# Efficient Scenario Generation for Chance-constrained Economic Dispatch
Considering Ambient Wind Conditions

Qian Zhang, Apurv Shukla, and Le Xie This work is supported in part by NSF
ECCS-2038963, the U.S. Department of Energy (DoE) Office of Energy Efficiency
and Renewable Energy (EERE) under the Solar Energy Technologies Office (SETO)
Award Number DEEE0009031, and Texas A&M Energy Institute. The authors are with
the Department of Electrical and Computer Engineering, Texas A&M University,
College Station, TX, 77843 USA (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>le.xie@tamu.edu).

###### Abstract

Scenario approach is an effective data-driven method for solving chance-
constrained optimization while ensuring desired risk guarantees with a finite
number of samples. Crucial challenges in deploying this technique in the real
world arise due to non-stationarity environments and the absence of
appropriate risk-tuning models tailored for the desired application. In this
paper, we focus on designing efficient scenario generation schemes for
economic dispatch in power systems. We propose a novel scenario generation
method based on filtering scenarios using ambient wind conditions. These
filtered scenarios are deployed incrementally in order to meet desired risk
levels while using minimum resources. In order to study the performance of the
proposed scheme, we illustrate the procedure on case studies performed for
both 24-bus and 118-bus systems with real-world wind power forecasting data.
Numerical results suggest that the proposed filter-and-increment scenario
generation model leads to a precise and efficient solution for the chance-
constrained economic dispatch problem.

###### Index Terms:

Scenario generation, chance-constrained programming, economic dispatch, wind
power forecasting error, scenario approach

##  I Introduction

The integration of intermittent renewable sources of generation into the
existing power grids has posed significant challenge due to the inherent
uncertainty associated with these sources. While both solar and wind
generators introduce uncertainty, wind power, in particular, presents unique
challenges. Unlike solar, wind energy lacks diurnal patterns, making it more
difficult to predict. Furthermore, the scale of installed and consumed wind
energy often surpasses that of solar power in many dispatch areas, leading to
a higher level of uncertainty in the overall power system. For instance, wind
power accounts for 25% of the total electricity consumed by the Electric
Reliability Council of Texas (ERCOT) region during 2022, compared to $6\%$ for
solar [1].
In response to these challenges, stochastic optimization (SO) and robust
optimization (RO) are widely used methods for power engineers to deal with
uncertainties [2]. SO utilizes probabilistic models to handle randomness [3],
while RO employs set-based and deterministic uncertainty models [4]. We focus
on chance-constrained optimization (CCO) which bridges the gap between
probabilistic and deterministic approaches, providing explicit probabilistic
guarantees on the feasibility of optimal solutions [5].
Over the past decade, many attempts have been made at reformulating CCO into a
more computationally tractable form. Bienstock et al. [6] reformulate the
chance-constrained DC-optimal power flow (DC-OPF) under affine control as a
second-order cone program by a moment-based reformulation. Roald et al. [7]
extend similar reformulations into the AC-OPF model with some efficient
algorithms.  Other reformulations have also been proposed considering the
distributionally robust property, especially in economic dispatch problem [8,
9].
However, most of the above approaches require the explicit or approximated
distribution function of uncertainty, which is hard to validate with streaming
data. Data-driven optimization methods, unconstrained by specific underlying
uncertainty distributions, have received substantial attention in recent years
[5], especially the Sample Average Approximation (SAA) [10] and the Scenario
Approach [11]. In the realm of power systems, applications of SAA are evident
in day-ahead unit commitment [12], capacity planning [13], and other offline
domains [14]. In real-time cases, the scenario approach proves advantageous
due to its rapid computational speed, avoiding the long solving time caused by
binary variables in SAA. Despite many recent works that have tried to apply
the scenario approach in power systems [2, 5, 15, 16, 17, 18, 19, 20], there
still exist two main limitations.
A. Scenario Generation Limitation
Several studies validate scenario approach in the economic dispatch by using
synthetic data created from representative distribution [15, 16]. These
techniques do not utilize and exploit the fact that the scenario approach can
be distribution-agnostic since the scenarios can be directly extracted from
previous experience. In most real-world settings, empirical data is collected
from non-stationary environments, wherein the distribution of the random
variable depends on environmental conditions [21]. Directly sampling from the
past will incorporate all environmental factors from empirical data in
scenario approach. For example, the wind power forecasting error addressed in
this paper, is the primary source of uncertainty in the economic dispatch
process [22, 23], while the wind forecast error distribution varies based on
forecasting techniques, power output levels, and ambient conditions, as noted
in studies by [24, 25].
B. Risk Tuning Limitation
After choosing the scenario space, the next step is to decide the number of
scenarios. The conventional sample and discard approach [26] requires the
decision-makers to first create an estimate of the sample size and calculate
the exact risk level ex-ante. Then, it is decided whether scenarios need to be
discarded to trade off risk and performance. To ensure meeting the risk
requirements, a conservative sample size is used, which could be astronomical
for problems with a large number of decision variables. This inefficient risk-
tuning process consumes superfluous data and results in a long computation
time.
The main contributions of this paper are dealing with these two limitations,
which can be summarized as follows:

  1. 1.

To enhance scenario accuracy, the conditional distribution is considered
without assuming any knowledge of the true distribution. Utilizing correlation
analysis, wind forecast error scenarios are generated from empirical data
similar to the present environment, ensuring a more precise representation of
real-world conditions.

  2. 2.

The incremental risk tuning method is introduced to meet the risk requirement
with minimum data resources [27]. After declaring a desired risk level,
scenarios are generated iteratively to eventually hit a desired level of risk.

  3. 3.

Algorithms are designed to incorporate scenario generation with risk-tuning
processes efficiently.

