Abstract— Speed governors are key elements in the dynamic
performance of electric power systems. Therefore, accurate
governor models are of great importance in simulating and
investigating the power system transient phenomena. Model
parameters of such devices are, however, usually unavailable
or inaccurate, especially when old generators are involved.
Most methods for speed governor parameter estimation are
based on measurements of frequency and active power
variations during transient operation. This paper proposes a
genetic algorithm based optimization technique for parameter
estimation, which makes use of such measurements. The
proposed methodology uses a real-coded genetic algorithm. The
paper estimates the parameters of all system generators
simultaneously, instead of every machine independently, which
is fully in line with the interest to treat the electric power
system as a whole and study its comprehensive behaviour.
Moreover, the methodology is not model-dependent and,
therefore, it is readily applicable to a variety of model types
and for many different test procedures. The proposed
methodology is applied to the electric power system of Crete
and the results demonstrate the feasibility and practicality of
OWER system simulation results depend greatly on the
accuracy of system model parameters. This is especially
true for synchronous generators and their control
subsystems, such as governors, exciters, limiters and
stabilizers. Dynamic data of generating units are, however,
usually inaccurate, incomplete, or even unavailable,
especially when old generators are involved. Therefore,
typical parameters are frequently used, leading to results of
reduced credibility. Thus, the estimation and verification of
these parameters are necessary for acquiring accurate system
Most techniques employed for the estimation of the
unknown parameters are based on processing suitable actual
measurements of the system dynamic behavior, recorded
Manuscript received March 7, 2005.
G. K. Stefopoulos and A. P. Sakis Meliopoulos are with the School of
Electrical and Computer Engineering, Georgia Institute of Technology,
Atlanta, GA 30332 USA
P. S. Georgilakis is with the Department of Production Engineering and
Management, Technical University
N. D. Hatziargyriou is with the School of Electrical and Computer
Engineering, National Technical University of Athens, Greece (e-mail:
of Crete, Greece (e-mail:
during appropriate tests [1-11]. These measurements are
used as input to an identification procedure to estimate the
model parameters. However, as noticed in literature ,
many of the existing methods may not be adequate. For
example, several methods are based on linear system
techniques (like transfer function identification), therefore,
have limited applicability when nonlinearities are present
[4,10]. Many methods require cumbersome symbolic
manipulations of dynamic models and therefore may be
limited mainly to simpler models . Furthermore, several
of the existing approaches are model-specific .
This paper presents a methodology for estimating the
dynamic data of generating units that is based on genetic
algorithms and makes use of measurements of transient
system response. It should be emphasized that the
methodology is not model-dependent and, therefore, it is
readily applicable to a variety of model types and different
test procedures. The work presented in the paper estimates
the governor and the electromechanical dynamic parameters
of a generating unit; however the methodology can be easily
expanded to any dynamic model, provided that appropriate
measurements are available.
Evolutionary computation techniques and particularly
genetic algorithms (GAs) are computational-intelligence-
based optimization methods. They are used in several
scientific fields, mainly in hard, large-scale optimization
problems, where other classical analytical optimization
techniques may prove inadequate. In the power engineering
area, such problems include operation optimization (unit
commitment , economic dispatch , optimal power
flow, optimal allocation of reactive resources ) [12-15],
parameter estimation [8-9], etc. A comprehensive literature
survey on such applications is presented in .
The paper investigates the parameter identification
problem from a power system point of view, rather than
from the electric machinery side. This means that the
identification procedure is not applied to every machine
independently, but it attempts the simultaneous parameter
estimation of all system generators. This is because it is of
interest to study the comprehensive behaviour of the system
as a whole, rather than of a single machine. It should be
noted that the methodology can be readily applied in a
machine-oriented approach, if appropriate measurements are
The paper is organized as follows. Section II presents a
general overview of the parameter estimation framework.
A Genetic Algorithm Solution to the Governor-Turbine Dynamic
Model Identification in Multi-Machine Power Systems
George K. Stefopoulos, Student Member, IEEE, Pavlos S. Georgilakis, Member, IEEE, Nikos D.
