# Application of Stochastic Optimization Techniques to the Unit Commitment
Problem — A Review

Vincent Meilinger  Technische Universität Berlin
Berlin, Germany
<EMAIL_ADDRESS>
###### Abstract

Due to the established energy production methods contribution to the climate
crisis, renewable energy is to replace a substantial part of coal or nuclear
plants to prevent greenhouse gases or toxic waste entering the atmosphere.
This relatively quick shift in energy production, primarily pushed by
increasing political and economical pressure, requires enormous effort on the
part of the energy providers to balance out production fluctuations.
Consequently, a lot of research is conducted in the key area of stochastic
unit commitment (UC) on electrical grids and microgrids. The term unit
commitment includes a large variety of optimization techniques, and in this
paper we will review recent developments in this area. We start by giving an
overview over different problem definitions and stochastic optimization
procedures, to then assess recent contributions to this topic. We therefore
compare the proposals and case studies of several papers.

###### Index Terms:

unit commitment, stochastic optimization, microgrids, energy generation
planning, uncertainty

##  I Introduction

Since the advent of renewable energy sources the need for scheduling power
production and energy distribution has become more important than ever. The
energy market is shifting from inflexible power production with coal or
nuclear power plants to the highly fluctuant wind or solar energy plants that
require more flexible control mechanisms [1]. Especially grid balancing, which
includes controlling available generation units and prediction of energy
production and load, is an increasingly difficult task, due to the stochastic
nature of renewable resources [2]. The process used by power generation
companies to create predictions about the energy production needs days or
weeks ahead of time is called unit commitment (UC) [1]. The actual real time
allocation of units minutes or hours before is referred to as economic
dispatch (ED). The focus of this review lies on the prior research topic.

UC decisions define, for each time step over a scheduling horizon, which
generation unit should be used, while accounting for market clearing,
reliability assessment, intra-day operations, expected load and operational
policies [1], [3]. Energy grids mainly comprise multiple distributed
generation resources, electric storage resources and loads. [4]. Resource
generation units can generally divided into several groups: thermal generation
resources (TGR) like coal or nuclear, renewable resources (RR) like wind and
solar, and resource storage, like batteries or gas storage. There are many
groups of optimization algorithms that try to solve the complex production and
demand balancing. [1] Since many parameters of energy production are
uncertain, especially with wind and solar, stochastic optimization methods are
a popular choice to optimize for cost, reliability, equipment failure and more
[5].

UC therefore represents a optimization problem under constraints and
uncertainty. UC is proven to be an NP-hard problem [1]. In this review we
compare three approaches to solve the UC problem: stochastic programming (SP),
stochastic dynamic programming (SDP) and distributionally robust optimization
(DRO). The paper is structured as follows.
In Section II we define the basic objective of SO problems in UC. We summarize
various approaches to solve UC. Thereafter, in Sections III, IV and V, we
review different DRO, SP and SDP models, respectively. To evaluate
performance, the case studies are analyzed. Section VI the advantages and
disadvantages of the optimization techniques are evaluated. A short discussion
regarding future development is held in Section VII, with a focus on
incorporating green house gas emissions in UC.

In the next section we give an overview over the different UC modeling and
solving approaches.

##  II Stochastic optimization problems

The term ’stochastic optimization’ includes a variety of optimization
techniques. When applied to UC for power grids, the problem can be defined as
an optimization of an objective function under a set of constraints [2].
Underlying the function parameters are probability distributions, which in
case of stochastic programming are assumed to be known prior to realization
[3].

To formalize the problem, we need to give an overview of suitable methods to
model the uncertainty of the problem first. UC models are often defined as
two-stage models. The first stage contains commitment decisions u, while
dispatch decisions are incorporated in the second stage after the uncertain
parameters are realized [5]. In two stage models the objective function is
often quadratic or piecewise linear [5]. Uncertainty is realized only once
[1].

