Fourier Series-based Kernel for Frequency Control Performance
Monitoring of Hydroelectric Generating Units
G. Robert12 , G. Besançon2

Abstract— An identification algorithm is proposed to estimate
varying parameters and some unknown input in a MISO
continuous-time system, for the purpose of monitoring
frequency control performance in hydroelectric generating
units. In the proposed algorithm, an integral transform with a
Fourier series-based kernel is developed from irregularly
sampled data and its efficiency is shown by comparison with real
hydropower plant parameters and with a S avitzky-Golay filter.
I. INT RODUCT ION
To guarantee the frequency stability of an electric power
system, power plants are requested by the power grid operator
(generally the transmission system operator) to provide some
ancillary services related to the system frequency control.
Primary Frequency Control (PFC) and Secondary Frequency
Control (SFC) refer to these services. Good explanations of the
power generation operation can be found in [1]. Basically, an
imbalance between production and consumption in the power
network causes a frequency deviation which is compensated
by control systems equipping power generation units and
having PFC and SFC functions.
In France, power plants which participate to ancillary
services have a contractual commitment with the power grid
operator to modulate the generation power with specific
dynamic performance criteria (power response time, static
error, damping…). These requirements become more and
more severe with respect to the future European grid code [2].
The issue for the producer is to check the power plant
frequency control performance, that is to assess the
contribution to PFC and SFC in order to fulfill contractual
commitments and possibly to detect faults in the power
generation process.
Few papers exist in the literature concerning the power grid
frequency control performance monitoring [3] [4]. In [3],
Wang uses a discrete linear time invariant (LTI) model to
identify parameters related to PFC provided by a thermal
generating unit. In [4], for wind turbines, again a LTI model is
used for parameter estimation. None of them identify SFC
parameters.
To that end, a continuous -time identification approach is
considered via integral transforms using Fourier-series-based
kernels. This approach shows good identification results, in
particular when dealing with irregularly sampled data, as those
that one has to face in real situations of power plant
monitoring.
In section II, the considered MISO system is presented
from a power generation model, and declined into a MISO
linear continuous-time model with varying parameters. The
problem of identification for such a continuous -time system
from irregularly sampled data is addressed in section III. An
application to a hydroelectric generating unit is then proposed
in section IV where the unknown input is estimated as well,
and section V finally concludes the paper.
II. PROBLEM ST AT EMENT
The nonlinear model of a power generation unit is represented
hereafter by a LTV model. Varying parameters of this model
are gains, time constant and damping ratio which characterize
the expected dynamic behavior in compliance with grid
operator requirements.
A. Power Generation Model
The power response P(t) of a generating unit participating
to PFC and SFC is usually the sum of 3 terms:
-
the power response P0 (t) of the production program or
a particular setpoint Pco (t),
-
the power response P1 (t) of the primary frequency
control steered by the network frequency deviation
ΔF(t) = F(t) - Fn ,
-
and the power response P2 (t) of the secondary
frequency control steered by N 2 (t ) the remote
setpoint sent by the grid operator.
Around an operating point, we can write deviations with
Laplace formalism and transfer functions H0 , H1 , H2 :
