources (RES) as well as system
spinning reserve (SR) and non-spinning reserve (NR) requirements are predicted
in the day-ahead stage.

  * •

Taking the predictions as inputs, system operators solve a deterministic _day-
ahead unit commitment_ (UC) [2] problem to determine the day-ahead operation
plans, including startup/shutdown schedules and power output baselines.

  * •

After the actual RES realizations are revealed by the operators, the best
operation plans against the realizations and day-ahead UC plans can be
obtained by solving an economic dispatch (ED) [3] problem. As this ED problem
is modeled and solved in the post-event stage after the RES realizations are
revealed, it is referred to as _hindsight ED_ in this paper.

  * •

With respect to the day-ahead predictions and the afterward RES realizations,
the startup and no-load costs determined by day-ahead UC and the dispatch
costs determined by hindsight ED constitute the actual system total operation
cost. This is referred to as _operation economics_ in this paper.

Figure 1: Comparison of traditional O-PO and proposed C-PO.

Although the O-PO process seems practically reasonable, its operation
economics suffers from two flaws: (_i_) The RES predictors myopically pursue
statistical accuracy, such as mean absolute percentage error (MAPE), mean
absolute error (MAE), and root mean square error (RMSE). However, due to the
inherent nonlinearity of the UC-ED optimization process, a statistically more
accurate prediction may not necessarily induce a lower system cost (as
illustrated in Section III-A1); (_ii_) The reserve requirements sized by
traditional rule-of-thumb methods may be redundant or insufficient, further
worsening the economics. With this, the myopic predictions in O-PO are
referred to as raw predictions in this paper.

###  I-B Literature Review

To improve the operation economics, one approach is to augment the
deterministic UC to its two-stage stochastic programming (T-SP) [4] or two-
stage robust optimization (T-RO) [5] counterpart for capturing the inaccuracy
of raw predictions. Another emerging way is to form a closed loop between
prediction and optimization through a feedback path, referred to as the
closed-loop predict-and-optimize (C-PO). Pioneering methodologies have been
explored in this direction, such as predictive prescription [6], smart
predict-then-optimize (SPO) [7], and regression [8]. In a nutshell, these
approaches achieve C-PO by training _cost-oriented predictors_ to improve the
performance of a specific optimization task [6, 7], or by forming
_prescriptive models_ to simultaneously realize prediction and optimization in
a single step [8].

Indeed, the C-PO idea has been recently applied to several power system
applications. References [9] and [10] train decision trees to predict
available RES power for RES producers, which use the RES trading profit
instead of entropy to evaluate prediction quality. Neural networks are trained
to predict RES power for RES trading [11], load [12] and RES quantile [13] for
ED, and electricity price for energy storage system arbitrage [14], in which
the training loss functions explicitly quantify profit losses induced by
predictions. All the references [13, 9, 10, 11, 12, 14] demonstrate that
tailoring raw predictions for targeted optimization tasks can improve the
economics noticeably, although the statistical prediction accuracy may be
compromised to a certain extent.

Rather than grounding on heuristic methods as [13, 9, 10, 11, 12, 14],
references [15, 16, 17, 18] use mathematical programming as the underlying
methodology to realize C-PO. The core is to integrate predictor training and
optimization tasks into an empirical risk minimization (ERM) problem, which is
then solved to obtain optimal predictors. By solving the ERM problem, [15]
trains an extreme learning machine to size SR requirements economically. By
solving a regression-based ERM, [16] obtains a profit-oriented prescriptive
model that can map feature data to RES trading decisions. The same authors
further use a similar idea to formulate a value-oriented prescriptive ED in
[17]. Different from [15, 16, 17] which focus on linear programming (LP)
tasks, our previous work [18] customizes the SPO approach [7] for the mixed-
integer programming (MIP)-based UC.

Although the methods in [15, 16, 17, 18] are regarded as stable and
interpretable, their single-level ERM structures require predictor training
and optimization sharing the same objective function. This restriction
prevents the optimization model from meeting the practical decision-making
needs of system operators, who aim to minimize the _system cost_ instead of
the _training loss_.

In fact, this issue can be resolved by constructing a bilevel programming-
based ERM—modeling the predictor training at the upper level while
uncompromisingly formulating the optimization tasks at the lower level.
Following this spirit, [19] and [20] enhance their single-level ERMs in [15]
and [16] to bilevel LP. Similarly, [21, 22, 23] utilize bilevel LP to
construct ERM and train cost-oriented SR predictors for ED. Nevertheless,
existing bilevel ERM studies are limited to LP-based ED problems.

###  I-C The Proposed Bilevel MIP-Based C-PO Framework

TABLE I: Review of C-PO Applications in Power Systems Methodology Property | ERM Construction | Reference | Methodology Foundation | Prediction Term & | Optimization Task
---|---|---|---|---|---
Heuristic | - | [9] | Regression tree | RES power & | RES trading
[10] |  Classification tree [6] | RES power & | RES trading
[11] | Extreme learning machine | RES power & | RES trading
[12] | Deep neural network | Load & | ED
[13] | Contextual bandit | RES power & | ED
[14] | Deep neural network | Energy price & | Storage arbitrage
Mathematical programming | Single-level LP | [15] | Extreme learning machine | RES power and SR & | Reserve sizing
[16] |  Linear regression [8] | RES power & | RES trading
[17] | Linear regression | Load & | ED
Single-level MIP | [18] |  Linear regression [7] | RES power & | UC
Bilevel LP | [19] | Extreme learning machine | RES power & | ED
[20] | Linear regression | RES power & | RES trading
[21] | Linear regression | Load and SR & | ED
[22] | Linear regression | SR & | ED
[23] | Linear regression | SR and transmission ability & | ED
Bilevel MIP | This paper | Linear regression | RES power, SR, and NR & | UC

Table I compares the proposed work with existing C-PO studies [13, 9, 10, 11,
12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. It is noteworthy that most
mathematical programming-based studies focus on LP tasks. Differently, this
paper presents a bilevel MIP-based C-PO framework, as shown in Fig. 1(b), to
accurately capture system operators’ practice and deliver realizable economic
improvements. Specifically, the ERM problem is constructed in the bilevel-MIP
form: the upper level trains the predictors for RES power as well as SR and NR
requirements; the lower level, with given upper-level predictions, includes
two MIP subproblems to mimic the sequential UC-ED operation process and
calculates the best possible system operation cost, which is fed back to the
upper level for improving prediction quality. The ERM is converted into a
bilevel MIP with only one lower-level subproblem, which is then solved by a
cutting plane method [24] to deliver the optimally trained predictors of RES,
SR, and NR. Finally, the predictors are integrated into the deterministic UC,
forming a prescriptive UC model that delivers cost-oriented RES-SR-NR
predictions and UC decisions simultaneously.

Motivated by the underlying idea of [18, 19, 20, 21, 22, 23], the proposed
C-PO framework presents pioneers with MIP-based operation tasks and a
tractable solution approach to accurately model operation practice for
delivering realizable economic improvements. Specifically, the proposed work
presents the following major distinctions:

  * •

Bilevel-MIP Based ERM Model and Solution: References [19, 20, 21, 22, 23]
exclusively focus on LP tasks that render bilevel-LP ERM problems, which can
take advantage of efficient bilevel LP solution methods by substituting the
lower-level LP with its Karush–Kuhn–Tucker (KKT) conditions. In comparison, in
recognizing that many practical operation tasks are MIP problems, this paper
presents a valuable extension of the bilevel MIP-based C-PO framework with one
upper-level LP problem and two lower-level MIP subproblems, and explores its
efficient solution approach. Specifically, we convert the ERM into a more
tractable bilevel MIP with only one lower-level MIP subproblem, and customize
the cutting plane method [24] to efficiently solve the ERM. To improve the
transparency and reproducibility of the C-PO framework, all testing data and
source codes are publicly accessible [25].

  * •

Versatility, Generalization Ability, and Compatibility of Trained Predictors:
Compared to our previous work [18], this paper presents three non-trivial
improvements: (_i_) The SPO method [7] cannot predict multiple terms
simultaneously. Thus, [18] can only predict RES availability or reserve
requirements. In comparison, the bilevel structure enables predicting RES and
multiple reserve requirements simultaneously, i.e., the presented C-PO is more
versatile; (_ii_) Due to the single-level construction, ERM of [18] can only
model the UC process with certain compromises. As a result, it can only
integrate UC costs in the closed loop and remains myopic to the ED process;
moreover, the feedback UC information cannot represent operators’ real
decisions (Section III-A2 will explain the reasons after introducing necessary
preliminaries), which is adverse to the generalization ability of predictors.
In comparison, the presented bilevel ERM can feed back both UC and ED costs,
forming a tighter closed loop. More importantly, the bilevel ERM enables the
optimization tasks to be modeled uncompromisingly, effectively guaranteeing
the generalization of predictors; (_iii_) Different from [18] which focuses on
the comparison with O-PO, this paper additionally compares C-PO with T-SP/T-RO
from an operator perspective with real-world data from a Belgian system [26].
The comparisons reveal the compatibility of the C-PO framework with the
current practice and its great potential to bridge the gap between T-SP/T-RO
and the practice.

The rest of the paper is organized as follows: Section II details the
necessary preliminaries; Section III expounds on the C-PO framework; Section
IV analyzes the experimental results; and Section V concludes this paper.

##  II Preliminaries

###  II-A Operation Models

Taking the predictions $\hat{\boldsymbol{w}}$ and $\hat{\boldsymbol{r}}$ as
inputs, the operators solve the MIP-based UC model to determine the day-ahead
operation plans. The compact form of UC is shown as in (1), and its detailed
formulation is presented in Appendix -A. Here, $\boldsymbol{F}(\cdot)$ and
$\boldsymbol{G}(\cdot)$ respectively represent the sets of equality and
inequality UC constraints, forming the feasible region $\mathcal{X}$. With the
predictions $\hat{\boldsymbol{w}}$ and $\hat{\boldsymbol{r}}$ as inputs, UC is
solved to deliver the optimal solutions of startup, commitment, base-point
generation, SR and NR schedule, as well as _anticipated system cost_
$c^{ant\star}$ in the day-ahead stage.

$\displaystyle
c^{ant\star}=\textstyle{\min_{\boldsymbol{x},\boldsymbol{y}}\boldsymbol{b}^{\top}\boldsymbol{x}+\boldsymbol{c}^{\top}\boldsymbol{y}}$
(1a)
$\displaystyle\textstyle{s.\,t.\,\boldsymbol{x},\boldsymbol{y}\in\mathcal{X}(\hat{\boldsymbol{w}},\hat{\boldsymbol{r}})=\left\\{\boldsymbol{x},\boldsymbol{y}\,\,\begin{array}[]{|l}\boldsymbol{F}(\boldsymbol{x},\boldsymbol{y})=0;\\\
\boldsymbol{G}(\boldsymbol{x},\boldsymbol{y},\hat{\boldsymbol{w}},\hat{\boldsymbol{r}})\leq
0\end{array}\right\\}}$ (1d)

Next, with respect to the operation plans
$\\{\boldsymbol{x}^{\star},\boldsymbol{y}^{\star}\\}$ from UC, the best
possible system operations against actual RES realization
$\tilde{\boldsymbol{w}}$ can be revealed by solving the MIP-based hindsight ED
model. The compact form of the hindsight ED is shown as in (2), and its
detailed formulation is presented in Appendix -B. Here,
$\boldsymbol{M}(\cdot)$ and $\boldsymbol{N}(\cdot)$ respectively represent the
sets of equality and inequality ED constraints, and $\mathcal{Z}$ is the
feasible region. Solving ED provides optimal solutions to status switch of
quick-start units, generation of all units, and slack variables against the
RES realization $\tilde{\boldsymbol{w}}$.