Multistage models also exist, where commitment decisions are modeled as nodes
of a scenario tree [5]. Uncertainty can therefore be realized over time,
decisions can hence be adjusted dynamically [1]. Therefore, these models
define the uncertain future as a scenario tree that includes many possible
simulated outcomes [1]. This method is common for stochastic programming
solutions. The scenario tree can be searched using Monte Carlo simulations
[2]. A disadvantage of this technique is the exponentially increasing
computational complexity when growing the number of modeled scenarios [2]. In
[5], algorithms used to cope with these enormous scenario trees are listed,
among them for example Benders decomposition, Lagrangian relaxation, dual
decomposition or Progressive hedging as used in [6]. The latter approach will
be explored in more detail later on. It is also possible to use uncertainty
sets or probabilistic constraints. Uncertainty sets are often used to model
uncertainty for robust optimization models [1]. Confidence intervals are the
most basic uncertainty sets, they are defined by the sample mean and quantiles
of an according distribution [7]. A general definition of some SP solving
approaches can be found in [1]. The authors show that stochastic programming
models often take the form

$\underset{\mathbf{u}\in\mathbb{U}}{\text{min}}\;\mathbf{c}^{\intercal}\mathbf{u}\;+\;E_{\xi}[F(\mathbf{u},\xi)]$
(1)

where $\mathbf{u}$ are the commitment decisions of slow reacting power units
like coal and nuclear and $\mathbb{U}$ is the set of feasible commitment
decisions, startup and shutdown costs are represented by the vector
$\mathbf{c}$ and $\xi$ is a vector of uncertain parameters drawn from a known
probability distribution [1]. The first term therefore models day-ahead
planning, which includes rather unresponsive commitment decisions e.g. of TGR
generators like coal and nuclear, while the second term models real time
decisions [1]. The latter ED phase is needed to adjust for fluctuations of
renewable energy production or outages. While stochastic programming models
have the advantage of cost saving and reliability improvements when compared
to simply using reserve constraints [1], a major drawback is the assumption
that the distribution of the data points used for the model is known a priori
[3]. In realistic scenarios, this is often not the case.

Another wide spread technology is robust optimization (RO) [7]. Robust
optimization models uncertainty through uncertainty sets. This method bases
its decisions on the worst case outcome, its predictions are therefore rather
conservative [7] [3]. The general robust optimization problem is defined as

$\underset{\mathbf{u}\in\mathbb{U}}{\text{min}}\left\\{\mathbf{c}^{\intercal}\mathbf{u}\;+\;\underset{\mathbf{v}\in\mathbb{V}}{\text{max}}[F(\mathbf{u},\mathbf{v})]\right\\}$
(2)

where $\mathbf{u}$ and $\mathbb{U}$ are defined analogous to (1), $\mathbf{v}$
again represents the uncertainty parameter with deterministic uncertainty set
$\mathbb{V}$ [1]. The function $F(\mathbf{u},\mathbf{v})$ models the real-time
dispatch cost for the decisions $\mathbf{u}$ and an uncertain variable
$\mathbf{v}$ [1]. The difference between the vectors $\mathbf{v}$ of SO and RO
is that while for SO the vector represents a vector of uncertain parameters,
for RO it is a variable vector. For the latter, $\mathbf{v}$ needs to be
adjusted to minimize the worst scenarios total cost [1].

In both SP and RO models second stage problem, function $F$ represents a
minimization task. The problem definition is very similar in both methods, as
shown in [1], where the general SP function is defined as

$\displaystyle
F(\mathbf{u},s)=\underset{\textbf{p}_{s},\textbf{f}_{s}}{\min}\;f(\textbf{p}_{s})$
(3) $\displaystyle\text{s.t.
}A_{\textbf{s}}\textbf{u}+B_{\textbf{s}}\textbf{p}_{s}+H_{\textbf{s}}\textbf{f}_{s}\geq\textbf{d}_{\textbf{s}}$
(4)

for each realization $s$ of random vector $\xi$, and the RO function as

$\displaystyle F(\mathbf{u},\mathbf{v})=$ $\displaystyle\underset{\textbf{p,
f}}{\min}\;\textbf{q}^{\intercal}\textbf{p}$ (5) $\displaystyle\text{s.t.
}A_{\textbf{v}}\textbf{u}+B_{\textbf{v}}\textbf{p}+H_{\textbf{v}}\textbf{f}\geq\textbf{d}_{\textbf{v}}$
(6)

