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Comparative performance study of QN and LM algorithms in predictive
control for NNARX-identified model of hydro-power plant
Article in Engineering with Computers · June 2007
DOI: 10.1007/s00366-006-0024-z
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Engineering with Computers (2007) 23: 71–78
DOI 10.1007/s00366-006-0024-z
O R I GI N A L A R T IC L E
Nand Kishor Æ S. P. Singh Æ A. S. Raghuvanshi
P. R. Sharma
Comparative performance study of QN and LM algorithms
in predictive control for NNARX-identified model of hydro-power plant
Received: 11 January 2005 / Accepted: 26 January 2006 / Published online: 30 August 2006
Springer-Verlag London Limited 2006
Abstract In this paper input-constrained predictive
control strategy for NNARX (neural network non-linear auto-regression with exogenous signal) model of
hydro-turbine is presented. The input (gate position) and
output (turbine power) data are generated by means of
dynamic plant model. The collected data are utilized to
develop the NNARX model of the plant. Then NNbased predictive control (NNPC) scheme is applied to
control the turbine power. The control cost function
(CCF) includes the squared difference between the
model predicted output and desired response and a
weighted squared change in the control signal. The CCF
is minimized with both Quasi-Newton and Levenberg–
Marquardt iterative algorithms. To demonstrate the
suitability of the strategy, the plant has been simulated
on two different reference signals.
Keywords Cost function Æ Hydro-plant Æ
Identification Æ Neural network Æ Predictive control
List of symbols
surge ; H
turbine
H
head available in riser of surge
tank, turbine (p.u.)
penstock
tunnel ; H
head loss in tunnel, penstock
H
(p.u.)
turbine ; U
surge velocity of water in tunnel,
tunnel ; U
U
turbine, surge tank (p.u.)
noload
velocity of water at no load (p.u.)
U
ftunnel, fpenstock
head loss coefficient in tunnel,
penstock (p.u.)
Csurge
storage constant of surge tank (s)
devia;tunnel
head deviation in tunnel (p.u.)
H
devia;penstock
head deviation in penstock (p.u.)
H
S. P. Singh
Department of Electrical Engineering, Motilal Nehru National
Institute of Technology, Allahabad, India
N. Kishor (&) Æ A. S. Raghuvanshi Æ P. R. Sharma
Department of Electrical Engineering, Royal Bhutan Institute of
Technology, Phuentsholing, Bhutan
E-mail: nand_research@yahoo.co.in
Zp
Tw
Tep
Twc
L
U0
g
H0
Dx
Gopening
Dturbine
Again
hydraulic surge impedance of
penstock (=Tw/Tep)
water time constant in penstock
(s) ð¼ LU0 =gH0 Þ; Tw varies
with load
elastic time constant of
penstock (s)
water time constant in tunnel (s)
length of penstock
velocity of water in penstock at
rated head (p.u.)
acceleration due to gravity
rated turbine head (p.u.)
deviation of rotor speed (p.u.)
gate opening (p.u.)
damping factor or coefficient in
p.u. torque/p.u. speed deviation
turbine gain
1 Introduction
A hydro-turbine is non-linear, non-stationary system
whose characteristics vary significantly with unpredictable load. The turbine model considered in the design of
the governor plays an important role in the efficient
operation of the power plant. At present, linear control
theory based PID controller finds its application in the
power plant.
Numerous methods for PID tuning are reported in
the literature [1]. To realize parameter optimization of
PID controller, an orthogonal test strategy is adapted in
[2–5] for hydro-turbine control application. In this
approach, control performance index is defined, which
depends on control parameters Kp, Ki and Kd. Each of
these parameters is considered under various levels as
discrete variable. An optimization algorithm is developed to search for better control parameters in the
neighboring space of the present ones. Lansberry and
Wozniak [6] have used genetic algorithm (GA) approach
72
for optimal governor tuning. The work investigates
the GA as one possible means of adaptively optimizing the
gains of proportional-plus-integral governors. Yamamoto et al. [7] and Cheng et al. [8] present the concept of
intelligent tuning of PID controller. The former authors
have discussed the use of adaptive and learning control
scheme—which is neural network techniques—while the
latter authors have presented an improved dynamic performance of the intelligent PID controller over the conventional PID. The developed intelligent PID controller is
based on an anthromorphic intelligence.