Hatziargyriou, Senior Member, IEEE, and A. P. Sakis Meliopoulos, Fellow, IEEE
Proceedings of the
44th IEEE Conference on Decision and Control, and
the European Control Conference 2005
Seville, Spain, December 12-15, 2005
0-7803-9568-9/05/$20.00 ©2005 IEEE
identification procedure of generator parameters. Section IV
presents results from the application of the proposed
methodology to a single-machine test system. Section V
describes the application of the proposed methodology to the
electric power system of Crete. Section VI concludes the
III describes the genetic-algorithm-based
The proposed identification procedure is a simulation-
based process that uses a genetic algorithm as optimization
tool, as presented in Fig. 1. The physical system and the
mathematical model of the system are excited by the same
input. The output of the physical system, which is the set of
available measurements, is compared to the simulated output
of the model. The error between the two outputs is used as
input to a genetic algorithm optimization module, which
updates the model parameters in such a way that this error is
of the system model is a function of the
system state, the input and the model parameters, as
described by the set of differential-algebraic equations (1):
where yˆ? is the vector of the system model outputs, x? is the
vector of the dynamical states of the system, z? is the vector
of the algebraic states, u? is the vector of the system inputs,
and a? is the vector of the model parameters. The global
,),( ),() (ˆy
state vector is denoted by
denotes the initial condition vector.
The identification procedure estimates the unknown
vector of model parameters, a?, so that the deviation
between the model and the real system responses to the same
input u? is minimized. The error to be minimized is the
square error between the measured and the simulated output
waveforms defined as (assuming discrete-time signals):
), ( ˆaty
values of the outputs at time instant
the time sample
observations are made on the real system, and i is the
, N being the number of outputs.
The vector of the unknown, constant, system parameters is
denoted by a?. The values of these parameters are
constrained in some specific intervals.
A key feature of the approach is that the estimation
? are the measured and simulated
kt , respectively;
, given that T discrete
process is not model-specific and it is therefore
straightforward to switch between a large variety of models.
This advantage results from the fact that the simulation-
based optimization method uses only the model output. It
does not require any knowledge of the specific model
structure. The use of GAs as optimization tool enhances this
feature, since one of the main attributes of genetic
algorithms is that they do not require any auxiliary
knowledge on the objective function, such as gradient
information. Therefore, the proposed method is, in fact, a
black-box identification method, which automatically
adjusts the parameters of the model until the model output
matches the measurements.
A. Fundamentals of Genetic Algorithms
Genetic algorithms are optimization methods inspired by
natural genetics and biological evolution. They manipulate
strings of data, each of which represents a possible problem
solution. These strings can be binary strings, floating-point
strings, or integer strings, depending on the way the problem
parameters are coded into chromosomes. The strength of
each chromosome is measured using fitness values, which
depend only on the value of the problem objective function
for the possible solution represented by the chromosome.
The stronger strings are retained in the population and
recombined with other strong strings to produce offspring.
Weaker ones are gradually discarded from the population.
The processing of strings and the evolution of the population
of candidate solutions are performed based on probabilistic
rules. References [17-19] provide a comprehensive
description of genetic algorithms.
B. Chromosome Representation
Two types of representations have been investigated in
this work, binary and real (floating-point).
C. Creation of Initial Population
The initial population of candidate solutions is created
D. Evaluation of Candidate Solutions
Each candidate solution represents a parameter vector, a?.
Fig. 1. Block diagram of estimation procedure.
The evaluation of each candidate solution is based on the
objective function value,
function value is obtained after system simulation. The
purpose of the process is to solve a minimization problem,
or equivalently, a maximization problem that maximizes a
transformed objective function. In this paper, the objective
function to be maximized is defined as
)e?. Note that the objective
where K is a small positive real number used as scaling
coefficient, in order to avoid problems that may arise as
e? approaches zero, and to control problems like
Reproduction refers to the process of selecting the best
individuals of the population and copying them into a
“mating pool.” These individuals form an intermediate
population. Three types of the reproduction process are
implemented in this work:
1) Roulette-wheel selection,
2) Tournament selection with user-defined window,
3) Deterministic sampling
proportionate selection scheme.
No significant differences in the results were observed
between the different types of reproduction in this problem.
The reported results are obtained using deterministic
sampling, i.e. each individual is assigned an expected
number of appearances in the “mating pool,” according to its
calculated fitness. Subsequently, the individuals in the
“mating pool” are randomly grouped in pairs, each of which
produces two offspring.
based on the fitness-
F. Crossover Operation
In binary representation the following four types of
crossover are used:
3)Uniform crossover, which is a crossover operator that
swaps only single bits between the two parent binary strings.
4)Multi-point crossover, in which one crossover point is
selected, randomly, for each parameter represented in the
chromosome, and, thereafter, 1-point crossover is performed
in each parameter.