In (3) Matrices $A_{s},B_{s},H_{s}$ usually model contingencies, while
$\textbf{d}_{s}$ models uncertain demand and renewable energy outputs [1].
Vectors $\textbf{p}_{s},\textbf{f}_{s}$ model dispatches and reserves of
multiple periods and other second stage decisions, respectively [1]. Function
$f$ models fuel cost and is usually quadratic and non-convex. The variables
and parameters are similarly defined in (5), with the only difference that the
matrices and vectors are functions of the uncertain parameters, and the
objective function uses piecewise linear approximation with coefficient vector
q to approximate the original SO cost function $f$ [1].

A newer optimization method, the distributionally robust optimization (DRO)
could solve the problems of RO and SO [3], [7]. DRO takes into account the
shortcomings of RO, in that it considers the uncertainty of the parameter
probability distribution [3]. The probability distributions here belong to
ambiguity sets that are constructed from historical data using methods like
Kullback-Leibler divergence, moment information or the Wasserstein distance
[3].

SDP solves the problem by breaking it down into a sequence of steps over time
[2]. In [1], a general finite-horizon, discrete-time SDP framework is defined
as follows:

$\underset{\pi\in\Pi}{\text{inf}}\;V_{\pi}(s_{0}):=\mathbb{E}\left[\underset{t=0}{\overset{T-1}{\sum}}C_{t}(s_{t},\mu_{t}(s_{t}),\xi_{t})+C_{T}(u_{T})\right].$
(7)

Here, $\xi_{t}$ are random variables, $C_{t}(\cdot)$ models system cost at
time period $t$, and $V_{\pi}$ is referred to as the value function. $s_{t}$
is the state of a system at time period $t$, $u_{t}$ are the commitment and
dispatch decisions according to policy function $\mu(\cdot)$ which maps from
system state $s_{t}$ At each time step, the best route is found based on the
possible optimum subsequences in the previous step [2]. The main advantage of
dynamic programming (DP) is that it can maintain the feasibility of the
solution, as it is able to find the optimal subsequence while searching for
the optimal sequence [2].

In the following sections we will summarize several papers using the
previously listed methods.

##  III Distributionally robust optimization

In [3], the authors present a distributionally robust optimization (DRO)
technique that minimizes the expected cost from the worst case scenario over
an ambiguity set of probability distributions constructed by Kullback-Leibler
(KL) divergence.

Kullback-Leibler divergence is a measure of distance between probability
distributions [7]. The nominal distribution $P_{0}$ can be constructed using
parametric methods like maximum likelihood estimators or non-parametric
methods. The latter are often used to handle data driven problems, because
they do not assume anything about the underlying datapoint distribution [7].

The DRO approach is applied to microgrids and incorporates the evaluation of
microgrid net load uncertainty and electricity market prices [3]. The proposed
approach is modeled as a two-stage problem, where the first stage determines
the binary commitment and startup variables of the thermal generation
resources (TGR) over a set period, incorporating typical TGR constraints like
uptime and downtime [3]. The uncertain net load and the electricity price are
not realized yet, since the objective of the first stage is to minimize fixed
costs and worst case expected power costs. The second stage problem minimizes
the power generation and purchase costs. It is constrained by the TGR output
limits, up/-downtime constraints and power balancing. The authors decided to
model the uncertainty through ambiguity sets, which is a common choice for RO
models [1].