Fangtong et al. [9] introduced the dynamic modeling
of hydro-turbine generating set as a single machine with
region load using recursive least square estimation
algorithm. Also, non-linear model with NN structure
3-2-1 was discussed in the study [10]. The non-linear
simulation of hydro-turbine governing system based on
NN is described in [11]. Two three-layered perceptrons
NN1 and NN2 of structures 1-4-1 and 2-12-2, respectively, are considered. The former structure forms a nonlinear relation between servomotor stroke and guide
vane opening while the latter structure between guide
vane, speed efficiency and discharge.
The modeling of dynamic interaction between the
gate (input) and turbine mechanical power (output) is
important as it regulates the operation of the governor
action. It is difficult to obtain a non-linear simulation
model of the hydro-turbine. The conventional characteristic curves of hydro-turbine do not provide sufficient
insight into non-linear simulation.
A great deal of attention has been drawn towards its
linearized modeling. However, detailed hydraulic system
model, which should include compressible and elastic
effects of penstock, is required to ensure effective control. These effects represent a delay e2sTe in the
hydraulic structure, which is irrational. As a result a
dynamic interaction between the hydraulic system and
the electrical system exists. Also while ensuring an
effective control action, it becomes necessary to use a
reduced order turbine–penstock model including delay
effects, especially of long penstock in hydro-power plant.
In recent years, many different types of NN-based
control systems have attracted increased interest. NN is
either trained to operate as a controller or a conventional control scheme is utilized with NN model of the
plant/process. An example of the latter is the NN-based
predictive control (NNPC) strategy. With NNPC, a
neural network model is developed to identify the nonlinear dynamics of the plant and a predictive control
strategy is implemented, being the sequence of control
actions computed iteratively by solving the control cost
function (CCF) [12]. An important feature of predictive
control is the ability to handle constraints of actuated
variables and internal variables [13].
This paper presents predictive control scheme adapted for hydro-plant NNARX (neural network non-linear
auto-regression with exogenous signal) model. A feedforward multilayer perceptron neural network
(MLPNN) with 15 hidden neurons is proposed for
identification. The CCF in non-linear predictive control
is minimized iteratively by both Quasi-Newton and
Levenberg-Marquardt approaches. The minimization is
computed assuming input constraints as (1) umin=¥
and umax=¥ and (2) umin=0 and umax=1.0. An
appropriate selection of control parameters is illustrated.
The predictive control technique based on identified
NNARX turbine model is investigated on two different
set-point reference signals (turbine power).
2 Basic power plant equations
As shown in Fig. 1, when the turbine-generator unit is
started, water from the reservoir flows into the tunnel
and then to a high-pressure conduit called penstock. The
water from the penstock flows into a scroll casing, which
distributes it evenly on the runner blades, mounted on
the common shaft with the generator. This results in
electro-mechanical conversion of power. The wicket gate
is operated by means of the governor to regulate the flow
as a function of variable electrical power. The sudden
closure of gates leads to pressure wave set-up moving
upstream in the penstock and increase in pressure at
turbine input (water hammer). The pressure wave
occurrence is due to penstock-wall elasticity and water
compressibility properties. In the case of hydro-plant
layout having a long penstock, the travel time of wave is
significant [14]. The riser in the surge tank helps to
suppress water hammer effect and the tank acts to store/
supply water temporarily. Practically, the water levels in
the riser and tank remain the same. The dynamic
equations of hydraulic and mechanical/electrical systems
in power plant are described [15].
Dynamics of tunnel:
Water flow through the tunnel subjected to frictional
forces leads to head loss given as
tunnel ¼ ftunnel U
tunnel jU
tunnel j:
H
ð1Þ
The dynamics of head variation in the tunnel is
devia;tunnel ¼ Twc dUtunnel :
ð2Þ
H
dt
Dynamics of surge tank:
Riser (surge tank) head:
surge ¼ 1:0 H
devia;tunnel ;
H
H
Z tunnel
surge dt:
surge ¼ 1
U
H
Csurge
ð3Þ
ð4Þ
Dynamics of penstock:
The water flow in the penstock being subjected to
friction causes a head loss given as
penstock ¼ fpenstock U
2
;
ð5Þ
H
turbine
Hdevia;penstock ¼ Zp tanh Tep s Uturbine :
ð6Þ
The available head at the turbine inlet is
surge H
penstock H
devia;penstock :
Hturbine ¼ H
ð7Þ
73
Reservoir
Surge tank
fsigmoid ðUðtÞÞ ¼
Tunnel
ð10Þ
while the closely related hyperbolic tangent function is
defined as
Generator
Penstock
Hydro-turbine
ftanh ðUðtÞÞ ¼ 2fsigmoid ðUðtÞÞ 1:
ð11Þ
These functions being non-linear thus describe the
nature of the NN. Cybenko [17] and Funahashi [18]
have proved that a MLPNN with a single hidden layer
of sigmoidal or hyperbolic tangent function can
approximate any continuous function. In the present
study, tanh is used as activation function for neurons in
hidden layer. A typical block diagram of NNARX
model for identification is shown in Fig. 2.