The remainder of this article is organized as follows: Section II formulates
the chance-constrained economic dispatch problem and highlights the challenges
associated with solving the problem with conditional wind power forecast
error. Section III introduces our incremental scenario approach, and Section
IV discusses the assumptions and limitations of the proposed method. We
demonstrate the efficacy of the proposed approach on 24 and 118-bus systems in
Section V.

##  II Problem Statements

###  II-A Chance-Constrained Economic Dispatch

We consider the chance-constrained DC-OPF formulation in the presence of wind-
forecasting uncertainty [6, 18]:

$\displaystyle\min_{g,\eta}\;$ $\displaystyle c(g)$ (1a) s.t
$\displaystyle\mathbf{1}^{\top}g=\mathbf{1}^{\top}d-\mathbf{1}^{\top}\hat{w}$
(1b) $\displaystyle\underline{g}\preceq g\preceq\bar{g}$ (1c) $\displaystyle
f(\hat{w},\tilde{w})=H_{g}\left(g-\mathbf{1}^{\top}\tilde{w}\eta\right)-H_{d}d+H_{w}(\hat{w}+\tilde{w})$
(1d)
$\displaystyle\mathbb{P}_{\tilde{w}}\left(\begin{array}[]{l}\underline{f}\preceq
f(\hat{w},\tilde{w})\preceq\bar{f}\\\ \underline{g}\preceq
g-\mathbf{1}^{\top}\tilde{w}\eta\preceq\bar{g}\\\
R_{d}\preceq-\mathbf{1}^{\top}\tilde{w}\eta\preceq R_{u}\end{array}\right)\geq
1-\epsilon$ (1h) $\displaystyle\mathbf{1}^{\top}\eta=1$ (1i)

The decision variables are generator output levels $g\in\mathbf{R}^{n_{g}}$,
and an affine control policy $\eta\in\mathbf{R}^{n_{g}}$ proportionally
allocating total wind fluctuation $\mathbf{1}^{\top}\tilde{w}$ to each
generator111 In this paper, $\mathbf{1}^{\top}$ denotes a row vector of all
ones, with its dimension adjustable to match the vector it multiplies.. The
objective function is the total generations cost $c(g)$. The load level is
$d\in\mathbf{R}^{n_{d}}$, and the wind generation $w=\hat{w}+\tilde{w}$
consists of deterministic wind forecast value $\hat{w}\in\mathbf{R}^{n_{w}}$
and the uncertain forecast error $\tilde{w}\in\Delta$, where
$\Delta\subseteq\mathbf{R}^{n_{w}}$ is the uncertainty set. Transmission line
flows $f\in\mathbf{R}^{n_{f}}$ are calculated using (3d), where $H_{g}$,
$H_{d}$, and $H_{w}$ are the corresponding sub-matrix of the power transfer
distribution factor (PTDF) matrix $H$. Constraints include transmission line
flow limits
$[\underline{f},\bar{f}]\in\mathbf{R}^{n_{f}}\times\mathbf{R}^{n_{f}}$,
generation capacity limits
$[\underline{g},\bar{g}]\in\mathbf{R}^{n_{g}}\times\mathbf{R}^{n_{g}}$ and the
ramp up(down) rate limits
$[R_{d},R_{u}]\in\mathbf{R}^{n_{g}}\times\mathbf{R}^{n_{g}}$ are modeled as a
chance-constraint form under risk $\epsilon$ in (1h).
As mentioned in [18], the affine control policy $\eta$ only focuses on the
steady-state behavior of the Automatic Generation Control (AGC) action in
dispatch time scale, i.e. $5$ to $15$ minutes, but not the $2$ to $6$ seconds
fast time-scale regulation process. The system’s active power deviation is
allocated to generators based on $\eta$, which is also well known as
participation factors in the conventional AGC scheme [28]. It is easy to
confirm that constraints (3b) and (3i) imply the supply-demand balance in the
presence of wind uncertainties:

$\mathbf{1}^{\top}(g-\mathbf{1}^{\top}\tilde{w}\eta)=\mathbf{1}^{\top}d-\mathbf{1}^{\top}(\hat{w}+\tilde{w})$
(2)

Setting the constant affine control policy $\eta$ prior to the next dispatch
interval will unavoidably be economically inefficient if the net load’s
fluctuation or forecasting error is large. Changing the $\eta$ more frequently
within dispatch interval, or incorporating the optimization program into the
AGC control policy may improve the economic efficiency [29, 30], but these
methods ignore the network constraints and are hard to apply to the bulk power
system due to the communication delay or solving time. In this paper, we
mainly concentrate on improving the dispatch performance in the chance-
constrained problem. The description of affine control policy $\eta$ in (1) is
based on two assumptions: 1) All the traditional generators participate in the
AGC actions; 2) The whole system is regarded as one control area.

###  II-B Conditional Wind Power Forecast Error

Due to the nonlinear wind power curve, the wind power forecast error is
observed to vary with the level of its output [31, 32], while spatial and
weather parameters also indirectly affect the forecasting quality [33, 34].
Based on these facts, the wind power forecast error measure
$\mathbb{P}_{\tilde{w}}$ in (1h) should be modeled as a conditional
probability distribution from similar environments. Previous studies have
focused on approximating the measure of conditional forecast error, eg., [31]
generate the conditional error model based on the wind turbine power curve,
[24] employ a fuzzy inference model to obtain conditional prediction intervals
and [25] calculate the conditional forecasting error from joint distributed
data using copula theory. All the above approximation methods for the
distribution $\mathbb{P}_{\tilde{w}}$ under similar conditions are
incompatible with the data-driven approach in chance-constraint optimization,
which uses distribution agnostic empirical data. Furthermore, the wind power
output, temperature, and weather parameters are continuous variables, meaning
the past scenarios’ observations will be distinct with probability 1, which
makes it impossible to generate scenarios in the data-driven scheme from an
identical environment but similar environments. Scenario selection from
similar environments presents a major engineering obstacle: a large search
space results in conservative decision-making while a smaller search space
results in limited data [21].