$\displaystyle\textstyle{\min_{\boldsymbol{z}}\boldsymbol{d}^{\top}\boldsymbol{z}}$
(2a)
$\displaystyle\textstyle{s.\,t.\,\boldsymbol{z}\in\mathcal{Z}(\boldsymbol{x}^{\star},\boldsymbol{y}^{\star},\tilde{\boldsymbol{w}})=\left\\{\boldsymbol{y}\,\,\begin{array}[]{|l}\boldsymbol{M}(\boldsymbol{x}^{\star},\boldsymbol{z})=0;\\\
\boldsymbol{N}(\boldsymbol{x}^{\star},\boldsymbol{y}^{\star},\boldsymbol{z},\tilde{\boldsymbol{w}})\leq
0\end{array}\mspace{-7.0mu}\right\\}}\mspace{-23.0mu}$ (2d)

The operation economics is evaluated after sequentially solving the day-ahead
UC and the hindsight ED. Specifically, since $\tilde{\boldsymbol{w}}$ remains
unknown in the day-ahead stage, operators first conduct UC based on
$\hat{\boldsymbol{w}}$ and $\hat{\boldsymbol{r}}$, obtaining the actual UC
cost (i.e., startup and no-load costs
$\boldsymbol{b}^{\top}\boldsymbol{x}^{\star}$). After $\tilde{\boldsymbol{w}}$
is revealed, hindsight ED is executed based on the UC decisions, providing the
hindsight ED cost $\boldsymbol{d}^{\top}\boldsymbol{z}^{\star}$ (i.e., startup
and no-load costs of quick-start units additionally committed in ED,
generation costs of all units, and slack penalties). With this, equation
(II-A) represents the best possible system cost one can achieve in reality
with respect to the UC solution and actual RES realization. This is referred
to as the operation economics in this paper.

$\displaystyle c^{act}=$
$\displaystyle\textstyle{\sum\limits_{t\in\mathcal{T},i\in\mathcal{I}}}(C^{su}_{i}U_{it}^{\star}+C^{nl}_{i}I_{it}^{\star})$
$\displaystyle\mspace{-82.0mu}\textbf{\\}}\,\boldsymbol{b}^{\top}\boldsymbol{x}^{\star}$
$\displaystyle\left.\begin{array}[]{l}\mspace{-28.0mu}+\textstyle{\sum\limits_{t\in\mathcal{T},i\in\mathcal{I}}}(C_{i}^{su}U_{it}^{ed\star}+C_{i}^{nl}I_{it}^{ed\star}+\sum\limits_{k\in\mathcal{K}}C_{ik}^{sg}P_{itk}^{ed,sg\star})\\\
\mspace{-28.0mu}+\textstyle{\sum\limits_{t\in\mathcal{T}}[C^{1}S_{t}^{1\star}+C^{2}S_{t}^{2\star}+C^{bs}\sum\limits_{{b}\in\mathcal{B}}(S_{bt}^{3\star}+S_{bt}^{4\star})]}\end{array}\mspace{-10.0mu}\right\\}\boldsymbol{d}^{\top}\boldsymbol{z}^{\star}$
(5) $\displaystyle=$
$\displaystyle\,\,\textstyle{\boldsymbol{b}^{\top}\boldsymbol{x}^{\star}+\boldsymbol{d}^{\top}\boldsymbol{z}^{\star}}$
(6)

##  III Bilevel MIP-Based C-PO Framework

This section first illustrates the necessity of the bilevel MIP-based C-PO
framework to train the cost-oriented predictors, followed by its formulation
and solution method. Finally, the cost-oriented predictors obtained from the
bilevel MIP-based ERM are integrated into UC to form a prescriptive UC model
that can derive operation plans with improved operation economics.

###  III-A Why Bilevel Closed-Loop Predict-and-Optimize?

####  III-A1 The Asymmetric Impacts of RES Prediction Errors on Operation
Economics Necessitate the C-PO

Figure 2: Asymmetric impact of RES prediction MAPE on operation economics.

To show the asymmetry, multiple raw predictions $\hat{\boldsymbol{w}}$ on
December 17, 2020 are generated based on the actual Belgian system data. These
raw predictions are applied on an IEEE 14-bus system to simulate the O-PO
process as shown in Fig. 1(a). Finally, the loss of operation economics is
evaluated by (7), where $\hat{c}^{act}$ and $\tilde{c}^{act}$ are actual
systems costs according to (II-A) with
$(\hat{\boldsymbol{w}},\hat{\boldsymbol{r}})$ and
$(\tilde{\boldsymbol{w}},\hat{\boldsymbol{r}})$ as inputs to UC.

$\text{Loss of Operation
Economics}=({\hat{c}^{act}-\tilde{c}^{act}})/{\tilde{c}^{act}}\times 100\%$
(7)

Fig. 2 shows the induced results of all raw prediction points, and the red
line fits the relationship between MAPE and loss of operation economics. This
line clearly illustrates the asymmetry from a macro perspective. The loss
caused by the over-predictions is fairly small in small-MAPE cases (MAPE of 5%
to 15%), but becomes significant in large-MAPE cases (MAPE of 25% to 30%); in
comparison, the loss induced by the under-predictions is relatively stable.
Furthermore, two specific cases are also noteworthy:

Case 1: Raw predictions of the same MAPE may lead to different losses of
operation economics. Note that points A and B in Fig. 2 have the same MAPE
(23.6%) but different losses (2.2% vs 3.8%). This is because the under-
prediction of point A merely causes downward generation adjustment and RES
curtailments in the ED; in comparison, the over-prediction of point B requires
more expensive NR deployment from quick-start units in the ED.

Case 2: Raw predictions of a worse MAPE may even lead to a smaller loss of
operation economics. Although MAPE of point C is noticeably high (19.0%), it
achieves the lowest loss (0.1%) among all raw predictions. This is because
each $\hat{\boldsymbol{w}}$ shall be associated with a just-enough reserve
level $\hat{\boldsymbol{r}}^{\star}$ with respect to $\tilde{\boldsymbol{w}}$
(i.e., a proper $\hat{\boldsymbol{r}}^{\star}$ depends on the difference
between $\hat{\boldsymbol{w}}$ and $\tilde{\boldsymbol{w}}$). Consequently, if
$(\hat{\boldsymbol{w}},\hat{\boldsymbol{r}}^{\star})$ is taken as input to
(1), the induced UC decisions can deploy just enough reserve to enable the
full utilization of available RES in actual operation, achieving the best
operation economics.

As a result, the asymmetric impacts inspire that it is more proper to train
the predictors based on the ultimate economic consequence of the sequential
UC-ED optimization process.

####  III-A2 Compromised UC Modeling in the Single-Level ERM Necessitates the
Bilevel MIP ERM

In the single-level ERM (8) of [18], the UC objective has to be packed into
the objective function of the predictor training model as in (8a), in which
$\tilde{c}^{ant}_{s}$ is the anticipated system cost (1a) induced by
$(\tilde{\boldsymbol{w}}_{s},\hat{\boldsymbol{r}}_{s})$. Indeed, since (8)
minimizes the in-sample training loss (8a) instead of system cost (1a), its
in-sample UC solutions $\boldsymbol{x}_{s}$ may violate the least-cost merit-
order principle of the system operators (i.e., using more expensive units to
pursue the minimum loss function (8a)), leading to compromised UC decisions.

$\displaystyle\textstyle{\min_{\mathcal{W},\boldsymbol{x}_{s},\boldsymbol{y}_{s}}}$
$\displaystyle\mspace{-20.0mu}\textstyle{\frac{1}{|\mathcal{S}|}\sum_{s\in\mathcal{S}}}||\boldsymbol{b}^{\top}\boldsymbol{x}_{s}+\boldsymbol{c}^{\top}\boldsymbol{y}_{s}-\tilde{c}^{ant}_{s}||_{1}$
(8a) $\displaystyle s.\,t.$
$\displaystyle\mspace{-45.0mu}\mathcal{W}(\cdot)\in\mathcal{P}^{w};\,\,\boldsymbol{x}_{s},\boldsymbol{y}_{s}\in\mathcal{X}(\mathcal{W}(\boldsymbol{f}_{s}^{w}),\hat{\boldsymbol{r}}_{s});\,\,\,\forall
s\in\mathcal{S}$ (8b)

We draw on [20] to explain this. Indeed, given a feature
$\boldsymbol{f}_{s}^{w}$, (8) cannot guarantee the induced in-sample UC
decision $\\{\boldsymbol{x}_{s}^{\star},\boldsymbol{y}_{s}^{\star}\\}$ being
the optimal solution to (1), because of the difference in objective functions
(1a) and (8a). Nevertheless, as the operators are responsible for delivering
the least-cost operation plans as implicated in (1a),
$\\{\boldsymbol{x}_{s}^{\star},\boldsymbol{y}_{s}^{\star}\\}$ does not reflect
the true decisions of operators. As a result, the predictor
$\mathcal{W}^{\star}$ trained via these compromised UC solutions would fail to
generalize well, although presenting good in-sample losses, i.e., overfitting.

In comparison, constructing ERM as a bilevel MIP can uncompromisingly
integrate the exact UC model in the lower level. In this way, the in-sample UC
solutions always minimize the system cost (1a), strictly following the least-
cost principle. With this, using accurate feedback information can effectively
avoid the overfitting issue, especially when the number of training scenarios
is limited.

###  III-B Constructing ERM Problem

Based on the prespecified scenario set $\mathcal{S}$ (the scenario selection
will be detailed in Section IV) as well as features $\boldsymbol{f}^{w}_{s}$
and $\boldsymbol{f}^{r}_{s}$, the bilevel ERM problem is constructed as in
(9). The upper level (9a)-(9c) trains the predictors $\mathcal{W}$ and
$\mathcal{R}$ for RES, SR, and NR , taking actual system costs as the loss
function. Thus, the ERM is to maximize the weighted sum of operation economics
for all scenarios. Constraint (9b) limits the predictor parameters within
their feasible regions $\mathcal{P}^{w}$ and $\mathcal{P}^{r}$, and (9c)
defines $\mathcal{W}$ and $\mathcal{R}$ which respectively take features
$\boldsymbol{f}^{w}_{s}$ and $\boldsymbol{f}^{r}_{s}$ as inputs. The two
lower-level subproblems (9d) and (9e) respectively model UC (1) and ED (2)
following their original formulations exactly. Solving (9) can provide cost-
oriented predictors $\mathcal{W}^{\star}$ and $\mathcal{R}^{\star}$.