Turbine gate
Fig. 1 A general layout of hydro-power plant
Mechanical power:
The mechanical power developed is
noload Þ Pdamping ;
turbine ðU
turbine U
Pmech ¼ Again H
Pdamping ¼ Dturbine Gopening Dx:
1
;
1 þ e/ðtÞ
ð8Þ
ð9Þ
The term Pdamping represents the damping effect due to
friction and is proportional to the rotor speed deviation
and the gate opening. This term is important in the
modeling of the turbine for generator start-ups and
other emergency situations. Non-linear models of power
plant are used in study involving large perturbations,
which normally occur during an islanding, load rejection
and system restoration conditions. In the present work,
the aim is to identify the NNARX model showing the
turbine power Pmech dynamics with the random variation
in gate position and develop a predictive controller to
track the given reference signals.
3 NNARX model identification
Over the last one decade, NN has been considered a
promising approach in system identification. A blackbox non-linear mathematical model that relates the inputs and outputs of any system may be developed with
NN. This approach does not need an exact mathematical model of the system. Narendra and Parthasarathy
[16] proposed the use of neural network in conjunction
with system theory to develop realizable models. With
the availability of such NN model, various control
techniques may be applied for designing an efficient
controllable system.
A MLPNN structure is developed to model the nonlinear dynamic relationship between the gate position
and turbine mechanical power. A feedforward MLPNN
is formed with layers of neurons between the input and
output layers called hidden layers. The hidden layer
neurons act as connecting element between neurons of
intermediate layers. The inputs to each neuron are
combined along with a bias (if any) and the neuron
produces an output if the sum of inputs exceeds a
threshold value of activation function. Generally, the
sigmoid function/hyperbolic tangent function is used as
activation function.
The sigmoid function is given as
3.1 Methodology
The dynamic model represented by (1)–(9) is simulated
to generate the data. The hydro-power plant parameters
used in the study are mentioned in Appendix. The
absolute value of pseudorandom binary signal is applied
to the input to represent the variation of gate position.
And the corresponding turbine mechanical power is
computed. The collected data (input and output) are
divided into two sets, one for training the NN and the
other for validation. Thus an input–output model is
established, resulting in black-box identification approach. The hydro-power plant can be represented in
discrete input and output form by the identification
structure:
^y ðk; hÞ ¼ f ðUðk; hÞÞ þ nðkÞ:
ð12Þ
The NNARX regressor vector is expressed as
UðkÞ ¼ ½yðk 1Þ; . . . ; yðk na Þ;
uðk nk Þ; . . . ; uðk nb nk þ 1ÞT ;
ð13Þ
where
na, nborder of the model structure
nk delay in the input signal
nn=[ na, nb, nk ] is the NNARX model structure.
y (k−1)
y (k−na)
u(k−nk)
NNARX model
structure
(Feedforward
multilayer neural
network)
u(k−nk − nb + 1)
Fig. 2 A typical non-linear neural network ARX structure
ŷ(k)
74
The identification cost function (ICF) formulated as
the mean square error on a training data is minimized
with respect to network weights. This is defined by
J¼
N2
X
½rðk þ iÞ ^y ðk þ iÞ2 þ
i¼N1
Nu
X
q½Duðk þ i 1Þ2
i¼1
ð15Þ
N
N
1 X
1 X
VN ðh; ZeN Þ ¼
½yðtÞ ^y ðtjhÞ2 ¼
n2 ðtjhÞ;
2N t¼1
2N t¼1
ð14Þ
where
N number of training data samples
Ze Na vector which contains the system output and
regression vector U (t), =[y(t)U (t)]
The parameter estimate ^
h ¼ arg min VN ðh; ZeN Þ:
In the development of NNARX model structure, the
NN System Identification Toolbox for MATLAB
developed by Nørgaard [19] is utilized.