##  III Method

###  III-A Scenario Approach

The scenario approach randomly extracts $N$ independent and identically
distributed (i.i.d.) scenarios to approximate the chance-constrained program.
Supposing we have the random wind forecasting error scenarios set
$\mathcal{N}:=\left\\{\tilde{w}_{1},\tilde{w}_{2},\cdots,\tilde{w}_{N}\right\\}$,
the chance-constrained inequalities (1h) in DC-OPF problem can be replaced by
scenario-based inequalities (3h):

$\displaystyle\min_{g,\eta}\;$ $\displaystyle c(g)$ (3a) s.t
$\displaystyle\mathbf{1}^{\top}g=\mathbf{1}^{\top}d-\mathbf{1}^{\top}\hat{w}$
(3b) $\displaystyle\underline{g}\preceq g\preceq\bar{g}$ (3c) $\displaystyle
f(\hat{w},\tilde{w})=H_{g}\left(g-\mathbf{1}^{\top}\tilde{w}\eta\right)-H_{d}d+H_{w}(\hat{w}+\tilde{w})$
(3d) $\displaystyle\begin{array}[]{l}\underline{f}\preceq
f(\hat{w},\tilde{w}_{i})\preceq\bar{f}\\\ \underline{g}\preceq
g-\mathbf{1}^{\top}\tilde{w}_{i}\eta\preceq\bar{g}\\\
R_{d}\preceq-\mathbf{1}^{\top}\tilde{w}_{i}\eta\preceq R_{u}\end{array}\quad
i=1,2,3,...,N$ (3h) $\displaystyle\mathbf{1}^{\top}\eta=1$ (3i)

To distinguish from the original optimization problem (1), we name the above
scenario problem as $\mathrm{SP}(\mathcal{N})$.

######  Definition 1 (Violation Probability):

The vilolation probability of a candidate solution $(g^{*},\eta^{*})$ is
defined as the probability that $(g^{*},\eta^{*})$ is infeasible, i.e.,
$\mathbb{V}_{\tilde{w}}(g^{*},\eta^{*}):=\mathbb{P}_{\tilde{w}}((g^{*},\eta^{*})\notin\mathcal{X}_{\tilde{w}})$,
where $\mathcal{X}_{\tilde{w}}$ is the decision set generated by
$\mathrm{SP}(\mathcal{N})$.

######  Definition 2 (Support Constraint):

The scenario-dependent constraint corresponding to sample
$\tilde{w}_{s},s\in\\{1,2,...,\mathcal{S}\\}$, is a support constraint or
support scenario if its removal improves the solution of
$\mathrm{SP}(\mathcal{N})$, i.e., if it decreases the optimal cost (3a).

######  Definition 3 (Sample Complexity):

The number of support scenarios in $\mathrm{SP}(\mathcal{N})$ is defined as
the sample complexity.

######  Definition 4 (Helly’s Dimension):

Helly’s dimension of the scenario problem $\mathrm{SP}(\mathcal{N})$ is the
smallest integer $h$ that $h\geq_{\text{ess
sup}\mathcal{N}\subseteq\Delta^{N}}|\mathcal{S(N)}|$ holds for any finite
$N\geq 1$, where $|\mathcal{S(N)}|$ is the number of support constraints or
sample complexity 222Because the randomness of sampling, the number of support
constraints might be different especially when sample size is small. Here
${}_{\text{ess sup}}$ means the essential supremum to ignore some exceptional
cases..

The most important contribution of the scenario approach is the relationship
between violation probability $\mathbb{V}_{\tilde{w}}(g^{*},\eta^{*})$, the
number of scenarios $N$ and the sample complexity.

######  Theorem 1 (Exact Feasibility [11, 35]):

Under the assumptions of nondegeneracy and feasibility of the optimization
problem, the deepest results show that the distribution of
$\mathbb{V}_{\tilde{w}}(g^{*},\eta^{*})$ is dominated by a Beta distribution,
namely:

$\mathbb{P}_{\tilde{w}}^{N}\left(\mathbb{V}_{\tilde{w}}(g^{*},\eta^{*})>\epsilon\right)\leq\sum_{i=0}^{h-1}\left(\begin{array}[]{c}N\\\
i\end{array}\right)\epsilon^{i}(1-\epsilon)^{N-i}:=\beta$ (4)

where $h$ is the Helly’s dimension of $\mathrm{SP}(\mathcal{N})$, and
$1-\beta$ is defined as the confidence bound for the solution based on any $N$
i.i.d scenarios.

######  Theorem 2 (Property for Convex Problem [11]):

Supposing all the constraints in $\mathrm{SP}(\mathcal{N})$ is convex for
every instance of $\tilde{w}$, the sample complexity $|\mathcal{S(N)}|$ for
$\mathrm{SP}(\mathcal{N})$ is at most $n$. In other words, $h\leq n$, where
$n$ is the number of decision variables after eliminating the equality
constraints [36].

For the convex problem, Helly’s dimension $h$ can be replaced by $n$ to
simplify the problem by applying Theorem 2, but it often causes extremely
conservative results. To compute the lower bound of $h$, we suggest using the
dual-based Algorithm 1 proposed in [17].