In floating-point representation the crossover types used
3) Uniform crossover,
4) Arithmetical crossover.
The arithmetical crossover operator creates offspring with
new parameters values, defined as a linear combination of
the two parents. If
w s are to be crossed, the
interval [0, 1] .
, where a is a random number of the
G. Mutation Operation
When binary coding is used, the genetic algorithm
mutation simply changes a bit from "0" to "1" or vice versa.
The bits that undergo mutation are chosen based on a
probability test. The probability of mutation is generally set
to a small value, about 0.001 to 0.01.
In real representation, two mutation operators are
implemented: uniform and non-uniform mutation.
1)Uniform mutation: This operator is analogous to the
binary operator, but it applies to real values instead of binary
bits; it randomly replaces the parameter value with another
one from the appropriate interval;
2)Non-Uniform mutation: This mutation type is described
in  and it is responsible for the fine-tuning capabilities of
the real-coded GA. If a parameter k of value
candidate solution is selected for mutation, its value is
depending on whether a random binary digit is 0 or 1. LB
and UB are the lower and upper bounds of the interval
parameter k belongs to. The function
value in the range ?? y , 0
such that the probability of
being close to 0 increases as the current generation number,
t , increases. This property causes this operator to uniformly
search the space at initial stages, when t is small, and very
locally at later stages. The function used is
where r is a random number in [0, 1], T is the maximal
generation number, and b is a parameter determining the
degree of non-uniformity .
In real representation, since parameters do not change
during crossover, but are just recombined differently (except
for the arithmetical crossover), the only way of affecting
their values is by the mutation operator. So, the mutation
probabilities used are greater than the ones in binary
representation and may reach up to 5%.
k u of a
k u , where
) 1 (),(
) 1 (
H. Creation of the Next Generation
After mutation is completed, the children population is
created and the previous population is replaced by the new
generation. Children are evaluated and the fitness function
for each individual is calculated. The procedure is repeated
until the termination criterion is met, defined by a maximum
number of generations.
As an option, an elitist operator is also used. If this option
is selected, the new population is not the children
population, but is created by the best N individuals from
the children and the previous population, where N is the
population size. The aim of this elitist strategy is to
eliminate the possibility of destruction of good solutions that
may appear in early generations and to aid in achieving good
solutions quite fast and to subsequently be able to fine-tune
them. Additionally, it is expected that the best individuals
will provide the best offspring after crossover. The risk of
premature convergence to a sub-optimal solution is
increased with this operation, but this can be controlled with
the parameter K of the fitness function and with a slightly
increased mutation probability.
A. Problem Formulation
The identification procedure was tested using a single-
machine test-system, to investigate the feasibility of the
approach and to configure the genetic algorithm parameters
for the specific problem. The model used for the governor-
turbine subsystem representation is shown in Fig. 2.
The following values are assumed for the five parameters
of the model that are to be estimated:
sTG2 . 0
7 . 0
A step input
the pre-disturbance power demand, is applied at
representing a load increase. The system is simulated in the
time interval from -2s to 10s. The frequency variation (in
Hz) and the mechanical power deviation (in p.u.) are
calculated every 0.05s, and these results are assumed to
represent the measured input data for the identification
procedure. This way the estimated parameters obtained can
be directly compared with the actual ones.
The optimization problem is defined as
05. 0, 2 . 001. 0
is the system
of 0.2 per unit (p.u.), i.e. 20% change of
, 20,1001. 0
6 . 0
,5 . 0
is the assumed system
the unknown parameter
B. Numerical Experiment Results
A number of numerical experiments were conducted on
this problem, testing the effect of the various parameters of
the genetic algorithm on the results. Results, using binary
and real representation, are presented in Tables I and II.
They reveal the fact that the proposed methodology for
model identification can provide satisfactorily accurate
results. Furthermore, by comparing Tables I and II, it is
concluded that real-coded GA performs better than the
TYPICAL RESULTS OF BINARY CODING
Binary coding with 20 bits per parameter
Population size: 200, Number of generations: 1000
Uniform crossover with probability 0.6
pm = 0.05, K = 0.01, Elitist operator: On
Mean error of final population = 9.21e-4, Best solution error = 9.21e-4
Real Values Estimated Values
C. Determination of Method Parameters
Results obtained using floating-point coding were
repeatedly much closer to the optimal solution compared to
binary coding. Furthermore,
representation was faster and more consistent from run to
A population size of one to two hundred, and about one
thousand generations proved to be sufficient for this
problem, providing very good or even excellent results.