The ambiguity set $\mathbf{P}$ in [3] is constructed with KL divergence
through k-means clustering and is defined as

$\displaystyle\mathbf{P}:=\\{\mathcal{P}:$
$\displaystyle\underset{\omega=1}{\overset{S}{\sum}}\pi^{\omega}_{\mathcal{P}}\log\left(\frac{\pi^{\omega}_{\mathcal{P}}}{\pi^{\omega}_{\mathcal{P}_{o}}}\right)\leq\rho,$
(8)
$\displaystyle\underset{\omega=1}{\overset{S}{\sum}}\pi^{\omega}_{\mathcal{P}}=1,$
(9) $\displaystyle\pi^{\omega}_{\mathcal{P}}\geq
0\;\;\;\forall\omega\in\Omega\\}.$ (10)

Divergence tolerance $\rho$ is used to adjust the ambiguity set size and
therefore its conservatism [3]. $\omega$ represents a cluster, cluster
centroids represent realizations $\xi^{\omega}$. Scenario $\omega$ has nominal
probability $\pi^{\omega}_{\mathcal{P}_{o}}$ for the nominal probability
distribution $\mathcal{P}_{o}$. The number of scenarios is denoted as
$\mathcal{S}$. The resulting KL divergence based microgrid UC (KL-MUC) is
reformulated to a convex mixed integer non-linear problem (RKL-MUC) defined as
follows:

$\displaystyle\text{RKL-MUC}:$ (11)
$\displaystyle\underset{x,\mu,\zeta}{\text{minimize}}\;\;\;$
$\displaystyle\;\;c\cdot
x+\mu+\rho\zeta+\zeta\underset{\omega=1}{\overset{\mathcal{S}}{\sum}}\pi^{\omega}_{\mathcal{P}_{o}}e^{\overline{\mathcal{K}}^{\omega}(x,\mu,\zeta)-1}$
(12) $\displaystyle s.t.\;\;\;$ $\displaystyle\;\;x\in\mathbf{X},$ (13)
$\displaystyle\;\;\zeta\geq 0,$ (14)

where $\mathbf{X}$ represents the feasibility region of x defined by the first
stage problem constraints, and
$\overline{\mathcal{K}}^{\omega}(x,\mu,\zeta)=\frac{\mathcal{Q}(x,\xi^{\omega})-\mu}{\zeta}$
[3]. $\mathcal{Q}(x,\xi^{\omega})$ is a function of the uncertain power
generation and purchase cost for variable $x$ and realization $\xi^{\omega}$
of uncertain values [3]. Dual variables $\zeta,\mu$ are are assigned to the
constraints (8) and (9), respectively [3]. Constraint (8) controls the
divergence between $\mathcal{P}_{o}$ and distribution $\mathcal{P}$
incorporated into ambiguity set P based on KL divergence, (9) ensures the sum
of probabilities assigned by $\mathcal{P}$ is one [3]. The RKL-MUC problem is
solved using an algorithm based on Benders decomposition, where it is
decomposed into a lower bounding master problem and an upper bounding
subproblem [3]. The approach enables hedging against adopting misrepresenting
probability distributions [3]. When benchmarked against a SUC problem
equivalent to the formulation of KL microgrid unit commitment problem that is
solved by the L-shaped algorithm, the total cost of RKL-MUC is observed to be
less than or equal to the cost under the SUC formulation [3]. Furthermore,
while both problem formulations give similar results when divergence parameter
$\rho=0$, the total cost of the RKL-MUC solution gets smaller compared to the
SUC solution. The authors state that this result can be traced back to the
fact that it is beneficial to take additional probability distributions other
than the nominal distribution into account [3].

Later in [8], Yurdakul et al. further build on their previous research in
distributionally robust unit commitment under net load uncertainty using KL
divergence. Improvements in the short term planning are achieved by
reformulation of the KL distrubutionally robust unit commitment (KL-DRUC),
obtaining a convex mixed-integer nonlinear reformulated KL-DRUC (RKL-DRUC)
problem where the objective function is defined in a similar way as the
objective function (11) of the previous paper, with the difference of more
constraints incorporated and the focus shifted from microgrids to power
systems under net load uncertainty in general [8]. Using a similar SUC model
as the previous paper, the focus here lies on investigating the influence of
different k-means clustering distance measures, the size of the dataset used
for ambiguity set construction and the influence of different values of
divergence parameter $\rho$ on the results [8]. Furthermore, more constraints
are incorporated into the second stage, namely the load curtailment cost,
ramping constraints [8]. The authors this time present the solution in the
form of a real world data case study, where net load values of the California
System Operator Grid are used. To gain an insight into the impact of $\rho$,
the authors optimize the model for different values and different distance
functions. A increase in total cost can be observed for larger values of
$\rho$ for all tested distance measures. Furthermore, a coupling between
optimal cost and number of datapoints can be noted, in that the total cost
decreases with an increasing count of time-series data points. The research
provided in this paper could aid grid operators with the UC decision process
for grids with an increasing amount of renewable energy generators.