3.2 Training parameter
The collected data set is divided into two sets, one for
training and the other for validation of the model.
During network training, the hidden layer weights were
initialized automatically. The learning rate g=1e 4
while the momentum coefficient a=0. The network
training is terminated when either the maximum error
gradient is less than 1e 4 or when the number of iteration exceeds 500. A Gauss–Newton based Levenberg–
Marquardt method is utilized for minimization of meansquare error criteria (VN).
3.3 Optimal network structure
The non-linear model representations have great difficulty in selection of appropriate model order. To select
an optimum NN structure, the influence of variation in
vectors: past input, past output data and number of
hidden layer neurons (HLNs) on ICF is studied. After a
few simulation trials, the NN configuration, HLNs,
H=15 and nn=[ 1,10,1 ] has been determined.
and identified NNARX model, it is possible to calculate
the optimal control strategy for a non-linear turbine
control. The term r(k+i) is the required reference turbine power, ^y ðk þ iÞ the predicted NN model output,
Du(k+i 1) the controlled gate position, N1 and N2 the
minimum and maximum prediction horizons, respectively, Nu the control horizon and q a control penalty
factor. The control approach uses a receding horizon
strategy. The basic concept of the receding horizon
control is to solve an above defined (15) optimization
problem for a finite future at a current time and to
implement the first optimal control input as the current
control input. At each sample time k, minimizing CCF
for the selected values of the control design parameters [
N1, N2, Nu, q ] calculates the vector u=[ Du(k),Du(k+1),...,Du(k+Nu 1) ]. The performance of predictive control is largely affected by these parameters. N1
is usually set to a value 1 and N2 is set to define the
prediction horizon (i.e. the number of time-steps in the
future for which the plant response is recursively predicted).
4.1 Control law
With identified NNARX plant model, an iterative
search method is applied to minimize the CCF. The
Quasi-Newton and Levenberg–Marquardt algorithm is
used in the calculation of suitable controlled gate position [21]. The minimization is computed with input
constraints as (1) umin= ¥ and umax=¥ and (2)
umin=0 and umax=1.0.
4.2 Selection of control parameters
In order to determine the appropriate value of {N1, N2,
Nu, q }, the influence of control penalty factor q and
maximum prediction horizon N2 on CCF is analyzed in
the following sections.
4.2.1 Quasi-Newton
4 Non-linear predictive control
Taking advantage of the recognized universal
approximation properties of NNs, a non-linear plant
in NNARX form is obtained using feedforward
MLPNN. Based on the neural model, a predictive
control strategy is implemented, being the sequence of
control actions computed by solving an optimization
problem iteratively.
By the methodology adapted using a weighted sum of
quadratic CCF [20]:
With input constraints as umin= ¥ and umax=¥ it is
observed after several trails that a minimum value of
CCF is obtained at q=0.7, q=0.8, q=0.9 and q=1.0
with least horizon values of N1=1,N2=10 and Nu=1.
This is presented in Fig. 3a. Next the same is determined
with variation of maximum prediction horizon N2.
Figure 3b illustrates a minimized CCF value for
N1=1,N2=12,Nu=1 and q=0.7. Selecting umin=0 and
umax=1.0 as input constraint the CCF surface is
illustrated in Fig. 4. The appropriate parameters are
75
Fig. 3 The CCF optimization
(umin= ¥ and umax=¥) using
QN algorithm. a Effect of
control penalty factor. b Effect
of maximum prediction horizon
Fig. 4 The CCF optimization (umin=0 and umax=1.0) using QN
algorithm
Fig. 6 The CCF optimization (umin=0 and umax=1.0) using LM
algorithm
observed as: N1=1,N2=12,Nu=1 and q=1.0 to
q=0.09.
with least horizon values of N1=1,N2=10 and Nu=1 is
obtained. This is shown in Fig. 5a. And Fig. 5b presents
an optimized value for N1=1,N2=16,Nu=2 and q=0.9.
The CCF surface with variation in control parameters
selecting umin=0 and umax=1.0 is illustrated in Fig. 6.
The parameters are appropriately determined as
N1=1,N2=12,Nu=1 and q=1.0 to q=0.09.
