Inclusion of Frequency Nadir constraint in the Unit Commitment Problem of Small Power Systems Using Machine Learning

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Inclusion of Frequency Nadir constraint in the

Unit Commitment Problem of Small Power

Systems Using Machine Learning

Mohammad Rajabdorri, Behzad Kazemtabrizi, Matthias Troﬀaes, Lukas Sigrist, Enrique Lobato

July 15, 2024

Abstract

As the intention is to reduce the amount of thermal generation and to

increase the share of clean energy, power systems are increasingly becom-

ing susceptible to frequency instability after outages due to reduced levels

of inertia. To address this issue frequency constraints are being included

in the scheduling process, which ensure a tolerable frequency deviation in

case of any contingencies. In this paper, a method is proposed to integrate

the non-linear frequency nadir constraint into the unit commitment prob-

lem, using machine learning. First a synthetic training dataset is gener-

ated. Then two of the available classic machine learning methods, namely

logistic regression and support vector machine, are proposed to predict

the frequency nadir. To be able to compare the machine learning meth-

ods to traditional frequency constrained unit commitment approaches,

simulations on the power system of La Palma island are carried out for

both proposed methods as well as an analytical linearized formulation of

the frequency nadir. Our results show that the unit commitment problem

with a machine learning based frequency nadir constraint is solved con-

siderably faster than with the analytical formulation, while still achieving

an acceptable frequency response quality after outages.

Nomenclature

Data-Driven Approach

ℓ(.) loss function

ˆypredicted label

Xset of all features

Yset of all labels

Θ set of θparameters

θcoeﬃcients in the linear model

Cregularization coeﬃcient

arXiv:2210.01540v2 [eess.SY] 12 Jul 2024

fθ(x) hypothesis function

F C set of feasible combinations

Kinumber of the steps

Mnumber of features

mindex of features

Nnumber of data samples

nindex of data samples

xfeatures of the dataset

ylabels of the dataset

Frequency Dynamics

α, β normalizing coeﬃcients

∆f′

crit critical rate of change of frequency

∆fnadir

crit critical frequency nadir [Hz]

∆fss

crit critical steady state frequency [Hz]

ℓindex of the lost generator

γjbinary operator of aﬃne segments [∈{0,1}]

λjweight associated with breaking point j

Mbase power of the unit [MW]

ajbreaking point

Dload damping factor [%/Hz]

f(t) frequency [Hz]

f0nominal frequency [Hz]

Hinertia [MW.s]

Jnumber of the breaking points

jbreaking point index

Pℓlost power [MW]

Peelectrical power [MW]

Pmmechanical power [MW]

Tgdelivery time of units [s]

z1, z2auxiliaries for changing variables

Unit Commitment

Iset of all generators

Dmaximum yearly thermal generation

Pimaximum power output of generator i[MW]

Rimaximum ramp-up of generator i[MW/h]

Dtdemand at hour t

Dminimum yearly thermal generation

Piminimum power output of generator i[MW]

Rimaximum ramp-down of generator i[MW/h]

DT minimum down-time of generators [hours]

gc generation costs [e]

Inumber of generators

iindex of generators

ii alias index for generators

ppower generation variable [MW]

ronline reserve power variable [MW]

salias index for time intervals

sg solar generation variable [MW]

suc(.) start-up costs [e]

Tset of all time intervals

tindex of time intervals

ucommitment variable [∈{0,1}]

UT minimum up-time of generators [hours]

vstart-up variable [∈{0,1}]

wshut-down variable [∈{0,1}]

wg wind generation variable [MW]

1 Introduction

The share of renewable energy sources (RES) is growing steadily in power

systems. It is essential to facilitate the growth of RES penetration to reduce

the carbon emission from fossil fuel based generators. There are however some

obstacles that limit the applicability of RES. RES are volatile in nature and

forecasting them is subject to uncertainties. Hence, integrating them in the

power system is challenging. Moreover RES are usually decoupled from the

system, and therefore they do not add any inertia to the system. This is par-

ticularly important in small power systems like islands, as they typically suﬀer

from inertia scarcity, and are therefore more prone to frequency volatility. For

that reason, when integrating RES in such systems, it can be very challenging

to maintain frequency stability in case of contingencies.

To address this issue, researchers have included frequency dynamics in short

term scheduling processes like Unit Commitment (UC) to form a frequency

constrained UC (FCUC), in [1], [2], [3], and etc. The standard (non-frequency

constrained) UC problem can be formulated as a mixed integer linear program-

ming (MILP) problem, which can be solved eﬃciently using standard solvers.

Unfortunately, the frequency dynamics of a power system is highly nonlinear

and non-convex, complicating how the UC problem can still be formulated as

a MILP problem. There is valuable research work in the literature, addressing

this very issue ([4], [5], [6], and [7]). Frequency dynamics after outages are usu-

ally described by the rate of change of frequency (RoCoF), frequency nadir, and

steady-state frequency. RoCoF and steady-state frequency can be formulated

linearly, but frequency nadir cannot. In previously mentioned studies, the non-

linear constraint on the frequency nadir (derived from the well-known swing

equation) has been simpliﬁed or approximated so that it still can be used in

the MILP formulation of UC problem. These formulations are based on simpli-

fying assumptions and usually are computationally demanding. More recently,

data-driven approaches are being introduced to more accurately model the fre-

quency dynamics in the UC problem, instead of relying on analytical methods

([8], [9], [10], [11]). These methods try to estimate the dynamics of the system

accurately, while keeping the solution time of UC reasonably low.

