Fuzzy Models for the Study of Hydro Power Plant Dynamics

2006 International Symposium on Evolving Fuzzy Systems, September, 2006

Fuzzy Models for the Study of Hydro Power Plant Dynamics

Nand Kishor, Student Member, IEEE, S. P. Singh, A. S. Raghuvanshi and P. R. Sharma

Abstract- In this paper, the hydro power plant dynamics isidentified using fuzzy models. The plant data is generatedfrom Pade and H-infinity approximated first, second, thirdand fourth-order rational transfer function models. Themodels are simulated as (i) gate-servo motor position andturbine speed with random load disturbance and (ii) gateposition and developed turbine power. Takagi-Sugeno fuzzymodel structures are identified with smooth stepped wavesignal input and the identified model is generalized on itsvalidation data set and with random stepped wave signal asinput. The fuzzy rules are extracted from data by means ofGustafson-Kessel clustering with antecedents determinedusing product-space and point-wise projection techniques.

Keywords- Fuzzy model, Hydro plant, Identification

I. INTRODUCTION

The power system dynamic response can be representedby different types of models, which in turn are required forstudy of low frequency oscillations, islanding and isolatedoperation, system restoration following a break-up, loadacceptance and water-hammer dynamics in penstock, etc.The approximation of a high order system by a low orderseeks importance as it involves less computation time of thetransient response [1] and thus suitable in the controllerdesign and control system analysis. The cost andcomplexity ofthe controller increase with the system order.Of the numerous literatures [2-4] on the governor design

or stability studies of the hydro plant, a simple linear modelneglecting the presence of water column elasticity effect inpenstock has been considered. The system identification isan established field in the area of system analysis andcontrol. It deals with the problem to determine simplifiedmathematical model of dynamical systems, based on input-output time-series measurement data [5]. An optimizationtechnique is applied to search a model from a space ofpossible models. There are numerous methods reported inthe literature to construct a fuzzy model from availableinput-output data [6].To mention here, neural networks (NNs), neuro-fuzzy [7],data clustering [8] etc are some of the methods.

Nand Kishor is as research scholar in the Alternate Hydro Energy Centre,Indian Institute ofTechnology Roorkee, IndiaS. P. Singh works as Assistant Professor in the department of electricalengineering, Indian Institute ofTechnology Roorkee, India.A. S. Raghuvanshi is as a lecturer in the department of electricalengineering, Royal Bhutan Institute of Technology, BhutanP. R. Sharma is associated to department of electrical engineering, RoyalBhutan Polytechnic, Bhutan

In a number of engineering and non-engineering applications [9-11] fuzzy modeling approach has been applied in last fewdecades. This paper presents an application of fuzzy modelingfor prediction of hydro turbine developed power and speed withvariation in its gate position. Input-output relationship by fuzzyif-then rules is used in fuzzy models.

II HYDRO-PLANT MODELING

For a small variation around an operating point, the linearisedequation of turbine using Taylor series approximation can berepresented as given below [ 12]:Aq = allAh +a 12Ag +a3w (1)

APm = a21Ah+a22Ag + a23Aw, (2)

The turbine constants a, are the partial derivatives of flowand torque with respect to head, guide-vane position and turbinespeed. The a, coefficients depend on turbine loading and maybe evaluated from the turbine characteristics at the operatingpoint.

Their values remain constant for variation near the turbinerated operating point ( q0, p9). These values have to bemeasured accurately or taken from turbine model tests. Thepenstock transfer function (TF) relating the incremental headand flow in terms of complex frequency s can be written as[12]:H(s) = -Z tanh(sTe + F) (3)Q(S) e

The elastic effect in hydro plant is represented by a delaye 2STe in the hydraulic structure. H-infinity [13] and Padeapproximation methods [14] have been utilized in thedevelopment of rational low order transfer function (TF) andtheir response was simulated to find its suitability in the systemstudies [15]. An irrational equation of penstock-turbine havingneglected the hydraulic friction losses from eqn (1)-(3) thatrelates the ratio of incremental torque to changes in turbine gateposition, approximated using Pade and H-infinity method isgiven as: [15]A. Pade method ofapproximation