Algorithm 1 Searching Support Scenarios Using Dual Variables

Solving the scenario problem $\mathrm{SP}(\mathcal{N})$
Generate the primal solution $(g^{*}_{\mathcal{N}},\eta^{*}_{\mathcal{N}})$
and the constraints (3h) related dual solution $\mu^{*}_{i}$ ($i=1,2,3,...,N$)
Let $\mathcal{M}={i\in\mathcal{N}:\|\mu^{*}_{i}\|>0}$. Set
$\mathcal{S}\leftarrow\O$

for $i\in\mathcal{M}$ do

Solve $\mathrm{SP}_{\mathcal{M}-i}$ and compute
$(g^{*}_{\mathcal{M}-i},\eta^{*}_{\mathcal{M}-i})$

if $c(g^{*}_{\mathcal{M}-i})<c(g^{*}_{\mathcal{N}})$ then

$\mathcal{S}\leftarrow\mathcal{S}+i$

end if

end for

OUTPUT: The support scenarios $\mathcal{S}$

###  III-B Sampling Scenarios from Parameter Space

Many environmental parameters, such as location, wind speed, temperature, wind
direction, and relative humidity affect the wind power forecasting quality
[34]. In this paper, we suppose the wind generators are from the same area
with the same forecasting algorithm, while the deterministic wind forecast
value, the wind power changing rate, temperature, and relative humidity are
selected as the four main parameters that affect the forecasting error.
Remark: The deterministic wind forecast value and the wind power ramp rate are
parameters that are integrated with other environmental information,
especially the wind speed and wind speed ramp rate, which are observed having
a close relationship with the wind power forecasting quality.

######  Definition 5 (Parameter Space for Wind Power Forecasting Error):

The parameter space $\mathcal{V}_{\mathcal{N}}$ is defined as the set of
environmental parameters which the past scenarios $\mathcal{N}$ are extracted
from. For instance, the temperature between $70^{\circ}F$ and $80^{\circ}F$ is
a temperature parameter space.

######  Definition 6 (Probability Distribution Over Parameter Space):

Let $\mathbb{P}_{\tilde{w}|\mathcal{V}_{\mathcal{N}}}$ be a probability
distribution over the parameter space $\mathcal{V}_{\mathcal{N}}$.

Remark: Strictly speaking, $\mathcal{V}_{\mathcal{N}}$ should be identical
with the environment parameter at the forecasting moment
$\mathcal{V}_{\text{Now}}$ to acquire more precise risk guarantee. However,
the input data in the scenario approach is directly extracted from the
empirical experience with continuous environment parameters. Finding the past
scenarios under the same environment parameter as the future is impossible
with probability 1. Even in some frontier probabilistic prediction methods, it
is also hard to guarantee the accuracy of predicted conditional distribution
but check how close it approximates to the real distribution based on testing
data [37].
Based on the definitions above, the barrier of bringing the scenario approach
to the real world is to find the parameter space $\mathcal{V}_{\mathcal{N}}$
which includes both the potential environment parameter in the future and
enough number of empirical scenarios to meet the risk requirement in Theorem
1.

###  III-C Main Result I: Correlation-Based Scenario Generation

The Pearson correlation coefficient is used to quantify the relationship
between each environmental parameter and the wind power forecasting error from
the past forecasting data, for example, the past half-year data before the
decision-making day. Let random variable $P$ represent one of the
environmental parameters, and random variable $\tilde{W}$ denote the wind
power forecasting error, then the Pearson correlation coefficient between
these two variables is given by:

$\rho_{P,\tilde{W}}=\frac{\mathbb{E}\left[(P-\mu_{P})(\tilde{W}-\mu_{\tilde{W}})\right]}{\sigma_{P}\sigma_{\tilde{W}}}$
(5)

where $\sigma_{P}$ and $\sigma_{\tilde{W}}$ are the standard deviation of $P$
and $\tilde{W}$, while $\mu_{P}$ and $\mu_{\tilde{W}}$ are the mean value of
$P$ and $\tilde{W}$, respectively. After calculating the correlation
coefficient of each parameter, we combine the penalized parameters as the
indicator vector for estimating the difference in environmental conditions.
Suppose $\rho_{1},\rho_{2},\rho_{3},\rho_{4}$ are the correlation coefficients
of the four normalized parameters $p_{1},p_{2},p_{3},p_{4}$ that affect the
past forecasting error respectively, then the indicator vector $v$ is
constructed as:

$v=[\rho_{1}p_{1},\rho_{2}p_{2},\rho_{3}p_{3},\rho_{4}p_{4}]^{T}$ (6)

Based on the indicator vector, we can define the wind power forecasting
environment difference $d_{ij}$ as the distance between two forecasting
environments $i$ and $j$, that is:

$d_{ij}=\|v_{i}-v_{j}\|$ (7)

where $\|\cdot\|$ can be any norm distance and Euclidean norm is used in the
case study part.
After calculating the environment difference $d_{ij}$ between the now and the
past, we can pick scenarios from the empirical data under more similar
decision-making environments. The overview of the proposed scheme and the
conventional scenario approach is drawn in Fig.1.

Figure 1: The comparison of the conventional scenario approach (left) and the
proposed scenario generation model (right)

Except for correlation analysis, many statistical or learning methods can be
used as the similar environment filter represented in Fig.1, such as the
coefficient of determination ($R^{2}$), and other learning-based ranking
algorithms [38]. Furthermore, some generative models, such as generative
adversarial networks (GAN), are another path to get the scenarios under a
similar environment. In this paper, we do not compare the accuracy of
different filters but rather focus on illustrating this scheme and its
performance on the economic dispatch problem.

###  III-D Main Result II: Incremental Scenario Optimization

To find the proper number of scenarios meeting the risk requirement, two main
scenario-based algorithms exist for convex problems. The simple one is called
a-priori approach [11], where Helly’s dimension $h$ is approximated by its
upper bound: the number of decision variables $n$. In a-priori approach, the
number of the needed scenarios can be directly given by Theorem 1 before
solving the optimization program. The scenario sampling from a similar
environment embedded in a-priori scenario optimization approach is summarized
in the Algorithm 2.
The a-priori approach has less computational complexity and risk guarantee
before solving the optimization problem. When the number of decision variables
is small, the a-priori approach is an efficient choice, but as the size of the
system increases it leads to extreme conservatism. For instance, the sample
complexity of the look-ahead economic dispatch problem [16] is $3-5$ while the
decision variables $n$ is 864.