Uniform and two point crossover provided better results
compared to other crossover types and the use of the non-
uniform mutation operator proved to be an important factor
when floating-point representation was used. Finally, results
obtained using the elitist operator were superior compared to
cases where no elitism was used.
The described floating-point configuration provides
results with an error less than 3% for every parameter. The
largest errors appear in the estimation of the time constants,
especially the governor time constant, while the other
parameters are estimated with a much higher precision.
However, simulation tests proved that simulation results are
much less sensitive to the values of the time constants
compared to the droop values, therefore, less accuracy for
Fig. 2. Unit speed control and turbine (engine) dynamic model.
TYPICAL RESULTS OF FLOATING-POINT CODING
Population size: 200, Number of generations: 1000
Uniform crossover with probability 0.6
Non-uniform mutation (b=4) pm = 0.05, K = 0.01, Elitist operator: On
Mean error of final population = 7.16e-4, Best solution error = 7.16e-4
Real Values Estimated Values
these parameters can be tolerated.
D. Effects of Measurement Noise
The work presented so far tested the capability of the GA
based estimation methodology in an ideal situation, where
the mathematical model was able to describe precisely the
actual system. This is not the case, when actual field
measurements are used. In a realistic situation the model
output cannot match precisely the actual system output,
especially if simplified models are used to facilitate
calculations. Moreover, field measurements may be severely
corrupted by noise, or unmodeled dynamics may be present,
having similar effects as noise.
In order to investigate the behavior of the methodology
under such conditions, numerical experiments were carried
out assuming the presence of random noise in the
measurements. The assumed noise was zero-mean,
uniformly or normally distributed.
Results obtained from a case with noise uniformly
) 15 . 0 ,15 . 0(?
for the frequency and in
)04 . 0 ,04 . 0(?
for the power deviation are presented in Fig. 3
and Table III, assuming the same real-coded GA
configuration as in Table II. The “measured” waveform in
Fig. 3 refers to the simulated results that are assumed to
represent the measurements as described in section IV.A
These numerical experiments reveal that, even with
heavily corrupted measurements from random noise, the
methodology provides results of satisfactory accuracy.
Furthermore, the maximum errors appear in the parameters
that least affect the outputs of the model, therefore, the error
in simulation studies using these parameter values is
In several cases, the measurement noise may not be
completely random, but it may follow some deterministic
pattern. Such a situation may arise if unmodeled dynamics
are present. To investigate this condition, numerical
experiments were carried out assuming the presence of
additive deterministic noise in the measurements of the form
of one or two sinusoidal signals. The total amplitude of the
disturbance was up to 0.15 Hz for the frequency and 0.02
p.u. for the power deviation. The numerical test showed that
the GA could filter out the deterministic noise almost
perfectly. Results from a test with a 2 Hz sinusoidal noise
are presented in Table IV and in Fig. 4, assuming the same
real-coded GA configuration as in Tables II and III.
A. Test-Case System of Crete
The estimation methodology was applied to the
autonomous power system of the Greek island of Crete. The
power system of Crete is a relatively large, isolated system
consisting mainly of oil-fired generators. It consists of 52
buses, 66 branches and 18 thermal units. Six of them are
steam units, four are diesel engines, seven are gas turbines
and there is a combined cycle plant. The total installed
capacity is about 400MW, while the system peak load is
approximately 360MW. The
corporation has conducted real time measurements of
frequency and unit active power variations during
intentional machine trip tests; these data were used for the
identification of the governor and the unit electromechanical
dynamic model parameters of each conventional generating
Greek public power
Noisy Frequency Variation Measurements Actual Frequency Variation
Simulated Frequency Variation Noisy Power Deviation Measurements
Actual Power DeviationSimulated Power Deviation
Fig. 3. Comparison of “measured” and simulated waveforms (using
estimation results), with additive stochastic noise in the measurements.
-2.00 0.002.004.00 6.008.00 10.00
Noisy Frequency Variation Measurements
Noisy Power Deviation Measurements
Simulated Frequency Variation
Simulated Power Deviation
Fig. 4. Comparison of “measured” and simulated waveforms (using
estimation results), with additive deterministic noise in the measurements.