Upper Level (Predictor Training):
$\displaystyle\textstyle{\min_{\mathcal{W},\mathcal{R}}}\textstyle{\frac{1}{|\mathcal{S}|}\sum_{s\in\mathcal{S}}}\boldsymbol{b}^{\top}\boldsymbol{x}_{s}+\boldsymbol{d}^{\top}\boldsymbol{z}_{s}$
(9a)
$\displaystyle\mathcal{W}(\cdot)\in\mathcal{P}^{w};\,\,\mathcal{R}(\cdot)\in\mathcal{P}^{r}$
(9b)
$\displaystyle\hat{\boldsymbol{w}}^{\diamond}_{s}=\mathcal{W}(\boldsymbol{f}^{w}_{s});\,\,\hat{\boldsymbol{r}}^{\diamond}_{s}=\mathcal{R}(\boldsymbol{f}^{r}_{s});$
$\displaystyle\mspace{-85.0mu}\forall s\in\mathcal{S}$ (9c) Lower Level (Day-
Ahead UC):
$\displaystyle\boldsymbol{x}_{s},\boldsymbol{y}_{s}\in{\arg\min}_{\boldsymbol{x}_{s},\boldsymbol{y}_{s}\in\mathcal{X}(\hat{\boldsymbol{w}}^{\diamond}_{s},\hat{\boldsymbol{r}}^{\diamond}_{s})}\boldsymbol{b}^{\top}\boldsymbol{x}_{s}+\boldsymbol{c}^{\top}\boldsymbol{y}_{s};$
$\displaystyle\mspace{-85.0mu}\forall s\in\mathcal{S}$ (9d) Lower Level
(Hindsight ED):
$\displaystyle\boldsymbol{z}_{s}\in{\arg\min}_{\boldsymbol{z}_{s}\in\mathcal{Z}(\boldsymbol{x}_{s},\boldsymbol{y}_{s},\tilde{\boldsymbol{w}}_{s})}\boldsymbol{d}^{\top}\boldsymbol{z}_{s};$
$\displaystyle\mspace{-85.0mu}\forall s\in\mathcal{S}$ (9e)

In this paper, affine linear functions are used to form the predictors, as
shown in (10), because of their interpretability [16] and numerical stability
[17]. The symbol $\circ$ indicates the element-wise product. As for features,
the raw prediction $\hat{\boldsymbol{w}}$ is used for $\boldsymbol{f}^{w}_{s}$
as in (10a), which is suggested by a standard coefficient regression method
[27]. In this way, the C-PO is an upgraded version based on the O-PO, i.e.,
the operators can continue using the original predictors and improve them. In
addition, the load prediction $\hat{\boldsymbol{l}}$ and RES prediction
$\hat{\boldsymbol{w}}$ are used to form $\boldsymbol{f}^{r}$, as in (10b).
This is because (_i_) Common feature selection methods (e.g., standard
coefficient regression and principal component analysis) are not applicable to
our case, as they need the just-enough reserve level
$\hat{\boldsymbol{r}}^{\star}$ as input labels which however cannot be
obtained until $\hat{\boldsymbol{w}}$ is revealed; and (_ii_) It is well
recognized that reserve requirements are mainly driven by load and RES
uncertainties. With this, the feasible regions $\mathcal{P}^{w}$ and
$\mathcal{P}^{r}$ are set as the positive real-number region $\mathbb{R}_{+}$
of appropriate dimensions. The training is basically to determine the optimal
vectors $\boldsymbol{m}$ and $\boldsymbol{n}$.

$\displaystyle\mspace{-3.0mu}\mathcal{W}(\boldsymbol{f}^{w}_{s})=\boldsymbol{m}\circ\boldsymbol{f}^{w}_{s}\Rightarrow\mathcal{W}(\hat{\boldsymbol{w}}_{s})=\boldsymbol{m}\circ\hat{\boldsymbol{w}}_{s};\boldsymbol{m}\in\mathbb{R}_{+}^{|\mathcal{T}|\times|\mathcal{J}|}$
(10a)
$\displaystyle\mspace{-3.0mu}\mathcal{R}(\boldsymbol{f}^{r}_{s})=\boldsymbol{n}\circ\boldsymbol{f}^{r}_{s}\Rightarrow\mathcal{R}(\hat{\boldsymbol{w}}_{s},\hat{\boldsymbol{l}}_{s})=\boldsymbol{n}\circ(\hat{\boldsymbol{w}}_{s},\hat{\boldsymbol{l}}_{s});\boldsymbol{n}\in\mathbb{R}_{+}^{|\mathcal{T}|\times
2}$ (10b)

Figure 3: Illustration of the closed-loop mechanism in bilevel ERM.

As a specific form of Fig. 1(b), Fig. 3 depicts how the bilevel ERM (9)
enables the interaction between the predictions and optimizations to form the
closed loop. The upper level generates the tailored predictions
$\hat{\boldsymbol{w}}^{\diamond}$ and $\hat{\boldsymbol{r}}^{\diamond}$, but
cannot directly assess their ultimate economic consequence. Thus,
$\hat{\boldsymbol{w}}^{\diamond}$ and $\hat{\boldsymbol{r}}^{\diamond}$ are
first passed to the lower-level day-ahead UC (9d), forming the UC feasible
region $\mathcal{X}$. Solving (9d) provides the actual UC cost
$\boldsymbol{b}^{\top}\boldsymbol{x}^{\star}$ to the upper level as well as
the UC decisions $\\{\boldsymbol{x}^{\star},\boldsymbol{y}^{\star}\\}$ to the
lower-level hindsight ED (9e). Taking the UC decisions as inputs, (9e) is
solved to further provide the hindsight ED cost
$\boldsymbol{d}^{\top}\boldsymbol{z}^{\star}$ to the upper level. With this,
the upper level can assess the prediction quality via actual system cost
$\boldsymbol{b}^{\top}\boldsymbol{x}^{\star}+\boldsymbol{d}^{\top}\boldsymbol{z}^{\star}$,
and tune the predictor parameters accordingly. This process repeats until the
optimally trained predictors $\mathcal{W}^{\star}$ and $\mathcal{R}^{\star}$
(i.e., vectors $\boldsymbol{m}$ and $\boldsymbol{n}$ in (10)) are obtained.
Since $\mathcal{W}^{\star}$ and $\mathcal{R}^{\star}$ are trained to optimize
the operation cost, they are cost-oriented and tailored for system operations.

###  III-C Solving ERM Problem

####  III-C1 Convert (6) with Two MIP Subproblems in the Lower Level to (8)
with One MIP in the Lower Level

The bilevel ERM (9) includes two MIP subproblems in the lower level,
introducing extra computational complexity. With this, we first convert (9)
into a more tractable bilevel MIP (11) with one MIP in the lower level.
Solving (11) provides the optimal solution of the original ERM (9), as stated
in Proposition 1.

###### Proposition 1

The optimal solution of the bilevel MIP (11) is also optimal to the original
bilevel MIP (9).

$\displaystyle\textstyle{\min_{\mathcal{W},\mathcal{R},\boldsymbol{z}_{s}}}$
$\displaystyle\mspace{-70.0mu}\textstyle{\frac{1}{|\mathcal{S}|}\sum_{s\in\mathcal{S}}}\boldsymbol{b}^{\top}\boldsymbol{x}_{s}+\boldsymbol{d}^{\top}\boldsymbol{z}_{s}$
(11a) $\displaystyle\,\,s.\,t.$
$\displaystyle\mspace{-70.0mu}\mathcal{W}(\cdot)\in\mathcal{P}^{w};\,\,\mathcal{R}(\cdot)\in\mathcal{P}^{r}$
(11b)
$\displaystyle\mspace{-70.0mu}\hat{\boldsymbol{w}}^{\diamond}_{s}=\mathcal{W}(\boldsymbol{f}^{w}_{s});\,\,\hat{\boldsymbol{r}}^{\diamond}_{s}=\mathcal{R}(\boldsymbol{f}^{r}_{s});$
$\displaystyle\mspace{-50.0mu}\forall s\in\mathcal{S}$ (11c)
$\displaystyle\mspace{-70.0mu}\boldsymbol{z}_{s}\in\mathcal{Z}(\boldsymbol{x}_{s},\boldsymbol{y}_{s},\tilde{\boldsymbol{w}}_{s});$
$\displaystyle\mspace{-50.0mu}\forall s\in\mathcal{S}$ (11d)
$\displaystyle\mspace{-70.0mu}\boldsymbol{x}_{s},\boldsymbol{y}_{s}\in\underset{\boldsymbol{x}_{s},\boldsymbol{y}_{s}\in\mathcal{X}(\hat{\boldsymbol{w}}^{\diamond}_{s},\hat{\boldsymbol{r}}^{\diamond}_{s})}{\arg\min}\boldsymbol{b}^{\top}\boldsymbol{x}_{s}+\boldsymbol{c}^{\top}\boldsymbol{y}_{s};$
$\displaystyle\mspace{-50.0mu}\forall s\in\mathcal{S}$ (11e)

###### Proof:

See Appendix -C. ∎

####  III-C2 Solve (8) by C&CG Algorithm

The bilevel MIP (11) is solved via a column-and-constraint generation (C&CG)
algorithm [24], which iterates between the master problem MP and two
subproblems SP1 and SP2. The process is summarized in Algorithm 1.

The first subproblem SP1 is shown as in (12). It is a duplication of the UC
problem (11e) while taking the incumbent predictors $\mathcal{W}_{e}$ and
$\mathcal{R}_{e}$ as inputs. Solving SP1 can reveal the anticipated system
cost $c_{s}^{ant\star}$ induced by the incumbent predictors.

SP1: $\displaystyle c^{ant\star}_{s}=$
$\displaystyle\mspace{-20.0mu}\textstyle{\min_{\boldsymbol{x}_{s},\boldsymbol{y}_{s}}\boldsymbol{b}^{\top}\boldsymbol{x}_{s}+\boldsymbol{c}^{\top}\boldsymbol{y}_{s}}$
(12a) $\displaystyle s.\,t.\,$
$\displaystyle\mspace{-20.0mu}\hat{\boldsymbol{w}}^{\diamond}_{s}=\mathcal{W}_{e}(\boldsymbol{f}^{w}_{s});\,\,\hat{\boldsymbol{r}}^{\diamond}_{s}=\mathcal{R}_{e}(\boldsymbol{f}^{r}_{s})$
(12b)
$\displaystyle\mspace{-20.0mu}\boldsymbol{F}(\boldsymbol{x}_{s},\boldsymbol{y}_{s})=0;\,\,\boldsymbol{G}(\boldsymbol{x}_{s},\boldsymbol{y}_{s},\hat{\boldsymbol{w}}^{\diamond}_{s},\hat{\boldsymbol{r}}^{\diamond}_{s})\leq
0$ (12c)

Note that multiple optimal binaries $\boldsymbol{x}_{s}^{\star}$ for SP1 may
exist, which however could lead to different system costs evaluated via (11a).
Thus, the second subproblem SP2, formed as in (13) with $c^{ant\star}_{s}$
from SP1 as input, is further solved to select a solution among multiple
optimal binaries $\boldsymbol{x}_{s}^{\star}$ of SP1 in favor of the objective
(11a). This is referred to as optimistic bilevel programming [24].