Algorithm 2 A-priori scenarios approach sampling from similar environment

INPUT1: The past forecasting error data with environment parameters
INPUT2: The environment parameter at decision-making time INPUT3: The risk and
confidence level $\epsilon$ and $\beta$

$N\leftarrow(\ref{ExactSP})$

if $N>$ empirical data size then

Reset risk and confidence level

else sampling the scenarios $\mathcal{N}$ under similar environments

$d_{ij}\leftarrow(\ref{correlation1})(\ref{correlation2})(\ref{correlation3})$

$\mathcal{N}\leftarrow$ the scenarios with the $N$ smallest $d_{ij}$

end if

Solve the scenario-based optimization problem $\mathrm{SP}(\mathcal{N})$ with
optimal solution $(g^{*},\eta^{*})$.

OUTPUT: The optimal solution $(g^{*},\eta^{*})$

Leveraging on support scenario searching algorithms, such as Algorithm 1, the
a-priori approach can be improved if the risk level is updated after solving
the optimization problem, called the a-posteriori approach [21]. In many
practical engineering situations, the decision makers are interested in
seeking the optimal solution given a certain risk level, implying that if the
updated risk level is too conservative, the decision maker trades off risks
for better performance by discarding some scenarios [26]. The computation
complexity of this risk-tuning process can be reduced by using an incremental
scenario optimization algorithm [27]. In this paper, we propose an improved
incremental scenario optimization algorithm based on sampling from a similar
environment, see Algorithm 3. In practice, this algorithm can generate the
optimal solution with the given risk guarantee and provide some higher-risk
solutions with better economic performance to meet the risk-tuning needs of
system operators.

Algorithm 3 Incremental scenario optimization sampling from similar
environment

INPUT1: The past forecasting error data with environment parameters
INPUT2: The environment parameter at decision-making time INPUT3: The risk and
confidence level $\epsilon$ and $\beta$

1:Set $j:=1$ and $N_{0}=0$

2:Suppose the problem has $j$ support scenarios, i.e. $h=j$, and calculate the
number of needed scenarios $N_{j}$ from (4).

3:Collect a sample of scenarios
$\tilde{w}_{N_{j-1}+1},\tilde{w}_{N_{j-1}+2},...,\tilde{w}_{N_{j}}$ with
$N_{j}-N_{j-1}$ smallest environment difference $d$ without replacement.

4:Solve the scenario-based optimization problem $\mathrm{SP}(\mathcal{N}_{j})$
with optimal solution $(g^{*},\eta^{*})_{N_{j}}$.

5:Compute the exact sample complexity $h_{j}$ of the solution
$(g^{*},\eta^{*})_{N_{j}}$ based on Algorithm 1.

6:(Optional) Compute the exact risk level $\epsilon_{j}$ corresponding to
$(g^{*},\eta^{*})_{N_{j}}$ after updating $h=h_{j}$ in (4).

7:if $h_{j}\leq j$ then

8: halt the algorithm and return $(g^{*},\eta^{*}):=(g^{*},\eta^{*})_{N_{j}}$

9:else set $j:=j+1$ and GOTO step 2

10:end if

OUTPUT1: The optimal solution $(g^{*},\eta^{*})$.
OUTPUT2: (Optional) The solution $(g^{*},\eta^{*})_{N_{j}}$ with higher risk
level $\epsilon_{j}$.

In the previous sample and discard risk tuning method [26], the sample
complexity is first supposed to be its upper bound, i.e. the number of
decision variables, which means a large sample size may be used to solve the
problem. After calculating the true sample complexity and risk level based on
the solution, the decision-makers can trade off the risk and performance by
gradually discarding some support scenarios. Compared to the incremental
optimization method, the sample and discard method tunes the risk in a
decremented way, which is inefficient when the true sample complexity is much
smaller than the number of decision variables.

##  IV Discussion

The most important assumption in the scenario approach is the samples are
identical independent distributed (i.i.d.) random variables. In this section,
we will discuss how the proposed method meets this i.i.d. property and our
weakness.

###  IV-A Identical Distribution

In the chance-constrained problem, the uncertainties are modeled as random
variables sampled from the identical probability distribution. For example,
the uncertainty of wind forecasting error in chance-constrained DC-OPF problem
(1) is sampled from the fixed probability distribution
$\mathbb{P}_{\tilde{w}}$. Because each forecasting happens in a complex and
varying environment, the precise description of uncertainty should be under
the environment condition, i.e.
$\mathbb{P}_{\tilde{w}|\mathcal{V}_{\text{Now}}}$. For data-driven
optimization methods, if we apply the whole empirical data to describe
uncertainty, the identical probability distribution $\mathbb{P}_{\tilde{w}}$
should be regarded as the marginal distribution over the (past) environment
parameters.
Existing papers focusing on modeling the gap between the solution under
empirical distribution $\mathbb{P}_{\tilde{w}}$ and exact distribution
$\mathbb{P}_{\tilde{w}|\mathcal{V}_{\text{Now}}}$ require prior knowledge of
the distance or mean value between two probability measures, which is hard to
calculate in the wind forecasting scenario. Some machine learning approaches
[39] may be useful to approximate this exact distribution, but their results
are not compatible with the direct data-driven program, especially the
scenario approach.
In this paper, instead of quantifying the gap between empirical distribution
and exact distribution, we seek to filter empirical data through the similar
environment parameter space $\mathcal{V}_{\mathcal{N}}$. The proposed
solution’s risk guarantee is based on the conditional distribution
$\mathbb{P}_{\tilde{w}|\mathcal{V}_{\mathcal{N}}}$, but the simulations in the
next section show that the testing results also meet the setting risk
threshold even in the real-world data.