Among the analytical methods, in [1], a linear formulation of inertial re-

sponse and the frequency response of the system is added to the UC problem,

which makes sure that in case of the largest outage, there is enough ancillary

service to prevent under frequency load shedding (UFLS). To linearize frequency

nadir constraint, ﬁrst-order partial derivatives of its equation with respect to

higher-order non-linear variables are calculated. Then the frequency nadir is

presented by a set of piecewise linear constraints. In [2], diﬀerent frequency ser-

vices are optimized simultaneously with a stochastic unit commitment (SUC)

approach, targeting low inertia systems that have high levels of RES penetra-

tion. The stochastic model uses scenario trees, generated by the quantile-based

scenario generation method. To linearize frequency nadir, an inner approxima-

tion method is used for one side of the constraint, and for the other side, a

binary expansion is employed to approximate the constraint as a MILP using

the big-M technique. In [3], a stochastic unit commitment approach is intro-

duced for low inertia systems, that includes frequency-related constraints. The

problem considers both the probability of failure events and wind power uncer-

tainty to compute scenario trees for the two-stage SUC problem. An alternative

linearization approach is used to make sure the nadir threshold is not violated.

Instead of piece-wise linearizing the whole equation, relevant variables includ-

ing the nonlinear equation are conﬁned within a plausible range that guarantees

frequency drop after any contingency will be acceptable. In [4], a reformulation

linearization technique is employed to linearize the frequency nadir limit equa-

tion. Results show that controlling the dynamic frequency during the scheduling

process decreases the operation costs of the system while ensuring its frequency

stability. In [5], ﬁrst, a frequency response model is developed that provides

enough primary frequency response and system inertia in case of any outages.

All frequency dynamic metrics, including the RoCoF and frequency nadir are

obtained from this model, as analytic explicit functions of UC state variables

and generation loss. These functions are then linearized based on a pseudo-

Boolean theorem, so they can be implemented in linear frequency constrained

UC problem. To ﬁnd the optimal thermal unit commitment and virtual inertia

placement, a two-stage chance-constrained stochastic optimization method is

introduced in [6]. Frequency nadir is ﬁrst calculated with a quadratic equation

and then it is constrained with the help of the big-M approach. In [7], the

frequency nadir is approximated as a piece-wise linear function to good (and in

principle, arbitrary) precision, and the resulting constraint is then reformulated

as a MILP using separable programming. A common assumption in [1], [2], [3],

[6], [7], and many other similar works, is that the provision of reserve increases

linearly in time, and all units will deliver their available reserve within a given

ﬁxed time. This assumption is the key to calculate the frequency nadir as a

function of other variables.

Among the data-driven approaches, in [8] a multivariate optimal classiﬁca-

tion trees (OCT) technique is used to learn linear frequency constraints. A

robust formulation is proposed to address the uncertainties of load and RES.

OCT is deﬁned and solved as an MILP optimization problem, so if the training

dataset is big, optimizing the OCT becomes very hard. A dynamic model is pre-

sented in [9] to generate the training dataset. The generated dataset is trained,

using a deep neural network. Trained neural networks are formulated so they

can be used in an MIL problem and the frequency nadir predictor is developed,

to be used in UC problem. Then in [10] deep neural network (DNN) is trained

by high-ﬁdelity power simulation and reformulated as an MIL set of constraints

to be used in UC. The generated data samples in [10] are denser where the

frequency nadirs are closer to the UFLS threshold. In [11] linear regression is

applied on a synthetic training dataset to extract the relationship between fre-

quency response and frequency deviation during primary frequency response.

The obtained regression is then used as a constraint in a distributionally robust

economic dispatch model. The results of these data-driven methods is heavily

reliant on the quality of the training dataset. Also, deﬁning the DNN as MIL

constraints to the UC problem, adds so many variables and sets of constraints

to the formulation. A summary of the reviewed papers is presented in ﬁg. 1.

Following the same line of research, this paper generates a synthetic training

dataset, and then applies machine learning (ML) methods on the dataset to

derive a linear constraint which approximates the original non-linear frequency

nadir constraint for all scenarios in the dataset. To evaluate the eﬀectiveness of

the ML methods, weekly FCUC of La Palma island power system is solved for

seasonal sample weeks. The results are compared to one of the recent FCUC

formulations that employs a MILP formulation based on an analytical expression

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InclusionofFrequencyNadirconstraintintheUnitCommitmentProblemofSmallPowerSystemsUsingMachineLearningMohammadRajabdorri,BehzadKazemtabrizi,MatthiasTroffaes,LukasSigrist,EnriqueLobatoJuly15,2024AbstractAstheintentionistoreducetheamountofthermalgenerationandtoincreasetheshareofcleanenergy,powersystemsa...

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Inclusion of Frequency Nadir constraint in the Unit Commitment Problem of Small Power Systems Using Machine Learning

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