SP2: $\displaystyle c_{s}^{act\star}=$
$\displaystyle\mspace{-10.0mu}\textstyle{\min_{\boldsymbol{x}_{s},\boldsymbol{y}_{s},\boldsymbol{z}_{s}}\boldsymbol{b}^{\top}\boldsymbol{x}_{s}+\boldsymbol{d}^{\top}\boldsymbol{z}_{s}}$
(13a) $\displaystyle s.\,t.$
$\displaystyle\mspace{-30.0mu}\hat{\boldsymbol{w}}^{\diamond}_{s}=\mathcal{W}_{e}(\boldsymbol{f}^{w}_{s});\,\,\hat{\boldsymbol{r}}^{\diamond}_{s}=\mathcal{R}_{e}(\boldsymbol{f}^{r}_{s})$
(13b)
$\displaystyle\mspace{-30.0mu}\boldsymbol{b}^{\top}\boldsymbol{x}_{s}+\boldsymbol{c}^{\top}\boldsymbol{y}_{s}\leq
c^{ant\star}_{s}$ (13c)
$\displaystyle\mspace{-30.0mu}\boldsymbol{F}(\boldsymbol{x}_{s},\boldsymbol{y}_{s})=0;\,\,\boldsymbol{G}(\boldsymbol{x}_{s},\boldsymbol{y}_{s},\hat{\boldsymbol{w}}^{\diamond}_{s},\hat{\boldsymbol{r}}^{\diamond}_{s})\leq
0$ (13d)
$\displaystyle\mspace{-30.0mu}\boldsymbol{M}(\boldsymbol{x}_{s},\boldsymbol{z}_{s})=0;\,\,\boldsymbol{N}(\boldsymbol{x}_{s},\boldsymbol{y}_{s},\boldsymbol{z}_{s},\tilde{\boldsymbol{w}}_{s})\leq
0$ (13e)

Given the selected decisions $\boldsymbol{x}_{s}^{\star}$ from SP2, the master
problem MP (14) is formulated as the high-point problem [24] of (11). Three
sets of variants for variables $\\{\boldsymbol{x}_{s},\boldsymbol{y}_{s}\\}$
are used in MP, including duplicated variables
$\\{\boldsymbol{x}^{dv}_{s},\boldsymbol{y}^{dv}_{s}\\}$, enumerated binary
variables $\boldsymbol{x}^{ev}_{s,e}$, and generated continuous variables
$\boldsymbol{y}^{gv}_{s,e}$. The duplicated variables server as the proxy of
the original variables, but with a larger feasible region; the enumerated and
generated variables and their associated cutting planes (14f)-(14i) serve to
cut the feasible region of duplicated variables gradually. MP is built in
three parts:

  * •

The first part (14a)-(14d) is based on the duplicated variables. The objective
(14a) also includes two regularization terms
$+\lambda^{w}\lVert\mathcal{W}\rVert_{1}$ (avoid over-predictions) and
$-\lambda^{r}\lVert\mathcal{R}\rVert_{1}$ (avoid insufficient reserves) with
hyper-parameters $\lambda^{w/r}$. Note that the regularization terms could be
removed when the number of scenarios is sufficient. Constraints (14b)-(14d)
correspond to the original upper-level constraints (11b)-(11d).

  * •

The second part (14e) is a duplication of UC constraints to ensure the
feasibility of the duplicated variables.

  * •

The third part (14f)-(14i) servers as the optimality cuts for the duplicated
variables. Note that $\boldsymbol{x}^{ev}_{s,e}$ is a constant vector
enumerated by SP2 (i.e., $\boldsymbol{x}^{ev}_{s,e}$ is fixed as the selected
$\boldsymbol{x}_{s}^{\star}$) for scenario $s$ in iteration $e$. When
$\boldsymbol{x}_{s}$ is fixed as $\boldsymbol{x}^{ev}_{s,e}$, (11e)
degenerates to an LP and is equivalently substituted by the KKT condition
(14f)-(14h). Here, $\mathcal{L}(\cdot)$ denotes the Lagrangian function;
$\boldsymbol{\mu}$ and $\boldsymbol{\nu}$ are dual variables of equalities and
inequalities. The KKT conditions (14f)-(14h) ensure the optimality of the
generated variables given $\boldsymbol{x}^{ev}_{s,e}$. Finally, (14i) links
the duplicated variables with their optimality cuts.

$\displaystyle\textbf{MP:}\,\min_{\Xi}\textstyle{\frac{1}{|\mathcal{S}|}\sum_{s\in\mathcal{S}}}(\boldsymbol{b}^{\top}\boldsymbol{x}^{dv}_{s}+\boldsymbol{d}^{\top}\boldsymbol{z}_{s})+\lambda^{w}\lVert\mathcal{W}\rVert_{1}-\lambda^{r}\lVert\mathcal{R}\rVert_{1}\mspace{-19.0mu}$
$\displaystyle\Xi=\\{\mathcal{W}(\cdot),\mathcal{R}(\cdot),\hat{\boldsymbol{w}}^{\diamond}_{s},\hat{\boldsymbol{r}}^{\diamond}_{s},\boldsymbol{z}_{s},\boldsymbol{x}^{dv}_{s},\boldsymbol{y}^{dv}_{s},\boldsymbol{y}^{gv}_{s}\\}\mspace{-20.0mu}$
(14a) Original Constraints:
$\displaystyle\mathcal{W}(\cdot)\in\mathcal{P}^{w};\,\,\mathcal{R}(\cdot)\in\mathcal{P}^{r}$
(14b)
$\displaystyle\hat{\boldsymbol{w}}^{\diamond}_{s}=\mathcal{W}(\boldsymbol{f}^{w}_{s});\,\,\hat{\boldsymbol{r}}^{\diamond}_{s}=\mathcal{R}(\boldsymbol{f}^{r}_{s});$
$\displaystyle\,\,\mspace{-155.0mu}\forall s\in\mathcal{S}$ (14c)
$\displaystyle\boldsymbol{M}(\boldsymbol{x}^{dv}_{s},\boldsymbol{z}_{s})=0;\,\,\boldsymbol{N}(\boldsymbol{x}^{dv}_{s},\boldsymbol{y}^{dv}_{s},\boldsymbol{z}_{s},\tilde{\boldsymbol{w}}_{s})\leq
0;$ $\displaystyle\,\,\mspace{-155.0mu}\forall s\in\mathcal{S}$ (14d)
Duplication of UC Constraints:
$\displaystyle\boldsymbol{F}(\boldsymbol{x}_{s}^{dv},\boldsymbol{y}_{s}^{dv})=0;\,\,\boldsymbol{G}(\boldsymbol{x}^{dv}_{s},\boldsymbol{y}_{s}^{dv},\hat{\boldsymbol{w}}_{s}^{\diamond},\hat{\boldsymbol{r}}_{s}^{\diamond})\leq
0;$ $\displaystyle\,\,\mspace{-155.0mu}\forall s\in\mathcal{S}$ (14e)
Stationarity of KKT (Optimality Cut):
$\displaystyle\nabla\mathcal{L}(\boldsymbol{y}_{s,e}^{gv},\boldsymbol{\mu}_{s,e},\boldsymbol{\nu}_{s,e})=0;$
$\displaystyle\,\,\mspace{-155.0mu}\forall s\in\mathcal{S}$ (14f) Primal and
Dual Feasibilities of KKT (Optimality Cut):
$\displaystyle\boldsymbol{F}(\boldsymbol{x}_{s,e}^{ev},\boldsymbol{y}_{s,e}^{gv})=0;$
$\displaystyle\boldsymbol{G}(\boldsymbol{x}_{s,e}^{ev},\boldsymbol{y}_{s,e}^{gv},\hat{\boldsymbol{w}}_{s}^{\diamond},\hat{\boldsymbol{r}}_{s}^{\diamond})\leq
0;\,\,\boldsymbol{\nu}_{s,e}\geq 0;$
$\displaystyle\,\,\mspace{-155.0mu}\forall s\in\mathcal{S},e\in\mathcal{E}$
(14g) Complementary Slackness of KKT (Optimality Cut):
$\displaystyle\boldsymbol{\nu}_{s,e}\bot\boldsymbol{G}(\boldsymbol{x}_{s,e}^{ev},\boldsymbol{y}_{s,e}^{gv},\hat{\boldsymbol{w}}_{s}^{\diamond},\hat{\boldsymbol{r}}_{s}^{\diamond});$
$\displaystyle\,\,\mspace{-155.0mu}\forall s\in\mathcal{S},e\in\mathcal{E}$
(14h) Objective Cut (Optimality Cut):
$\displaystyle\boldsymbol{b}^{\top}\boldsymbol{x}_{s}^{dv}+\boldsymbol{c}^{\top}\boldsymbol{y}_{s}^{dv}\leq\boldsymbol{b}^{\top}\boldsymbol{x}_{s,e}^{ev}+\boldsymbol{c}^{\top}\boldsymbol{y}_{s,e}^{gv};$
$\displaystyle\,\,\mspace{-155.0mu}\forall s\in\mathcal{S},e\in\mathcal{E}$
(14i)

By linearizing the complementary slackness (14h) via the Big-M method [28], MP
can be converted into a MIP problem and directly solved by commercial MIP
solvers. The user-friendly KKT and export functions of YALMIP [29] are used to
facilitate the implementation of KKT constraints (14f)-(14h).

Input: Index of iterations $e=0$ and its limitation $E$, desired optimality
gap $\epsilon^{\star}$, initial predictors $\mathcal{W}_{e}$ and
$\mathcal{R}_{e}$, lower bound $LB$, and upper bound $UB$.

Output: Well-trained predictors $\mathcal{W}^{\star}$ and
$\mathcal{R}^{\star}$.

while  _$e\leq E$_ do

Solve SP1 and SP2 for each scenario $s\in\mathcal{S}$;

$UB\leftarrow\min\\{UB,\sum_{s\in\mathcal{S}}c_{s}^{act\star}/|\mathcal{S}|+\lambda^{w}\lVert\mathcal{W}\rVert_{1}-\lambda^{r}\lVert\mathcal{R}\rVert_{1}\\}$;

$\epsilon\leftarrow 100\%\times(UB-LB)/UB$;

if  _$\epsilon\leq\epsilon^{\star}$_ then

$\mathcal{W}^{\star}\leftarrow\mathcal{W}_{e}$,
$\mathcal{R}^{\star}\leftarrow\mathcal{R}_{e}$;

break;

else

$\boldsymbol{x}_{s,e}^{ev}$ $\leftarrow$ optimal solution of decision
$\boldsymbol{x}_{s}$ in SP2;

Generate new $\\{\boldsymbol{x}^{gv}_{s,e},\boldsymbol{y}^{gv}_{s,e}\\}$ and
cuts (14f)-(14i) in MP;

Solve MP;

$e\leftarrow e+1,LB\leftarrow$ optimal objective value of MP;

$\mathcal{W}_{e}$ and $\mathcal{R}_{e}$ $\leftarrow$ optimal $\mathcal{W}$ and
$\mathcal{R}$ from MP;

end if

end while

Algorithm 1 A C&CG Algorithm for Solving (11)

###  III-D Forming Prescriptive UC Model

The prescriptive UC model (15) is formed by integrating the trained cost-
oriented predictors $\mathcal{W}^{\star}$ and $\mathcal{R}^{\star}$ into the
deterministic UC model (1). It possesses two major favorable properties as
compared to the deterministic UC model (1):

  * •

Integrating the predictors enables (15) to simultaneously deliver the RES-
reserve predictions and UC decisions, referred to as prescription. In
comparison, in the O-PO framework, (1) exclusively focuses on the UC decision,
and most existing forecasting works mainly focus on the prediction.