P(s)  P0 (s)  P1 (s)  P2 (s)
(1.1)
P  H 0 (s)Pco  H1 (s) K F  H 2 (s) Pr N2
(1.2)
In the present paper, both PFC and SFC parameters will be
estimated for control performance monitoring purposes. A
hydroelectric powerplant will be considered with variant
setpoints. In this context, the paper addresses the problem to
estimate both an unknown input and varying parameters of a
multi-input single output (MISO) system.
The gain Pr represents the secondary power reserve in SFC
and K is linked with the controller speed droop ratio b p :
1
2
Gérard Robert is with EDF Hydro, 73373 Le Bourget du Lac, France
(gerard.robert@edf.fr).
Gildas Besançon and Gérard Robert are with Univ. Grenoble Alpes,
CNRS, Grenoble INP*, GIPSA-lab, 38000 Grenoble, France
(gildas.besancon@gipsa-lab.grenoble-inp.fr). * Institute of Engineering
K
Pmax
bp Fn
(1.3)
with Fn (Hz) the nominal network frequency and Pmax the
maximu m power of the generator.
The signal N2 (t) varies from -1 to 1. The output P(t) and
the frequency F(t) are corrupted by measurement noise (see
for instance Figure IV.1). The setpoint Pco (t) can be a
piecewise constant signal (case for nuclear, thermal or
reservoir hydropower plants) or a time-varying signal (run-ofthe-river power plants).
The process related to power generation consists of a
turbine controller (see Figure II.1), an actuator and a
generating unit (turbine + power generator). The turbine
controller has two inputs (R, ΔF) and two operating modes:
Power control mode in which the reference signal R(t) is a
power setpoint Pc and Gate opening control mode where R(t)
is a gate opening setpoint.
Due to nonlinear conversions in the control system, the
setpoint Pc is not always available in power format. It can yet
be assessed indirectly with the reference signal R(t) sent to the
turbine controller.
B. Equivalent LTV Model
The sampling frequency of the controller is very fast
compared with the acquisition of monitored variables. That’s
why a continuous-time model is considered for the process
model. Moreover, dynamics of the primary and secondary
frequency control are approximated by a 2nd order with a
natural pulsation n  1/  and a damping ratio  .
Regarding dynamic models for steam and hydro turbines
in power system studies [6], a second order continuous -time
state space representation is proposed with varying parameters
which all depend on the operating point R0 :
 2 x1  2 x1  x1  G1  u1  T1u1 
(1.10)
 2 x2  2 x2  x2  G2  u2  T2u2 
(1.11)
y  x1  x2
(1.12)
with G1  1/ bp , y  P / Pmax , u1  F / Fn, u2  R / Pmax
x1  P1 / Pmax and x2   P0  P2  / Pmax .
From this, coefficients of the model may be poorly known
since they are function of the operating point R0 .
Pc (t )  Pco (t )  Pr N (t )
(1.4)
Pc  f ( R)
(1.5)
Since initial conditions are unknown, we build an inputoutput LTV model by summing the two equations (1.10)
(1.11):
where f is a nonlinear uncertain function 1 representing the
turbine characteristic.
 2 y  2 y  y  G1  u1  T1u1   G2 u2  T2u2  (1.13)
Whatever the operating mode is, from the power system
viewpoint, the following linearized model can be stated in
equivalent power format around an operating point:
Pc (t )  Pco (t )  Pr N2 (t )  G2R(t )
(1.6)
Let us set now pco  Pco /Pmax and u3  N2 /Pmax , so from
(1.6) we can write:
where G2 denotes a conversion gain: G2   f / R R and
0
The conventional structure of turbine controller [1] [5]
allows us to write some properties for the transfer functions
appearing in (1.2):
H 0 ( s)  H 2 ( s)
(1.7)
H 0 (0)  H 2 (0)  1
(1.8)
H1 (0)  1
(1.9)
In addition, it is assumed that these 3 transfer functions
have the same pair of poles.
T urbine
Controller
Generating
Unit
(1.14)
pc  G2u2
(1.15)
with
where Pr and G2 depend on the operating point R0 .
ΔF
Δ
R
pc (t )  pco (t )  Pr u3 (t )
P