##  IV Stochastic Programming

A classical SP approach can be found in [9]. The authors provide a method to
integrate stochastic solar power into power grids. The fist stage solves the
problem of UC and economic dispatch based on predicted solar power generation.
The second stage handles thermal generation rescheduling when solar power is
generated. In order to cope with overestimation of solar production,
additional reserve purchasing is included in the model. All in all, the model
is constrained by power balance, minimum/maximum generation, operating
reserve, ramp rate, up-/downtime, operating costs and reserve buying/penalty
costs. The objective function to minimize is defined as

$\displaystyle\min\underset{t=1}{\overset{T}{\sum}}(S_{t}+$
$\displaystyle\overline{E}_{S}(O_{s,t}(\tilde{p}_{pv,s,t},\tilde{\mathbf{p}}_{s,t},\textbf{u}_{t}))+$
(15)
$\displaystyle\overline{E}_{S}(\omega_{t}|\tilde{p}_{pv,s,t}-\tilde{P}_{pv,s,t}|)),$
(16)

where $\textbf{u}_{t}$ are the UC decisions of thermal units,
$\tilde{\textbf{p}}_{t},\tilde{p}_{pv,s,t}$ are the scheduling decisions
depending on the actual and stochastic solar power generation.
$\overline{P}_{pv,s,t}$ is a stochastically simulated scenario and decision
variables with subscript $s$ are the corresponding scheduling decisions in a
scenario $s$. $\overline{E}_{S}$ is an approximation to expected value $E$
with S solar power scenarios. $S_{t}$ is the startup cost. The model is tested
on a simulation containing several loads and generators, solar and thermal
[9]. When comparing the proposed method to deterministic or robust
optimization, a significant improvement can be noticed [9]. With all three
methods, the average cost rises with rising solar uncertainty due to over-
scheduling or penalty cost, but the stochastic approach stays relatively
constant in comparison to the other models [9]. Especially the deterministic
model suffers from this uncertainty [9]. The robust model is only slightly
worse than the stochastic model, because its decisions are based on the worst
case scenario and, as mentioned earlier, therefore rather conservative [9].

When using ambiguity sets constructed from observed real world data as in [8],
DRO still has a main advantage over SP methods. SP can potentially suffer from
an inaccurate distribution of uncertain parameters [7]. Of course, even the
distribution of the observed data in the ambiguity set can differ from the
underlying real distribution when measurement errors occur, but in reality the
obtained distribution is often close to the ground truth [7]. The disadvantage
of distance based DRO is the high complexity of the problem, as the objective
function of these problems are often non-linear [7]. To counteract this
problem, in [7], the authors propose a distance based distributionally robust
unit commitment model (DB-DRUC) based on KL divergence, that is reformulated
into a mixed integer non-linear programming problem (RDB-DRUC). The RDB-DRUC
model is scenario based and is defined as

RDB-DRUC: $\displaystyle\text{min
}c^{\intercal}y+\alpha\log\left\\{\underset{s=1}{\overset{S}{\sum}}\rho_{s}e^{\theta_{s}/\alpha}\right\\}+\alpha\eta$
(17) $\displaystyle s.t.\;$
$\displaystyle\;\theta_{s}-q_{s}^{\intercal}x_{s}=0,\forall s=1,2,...,S,$ (18)
$\displaystyle\;Tx_{s}+Wy-h(v_{s})\leq 0,\forall s=1,2,...,S,$ (19)
$\displaystyle\;Ay-b\leq 0,$ (20)
$\displaystyle\;y\in\\{0,1\\}^{\mathcal{M}}.$ (21)