Gf(s) APml(s) 0.839 -ZP (0.3973d)s (4a)I

AG, (s) I + Zp (0.15d )s

0.839 Z (0.368475d)s-0.0862d2s22 1 + Z (0.139119d)s + 0.102747d2s2

Gp ( 0.839 - Zp (0.423006d)s + 0.082935d2'2 - Zp (0.010279d3)s31+ Zp (0.159705d)s + 0.098504d's' + Zp (0.003 8809d3)

(4b)

(4c)

p 0.839-Zp(0.373951d)s+0.09571d's' -Zp(0.009338d')s' +0.0009104d4s

1+ Zp(0. 141184d)s + 0.1 14077d's' + Zp(0.003525d')s' + 0.0010852d4s

(4d)

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B. H-infinity method ofapproximation(s) APm(s) 0.839-Zp (0.4454d) s

AG,(s) l+Zp (0.1681d)s

G2 (s) =0.839 -Zp(0.3973d)s+0.069913d2s21 + Zp (0.15d)s + 0.08333d S

(5a)

(5b)

H ( ) 0.839- Zp(0.3973d)s+0.0839d's' -Zp(0.0066214d )s3 (5c)

1 + Zp (0. 15d)s + 0. IdS22 + Zp (0.002499d )s

H 0.839-ZP(0.3973d)s+0.0898904d2s2 Zp(0.0094557d3)s3 +0.0004993d4s

1+ZP(0. 15d)s+0.10714d2s +ZP(0.00357d3)s3 +0.00059523d4s

(5d)The transfer function of gate servomotor may be given as:

G, (s)= l1+S1+s (6)

The generator-turbine rotational inertia equation can bewritten as:

AWv (Pm - Pi - DAw) (7)

III. SIMULATED PLANT DATAThe above-mentioned eqn (1-7) may be presented in

block diagram as illustrated in Fig. 1. The plant is simulatedfor input and output variable as:(i) Gate-servo motor position and turbine speed with

random load disturbance applied. The dynamic eqn(4)-(7) excited with a stepped wave signal.

(ii) Gate position and developed turbine power: Thedynamic eqn (4) and (5) excited with a stepped wavesignal as shown in Fig. 2.

The collected input and output data at a sample rate of O.9sis divided into identification and validation data set of equalsize.

iv. FuzzY MODEL FOR IDENTIFICATION-PRELIMINARIES

Many approaches have been proposed to obtain fuzzymodels from input-output measurement data. Few to namethem, local linear tree method, tree construction algorithms,or variants of neuro-fuzzy approaches. Takagi-Sugeno (TS)fuzzy model with Gustafson-Kessel (GK) data clusteringalgorithm is proven to be suitable for identification [16].Computationally the said method is fast as compared toother nonlinear identification methods.

Loaddeviation Ap,

Au /NAg Apm Aw,Gate servo Hydraulic Rotational

motor system inertia

Fig. I Block diagram of hydro plant

--

A

v~Y

500 1000Samples [sec]

Fig. 2 Smooth stepped wave signal as input signal

This algorithm employs an adaptive distance norm in order todetermine clusters of different geometric shapes on the data set[17]. This paper presents the use of fuzzy model as a tool formodel identification of a hydro plant.

A. TS model identification using GK algorithmThis model consists of if-then rules with fuzzy antecedents

and mathematical functions in the consequent part. Theantecedent fuzzy sets partition the input space into a number offuzzy regions, while the consequent functions describe thesystem's behaviour in these regions. The development of a TSmodel takes place in two steps; in the first step the fuzzy sets i.e.membership functions belonging to rule antecedents arecomputed and in the second step, the parameters of theconsequent functions are determined using least squaresestimation method.