  * •

$\mathcal{W}^{\star}$ and $\mathcal{R}^{\star}$ are affine linear functions
(10). Therefore, the tailored predictions are essentially the linear
combination of the raw predictions, which will not significantly complicate
the computation of (15).

$\displaystyle\textstyle{\min_{\boldsymbol{x},\boldsymbol{y},\hat{\boldsymbol{w}}^{\diamond},\hat{\boldsymbol{r}}^{\diamond}}\boldsymbol{b}^{\top}\boldsymbol{x}+\boldsymbol{c}^{\top}\boldsymbol{y}}$
(15a)
$\displaystyle\,\,s.\,t.\,\boldsymbol{x},\boldsymbol{y}\in\mathcal{X}(\hat{\boldsymbol{w}}^{\diamond},\hat{\boldsymbol{r}}^{\diamond});\,\hat{\boldsymbol{w}}^{\diamond}=\mathcal{W}^{\star}(\boldsymbol{f}^{w});\,\hat{\boldsymbol{r}}^{\diamond}=\mathcal{R}^{\star}(\boldsymbol{f}^{r})$
(15b)

Finally, the presented C-PO framework replaces the deterministic UC model (1)
with the prescriptive UC model (15) to boost operation economics.

##  IV Case Studies

###  IV-A Experimental Setting

####  IV-A1 C-PO Implementation Design

To leverage compatibility with the system operators’ practice and solution
quality of (15), the proposed C-PO is implemented via a weekly rolling scheme.
That is, for each week, the predictors are retrained to update (15) using the
information of the past $NT$ days.

Specifically, to study the dispatch week of days $(D+\text{1})$ to
$(D+\text{7})$, the past $NT$ days (i.e., days $(D-NT)$ to $(D-\text{1}))$ are
selected to build the training scenario set $\mathcal{S}$. Then, the ERM (9)
is solved to provide the cost-oriented predictors $\mathcal{W}^{\star}$ and
$\mathcal{R}^{\star}$, which are used to form the prescriptive UC models (15)
for each of days $(D+\text{1})$ to $(D+\text{7})$. At the end of this period,
the above process repeats to update the predictors and solve (15) for the next
week (i.e., days $(D+\text{8})$ to $(D+\text{14})$).

####  IV-A2 Experimental Setup

Based on the Belgian system data [26], four weeks in 2020 as shown in Table II
are selected, representing peak RES availabilities in the four quarters. The
Belgian data are scaled to build the RES and load data for the test systems.
The slack penalty prices $C^{1}$, $C^{2}$, and $C^{bs}$ are all set as
$2,000/MWh. The rule-of-thumb reserve requirement is set according to the
CAISO practice: system reserve requirement represented as a proper proportion
$\alpha$ of the load is met by non-RES units, of which at least 50% is SR.
Other data can be referred to from [25].

Table III lists all the models to be analyzed. The threshold on the optimality
gap of all models is set as 1%. For C-PO-NTs, the hyper-parameters
$\lambda^{w/r}$ are set as $\text{10}^{\text{6}}$. As the magnitude of the
actual system cost $c^{act}$ (11a) is between $\text{10}^{\text{5}}$ and
$\text{10}^{\text{6}}$, a hyper-parameter of $\text{10}^{\text{6}}$ would
present a sufficient, but not overly dominating, effect on the ERM problem.
All cases are solved by Gurobi 9.1, called by YALMIP on MATLAB, on a 3.5 GHz
PC.

TABLE II: Selective Weeks of 2020 Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4
---|---|---|---
Feb 04 – Feb 09 | Jun 28 – Jul 04 | Aug 20 – Aug 26 | Dec 12 – Dec 18
TABLE III: Methods to Be Discussed Acronym | Note
---|---
C-PO-NT |  Presented C-PO framework with $NT$ training scenarios.
C-PO-NT-R |  C-PO-NT that only predicts reserves $\hat{\boldsymbol{r}}^{\diamond}$ and keeps raw RES $\hat{\boldsymbol{w}}$.
C-PO-NT-W |  C-PO-NT that only predicts RES $\hat{\boldsymbol{w}}^{\diamond}$ and keeps raw reserves $\hat{\boldsymbol{r}}$.
O-PO |  Traditional O-PO framework with raw $\hat{\boldsymbol{r}}$ and $\hat{\boldsymbol{w}}$.
P-PO |  Perfect predict-then-optimize that uses RES realization $\tilde{\boldsymbol{w}}$ as raw
|  RES prediction $\hat{\boldsymbol{w}}$ and keeps raw reserves
$\hat{\boldsymbol{r}}$.
T-SP-NS |  Two-stage stochastic programming with $NS$ scenarios.
T-RO-NB |  Two-stage robust optimization with budget parameter $\Gamma$ = $NB$.

###  IV-B C-PO vs O-PO

Based on an IEEE 14-bus system (with $\alpha$ = 50%, 1,700MW non-RES capacity,
and 400MW RES capacity), this section compares C-PO and O-PO via two metrics:
economics improvement (EI) (16) to evaluate improvement in operation costs and
value of information (VoI) (17) to quantify the value of feedback information.

$\displaystyle\mspace{-8.0mu}\text{EI}=({c^{act,o\text{-}po}-c^{act,c\text{-}po}})/({c^{act,o\text{-}po}})\times
100\%$ (16)
$\displaystyle\mspace{-8.0mu}\text{VoI}=({c^{act,o\text{-}po}-c^{act,c\text{-}po}})/({c^{act,o\text{-}po}-c^{act,p\text{-}po}})$
(17)

Figure 4: Economics comparison of O-PO, C-PO, and P-PO.  TABLE IV: Breakdown of Actual System Cost |  Actual UC Cost/$\text{10}^{\text{3}}$$ |  Hindsight ED Cost/$\text{10}^{\text{3}}$$ | $c^{act}$/$\text{10}^{\text{3}}$$
---|---|---|---
| Startup | No-Load | Startup | No-Load | Gen.
O-PO | 1.1 | 2.1 | 0.1 | 0.1 | 237.9 | 241.3
C-PO-7 | 1.0 | 2.2 | 0.1 | 0.1 | 231.9 | 235.3
TABLE V: Comparison of Different C-PO Methods | Scheduled Reserve | Training | EI
---|---|---|---
| SR/MW | NR/MW | CPU Time/s | # of Iterations
O-PO | 113.2 | 113.2 | - | - | 0.00%
C-PO-1 | 110.8 | 75.6 | 19 | 2 | -0.09%
C-PO-7 | 92.9 | 65.4 | 786 | 6 | 2.54%
C-PO-1-W | 113.2 | 113.2 | 21 | 2 | -0.22%
C-PO-7-W | 113.2 | 113.2 | 750 | 3 | -0.02%
C-PO-1-R | 98.8 | 85.8 | 13 | 2 | -0.48%
C-PO-7-R | 88.8 | 63.3 | 1,857 | 4 | 1.77%
TABLE VI: Statistical Accuracy of RES Prediction | MAE/MW | RMSE/MW | MAPE | MOPE | MUPE
---|---|---|---|---|---
$\hat{\boldsymbol{w}}$ in O-PO | 15.0 | 21.0 | 36.4% | 29.6% | 6.8%
$\hat{\boldsymbol{w}}^{\diamond}$ in C-PO-7 | 16.5 | 23.2 | 35.9% | 27.7% | 8.2%

####  IV-B1 Results

Fig. 4 shows that all the six C-PO-NT models with $NT$ = 2 to 7 outperform
O-PO economically, in which C-PO-7 renders the best performance (i.e., 2.54%
average EI and 0.58 average VoI). As the ideal P-PO with the 1.00 VoI only
achieves 4.43% EI, the 2.54% EI of C-PO-7 could be regarded as non-trivial.

Table IV breakdowns the actual system costs of O-PO and C-PO-7. The main
difference between O-PO and C-PO-7 stems from the generation cost in ED. It is
also noteworthy that C-PO-7 has a lower startup cost and a higher no-load cost
in UC.

Table V compares different C-POs. It shows that C-PO-NTs and C-PO-NT-Rs render
lower SR and NR than the raw schedules, among which SR is always higher than
NR. Regarding the training, both computational time and the number of
iterations for solving the ERM problem increase with $NT$. Moreover, it is
noteworthy that four C-POs cause negative EI.

Finally, Table VI evaluates RES prediction accuracy via five metrics,
including MAE, RMSE, MAPE, mean over-prediction percentage error (MOPE), and
mean under-prediction percentage error (MUPE). Compared to O-PO, C-PO-7 has a
better MAPE but worse MAE and RMSE. Moreover, C-PO-7 has a better MOPE but a
worse MUPE, indicating that C-PO-7 tends to conservatively predict RES power
(i.e., $\hat{\boldsymbol{w}}^{\diamond}$ is statistically smaller than
$\hat{\boldsymbol{w}}$).

####  IV-B2 Discussions

The results clearly show that C-PO outperforms O-PO in terms of operation
economics. Still, the reasons behind this conclusion and potential issues
deserve further discussion.

Table IV shows that the economic difference mainly stems from the generation.
This is because C-PO-7 takes advantage of the units with cheaper generation
costs. Although these units have higher no-load costs, C-PO-7 keeps them
online longer enough by setting the predictions
$\hat{\boldsymbol{w}}^{\diamond}$ and $\hat{\boldsymbol{r}}^{\diamond}$
properly. As a result, C-PO-7 uses fewer units in ED, and operates them in the
most efficient condition.

According to Table V, C-POs believe a proper reserve deployment should have a
higher SR and a lower NR, instead of half-and-half. This is because the
bilevel ERM enables C-POs to learn that triggering quick-start units in ED is
more expensive. Moreover, although reserve requirements of C-POs are lower
than those of O-PO in general, our experiments show that (_i_) Reserve
requirements of C-PO-7 are 5.28% higher than those of O-PO in quarter 2,
indicating that the rule-of-thumb $\hat{\boldsymbol{r}}$ may be insufficient
in certain cases; and (_ii_) C-PO-NTs leverage lower
$\hat{\boldsymbol{w}}^{\diamond}$ and higher base-generation $P_{it}^{\star}$
in the day-ahead UC, allowing system operators to possess enough capacity and
avoid load shedding when faced with $\tilde{\boldsymbol{w}}$ in actual
operations. As a result, it can be pointed out that C-POs tailor the
predictions for the operation problem suitably. Last but not least, two more
points can be implied by Table V:

  * •

Negative EIs indicate that closed-loop learning may lead to an adverse impact
on operation economics. Indeed, C-PO possesses the common issue in wide
machine learning methods—it cannot fully guarantee that learning will always
bring in extra benefits. Nevertheless, this issue could be relieved with
sufficient training scenarios and well-tuned hyper-parameters.