The goal is then to estimate parameters in equation (1.13),
and from them gain K will be deduced with relation (1.3). Gain
Pr and the setpoint Pco(t) will be estimated in section IV.B from
(1.14).
The noisy measurements signals P(t), F(t) are known and
the setpoints R(t) and N2 (t) are available through irregular
sampling. Hence, u 1 , u 2 , u 3 and y are known.
The ordinary differential equation (ODE) given by (1.13)
can be rewritten in regression form:
y(t )  hT (t )
(1.16)
hT   u1 u1 u2 u2  y  y 
(1.17)
  1 ... 6 
(1.18)
T
Figure II.1. Power-Frequency Control of a generating unit
1
For hydropower plants, f depends also of the water head where the effect
is here neglected.
where the expressions of
 j are given in appendix.
III. PARAMET RIC IDENT IFICAT ION
The literature proposes many techniques to perform
identification of LTV systems (see for example [7]). To
estimate parameters of the LTV model, data are considered
over a narrow time-shifting window [t, t+T] where parameters
to identify are assumed invariant but with a time-varying
unknown input Pco (t).
This identification problem of such a linear continuous
time-varying system is difficult for several reasons. Firstly, it
is well known that calculations of the required derivatives give
rise to problems of accuracy and stability [8]. Secondly, a
narrow time window with an irregular sampling (i.e. a low
number of samples) leads to a poor estimation quality with
conventional methods [9]. Thus, a significant issue is to find
an algorithm able to estimate 6 parameters within a small time
window T.
To generalize the problem of equation (1.13), let us
consider a SISO process (extension to the MISO case will be
easy) characterized by a p th order ordinary differential equation
having p  q  1 unknown coefficients to be identified:
p
q
i 1
j 0
y   ai y (i ) (t )  b ju ( j ) (t )
with q  p, ai , b j 
(1.19)
, u the input and y the output.
The parametric identification procedure consists in
transforming ODE (1.19) in a set of algebraic equations. To
avoid noise amplification, many techniques exist either in
frequency domain [10], in Laplace domain or in time domain
[11]. For instance, we can find linear filter methods, numerical
integration-based methods [10] or integral transform methods
with different kernel [11] [12].
In the present paper, we will focus on an integral transform
method, tailored so as to overcome problems of irregular
sampling, measurement noise and eliminating the need for
identifying initial conditions of the ODE. It will be compared,
in the example of section IV, to an alternative filter-based
approach due to Savitzky and Golay.
B. Integral transform method
Integral transform methods are efficient techniques to
overcome noisy data. They consist in multiplying both sides of
ODE (1.19) by a kernel function and integrating both sides
over a time interval. Many choices are possible for the kernel:
Walsh functions [11], splines functions [12], Hilbert space
[16], modulating functions … A focus is made here on
modulating functions since the method we will propose is
similar to this one.
Modulating functions-based method has been introduced
in the fifties by Shinbrot [17] for aeronautics applications.
Several types of modulating functions have been proposed and
used, including sine functions [17], Hermit functions [18],
spline-type functions [19] [12], Poisson moment functionals
[20], and Hartley modulating functions [21]. The principle of
modulating function technique is to transfer the continuous
time derivatives signals to the derivative of a set of smooth
user-chosen “modulating functions” which constitute the
kernel of the integral transform applied to the ODE. The same
approach is used hereafter with a different insight and a new
kernel.
C. Fourier series-based kernel
Let us consider an integral transform I with the kernel