The following regression model represents nonlinear staticand dynamic systems:Yk =f (Xk ) (8)

where f (.) is the nonlinear function realized by fuzzy model,Xk represents the model input regression input vector andk = 1,... N represents the index of k - th available input-outputdata sample.

Model input regressor vector consists of past plant output Ykand plant input uk, i.e.

Xk = Yk-l 'Yk-, Ukk-l---k-,b] (9)Any N observations, each consisting of n measured

variables, may be grouped into a n - dimensional column vector

Zk [ZlkI... IZ,k]IT

ZkcE (I 0)The input-output data samples of gate position and turbine

output; power/speed are determined and a regression matrix Xand output vector y are formed as [ 18]:

X [Xl,...XN]T (11)

Y= [Y ...YN ] (12)HereN >> n is the number of samples used for identification.This results in approximation of plant by local linear models.The data set to be partitioned is written as:

236


Fuzzy partition of data set Z is a family of fuzzy subsets{A1l <.i<NC} (14)The subsets are defined by their membership functions,represented in the partition matrix form as:

U=[Aik]N1N (15)

The i- th row of this matrix contains values of themembership function of the i-th fuzzy subset Ai of Z.

Through clustering approach, the vectors zk, k =1,2, ...,N,contained in the columns of the data matrix Z, arepartitioned into NC clusters, represented by their

prototypical vectors Vi = [Vil' '... v E <in i = 1,..., NnIThe number of clusters is determined by testing errorcriteria. The GK algorithm is used to determine the fuzzypartition matrix U. The GK algorithm tends to find clustersof different shapes and orientations based on theminimization of objective function:

NC N

J(Z,V,U) E 2(ik)mDkA (16)i=l k=l

subjected to partition matrix U that satisfies the followingconditions:AikE[^ 1]^ I < i < N, I < k < N, (I17a)

N,E fLik = 1,i11

N0 < ZI'Uik < N,

k=l

1 < k < N

I < i < N-

In eqn (16), m > 1 is a parameter that controls thefuzziness of the clusters. The usual setting with m = 2 issuitable for most applications [19]. The interpretation ofeqn (17a) suggests that the membership degrees are realnumbers in the range [0,1]. Eqn (17b) constraints sum ofeach column equal to one, i.e. total membership of each Zkin the entire clusters equal one. It is understood from eqn(17c) that none of the fuzzy subsets is empty nor it containsall the data.

B. Algorithm.For a given data set Z, choose the number of clusters

1 < Nc < N, the weighting exponent m > 1 and the test

error criteria £ > 0 [18]. Initialize the partition matrix U0randomly, such that it satisfies the condition (17a)-(17c).Repeat for iteration I = 1,2,....

Step 1: Compute cluster means:N (1-1) m

V., i = 1,2,...,IN

Nc

Step 2: Compute the cluster covariance matrices:

E, Pik Zk -V I LZk- i

F = k=1F (1-1) m

"=.ik1= 1,2, ...,I NC

Step 3: Compute the distances:

D1 =1 -V,N kdet (F)F1 z -v

i = 1,2,...,I NC, k = 1,2,...,I N

Step 4: Update partition matrix:for 1 < k < Nif Dik > 0 for 1 < i < NC,

(1ik _

(19)

(20)

1 .k < N,

i

ZDk D 2k] (m-1)(21)

(22)

Otherwise, y() = ° if Dk > 0 and/'ik =0 ifD >k

(/) _ [0, 1] with -1,)= Ii=1

until test criteriau (/) - u Q - ) < £ (23)

The structure ofthe model i.e. na, nb and nk (delay betweenthe input sample) are determined by the user on the basis ofplant's prior knowledge and / or by comparison of differentstructures based on test error criteria. Also the user can specifythe number of clusters and thus the rules. In this work, with atrial and error procedure, different cluster numbers are assigned,and validation satisfying error criteria is assessed on thevalidation data

The determination of antecedent membership functions withfuzzy clustering in the Cartesian product-space of the inputs andoutputs of membership function has been widely accepted.Using product-space approach, antecedent membership functionalong with the consequent local linear models and implicitregularization can be represented simultaneously. Each clustercorresponds to a fuzzy if-then rule. Product-space membershipfunctions are faster but often less accurate models. Anothermethod to derive the antecedent MFs is point-wise projection ofthe antecedent variables [ 18]. Its main advantage is thatinterpretation of the fuzzy model by using membershipfunctions defined for input variables can be obtained.