  * •

Although EI monotonically increases with $NT$ in Table V when multiple
scenarios are used for training, Fig. 4 indicates that this relationship may
not hold in a 1-scenario grain. Two possible reasons are responsible for this
fluctuation: (_i_) A single predictor is used throughout one week, regardless
of weekdays and weekends; and (_ii_) The number of training scenarios is
limited. To relieve the fluctuation, the first approach is to train different
predictors for weekdays and weekends, and the second approach is to increase
$NT$—the best performance is generally achieved when $NT$ reaches about 10
times the number of predictor parameters. However, these two approaches would
significantly increase the training cost. Therefore, system operators should
weigh up the pros and cons according to the computational budget.

As shown in Table VI, C-PO-7 performs worse in MAE and RMSE. This is because
C-PO evaluates the prediction quality by actual operation cost $c^{act}$
instead of traditional statistical metrics. Nevertheless, it is interesting to
observe that C-PO-7 outperforms O-PO in MAPE. This is because C-PO-7 generally
under-predicts RES, hence the MAPE of $\hat{\boldsymbol{w}}^{\diamond}$ is
smaller than that of $\hat{\boldsymbol{w}}$ in the cases of scarce RES.
However, these cases have a limited impact on both MAE and RMSE, because they
are relatively insensitive to small values. Accordingly, Table VI implies a
fact—judging prediction quality with a single statistical metric could be one-
sided, as each metric owns a specific property. To this end, this paper
suggests evaluating the prediction quality with the ultimate operation goal.

In summary, by effectively leveraging the bilevel MIP structure to learn the
UC-ED optimization information, C-PO tailors raw
($\hat{\boldsymbol{w}},\hat{\boldsymbol{r}}$) to cost-oriented
($\hat{\boldsymbol{w}}^{\diamond},\hat{\boldsymbol{r}}^{\diamond}$) for
improving operation economics. Even though the tailored predictions are
slightly worse in certain statistical metrics, they enable the system to
operate more economically. As a result, C-PO significantly outperforms O-PO in
terms of operation cost. Nevertheless, the reduction in prediction accuracy,
as shown in Table VI, shall not be considered a compromise, because the
ultimate goal of the prediction is to improve the optimization performance
instead of simply improving the prediction accuracy. Moreover, the presented
C-PO framework is built on top of the system operators’ accuracy-oriented
predictors and thus is compatible with their current practice.

###  IV-C C-PO vs Two-Stage Optimization Models with MIP Recourse

Based on a modified IEEE 118-bus system (with 8,600MW non-RES capacity and
1,000MW RES capacity), C-PO, T-SP, and T-RO are compared on the first day for
each of the four weeks listed in Table II. For the sake of comparison, the
following settings on the T-SP and T-RO models with MIP recourse are adopted:

  * •

Load and RES are regarded as uncertainty factors, whose prediction and actual
realization are denoted as $\hat{\boldsymbol{u}}$ and
$\tilde{\boldsymbol{u}}$, respectively. The day-ahead point forecast and the
90% confidence interval for $\hat{\boldsymbol{u}}$ are built based on [26].

  * •

T-SP is modeled as a scenario-based model [4] via (18). To construct the
scenario set $\mathcal{H}$, 3,000 scenarios are generated via Latin hypercube
sampling within the 90% confidence interval, and then reduced to $NS$
scenarios via a scenario-tree method [30]. $p_{h}$ denotes the probability of
scenario $h$. (18) is solved by Gurobi 9.1 directly.

$\displaystyle\textstyle{\min_{\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}}}$
$\displaystyle\,\,\boldsymbol{b}^{\top}\boldsymbol{x}+\boldsymbol{c}^{\top}\boldsymbol{y}+p_{h}\textstyle{\sum_{h\in\mathcal{H}}\boldsymbol{d}^{\top}\boldsymbol{z}_{h}}$
(18a) $\displaystyle s.\,t.\,$
$\displaystyle\,\,\boldsymbol{x},\boldsymbol{y}\in\mathcal{X}(\hat{\boldsymbol{u}});\,\,\boldsymbol{z}_{h}\in\mathcal{Z}(\boldsymbol{x},\boldsymbol{y},\tilde{\boldsymbol{u}}_{h});\,\,\forall
h\in\mathcal{H}$ (18b)

  * •

T-RO is modeled as an adjustable RO [5] via (19). The box uncertainty set
$\mathcal{U}$ is constructed using the 90% confidence intervals, and the
budget parameter $\Gamma\in[\text{0},\text{48}]$ controls the robustness. We
use a nested C&CG method [31] to solve (19).

$\displaystyle\textstyle{\min_{\boldsymbol{x},\boldsymbol{y}}}$
$\displaystyle\,\,\boldsymbol{b}^{\top}\boldsymbol{x}+\boldsymbol{c}^{\top}\boldsymbol{y}+\textstyle{\max_{\tilde{\boldsymbol{u}}\in\mathcal{U}}\min_{\boldsymbol{z}}}\boldsymbol{d}^{\top}\boldsymbol{z}$
(19a) $\displaystyle s.\,t.$
$\displaystyle\,\,\boldsymbol{x},\boldsymbol{y}\in\mathcal{X}(\hat{\boldsymbol{u}});\,\,\boldsymbol{z}\in\mathcal{Z}(\boldsymbol{x},\boldsymbol{y},\tilde{\boldsymbol{u}},\Gamma)$
(19b)

  * •

The reserve requirement constraints are not explicitly included in (18) and
(19), because the reserve schedules are implicated in their second-stage ED
(i.e., feasible regions $\mathcal{Z}$).

####  IV-C1 C-PO vs T-SP

Figure 5: Comparison of C-PO and T-SP.

Fig. 5 sketches the operation economics of C-PO and T-SP solutions, leading to
the following observations:

  * •

T-SP-NSs require many scenarios to perform well, leaving a wide cost gap
between T-SP-1 and T-SP-99. Relatively, C-PO-NTs need fewer scenarios, with a
narrower cost gap between C-PO-1 and C-PO-7. It indicates that the bilevel ERM
enables C-PO to effectively mine the value of information, ensuring its
performance in case of limited scenarios.

  * •

Except for quarter 1, the costs of T-SP-NSs and C-PO-NTs eventually intersect.
It implies that the economics of T-SP and C-PO are mutually reachable with
suitable $NT$ and $NS$.

  * •

In quarter 1, even with $NS$ = 99, T-SP-NS still performs poorly. This is
because the confidence intervals cannot exactly cover the underlying
distributions of $\tilde{\boldsymbol{u}}$ and T-SP-NSs would encounter
significant slack penalties.

TABLE VII: Breakdown of Actual System Cost on 28 Jun 2020 |  Actual UC Cost/$\text{10}^{\text{3}}$$ |  Hindsight ED Cost/$\text{10}^{\text{3}}$$
---|---|---
| Startup | No-Load | Startup&No-Load | Generation | Slack
C-PO-7 | 6.7 | 9.0 | 0 | 894.9 | 0
T-SP-1 | 6.5 | 8.1 | 1.1 | 934.3 | 13.7
T-SP-4 | 5.1 | 7.7 | 1.0 | 926.4 | 0
T-SP-99 | 4.8 | 7.8 | 0.7 | 904.7 | 0

Table VII breaks down the cost on the _quarter-2 day_. It shows that the T-SP-
NSs bring lower startup and no-load costs. This is because the reserve
requirements are implicated in (18) rather than being explicit as in (15).
Thus, T-SP-NSs turn on fewer units and schedule less reserve in the day-ahead
stage, resulting in relatively more extensive usage of quick-start units in
the hindsight ED stage. Indeed, our results show that the average hourly
reserve schedules (SR plus NR) of T-SP-1 and T-SP-99 are respectively 89.9MW
and 207.2MW. In comparison, these average schedules are 1,265.8MW and
1,286.8MW in C-PO-1 and C-PO-7. Moreover, it should be pointed out that as
quick-start units have limited ramping ranges, they are operated away from the
most efficient points in ED, causing higher generation costs for T-SP-NSs.

Indeed, the difference in reserve schedules implies that T-SP and C-PO achieve
similar economics through different ways. Specifically, T-SP schedules fewer
reserves and utilizes more RES curtailments in place of SR in actual
operation. This may work well in case of ample RES, but becomes economically
risky when the actual RES availability is low, as expensive slack penalties
will be triggered (e.g., due to insufficient capacity, T-SP-1 is penalized as
in Table VII). This could also cause a lower RES utilization, i.e., the RES
utilization of T-SP-NSs on the four days is 4.2% lower than C-PO-NTs.

####  IV-C2 C-PO vs T-RO

Figure 6: Comparison of C-PO and T-RO.  TABLE VIII: Breakdown of Actual System Cost on 20 Aug 2020 |  Actual UC Cost/$\text{10}^{\text{3}}$$ |  Hindsight ED Cost/$\text{10}^{\text{3}}$$
---|---|---
| Startup | No-Load | Startup&No-Load | Generation | Slack
C-PO-7 | 5.9 | 12.6 | 0.2 | 1789.7 | 0
T-RO-24 | 6.5 | 11.3 | 1.2 | 1764.9 | 0
T-RO-36 | 6.7 | 11.4 | 1.2 | 1764.3 | 0
T-RO-48 | 6.5 | 11.3 | 1.2 | 1764.9 | 63.5

Fig. 6 sketches the operation economics of C-PO and T-RO. It shows that both
methods are relatively stable, i.e., the gap between the best and worst costs
is small. Moreover, multiple intersections exist in the curves, implying that
T-RO and C-PO could achieve similar economic performance with properly tuned
parameters.

Table VIII further compares the costs on the _quarter-3 day_. It shows that
the T-RO-NBs have a higher startup cost but a lower no-load cost in the UC
stage. The reason is that T-RO-NBs schedule SR for those hours with potential
large volatility, so that multiple units in the UC stage are turned on before
these hours and turned off afterward. On the other hand, C-PO-7 constantly
keeps more units online to meet the reserve requirements of individual hours,
with higher no-load costs. In addition, the penalty of T-RO-48 means that a
large $\Gamma$ may not lead to a sufficient reserve schedule. Indeed, the
hourly average reserve schedules of T-RO-48 and C-PO-7 are 278.4MW and
1,214.7MW, implying that C-PO-7 is more conservative. With this, it can be
pointed out that C-PO achieves similar economics as the state-of-the-art
methods in a more reliable way, i.e., schedule more reserves to avoid
potential slack penalties.

####  IV-C3 Discussions

The above comparisons indicate that the operation economics of C-PO is
comparable to T-SP and T-RO. Although a clear-cut economic benefit of C-PO is
yet to be observed, C-PO does present promising practical advantages.