TYPICAL RESULTS USING MEASUREMENTS WITH DETERMINISTIC NOISE
Real Values Estimated Values
TYPICAL RESULTS USING MEASUREMENTS WITH STOCHASTIC NOISE
Real Values Estimated Values
B. Transient-Response Measurements
Field tests involved a conventional machine rejection
under different operating conditions. Two outages were
performed of 10 MW and 19 MW, at a total load of 159
MW and 208 MW, respectively. The transient behavior of
the system was recorded in computers equipped with A/D
converter cards. The sampling rate was 20 Hz. Recordings
involved the active power response of the remaining thermal
units and the system frequency deviation, which was
measured at four points in the system. The total duration of
each recording was 3min, including some pre-disturbance
time. Data up to 10s after the disturbance were used for
estimation procedure, since the dynamics of interest had
reached steady state after 10s. Some typical recording are
shown in Fig. 5 and 6.
It is of interest to observe the active power oscillations in
Fig. 6. Such oscillations of frequency around 5 Hz were
observed in the output of all the diesel and steam units, even
in steady-state operation. They exist because the mechanical
system of the diesel units produces a pulsating torque on
their shaft. The steam units are physically installed on the
same power plant as the diesel units, and, therefore, they
also produce a pulsating active power to compensate for the
oscillations of the diesel units. Fig. 6 shows that the diesel
and steam unit oscillations are in fact in opposite phase.
Since modeling such oscillations would not provide any
additional information for the governor-model estimation
procedure, these oscillations are considered unmodeled
dynamics and are treated as noise. However, based on the
discussion on measurement noise, in section IV, the GA is
expected to be able to filter out the noise very adequately.
This was, indeed, observed in the estimation procedure
C. Estimation Results
The identification procedure is applied to both sets of
available measurements performing two independent
estimation procedures, for the different disturbances and
under different loading conditions.
The power system of Crete was modeled in the
EUROSTAG dynamic simulation program. Static network
data and pre-disturbance operating conditions were provided
by electric utility, along with any available generator
dynamic data. These data allowed a three-winding
representation of the synchronous generators . A
standard IEEE Type 1 voltage regulator-exciter model was
used for all units . The three parameter governor-turbine
model shown in Fig. 2 was used. Governor limits were set
based on the utility provided values of minimum and
maximum power output for each unit. The parameters to be
identified were constrained as follows:
2 . 0 01 . 0
5 . 005 . 0
5 . 1 5 . 0
constant of each unit,
)( j steam
each steam unit,
TD the mechanical time constant of each
diesel engine, and
the turbine time constant of each
Comparative graphs of the measured transients and the
simulated dynamic responses using the estimated parameters
are presented in Fig. 7 through 9. The results show a
considerably good agreement between the measured
response and the simulated waveforms using the estimated
i R is the droop of each unit,
TG the governor time
the turbine time constant of
Active Power (MW)
Diesel Unit No 4 (MW)
Steam Unit No 3
Fig. 6. Recordings of active power variations for a steam and a diesel unit
(19 MW rejection test).
19 MW Rejection10 MW Rejection
Fig. 5. Recordings of frequency variations of the system for the three
Recorded Frequency (19 MW Rejection Test)
Simulated Frequency (19 MW Rejection Test)
Recorded Frequency (10 MW Rejection Test)
Simulated Frequency (10 MW Rejection Test)
Fig. 7. Measured and simulated system frequency for the 10 MW and 19
MW rejection tests.
This paper investigates the application of genetic
algorithms for the identification of dynamic models of
generating units in power systems. The paper proposes the
use of a real-coded genetic algorithm as optimization tool
for the estimation procedure. The main advantages of the
proposed methodology are the few input data required, its
flexibility, and the simplicity of its mechanism.
The methodology proved to be able to provide accurate
results, even in the presence of measurement noise or
unmodeled dynamics. It is shown that the simulated system
response using the estimated parameter values can correctly
represent measurements, even if they are significantly
corrupted by noise. It was also shown that the simulated
system response using the estimated parameter values can
correctly capture the main features of the measurement even
with some deviation present in the parameter values.
The proposed method has been successfully applied to the
simultaneous identification of the turbine–governor models
of the units of the medium size, isolated power system of
Crete. The obtained results demonstrate the feasibility and
practicality of the proposed GA approach.
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Active Power (MW)
Recorded Active Power for Steam Unit 1
Simulated Active Power for Steam Unit 1
Fig. 9. Measured and simulated power output for steam unit 1, for the 19
MW rejection test.
Active Power (MW)
Recorded Active Power for Diesel 1 Unit
Simulated Active Power for Diesel 1 Unit
Fig. 8. Measured and simulated power output for diesel unit 1, for the 19
MW rejection test.