V. RESULTS AND DISCUSSION

To obtain TS fuzzy model based on GK clustering, the inputis excited with a smooth stepped wave signal as shown in Fig. 2.The data is collected at sample rate of 0.9 sec for (i) gate-servomotor position and turbine speed at random load disturbance

(18) and (ii) gate position and developed turbine power. The data setis divided into two halves; one as identification data (risingstepped wave) and remaining data (falling stepped wave) forvalidation to check the generalization capabilities of the model.The performance of fuzzy models is validated / assessed onvalidation data set. The following performance indices; the

237

z= [X, Y]T (13)


variance accounted for (VAF) and the root mean squareerror (RMSE) are used.

VAF = 1- var(y, -Y~i ) X100%var(y P) J

RMSE = i=l

N

(24)

(25)

where var denotes the variance, Yi is the actual plantoutput, 5' is the predicted output and N is the number ofdata sample.A higher VAF indicates the better closeness between the

two outputs. And lower the value of RMSE, the better theperformance ofmodel.

A. Gate-servo motor position and turbine speed atrandom load disturbance

The parameters of fuzzy models obtained through GKalgorithm for product-space and point-wise projection MFsevaluated to satisfy the performance indices are mentionedbelow:

(i) Product-space:Nc =3; m =5; test criteria

nk = 1.

(ii) Point-wise projection MFs:Nc =2; m =5; test criteria

0.01; na =2; nb = 2;

0.01; na =2; nb = 2;

nk =2.

Fig. 3 shows the performance of the fuzzy model for the4th order approximated plant model on validation data set.The regression analysis of the same model is shown in Fig.4. The quantitative analysis of performance index for first,second, third and fourth -order plant models are given inTable I(a). From the table it is observed that theperformance / response of the plant with point-wiseprojection MFs based on GK algorithm fuzzy model ismore accurate than the product-space. As can be observed,the prediction accuracy in terms of both performanceindices is better for the validation data sets than for thetraining data set. This concludes good generalizationcapabilities of the TS fuzzy model. It is also suggested thatthe performance of identified fuzzy model using datagenerated from both H-infinity and Pade transfer functionremain same.

To analyze the generalization property of the modelstructures considered above, the same identified structuresof TS fuzzy models are now excited with random steppedwave signal.

(a) Product-space (b) Point-wise projectionFig. 3Performance of TS fuzzy model with smooth stepped wave signal

(i) H-infinity (ii) Pade

(a) Product-space

>,./' '~~~~~~0.25

X1 005 _340.05 0.1 0.15 02 0.25 03 0.35 0.4 0 05 0validation data

(i) H-infinity(b) Point-wise projection

0.1 0 2 0.5 0 3 03

(H) Pad6

Fig. 4Regression analysis of TS fuzzy model with smooth steppedwave signal

Fig. 5 illustrates the performance of the fuzzy model for the 4thorder plant model when excited with a random stepped waveinput signal. The quantitative analysis of performance index forfirst, second, third and fourth-order transfer function models aregiven in Table I(b). A relatively better result is observed onvalidation data set for point-wise projection MFs.

B. gate position and developed turbine power.The details of fuzzy model obtained through GK algorithm forproduct-space MFs evaluated to satisfy the performance indicesis mentioned below:N =2; m=4;test-criterion=0.01; na =1; nb = 1; nk = 1.The quantitative analysis of performance index for first, second,third and fourth-order transfer function models are given inTable II. It is observed the property of identified fuzzy modelusing data generated from both H-infinity and Padeapproximated plant model remain same.