###  IV-B Independent Random Variables

The purpose of sampling scenarios from parameter space is to make the
conditional distribution $\mathbb{P}_{\tilde{w}|\mathcal{V}_{\mathcal{N}}}$
much closer to the exact one
$\mathbb{P}_{\tilde{w}|\mathcal{V}_{\text{Now}}}$, which do not affect the
independent property of scenarios under the two assumptions. The first is that
the forecasting algorithm does not use the previous forecasting error data,
which is true for most updated forecasting methods [34]. The second assumption
is that the dispatch decision itself will not affect future forecasting
errors, which is also justifiable because the dispatch decision does not
interact with the whole weather system in the short term.

##  V Case Study

The scenario approach formulation (4) has been validated in the chance-
constrained economic dispatch many times [5, 15, 16]. The scenarios in these
previous papers were all obtained by sampling from some particular
distribution, such as normal and beta distribution, which didn’t utilize and
exploit the distribution-free advantage of the original scenario theory.
In our simulation, the knowledge about uncertainty is acquired directly
through experience, i.e. the past recorded data. We focus on the economic
dispatch under the 5-min unit, where the 5-min ahead wind forecast uncertainty
plays an important part. The peak hours 16:00-18:00 in August 2022 is selected
as the testing  period with 744 dispatch intervals. For each testing interval,
the empirical forecasting error scenarios are generated from the past three or
six months, i.e. a dataset with 25920 or 51840 5-min ahead forecasting errors
with their corresponding environment parameters. All the data is acquired from
the ERCOT website  based on the five wind forecasting regions in Fig.2 [1],
while the weather data is from [40].

Figure 2: The wind forecasting region in ERCOT

All the problems are solved using 64 GB RAM on the Intel XEON-10885M CPU
(2.4GHz). The mathematical models were formulated using YALMIP on Matlab
R2023a and solved using Gurobi v9.5.

###  V-A Conditional Wind Forecasting Error

In this section, we mainly focus on the wind forecasting error affected by the
deterministic wind forecast value, the wind power ramp rate, temperature, and
relative humidity. To illustrate the wind forecasting error under different
wind forecast values, the empirical forecasting error data is extracted from
June to August 2022 in a similar geometric region in Texas. Fig.3. shows the
wind forecasting error density function under different normalized wind
forecasting levels. It is clear that the empirical distribution under high
wind output level ([0.8, 1], blue) is right-skewed to the lower wind output
level ([0, 0.8], green), which means directly using the whole past scenario
may make the decision aggressive under high wind weather. Similar results are
also found in [25, 41].

Figure 3: The scatter plot (left) and density function (right) of forecasting
error under different wind power forecasting value

Similarly, under different wind power ramp rates, the forecasting error has
different patterns. The ramping rate of wind power is an aggregated parameter
reflecting the stationary level of the environment. Typically, the high
absolute wind power ramp rate means the weather data used for wind power
forecasting is less precise than the stable environment, which results in
different forecasting error distributions. After normalizing the ramping rate
to [-1, 1] interval, the distribution of forecasting error conditions on a
high ramp-up rate ([0.6, 1]) from June to August 2022 compared with its
marginal distribution is shown in Fig.4.
Unlike the wind power forecast value and ramp rate, the temperature and
relative humidity have less influence on the wind power forecasting error. The
distribution of wind power forecasting errors in southern Texas during hot and
cold, dry and wet days of 2022 are compared in Fig.5. It is obvious that the
forecasting error under different temperatures or humidity shares a very
similar distribution.

Figure 4: The density function of forecasting error under different wind power
ramping rate

Figure 5: The density function of forecasting error under different
temperatures and relative humidity

To quantify the correlation level between the above four parameters and the
wind power forecasting error, we calculate the Pearson correlation coefficient
between each parameter and the forecasting error in southern Texas based on
the past half year’s data, i.e. February to July 2022. As shown in TABLE
LABEL:coefficient, the wind power forecasting value and changing rate have
more correlation with the forecasting error than the temperature and relative
humility.

TABLE I: The correlation coefficient between wind power forecasting error and some environmental parameters Correlation between forecasting error and | Coefficient
---|---
Wind Power Forecasting Value | 0.50
(Absolute) Wind Power Changing Rate | -0.17
Temperature | -0.07
Relative Humility | -0.03

Remark: Due to confidentiality concerns, the chosen four parameters are
publicly available information accessible online. Decision-makers also have
the option to incorporate locally measured data, including wind speed and
direction, air pressure, and freezing level, to enhance the resolution of
their analysis. By combining these diverse meteorological parameters, it
becomes possible to delineate specific scenarios for various extreme weather
events based on their respective thresholds.
After the above calculation, the scenarios under similar environments can be
generated by Algorithm 2 or 3, where the scenarios are selected from a smaller
parameter space rather than the conventional method whose scenarios are
randomly extracted from the whole past data. In other words, the distribution
of empirical scenarios is adaptive changing with the environment, while it is
fixed in the conventional method.

Figure 6: The future error values and the scenarios’ empirical distribution
(conventional method)

Supposing 200 scenarios are needed to meet the risk requirement, Fig.6 shows
some real future error values and the scenarios’ empirical distribution in the
whole last year, while the proposed method gives the adaptive empirical
distribution for each decision-making time in Fig.7.
The advantages of sampling from similar parameter space are obvious from the
above simulation. First, the distribution of scenarios will be compressed into
narrower intervals giving a higher-resolution description of the future
uncertainty. Second, the future error values are also covered by this
empirical distribution, which somewhat validates our method’s correctness.