Computational affordability to system operators: The day-ahead UC to determine
optimal operation plans has been widely regarded as one of the most
computationally challenging tasks. Concerning that T-SP/T-RO is more
computationally taxing than its deterministic counterpart, most system
operators still utilize the deterministic UC model (1). In comparison, C-PO is
trained offline, and the prescriptive UC model (15) keeps a similar online
computation burden as O-PO, thus being more affordable to system operators.

Compatibility with the goal of system operators: In most real-world power
systems, the system operators aim to identify the economically optimal UC
decisions given the predictions. However, UC decisions of T-SP/T-RO may not
fully follow the least-cost principle in the short term due to the uncertainty
consideration. Although better operation economics may be reached in the long
term, a short-term financial deficit would be unavoidable.

While pursuing to maximize both short- and long-term system economics, the
system operators must have a stably interpretable schedule plan (i.e., the
operation plan on which units to dispatch) so that both operation plans and
post-operation analyses can be effectively conducted. Regarding this, T-SP and
T-RO could expose intrinsic obstacles in practical applications. For T-SP, a
stable schedule result requires a sufficiently large number of scenarios,
which, however, will undoubtedly magnify the existing computational challenge
and ultimately end up with a trade-off between the number of scenarios and
schedule stability; while for T-RO, subject to the availability of tractable
reformulation, uncertainty budget forms are usually designed to be simple,
abstract, and heavily parameter-dependent. These together lead to volatility
and poor interpretability of schedule results.

Specifically, the scenario generation process (for constructing the scenario
set) inevitably introduces randomness. However, different scenario sets, even
with tiny differences, may lead to noticeable commitment changes, i.e., those
units are committed/decommitted from one set to another. These issues
adversely affect the interpretability of the schedule plan. Analogously,
uncertainty budget settings will suffer from the same issue. Moreover, an
abstract budget lacks a way to be intuitively linked to the scheduling result.
That is, the impacts of budget forms and settings on the schedule results are
difficult to interpret and quantify. As a result, the system operators have to
face the same issue on a daily basis—what is the best budget parameter for the
next operation day?

Last but not least, the difference between the scheduled reserves of C-PO and
T-SP/T-RO implies that the second stage of T-SP/T-RO may not be able to
implicitly secure enough reserves in the UC plan. Hence, the explicit
expression of reserve requirements is necessary for the UC model.

In comparison, the prescriptive UC (15) has the same model structure as (1),
thus following the least-cost principle and remaining strong interpretability.
In addition, the results have clearly shown that efficiencies of T-SP/T-RO and
C-PO are mutually reachable. Therefore, it is promising to harness the
compatibility of C-PO in bridging the gap between the state-of-the-art
T-SP/T-RO and the current practice.

##  V Conclusion

This paper proposes a bilevel MIP-based C-PO framework for improving the
operation economics as compared to the traditional O-PO framework. The idea is
to leverage the bilevel MIP structure to train the cost-oriented RES and
reserve predictors to be integrated into the optimal operation task, in which
the prediction quality is evaluated via operation costs. Case studies lead to
the following conclusions:

  * •

Benefiting from the bilevel MIP structure, C-PO can economically outperform
traditional O-PO with limited training scenarios. According to the results of
selective weeks in different seasons, C-PO with 7 training scenarios can lead
to a 2.54% improvement in daily operation economics on average, even though
the cost-oriented RES predictions present slightly worse MAEs and RMSEs.

  * •

The operation economics of C-PO, T-SP, and T-RO could be mutually reachable,
with sufficient scenarios for T-SP and comprehensively tuned parameters for
T-RO. Nevertheless, given its high compatibility with the current power system
operation practice, C-PO presents higher implementation values to serve as a
bridge between T-SP/T-RO-based UC and the current deterministic UC practice.

Future work could focus on accelerating the training process of C-PO to
further boost its implementation in power system practice.

###  -A Deterministic UC Model

The objective (20a) is to minimize the total system cost, including startup,
no-load, and generation costs. The generator constraints include generation
limits (20b), segment-based generation representation (20c)-(20d), SR (20e)
and NR (20f) capacity limits, startup-shutdown-commitment status logic (20g),
minimum on and off requirements (20h)-(20i), ramping limits (20j), and RES
power limitation (20k). Constraints (20l)-(20m) mean that offline non-quick-
start units cannot provide NR. Constraint (20n) is the integrality
requirement. System constraints include power balance (20o), DC power flow-
based transmission limits (20p) [4], and system reserve requirements (20q).

$\displaystyle\min\textstyle{\sum\limits_{t\in\mathcal{T},i\in\mathcal{I}}}(\textstyle{C_{i}^{su}U_{it}+C_{i}^{nl}I_{it}+\sum\limits_{k\in\mathcal{K}}C^{sg}_{ik}P_{itk}^{sg})}$
(20a) Generator Constraints:
$\displaystyle\textstyle{P_{it}-R_{it}^{sr}\geq\underline{P\mkern-6.0mu}\mkern
4.0mu_{i}I_{it}};\,\textstyle{P_{it}+R_{it}^{sr}\leq\bar{P}_{i}I_{it}};$
$\displaystyle\mspace{-197.0mu}\textstyle{\forall
t\in\mathcal{T},i\in\mathcal{I}}$ (20b)
$\displaystyle\textstyle{P_{it}=\sum_{k\in\mathcal{K}}P_{itk}^{sg}};$
$\displaystyle\mspace{-199.0mu}\textstyle{\forall
t\in\mathcal{T},i\in\mathcal{I}}$ (20c) $\displaystyle\textstyle{0\leq
P_{itk}^{sg}\leq\bar{P}_{ik}^{sg}I_{it}};$
$\displaystyle\mspace{-199.0mu}\textstyle{\forall
t\in\mathcal{T},i\in\mathcal{I},k\in\mathcal{K}}$ (20d)
$\displaystyle\textstyle{0\leq R_{it}^{sr}\leq\bar{R}_{i}^{sr}I_{it}};$
$\displaystyle\mspace{-199.0mu}\textstyle{\forall
t\in\mathcal{T},i\in\mathcal{I}}$ (20e)
$\displaystyle\textstyle{\underline{P\mkern-6.0mu}\mkern 4.0mu_{i}O_{it}\leq
R_{it}^{nr}\leq\bar{R}_{i}^{nr}O_{it}};$
$\displaystyle\mspace{-199.0mu}\textstyle{\forall
t\in\mathcal{T},i\in\mathcal{I}}$ (20f)
$\displaystyle\textstyle{U_{it}-D_{it}=I_{it}-I_{i,t-1}};$
$\displaystyle\mspace{-199.0mu}\textstyle{\forall
t\in\mathcal{T},i\in\mathcal{I}}$ (20g)
$\displaystyle\textstyle{\sum_{t^{\prime}=t-T_{i}^{su}+1}^{t}U_{it^{\prime}}\leq
I_{it}};$ $\displaystyle\mspace{-199.0mu}\textstyle{\forall
t\in\mathcal{T}^{su}_{i},i\in\mathcal{I}}$ (20h)
$\displaystyle\textstyle{\sum_{t^{\prime}=t-T_{i}^{sd}+1}^{t}D_{it^{\prime}}\leq
1-I_{it}};$ $\displaystyle\mspace{-199.0mu}\textstyle{\forall
t\in\mathcal{T}^{sd}_{i},i\in\mathcal{I}}$ (20i)
$\displaystyle\textstyle{P_{it}-P_{i,t-1}\leq\bar{P}_{i}(1-I_{it})+R_{i}^{\uparrow}I_{i,t-1}+R_{i}^{su}(I_{it}-I_{i,t-1})};\,\,$
$\displaystyle\textstyle{P_{i,t-1}-P_{it}\leq\bar{P}_{i}(1-I_{i,t-1})+R^{\downarrow}_{i}I_{it}+R^{sd}_{i}(I_{i,t-1}-I_{it})};$
$\displaystyle\mspace{-199.0mu}\textstyle{\forall
t\in\mathcal{T},i\in\mathcal{I}}$ (20j) $\displaystyle\textstyle{0\leq
W_{jt}\leq\hat{{W}}_{jt}};$ $\displaystyle\mspace{-199.0mu}\textstyle{\forall
t\in\mathcal{T},j\in\mathcal{J}}$ (20k)
$\displaystyle\textstyle{O_{it}+I_{it}\leq 1};$
$\displaystyle\mspace{-199.0mu}\textstyle{\forall
t\in\mathcal{T},i\in\mathcal{I}}$ (20l) $\displaystyle\textstyle{O_{it}=0};$
$\displaystyle\mspace{-199.0mu}\textstyle{\forall
t\in\mathcal{T},i\in\mathcal{I}^{ns}}$ (20m)
$\displaystyle\textstyle{I_{it},U_{it},D_{it},O_{it}\in\\{0,1\\}};$
$\displaystyle\mspace{-199.0mu}\textstyle{\forall
t\in\mathcal{T},i\in\mathcal{I}}$ (20n) System Constraints:
$\displaystyle\textstyle{\sum\nolimits_{i\in\mathcal{I}}P_{it}+\sum\nolimits_{j\in\mathcal{J}}W_{jt}=\sum\nolimits_{q\in\mathcal{Q}}\hat{L}_{qt}};$
$\displaystyle\mspace{-199.0mu}\textstyle{\forall t\in\mathcal{T}}$ (20o)
$\displaystyle\textstyle{-B_{b}\leq\mathcal{F}_{b}(P_{it},W_{jt},\hat{L}_{qt})\leq
B_{b}};$ $\displaystyle\mspace{-199.0mu}\textstyle{\forall
t\in\mathcal{T},b\in\mathcal{B}}$ (20p)
$\displaystyle\textstyle{\sum_{i\in\mathcal{I}}R_{it}^{sr}\mspace{-3.0mu}\geq\mspace{-3.0mu}\hat{R}_{t}^{sr}};\,\,\textstyle{\sum_{i\in\mathcal{I}}(R_{it}^{sr}+R_{it}^{nr})\mspace{-3.0mu}\geq\mspace{-3.0mu}\hat{R}_{t}^{sr}\mspace{-3.0mu}+\mspace{-3.0mu}\hat{R}_{t}^{nr}};$
$\displaystyle\mspace{-110.0mu}\textstyle{\forall t\in\mathcal{T}}$ (20q)

###  -B Hindsight ED Model

The objective (21c) is to minimize startup and no-load costs of quick-start
units that are not committed in UC but scheduled for providing NR, generation
costs of all units, and slack penalty costs. Constraints (21d)-(21e) mean that
only the quick-start units not committed in UC may change statuses. All units
are subjected to generation limits (21f)-(21h), dispatch adjustment limits
(21i), ramping limits (21j), RES power limits (21k), and integrality
requirement (21l). The system shall satisfy power balance (21m) and
transmission limits (21n)-(21o) against actual load and RES realizations.
Slack variables $\boldsymbol{S}^{1/2/3/4}$ are used to ensure ED feasibility
with respect to the given UC solutions.