238


04L15

0-4 ----- Identified fuzy model 04 ----- Identified fuy model

0.35 0.35

0.3 -0.3

0.25 025 -

00150.2-

00.15[ ' 0'] a0015 'K

01 , 01 J

005 . 005

0 0-

0.05 -W/eO0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800Time [sec] Time [secl

(a) Product-space (b) Point-wise projection

Fig. 5 Performance of TS fuzzy model with random stepped wave signal

TABLE IPERFORMANCE INDICES OF FUZZY MODELS FOR INPUT-OUTPUT VARIABLE: GATE-SERVO MOTOR AND TURBINE SPEED WITH

RANDOM LOAD DISTURBANCE

(a) Identified and validated with smooth stepped wave as input signal

(i) Product-spaceSystem: H-infinity PadeOrder: First Second Third Fourth First Second Third FourthRMSE (trained) 0.2592 0.2584 0.2675 0.2604 0.2592 0.2584 0.2675 0.2604RMSE (validated) 0.0170 0.0236 0.0366 0.0149 0.0170 0.0236 0.0366 0.0149VAF (0) 98.3253 96.7897 92.4007 98.7399 98.3253 96.7897 92.4007 98.799

(ii) Point-wise projection MFsSystem: H-infinity Pade'Order: First Second Third Fourth First Second Third FourthRMSE (trained) 0.2638 0.2638 0.2638 0.2638 0.2638 0.2638 0.2638 0.2638RMSE (validated) 0.0072 0.0072 0.0072 0.0072 0.0072 0.0072 0.0072 0.0072VAF (%) 99.7290 99.7264 99.7283 99.7266 99.7290 99.7264 99.7283 99.7266

(b) Identified and validated with random stepped wave as input signal

(i) Product-spaceSystem: H-infinity PadeOrder: First Second Third Fourth First Second Third FourthRMSE (trained) 0.2650 0.2638 0.2651 0.2638 0.2650 0.2638 0.2651 0.2638RMSE (validated) 0.0347 0.0363 0.0347 0.0364 0.0347 0.0363 0.0347 0.0364VAF (Oo) 97.0340 97.03 15 97.0502 97.0507 97.0340 97.03 15 97.0502 97.0507

(ii) Point-wise projection MFsSystem: H-infinity Pade'Order: First Second Third Fourth First Second Third FourthRMSE (trained) 0.2688 0.2684 0.2686 0.2685 0.2688 0.2684 0.2686 0.2685RMSE (validated) 0.0247 0.0250 0.0247 0.0250 0.0247 0.0250 0.0247 0.0250VAF 97.9064 97.9262 979175 979206 979064 97T9262 97L9175 979206

TABLE, II

239

0.45"'' Plant model I Plant model I


6

PERFORMANCE INDICES OF FUZZY MODELS FOR INPUT-OUTPUT VARIABLE: GATE-POSITION AND DEVELOPED TURBINE POWER

System:Order:RMSE (trained)RMSE (validated)VAF (0 o)

H-infinityFirst0.52620.057095.5736

Second0.54040.045298.0946

Third0.52630.057195.5465

VI. CONCLUSION

As observed with the discussion in the above sections, it isconcluded that TS fuzzy model based on GK algorithmprovides a good scope for modeling the hydro plant. Thedetermination of antecedent membership functions withfuzzy clustering by point wise projection MFs computedbetter results than product space. The performance indicesdetermined from TS fuzzy models for H-infinity and Padeapproximated first, second, third, fourth-order dynamictransfer functions do not vary substantially. The identifiedmodel structure is found to be generalized one.

REFERENCES

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[2] S. Hagihara, H. Yokota, K. Gode, K. Isobe, "Stability of a hydraulicturbine-generating unit controlled by PID governor", IEEE TransPower Syst Apparatus, vol. 98, pp. 2294-2298, 1979.

[3] L. N. Hannett, B. Fardanesh, "Field test to validate hydro turbine-governor model structure and parameters", IEEE Trans Power SystApparatus, vol. 9, pp. 1744-1751, 1994.