Figure 7: The future error values and the scenarios’ empirical distribution
(proposed method)

###  V-B 24-bus System

The 24-bus System is modified from the IEEE Reliability Test System (RTS-24)
[42] with an additional six wind farms to mimic the high renewable energy
penetration as illustrated in Fig.8. The detailed information, including
generator parameters, reactance and capacity of transmission lines, and the
load profile can be found in [43]. The forecasting profile of wind generators
located at bus 3, 5 and 7 are directly scaled from the west Texas region wind
forecasting results, while the wind farms at bus 16, 21 and 23 are from the
south Texas region. Each wind farm is assumed with 400MW capacity and a low
marginal price (3 $/MWh).
As suggested in [44], the capacity on the transmission lines connecting the
node pairs (15, 21), (14, 16), and (13, 23) is reduced to 400MW, 250MW, and
250MW, respectively. This is done to introduce bottlenecks or congestion in
this high wind penetration system, which shares a similar situation when comes
to the real power grids [45].
The number of decision variables $n$ in the 24-bus system is 22 after
eliminating the equation constraints. In this case, a rough approximation of
Helly’s dimension $h$ by $n$ will increase the needed number of scenarios to
meet the risk threshold, which results in great conservatism of the final
decision. To illustrate this, Fig.9 shows the relationship between the needed
number of scenarios and sample complexity under the risk $\epsilon=0.05$ and
confidence parameter $\beta=10^{-3}$. It can be seen that when the sample
complexity is 22, the needed number of scenarios is 779, which is much larger
than the 324 needed scenarios under 6 sample complexity.

Figure 8: Modified 24-bus power system integrated with wind Figure 9: The
relationship between the needed number of scenarios and sample complexity

Instead of using the a-prior approach, we apply Algorithm 3 incrementally
tuning risk for finding the exact sample complexity and the needed number of
scenarios. Because the wind output and demand are changing, the sample
complexity needs to be updated for each dispatch interval. Fig.10 shows the
empirical distribution of the sample complexity of the 744 dispatch intervals
during August 16:00-18:00 we studied, where the average sample complexity
(5.8) is much smaller than the number of decision variables (22). This
evidence implies that in the previous sample and discard risk-tuning method
[16], where the sample complexity is first supposed to be the number of
decision variables 22, the decision makers may need to discard some redundant
scenarios to achieve the same performance as our incremental method.

Figure 10: The density function of the sample complexity over the 744 dispatch
intervals TABLE II: Comparison of the risk tuning process

Risk Tuning Method | Sample and Discard [26] | Incremental Optimization
---|---|---
Initial Input Sample Size | 779 | 135
Intial Risk Level | 0.021 | 0.117
Final Input Sample Size | 771 | 324
Iteration Times | 8 | 6

TABLE II compares the initial and final input sample size of these two risk-
tuning methods when sample complexity is 6. Compared with sample and discard,
incremental optimization guarantees the minimum input sample size with fewer
iteration times, which is beneficial in the situation of limited high-accuracy
data. Fig.11 shows more details about the risk-tuning process of these two
methods. The incremental optimization method’s efficiency results in fewer
iteration times and less sample size in each iteration.
Remark: The final input sample size in the sample and discard method is much
larger than the incremental method because only support scenarios are
discarded in [26] but not randomly, and this risk bound is proved to be not
tight in recent research [46].

Figure 11: The risk tuning details of two different methods

So far, the simulation results have provided insights into the specifics of
scenario generation and risk-tuning processes. Subsequent simulations will
focus on examining the cost and actual violation outcomes following the
scenario-based optimization model (3).
After setting the same risk and confidence parameter as Fig.9, we can input
different size of scenarios to meet the risk requirement based on the sample
complexity of each dispatch interval, which is more efficient than the
traditional sample and discarding method. Fig.12 illustrates the adaptive
input sample size (green triangle) between 16:00 and 18:00 on August 2nd,
2022, where the dispatch cost of sampling from different parameter spaces is
also compared 333The dispatch cost is calculated after each dispatch with true
wind output data, and the searching space of a similar environment is set to
the past 3 months.. During these two hours, the average wind power output is
23% of the total wind generation capacity, which is a relatively high wind
period during August rush hours. It is clear that sampling from a similar
environment space results in lower costs in each dispatch interval than
others.

Figure 12: The dispatch cost by sampling from different parameter space and
input sample size between 16:00 and 18:00 on August 2nd, 2022 (24-bus system,
high wind day)

It needs to be clarified that because of the stochastic property, sampling
from a similar environment does not ensure a lower cost in each dispatch
interval, especially on a low wind day. For example, the average wind power
output is only 13% of the total wind generation capacity between 16:00 and
18:00 on August 30th, which makes the economic benefits of sampling from a
similar environment not comparable with the others (Fig.13).

Figure 13: The dispatch cost by sampling from different parameter space and
input sample size between 16:00 and 18:00 on August 30th, 2022 (24-bus system,
low wind day)

However, sampling from a similar environment builds a more precise model of
the uncertainty variables, which results in a more trustworthy risk guarantee
of the final solution. TABLE.III compares the testing risk and the average
cost of each 5-min dispatch interval by sampling from different parameter
spaces during the rush hours 16:00 to 18:00 in August, where sampling from
similar environment parameter space results in both lower testing violations
and less dispatch cost. Meanwhile, when randomly sampling from the past half
year, the testing violation (0.054) even exceeds the setting risk (0.05),
which invalidates the risk-guarantee property of the scenario approach.