and n  :
 (t, n)  C p where t 
T
I [ x](n)  X (n)  x, (n)    (t , n) x(t )dt (1.21)
0
The particularity of this transformation is that it sets up a
map between a continuous time signal x (t ) and a sequence of
numbers  X ( n) yielding a discrete variable in the frequency
domain.
In order to get rid of derivatives terms appearing in (1.19),
the following proposition is considered.
Proposition:
Let x(t ), (t , n)  C
p
[0,T ] . If:
 ( j ) (0, n)   ( j ) (T , n)  0 , j  0,1,..., p -1
A. Savitzky-Golay filter
Developed in 1964, the Savitzky-Golay filter also known
as polynomial smoothing filter is a very efficient algorithm to
smooth data through a polynomial fitting and to estimate time
derivatives [13]. In his book [14], Orfanidis gives a nice
description of the method which can be written as a
convolution equation:
ys(i ) (k )  i !
M

m M
gi (m) y(k  m)
then
x( p ) ,  (1) p x, ( p )
where x ( p ) 
(1.22)
(1.23)
d px
dt p
Proof:
x ( p ) ,     (t , n) x ( p ) (t )dt and by
T
(1.20)
with i  0,1,..., d and g i is given by a least squares algorithm
[13] [14] or through the Matlab function sgolay from the
Signal Processing Toolbox [15].
By definition,
0
successive integration by parts the right-hand side is equal to:
T
 (0) x( p1)   (1) x( p2)  ...  (1) p  ( p1) x(0) 
0
T
The d order polynomial used in this method is calculated on
a moving window with half-length M [14].
(1) p   ( p ) xdt
0
Thus, (1.23) comes from boundary conditions (1.22).
(1.24)
Our contribution is to propose a new kernel which is a
periodic function (period T/n) expanded in a finite Fourier
series where coefficients will be calculated to satisfy initial and
final values of condition (1.22):
L
 (t , n)   ak cos knt  bk sin knt
(1.25)
k 0
with
n  n2 /T .
For sake of simplicity, an even function is chosen giving a
cosine partial sum:
L
 (t , n)  a0   ak cos knt
(1.26)
k 1
with
The
  2 /T .
j order derivative is:
L
which can be determined by solving a set of
with m  1,2,..., M :
 L
  ak  a0
 k 1
L
 k 2ma  0
k

k 1
M  1 equations
,p
(1.32)
,p  2
If the coefficient a 0 is fixed to an arbitrary value, it can be
easily shown that the solution {a k} of (1.32) is proportional to
a 0 . That means that the kernel  is also proportional to a 0 and
that this coefficient disappears when the integral transform I
(1.21) is applied to equation (1.16). That’s why, we will take
a0  1 .
For a 1st or 2nd order dynamic system (p = 1 or 2), we find


 ( j ) (t , n)    kn  ak cos  knt  j
j
k 1
  (1.27)

L
 (t, n)  1  cos nt
2
Boundary conditions (1.22) applied to (1.26) for j  0
give:
a0   ak  0
a1  a0  1 and hence:
,p
(1.33)
With a period T/n, this simple kernel is illustrated in Figure
III.1 for n = 1 and n = 2.
(1.28)
k 1
And applied to (1.27) for
equations:
L
k a
j
k
k 1
with
j  0 , we
obtain a set of J
 
cos  j   0
 2
(1.29)
j  1,2,..., J .
 
  0 for odd integers j , two cases must
 2
be considered for p  2 :
Since cos  j
Figure III.1: Fourier series kernel (1.35)
 p  2 if p  1 is odd
J 
 p  1 if p  1 is even
(1.30)
In addition, we set j  2m and M  J / 2 in order to
have the same number of equations as the number of unknown
coefficients ak . So, equation (1.29) becomes:
L
k
k 1
with
2m
ak  1  0
m
,p  2
th
a1  a2  a0

 a1  4a2  0
(1.31)
whose result is:
m  1,2,..., M and L  M  1
Thus to summarize, the cosine series kernel
For a 3th or 4nd order dynamic system (p = 3 or 4), we have
to solve:

(1.34)
4
1
a1   a0 and a2  a0 .
3
3
The cosine kernel is therefore:
verifying
the p order relation (1.23) is characterized by coefficients
ak
4
1
 (t , n)  1  cos nt  cos 2nt
3
3
(1.35)
IV. A PPLICATION TO A HYDROELECTRIC GENERAT ING UNIT
In this section, let us consider a 100 MW run-of-river
hydroelectric generating unit in France as an application
example for the above approach. In this example, actual
parameters are:
G1 = 25, G2 = 0.8, T1 = 10 s, T2 = 20 s, 
 12 s,   0.8
Fn = 50 Hz, Pmax = 100 MW and K = 50 MW/Hz
Experimental measurement signals (see Figure IV.1) are
available with a non-uniform sampling period: Ts  1, 3  s .
A. Parameter Estimation
Parameters are assumed invariant in a narrow window
[0,T] where T (= 500 s) represents only 50 times the largest
time constant (T1 ).
We define the following notations:
X  X (n)  x, (n)
(1.36)
X i  X i (n)  x, (i ) (n)
(1.37)
Figure IV.1: Industrial process data
(1.38)
In tab. I, a comparison is given between the actual
parameters and parameters estimated for t   0, T  by the two
By applying the integral transform (1.21) with the kernel
(1.33) to the equation (1.16) and using notations (1.36)(1.38),
we can write :
discussed methods: Savitzky-Golay (SG) and Fourier series
(FS). To obtain good performance with respect to
measurement noise, the SG filter is here parameterized with
d = 2 and M = 41.
H (n)  h , (n)
T
n  1,..., N
Y (n)  H (n)
(1.39)
Numerical results show a better matching for the Fourier
series method especially for the time constant T1 .
with the regressor vector:
H (n)  U1  U11 U 2  U 21  Y 2 Y 1  (n)
Hence we have a linear regression problem in a T-large
window. Due to the number p + q + 1 of parameters to be
estimated in θ, we consider a sufficiently large 2 number N of
equations (1.39):
 Y (1)   H (1) 