[4] L. N. Hannet, J. W. Feltes, B. Fardanesh, W. Crean, "Modeling andcontrol tuning of a hydro station with units sharing a commonpenstock section", IEEE Trans Power Syst, vol. 14, pp. 1407-1414,1999.

[5] J. J. Flores, N. Pastor, Time-invariant dynamic systems identificationbased on the qualitative features of the response, Engg. ApplicationofArtificial Intell. in press.

[6] 0. Nelles, A. Fink, R. Isermann, "Local linear models trees(LOLIMOT) Toolbox for nonlinear system identification, 12th IFACSymposium on System Identification (SYSID), Santa Barbara, USA,2000.

[7] J.-S. R. Jang, "ANFIS: Adaptive-network based fuzzy inferencesystem", IEEE Trans Systems, Man and Cybernetics, vol 23, pp.665-685, 1993, 1993.

[8] R. Babuska, "Fuzzy modeling for control. Boston, MA: Kluwer,1998.

[9] T. Takagi, M. Sugeno, "Fuzzy identification of systems and itsapplications to modeling and control, IEEE Trans Syst ManCybernet, vol. 15, pp.15:116-132, 1985.

[10] M. Sugeno, K. Tanaka, "Successive identification of a fuzzy modeland its applications to prediction of a complex system", Fuzzy SetsSyst, vol. 42, pp.315-334, 1991.

[11] R. Babuska, H. B. Verbruggen, "A overview of fuzzy modeling forcontrol", ContrEnggPract, vol. 4, pp.1593-1606, 1996.

[12] C. K. Sanathanan, "Accurate low order model for hydraulic turbine-penstock", IEEE Trans Energy Conversion, vol. EC-2, pp.196-200,1987.

[13] S. H. Al.-Amer, F. M. Al-Sunni, "Approximation of delay systems",Proc. American Control Conference. Chicago, Illinois, June 2000,pp. 2491-2495.

[14] J. Lam, "Model Reduction of delay systems using Padeapproximations", Int. J. Control, vol. 57, pp.57: 377-391, 1993.

PadeFourth First Second Third Fourth0.5411 0.5262 0.5404 0.5263 0.54110.0446 0.0570 0.0452 0.0571 0.044698.0020 95.5736 98.0946 95.5465 98.0020[15] N. Kishor, R. P. Saini, S. P. Singh, "Most appropriate rationalized transfer

function with elastic water column effect", Proc. Instrumentation &Control Engg. Conference, Tiruchirappalli, India, Dec. 4-6, 2003.

[16] D. E. Gustafson, W. C. Kessel, ,,Fuzzy clustering with a fuzzy covariancematrix", Proc. ofIEEE CDC, San Deigo, CA, 1979, 761-766.

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[18] M. A. Grima, R. Babuska, "Fuzzy model for the prediction of unconfinedcompressive strength of rocks samples", Int. J of Rock Mechanics andMining Sciences, vol. 36, pp.339-349, 1999.

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NOMENCLATURE

Awlgrl\g

AhAqAp,APnAPC(s)AG(s)

AQ(s)AH(s)hgw

qzp

Tw

Te

DFd =2TTmTg

Incremental speed deviation

Incremental guide-vane / wicket gate positiondeviationIncremental head deviation

Incremental flow deviation

Load disturbance deviation

Incremental torque deviation

Laplace transform of APm

Laplace transform of AgLaplace transform of AqLaplace transform of Ah

Turbine headGuide-vane / Wicket gate position

Turbine speedTurbine flow

Hydraulic surge impedance, = T /ITWater starting time

Elastic time

Damping factorFriction loss in the hydraulic structureTime lag

Mechanical time constant

Servomotor gate constant

APPENDIX

Parameters of the system studied:Tm = 7s; T = 2.23s; D = 2.0; F=O.Op. T = 0.332s;T = 0.5sg

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Fuzzy Models for the Study of Hydro Power Plant Dynamics