4.2.2 Levenberg–Marquardt
With umin= ¥ and umax=¥ using Levenberg–Marquardt algorithm, the optimization at q=0.9 and q=1.0
Fig. 5 The CCF optimization
(umin= ¥ and umax=¥) using
LM algorithm. a Effect of
control penalty factor. b Effect
of maximum prediction horizon
76
Fig. 7 The QN-based predictive control of set-point trajectory (reference-1), turbine output and control action. a umin= ¥ and umax=¥.
b umin=0 and umax=1.0
Fig. 8 The QN-based predictive control of set-point trajectory (reference-2), turbine output and control action. a umin= ¥ and umax=¥.
b umin=0 and umax=1.0
5 Simulation results
A sequence of reference tests are conducted to investigate initial start-up and set-point following. The identified NNARX turbine model with control design
parameters as summarized in Table 1 is simulated on
two different set-point step reference signals. The model
output with NN-based predictive controller solved by
QN algorithm is illustrated in Fig. 7 (reference-1). It is
seen that a good set-point tracking is achieved for all
steps with both types of input constraints. The response
shows a non-minimum phase characteristic of hydro-
turbine. And the control signal is maintained within a
modest magnitude. Keeping the same control parameters, simulation results on another desired output of
different amplitudes step function (reference-2) is shown
in Fig. 8. A poor set-point tracking is optimized with
umin= ¥ and umax=¥ as input constraint. Also the
gate position variation does not remain in prescribed
limits. On the other hand the result obtained with
umin=0 and umax=1.0 exhibits a close and offset-free
reference tracking and gate position variation within
limits.
Next the controller performance using LM optimization approach is demonstrated in Figs. 9 and 10. An
Fig. 9 The LM-based predictive control of set-point trajectory (reference-1), turbine output and control action. a umin= ¥ and umax=¥.
b umin=0 and umax=1.0
77
Fig. 10 The LM-based predictive control of set-point trajectory (reference-2), turbine output and control action. a umin= ¥ and
umax=¥. b umin=0 and umax=1.0
Table 1 The control design parameters for predictive control approach
Algorithm
Quasi-Newton
Levenberg–Marquardt
Control parameters
Input constraint:
umin= ¥ and umax=¥
Input constraint:
umin=0 and umax=1.0
N1=1,N2=12,Nu=1,q=0.7
N1=1,N2=16,Nu=2, q=0.9
N1=1,N2=12,Nu=1, q=1.0
N1=1,N2=12,Nu=1, q=1.0
output response without overshoot and oscillation has
been obtained as similar to QN algorithm based strategy. However, the gate position variation is oscillatory
to some extent as can be seen in Fig. 9a. And in practice,
this would cause undue actuator wear. However, the
CCF optimized with umin=0 and umax=1.0 results in
good performance as illustrated in Fig. 9b.
When the CCF is optimized on reference-2 with
umin= ¥ and umax=¥ as input constraint, a poor setpoint tracking and gate position variation beyond the
prescribed limits are obtained. This is illustrated in
Fig. 10a. Thus suitable controlled gate position has not
been found to make the plant output follow the setpoints. The performance on second input constraint as
presented in Fig. 10b is in close resemblance to reference-1.
This suggest the controllers are capable to follow
adequately well the set-point trajectory, regardless of
set-points values and difference in level between each
reference.
6 Conclusion
Based on the identified turbine NNARX model, this
paper presented a non-linear predictive control scheme.
The CCF was solved using Quasi-Newton and Levenberg–Marquardt algorithms with input constraints as (1)
umin= ¥ and umax=¥ and (2) umin=0 and umax=1.0.
Also the study was carried out to show the influence of
control parameters on CCF. A relatively good set-point
tracking was shown with CCF having input constraint
umin= ¥ and umax=¥ minimized using QN approach.
On the other hand there was no significant difference
between the two optimization algorithms in set-point
tracking performance optimized with input constraint
umin=0 and umax=1.0. Thus both the approaches’ offer
as an effective computational method to drive the
NNARX model output follows the set-points.
7 Appendix
Parameters of the system studied:
Zp (Tw/Tep)=6.335; Tw=2.23 s; Tep=0.352 s;
Twc=39.15 s; Csurge=240 s; ftunnel=0.0475 p.u.;
noload ¼ 0:13 p.u.; D
fpenstock=0.089 p.u.; Again=1.67; U
x=0.05; Dturbine=2.0.
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