Here, the subscript $s$ represents the scenario index.
$\\{v_{1},\dots,v_{S}\\}$ are uncertainty variables in the scenarios with
their probabilities $\\{\rho_{1},\dots,\rho_{S}\\}$, respectively [7].
$\theta_{s}$ represents the generation cost of all units in scenario $s$,
$y,x$ are the first and second stage decisions of the model [7]. Variable
$\alpha$ is derived from the dual transformation [7]. RDB-DRUC is a large-
scale mixed integer non-linear problem that can be decomposed using
generalized Benders Decomposition [7]. The resulting subproblem can be further
decomposed into a second level master- and subproblem that can be solved
seperately [7]. In general the RDB-DRUC model outperforms the robust UC (RUC)
counterpart, as shown in the case study [7]. As $\eta\rightarrow\infty$, RDB-
DRUC performance is equal to basic RUC, for all other values it is less
conservative [7]. The authors further show that, by parallelizing the
optimization task with four threads, the solution of the second level problem
can be sped up by about $65\%$ [7]. From experiments with real world data it
can be concluded that the model scales well even with high scenario counts
[7].

Parallelization as in [7] is not always easy to achieve. In [6], a stochastic
UC model for electric-gas coupled grids is proposed. Coupled systems often
pose a heavy computational burden for traditional UC methods, since scenario
independent variables can be correlated across scenarios when using algorithms
like Benders Decomposition [6]. For this reason, parallelizing these
algorithms is difficult. The authors therefore apply a modified progressive
hedging (MPH) algorithm to accelerate the SUC optimization performance. The
model for the electricity-gas coupled system includes wind power and gas flow
scenarios. The objective function to minimize is defined as the total cost
over a time horizon $t\in\mathcal{T}$ and scenarios $sc\in\mathcal{SC}$. Gas
system constraints include gas well production limit, storage capacities,
injection and withdrawal limits and storage level variation time [6]. A
significant optimization acceleration is achieved when compared to commercial
algorithms through uniform decomposition of the stochastic model with the
modified progressive hedging algorithm, which can run on parallel hardware
[6]. In the case study, up to four workers are used in parallel [6].
Furthermore, instead of piecewise linear function approximation, second-order
cone relaxation is applied to model gas flow [6]. These changes are tested in
the case study. The MPH algorithm is compared to several other optimization
techniques, the most interesting for out purpose being Deterministic UC and
SUC solved using Benders decomposition. While deterministic UC only needs a
short amount of time to converge to a solution, the expected cost is in some
cases more then ten times higher than with the other approaches [6]. The
Benders decomposition SUC approach converges to a similar expected cost as the
MPH SUC model, but it is outperformed by MPH with computational speeds being
up to $17$ times faster in some cases due to the ability to be run in parallel
[6].

##  V Stochastic Dynamic Programming

In an attempt to improve UC modeling under renewable generation uncertainty
through SDP, the authors in [10] propose a SDP based reformulation of multi-
stage UC. To optimize the model, an algorithm called Stochastic Dual Dynamic
integer Programming (SDDiP) is used. SDDiP is a sampling based algorithm and
is scalable to even large-scale scenario trees [10]. The authors propose
several approaches to improve performance of SDDiP, a leveling method for
Lagrangian cuts, a hybrid model approach to faster approximate the expected
cost-to-go function in each stage and backwards parallelization to compute
multiple scenario problems at once [10]. After all the applied performance
enhancements, the authors are able to show that the proposed algorithm handles
multistage UC problems with a large number of scenarios [10]. All of the
proposed improvements on the SDDiP algorithm are tested separately [10].
Applying all those performance improvements and running the algorithm on 32
threads ultimately results in a speedup ratio of $4.8$ and an efficiency of
$15\%$ on average [10].