 

Y ( N )   H ( N ) 
 Y  H
(1.41)
Under sufficient excitation indeed, system (1.41) can
become invertible, and an estimate of θ be obtained as:
   H T H  H TY
1
(1.42)
Finally, physical parameters (gains and time constants) are
easily deduced from equations given in appendix, as well as
gain K (from G1 and equation (1.3)):
K  G1
2
Pmax
Fn
TABLE I.
(1.40)
(1.43)
Note that N should not be chosen too large either, in order to keep enough
accuracy in the computation of integrals from sampled measurements.
NUMERICAL RESULTS
G1
G2
T1
T2


Actual
25
0.8
10
20
12
0.8
SG
24.32
0.76
3.25
18.24
9.41
0.90
FS
24.96
0.80
11.16
20.39
13.05
0.76
From relation (1.43), we find K = 2G1 that is 49.92 MW/Hz
with FS method compared to the expected value 50 MW/Hz.
B. Estimation of Pco(t) and Pr
From estimation (1.42), the gain G2 is known and thus pc
can be calculated from (1.15).
The signal p co (t) is time bounded on [0, T] and we assume
that pco  L2 ([0, T ]) . In order to expand pco (t ) in a Fourier
series (see [22]), we extend it by parity on [-T, T] then we
extend it again on
with the period T0  2T .
Thus, with the angular frequency 0  2 / T0   / T , we
have the following finite Fourier series:
N0
pco (t )   ak cos k0t
(1.44)
k 0
For our application, we limit this expansion to 3 harmonics
in (1.14) we have a linear
 N0  3 , and by injecting (1.44)
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[3]
regression problem computed on the interval [0, T]:
pc  1 cos 0t cos 20t cos30t u3 
   a0 a1 a2 a3 Pr 
T
(1.45)
[4]
(1.46)
Least squares calculus yields  which permits to
reconstruct the signal Pco (t) shown in Figure IV.2. For the
parameter Pr , the estimated value is 4.8 MW compared to 5
MW for the actual value.
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
Figure IV.2: Comparison between pco actual and estimated
V. CONCLUSION
In this paper, a Fourier series approach has been considered
to solve a MISO continuous-time LTV system identification
problem and to estimate an unknown input from irregularly
sampled data. Experimental data from a hydro generating unit
allowed to evaluate the proposed estimator, which indeed has
given a good matching between expected and estimated
parameters. The proposed Fourier kernel has the advantage to
be simple, easy to implement and applicable for ODEs of any
order. This method is also valid for other types of power plants
(wind turbine, PV, nuclear, thermal power plants).
APPENDIX
Physical parameters and identification parameters are
linked with the following relations:
1  G1 ;  2  G1T1 ; 3  G2 ;
 4  G2T2 ; 5   2 ; 6  2
(1.47)
and reciprocally:
G1 =1 ; G2  3 ; T1   2 / 1 ;
T2   4 / 3 ;   5 ;  
6
2 5
(1.48)
[13]
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