TABLE III: The violation and average dispatch cost of sampling from different parameter spaces (24-bus system) | Sampling Space | Past Half Year | Past Three Months | Similar Environment
---|---|---|---|---
Whole August | Setting Violation | 0.05 | 0.05 | 0.05
Actual Violation | 0.054 | 0.040 | 0.036
Average Cost($10^{4}\$$) | 4.7522 | 4.7512 | 4.7444
Average Solving Time(s) | 0.181 | 0.166 | 0.165
Average Sampling Time(s) | 0.015 | 0.014 | 0.020
2nd August (high wind) | Average Cost($10^{4}\$$) | 4.2257 | 4.2154 | 4.1956
30th August (low wind) | Average Cost($10^{4}\$$) | 4.6414 | 4.6430 | 4.6433
TABLE IV: The violation and average dispatch cost of sampling from different parameter spaces (118-bus system) | Sampling Space | Past Half Year | Past Three Months | Similar Environment
---|---|---|---|---
Whole August | Setting Violation | 0.05 | 0.05 | 0.05
Actual Violation | 0.042 | 0.035 | 0.035
Average Cost($10^{5}\$$) | 1.7445 | 1.7454 | 1.7414
Average Solving Time(s) | 13.66 | 13.41 | 13.30
Average Sampling Time(s) | 0.022 | 0.021 | 0.028
2nd August (high wind) | Average Cost($10^{5}\$$) | 1.5294 | 1.5435 | 1.5173
30th August (low wind) | Average Cost($10^{5}\$$) | 1.7251 | 1.7255 | 1.7271

###  V-C 118-bus System

We apply our method to a larger system in this section, based on the test case
c118swf.m in MATPOWER [47]. This system includes 118 nodes, 210 lines, and 52
generators, 11 of which are modeled as wind farms. To address the influence of
wind uncertainty on the economic dispatch, we replace the 4 storage units with
the same capacity wind farms. Meanwhile, the transmission line capacities are
set to be 60% of the original value to introduce more congestion in the
system.
Similar to the simulation in the 24-bus system, the wind power forecasting and
real value are directly modeled by the data from ERCOT with 7 wind generators
from south Texas and 8 wind generators from west Texas. Fig.14 shows the
adaptive input sample size and the dispatch cost of sampling from different
parameter spaces between 16:00 and 18:00 on August 2nd, 2022. In this period,
the average wind power output accounts for 24% of the total wind generation
capacity, a relatively high wind output level during August peak hours.
Benefiting from the smaller uncertainty space (see Fig.6, Fig.7), sampling
from a similar environment helps the decision maker avoid some odd scenarios
when applying scenario approach in the real world. These odd scenarios
typically make the final decision more conservative, which can be illustrated
by the cost spikes in Fig.14 when we directly sample scenarios from the past.

Figure 14: The dispatch cost by sampling from different parameter space and
input sample size between 16:00 and 18:00 on August 2nd, 2022 (118-bus system,
high wind day)

TABLE.IV lists the testing risk and the average cost of 5-min dispatch during
August peak hours (16:00-18:00), where the result has a similar pattern with
the 24-bus system. The 5-min short-term wind power forecasting error is
relatively lower than other long-term forecasting errors, but a more efficient
scenarios generation model will make the chance-constrained dispatch solution
more trustworthy.
Although there’s no theory guarantee for a lower cost solution by sampling
from a similar environment, the simulation shows the economic advantage of the
proposed efficient scenarios generation model, especially during high wind
situations. To quantify the relationship between the accuracy of the scenario
generation model and economic benefits will be one possible direction of
future research.
Regardless of the selection of sample space, the number of needed scenarios is
the same, which results in a similar problem-solving time of three different
sample spaces in both TABLE.III and TABLE.IV. Meanwhile, computing indicator
vector (7) of similar environments increases the sampling time, but it’s far
less than the solving time by one or more orders of magnitude.

##  VI Conclusion

This paper studies the main two barriers to applying the scenario approach to
the economic dispatch with high penetration of renewable resources, i.e. lack
of accurate scenario generation models and inefficient risk tuning process.
Leveraging correlation analysis, we generate scenarios via an environment
filter with empirical distribution closer to the true probability measure.
After embedding this scenario generation model with the incremental scenario
optimization algorithm, we propose an efficient risk-tuning scheme, which can
solve the optimal solution meeting risk requirement with minimum data size and
provide other higher-risk solutions to system operators in meanwhile. Case
studies based on real-world wind data and modified IEEE benchmark systems show
the effectiveness and advantages of our methods.
Directly generating scenarios from past experience may be a naive approach,
but it works well in practice when the needed data size is much smaller than
the size of the past data pool. Future work includes (1) comparing the results
of using other scenario generation methods, such as the generative model in
the field of machine learning; (2) extending the economic dispatch problem to
a multi-stage framework, i.e. look-ahead economic dispatch, where varying risk
tolerance levels are allowed across different time horizons to enhance the
adaptability; and (3) applying the proposed scheme to unit commitment and
other non-convex decision-making processes in electric power systems,  which
requires efficient handling of discrete variables under incremental scenarios
to ensure timely solutions.

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|  Qian Zhang (Student Member, IEEE) received his B.E. and M.S. degrees in
Electrical Engineering from Zhejiang University, Hangzhou, China, in 2019 and
2022 respectively. He is currently working toward the Ph.D. degree at Texas
A&M University, College Station, TX, USA. His research interests include
machine learning, optimization in the electricity market, and power system
stability and control.
---|---
|  Apurv Shukla (Member, IEEE) received his bachelor’s degree in Manufacturing
Science and Engineering from IIT Kharagpur in 2016 and his Ph.D. degree in
Operations Research from Columbia University in 2022. He is a Postdoctoral
Associate at Texas A&M University. His research interest lies in systems and
control.
---|---
|  Le Xie (Fellow, IEEE) received the B.E. degree in electrical engineering
from Tsinghua University, Beijing, China, in 2004, the M.S. degree in
engineering sciences from Harvard University, Cambridge, MA, USA, in 2005, and
the Ph.D. degree from the Department of Electrical and Computer Engineering,
Carnegie Mellon University, Pittsburgh, PA, USA, in 2009. He is currently a
Professor with the Department of Electrical and Computer Engineering, Texas
A&M University, College Station, TX, USA. His research interests include
modeling and control of large-scale complex systems, smart grid applications
with renewable energy resources, and electricity markets.
---|---