Instead of trying to handle a large amount of scenarios through advanced
algorithms, there is also the option to use probabilistic constraints instead.
Nguyen et al. propose in [2] a new battery operating cost model that considers
battery cycle life and charge and discharge efficiencies of batteries. To
relax hard power balancing constraints, a probabilistic constraint is used
that incorporates all random variables of renewable energy generation and
loads [2]. One of the main achievements of the authors is to make economic
dispatch possible for several batteries without introducing more objective
functions to solve to the problem [2]. The main advantage of the proposed
method is that it accurately computes the actual performance of the energy
storage system performance, so resources can be managed better [2]. The SUC is
defined to minimize expected operation cost of a microgrid over a time horizon
[2]. The objective function is

$\displaystyle\min\;\;$ $\displaystyle
C=\underset{k=1}{\overset{N}{\sum}}(F_{k}+S_{k})$ (22) $\displaystyle
F_{k}=\mathbb{E}\\{F_{g,k}+F_{b,k}^{d}+F_{b,c}^{c}+F_{m,k}\\}$ (23)

where $N$ is the time horizon, $T$ the time step, $F_{k}$ the total expected
operation cost in period $k$, $S_{k}$ the total transition cost which accounts
for the start-up and shut-down cost of the generators in period $k$ [2].
$F_{g,k}$ is the total operation cost of generators during period $k$,
$F_{b,k}^{d}$ represents (dis-) charging batteries, $F_{m,k}$ is the cost due
to power mismatch [2]. The developed case study models a rural or developing
nation microgrid which consists of batteries and wind, photovoltaic and diesel
generators [2]. The study shows that the stochastic model can handle different
load and generation scenarios well [2]. It better models the performance of
energy storage systems and therefore better allocates resources that a
deterministic model [2]. It is further shown that batteries provide a more
cost effective choice than diesel generators, therefore increasing the energy
storage system size can provide economic advantages [2].

##  VI Optimization paradigm comparison

In real world use cases of electricity grid balancing, RO is still the most
widely used method of optimization [1]. This mostly comes down to the
simplicity of implementation and intuitiveness of this approach. While SP and
SDP show promising results, in some cases outperforming RO [9], large scenario
trees and therefore long optimization times hinder wide spread adoption of
these models. In addition, the complexity of the applied hedging algorithms to
cope with these large scenario trees, as in [10] and [7], can be discouraging
to power generation companies. DRO here has an advantage over the stated
methods. It builds on the well researched foundations of RO, but provides more
realistic results due to the use of ambiguity sets [1]. The conservativeness
can be controlled to some extent, as shown in [8], and DRO solutions do not
suffer as much from inaccurate distributions of uncertain parameters as SP and
SDP methods [7]. DRO further benefits from the fact that that the underlying
probability distributions of the model parameters do not need to be determined
a priori [8].

##  VII Renewable energy outlook

While the reviewed papers all provide improvements to models incorporating
renewable energy uncertainty, they do not take additional costs like green
house gas emissions and the associated rising environmental taxes into
account. This can be important, as can be seen in [4], where environmental
unit commitment (EUC) is proposed to incorporate carbon tax payments into
account. EUC therfore also minimizes green house gas emissions. The performed
case study clearly shows the advantages of EUC over CUC. It is shown that for
varying carbon tax rates, EUC outperforms CUC even in scenarios where the tax
is low. This is due to the increased cost of TGR generation which can only be
incorporated ex-post by CUC. Applying the modifications to other models can
help to more realistically predict actual generation cost of green house gas
producting generation units. Yurdakul, Sivrikaya and Albayrak plan to propose
models incorporating the gained experience [4]. We conclude that research in
this area is still in progress.

##  VIII Conclusion

In the preceding sections, we have seen several modern stochastic approaches
to the unit commitment problem. All of the discussed papers provide relevant
additions to the UC problem. The main approaches we reviewed are DRO, SP and
DSP. We summarized several promising methods in the field of DRO, which
propose multiple improvements to (micro) grid operation, which will be
increasingly important for the rising share of renewable energy. As two stage
models are still the more viable and applied solution to SUC, we suggest that
the DRO approach is especially worth to do further research in, since it
improves the already common robust optimization through the use of ambiguity
sets. It is therefore well suited to incorporate renewable generation
uncertainty. All of these advances help improving the efficiency of power
production, thus enabling a more sustainable future.

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