$\displaystyle\min\mspace{-5.0mu}\textstyle{\sum\limits_{t\in\mathcal{T}}}\mspace{-5.0mu}\left\\{\mspace{-5.0mu}\begin{array}[]{l}\textstyle{\sum\limits_{i\in\mathcal{I}}(C_{i}^{su}U_{it}^{ed,qs}+C_{i}^{nl}I_{it}^{ed,qs}+\sum\limits_{k\in\mathcal{K}}C_{ik}^{sg}P_{itk}^{ed,sg}})\\\
\textstyle{+[C^{1}S_{t}^{1}+C^{2}S_{t}^{2}+C^{bs}\sum\nolimits_{b\in\mathcal{B}}(S_{bt}^{3}+S_{bt}^{4})]}\end{array}\right\\}\mspace{-24.0mu}$
(21c) Generator Constraints:
$\displaystyle\textstyle{I_{it}^{ed}=I_{it}^{\star}+I_{it}^{ed,qs}};\,\,\textstyle{I_{it}^{ed,qs}\leq
O_{it}^{\star}};$ $\displaystyle\mspace{-200.0mu}\textstyle{\forall
t\in\mathcal{T},i\in\mathcal{I}}$ (21d)
$\displaystyle\textstyle{U_{it}^{ed,qs}-D_{it}^{ed,qs}=I_{it}^{ed,qs}-I_{i,t-1}^{ed,qs}};$
$\displaystyle\mspace{-200.0mu}\textstyle{\forall
t\in\mathcal{T},i\in\mathcal{I}}$ (21e)
$\displaystyle\textstyle{P_{it}^{ed}=\sum\nolimits_{k\in\mathcal{K}}P_{itk}^{ed,sg}};$
$\displaystyle\mspace{-200.0mu}\textstyle{\forall
t\in\mathcal{T},i\in\mathcal{I}}$ (21f) $\displaystyle\textstyle{0\leq
P_{itk}^{ed,sg}\leq\bar{P}_{ik}^{sg}I_{it}^{ed}};$
$\displaystyle\mspace{-200.0mu}\textstyle{\forall
t\in\mathcal{T},i\in\mathcal{I},k\in\mathcal{K}}$ (21g)
$\displaystyle\textstyle{P_{it}^{ed}\geq\underline{P\mkern-6.0mu}\mkern
4.0mu_{i}I_{it}^{ed}};\,\,\textstyle{P_{it}^{ed}\leq\bar{P}_{i}I_{it}^{\star}+R_{it}^{nr\star}I_{it}^{ed,qs}};$
$\displaystyle\mspace{-145.0mu}\textstyle{\forall
t\in\mathcal{T},i\in\mathcal{I}}$ (21h)
$\displaystyle\textstyle{-R_{it}^{sr\star}I_{it}^{\star}\leq
P_{it}^{ed}-P_{it}^{\star}\leq
R_{it}^{sr\star}I_{it}^{\star}+\bar{P}_{i}I_{it}^{ed}};$
$\displaystyle\mspace{-145.0mu}\textstyle{\forall
t\in\mathcal{T},i\in\mathcal{I}}$ (21i)
$\displaystyle\textstyle{P_{it}^{ed}-P_{i,t-1}^{ed}\leq\bar{P}_{i}(1-I_{it}^{ed})+R_{i}^{\uparrow}I_{i,t-1}^{ed}+R_{i}^{su}(I_{it}^{ed}-I_{i,t-1}^{ed})};$
$\displaystyle\textstyle{P_{i,t-1}^{ed}-P_{it}^{ed}\leq\bar{P}_{i}(1-I_{i,t-1}^{ed})+R^{\downarrow}_{i}I_{it}^{ed}+R^{sd}_{i}(I_{i,t-1}^{ed}-I_{it}^{ed})};$
$\displaystyle\mspace{-200.0mu}\textstyle{\forall
t\in\mathcal{T},i\in\mathcal{I}}$ (21j) $\displaystyle\textstyle{0\leq
W_{jt}^{ed}\leq\tilde{{W}}_{jt}};$
$\displaystyle\mspace{-200.0mu}\textstyle{\forall
t\in\mathcal{T},j\in\mathcal{J}}$ (21k)
$\displaystyle\textstyle{I_{it}^{ed},I_{it}^{ed,qs},U_{it}^{ed,qs},D_{it}^{ed,qs}\in\\{0,1\\}};$
$\displaystyle\mspace{-200.0mu}\textstyle{\forall
t\in\mathcal{T},i\in\mathcal{I}}$ (21l) System Constraints:
$\displaystyle\textstyle{\sum\limits_{i\in\mathcal{I}}\mspace{-5.0mu}P_{it}^{ed}+\mspace{-5.0mu}\sum\limits_{j\in\mathcal{J}}\mspace{-5.0mu}W_{jt}^{ed}+\mspace{-3.0mu}S_{t}^{1}=\mspace{-5.0mu}\sum\limits_{q\in\mathcal{Q}}\mspace{-5.0mu}\tilde{L}_{qt}+\mspace{-3.0mu}S_{t}^{2}};\,\textstyle{S_{t}^{1},S_{t}^{2}\geq
0};$ $\displaystyle\mspace{-95.0mu}\textstyle{\forall t\in\mathcal{T}}$ (21m)
$\displaystyle\textstyle{\mathcal{F}_{b}(P_{it}^{ed},W_{jt}^{ed},\tilde{L}_{qt})-S_{bt}^{3}\leq
B_{b}};\,\,\textstyle{S_{bt}^{3}\geq 0};$
$\displaystyle\mspace{-150.0mu}\textstyle{\forall
t\in\mathcal{T},b\in\mathcal{B}}$ (21n)
$\displaystyle\textstyle{\mathcal{F}_{b}(P_{it}^{ed},W_{jt}^{ed},\tilde{L}_{qt})+S_{bt}^{4}\geq-
B_{b}};\,\,\textstyle{S_{bt}^{4}\geq 0};$
$\displaystyle\mspace{-150.0mu}\textstyle{\forall
t\in\mathcal{T},b\in\mathcal{B}}$ (21o)

###  -C Proof of Proposition 1

###### Proof:

To prove Proposition 1, problem (22) is built to serve as a bridge between (9)
and (11), which is constructed by adding (22d) to (9) or adding (22f) to (11).

$\displaystyle\textstyle{\min_{\mathcal{W},\mathcal{R}}}$
$\displaystyle\mspace{-30.0mu}\textstyle{\frac{1}{|\mathcal{S}|}\sum\nolimits_{s\in\mathcal{S}}}(\boldsymbol{b}^{\top}\boldsymbol{x}_{s}+\boldsymbol{d}^{\top}\boldsymbol{z}_{s})$
(22a) $\displaystyle s.\,t.\,$
$\displaystyle\mspace{-70.0mu}\mathcal{W}(\cdot)\in\mathcal{P}^{w};\,\,\mathcal{R}(\cdot)\in\mathcal{P}^{r}$
(22b)
$\displaystyle\mspace{-70.0mu}\hat{\boldsymbol{w}}^{\diamond}_{s}=\mathcal{W}(\boldsymbol{f}^{w}_{s});\,\,\hat{\boldsymbol{r}}^{\diamond}_{s}=\mathcal{R}(\boldsymbol{f}^{r}_{s});$
$\displaystyle\mspace{-30.0mu}\forall s\in\mathcal{S}$ (22c)
$\displaystyle\mspace{-70.0mu}\boldsymbol{z}_{s}\in\mathcal{Z}(\boldsymbol{x}_{s},\boldsymbol{y}_{s},\tilde{\boldsymbol{w}}_{s});$
$\displaystyle\mspace{-30.0mu}\forall s\in\mathcal{S}$ (22d)
$\displaystyle\mspace{-70.0mu}\boldsymbol{x}_{s},\boldsymbol{y}_{s}\in{\arg\min}_{\boldsymbol{x}_{s},\boldsymbol{y}_{s}\in\mathcal{X}(\hat{\boldsymbol{w}}^{\diamond}_{s},\hat{\boldsymbol{r}}^{\diamond}_{s})}\boldsymbol{b}^{\top}\boldsymbol{x}_{s}+\boldsymbol{c}^{\top}\boldsymbol{y}_{s};$
$\displaystyle\mspace{-30.0mu}\forall s\in\mathcal{S}$ (22e)
$\displaystyle\mspace{-70.0mu}\boldsymbol{z}_{s}\in{\arg\min}_{\boldsymbol{z}_{s}\in\mathcal{Z}(\boldsymbol{x}_{s},\boldsymbol{y}_{s},\tilde{\boldsymbol{w}}_{s})}\boldsymbol{d}^{\top}\boldsymbol{z}_{s};$
$\displaystyle\mspace{-30.0mu}\forall s\in\mathcal{S}$ (22f)

First, we claim that (9) and (22) have the same optimal solutions because
their objective functions are identical and their feasible regions are also
the same. They have the same feasible regions because (_i_) A feasible
solution of (9) satisfies all constraints of (22) and thus is feasible to
(22); and (_ii_) A feasible solution of (22) also meets all constraints of (9)
and thus is feasible to (9). Thus, (9) and (22) have the same optimal
solutions.

Second, we claim that the optimal solution of (11) is also optimal for (22).
Note that as (11) is equivalent to (22a)-(22e), we only need to show that the
optimal solution of (11) also satisfies (22f). Specifically, if
$\Omega^{\star}$ (including $\boldsymbol{x}_{s}^{\star}$,
$\boldsymbol{y}_{s}^{\star}$, and $\boldsymbol{z}_{s}^{\star}$) is an optimal
solution of (11), $\boldsymbol{z}^{\star}_{s}$ must be optimal for (23) that
is constructed by fixing all variables in (11) to their optima except
$\boldsymbol{z}_{s}$. Problem (23) is equivalent to (22f), i.e.,
$\boldsymbol{z}^{\star}_{s}$ satisfies (22f). Thus, because $\Omega^{\star}$
is optimal for (22a)-(22e) and its $\boldsymbol{z}^{\star}_{s}$ also satisfies
(22f), $\Omega^{\star}$ must be optimal for (22).

$\displaystyle\textstyle{\min_{\boldsymbol{z}_{s}}}$
$\displaystyle\mspace{-35.0mu}\textstyle{\frac{1}{|\mathcal{S}|}\sum\nolimits_{s\in\mathcal{S}}}(\boldsymbol{b}^{\top}\boldsymbol{x}_{s}^{\star}+\boldsymbol{d}^{\top}\boldsymbol{z}_{s})$
(23a) $\displaystyle s.\,t.$
$\displaystyle\mspace{-35.0mu}\textstyle{\boldsymbol{z}_{s}\in{\arg\min}_{\boldsymbol{z}_{s}\in\mathcal{Z}(\boldsymbol{x}^{\star}_{s},\boldsymbol{y}^{\star}_{s},\tilde{\boldsymbol{w}}_{s})}\boldsymbol{d}^{\top}\boldsymbol{z}_{s}};\quad\forall
s\in\mathcal{S}$ (23b)

From the above claims, it is direct to conclude that the optimal solution of
(11) is optimal for (22), and thus also optimal for (9). ∎

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