differential evolution-based identification of the nonlinear kaplan turbine model

178 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 29, NO. 1, MARCH 2014

Differential Evolution-Based Identificationof the Nonlinear Kaplan Turbine Model

Dalibor Kranjcic and Gorazd Stumberger, Member, IEEE

Abstract—In this paper, various new models of double-regulatedKaplan turbines are proposed, whose parameters are determinedon the basis of field measurement data acquired during normaloperating conditions. The model being identified is an extensionof the nonlinear single-regulated turbine model obtained throughan approximation function that defines the relationship betweenthe wicket gate opening and runner blades angle. To determine theunknown parameters of the approximation function, a stochasticsearch algorithm called differential evolution (DE) is used. Thispaper focuses on the progressive development of the DE algorithm-based methods that are applied to determine different forms of theapproximation functions, until an optimal form is attained thatensures the best possible agreement between the measured andcalculated responses.

Index Terms—Function approximation, hydraulic turbines, in-terpolation, parameter estimation, power system modeling, powersystem simulation.

NOMENCLATURE

Pm Turbine mechanical power [per unit (p.u.)]Q Turbine discharge (p.u.).Q0 Turbine discharge, initial value (p.u.).U Velocity of water in conduit (p.u.).G Ideal gate opening (p.u.).g Real gate opening (p.u.).gF L Gate opening at rated load (p.u.).gN L Gate opening at no load (p.u.).β Runner blades angle/opening (p.u.).H Turbine head (p.u.).H0 Turbine head, initial value (p.u.).Hchar Turbine characteristic (TC) head (p.u.).GVOP Virtual gate opening (VGO) (p.u.).fVOP VGO function (p.u.).UN L No-load water velocity (p.u.).ω Turbine speed (p.u.).M Turbine torque (p.u.).A Cross-section area of conduit (m2).At Turbine gain.TW Water starting time (s).KR Runner blades influence parameter.ηt Turbine efficiency.

Manuscript received April 23, 2013; revised August 12, 2013; acceptedNovember 14, 2013. Date of publication December 11, 2013; date of currentversion February 14, 2014. Paper no. TEC-00241-2013.

D. Kranjcic is with DEM—Drava River Power Company, Maribor 2000,Slovenia (e-mail: [email protected]).

G. Stumberger is with the Faculty of Electrical Engineering and Computing,University of Maribor, Maribor 2000, Slovenia (e-mail: [email protected]).

Digital Object Identifier 10.1109/TEC.2013.2292927

F Mutation constant.Cr Crossover constant.NP Size of population.D Dimension of DE problem.GEN Maximal number of generations (iterations).L,H Boundary constraints—lower and upper limit.X Trial vector of parameters (X ∈ �D ).xi Parameters of trial vector (i = 1, . . . , D).Pop Population-potential solutions (Pop ∈ �[DxN P ]).Fit Fitness of population (Fit ∈ �N P ).f Objective function (f ∈ �).iBest Index of the best current vector (iBest ∈ N).i, j, q Loop variables (i, j, q ∈ N).Rd Mutation parameter (Rd ∈ N).r Indices of randomly selected vectors (r ∈ N).Pelmeas Measured electrical power (MW).Pcalc Calculated model power (MW).et Sum of squared model errors.Pwg “Gate” polynomial.Prun “Runner” polynomial.PVOP Surface polynomial.gk Coefficients of the “gate” polynomial (k = 0, . . . , 5).bk Coefficients of the “runner” polynomial (k = 0, . . . ,

5).ak Coefficients of the surface polynomial (k = 1, . . . ,

16).

I. INTRODUCTION

NOWADAYS, there are a variety of nonlinear turbine mod-els that are applicable for active power control system

analysis. A considerable number of works have been publisheddealing with the development and validation of these mod-els used for single-regulated turbines (SRT—turbines with ad-justable wicket gates and fixed runner blades). Oldenburger andDonelson [1] provided the theoretical foundations for nonlinearSRT models that rely on basic hydraulic equations (i.e., New-ton’s second law and continuity equation), which determine theflow of a compressible fluid through a uniform elastic pipe.These differential equations are of the same type as those em-ployed for electrical transmission lines, where the velocity ofwater and head are analogous to the transmission line currentand voltage, respectively.

After three decades, these ideas were used again by the IEEEworking group [2], Kundur [3], and Jaeger et al. [4]. An accuratereview and analysis of the existing models is presented in [5]including some modifications and improvements. Zeng et al. [6]present a critical analysis regarding the accuracy of the existingmodels with the definition of two new types of energy losses.Also, refinements of the IEEE working group model have been

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KRANJCIC AND STUMBERGER: DIFFERENTIAL EVOLUTION-BASED IDENTIFICATION OF THE NONLINEAR KAPLAN TURBINE MODEL 179

proposed in [7] in order to improve the capability of the existingmodels to simulate the transient operations of the power station.

However, the number of double-regulated turbine (DRT)models for active power control analysis (i.e., models for tur-bines with adjustable wicket gates and runner blades), whichcould lead to a satisfactory agreement between the measuredand calculated responses, is relatively small. A rare exampleof the DRT model derived mathematically from an analysis ofthe water velocity vectors acting on the runner blade is givenin [8]. This model gives quite satisfactory results, but a consider-able disagreement between the measured and calculated outputpower may also appear.

A different approach to the development of the DRT modelshas been taken in two recent works, published in the last decade,where the DRT model is introduced as an extension of the SRTmodel obtained through the determination of a functional re-lationship between the wicket gate opening and runner bladesangle (gate–runner relationship). The first work, presented byKosterev [9], determines the gate–runner relationship by mul-tiplying the wicket gate opening with the runner angle and theempirically deduced parameter, which represents the runner’sinfluence on the output power of the applied SRT model. Inthe second work, presented by Brezovec et al. [10], the gate–runner relationship is determined from the discharge and ef-ficiency functions interpolated from the turbine characteristics(TC) data. The interpolated functions also modify the structureof the applied SRT model.

By following these approaches, based on an extension of theSRT model, a new concept for the development of the DRTmodel is proposed in this paper, called the virtual gate open-ing (VOP). This concept is based on the assumption that thegate–runner relationship may be described by an approximationfunction (the VOP function) based on field measurement dataacquired during normal operating condition (NOC). The term“virtual” denotes the actual position of the VOP function, sinceits output becomes the input (i.e., the wicket gate opening) ofthe applied SRT model. The advantage of the VOP concept isthat the VOP function does not modify the structure (i.e., modelequations) of the applied SRT model; it only modifies its input.

This paper describes the identification process of differentDRT models developed according to the proposed VOP con-cept. The models are identified on the basis of measured timebehaviors, acquired during NOC at the hydropower plant (HPP)Mariborski otok over a period of two months. The possibilityof identifying the model from the measured time behaviors ac-quired during NOC constitutes the second advantage of the VOPconcept.

The identification focuses on the determination of the VOPfunction parameters by using the VOP concept-based dynamicmodel, the differential evolution (DE) algorithm and few pre-selected measured time behaviors that contain typical start-up/shut-down sequences. The proposed VOP concept-basedmodels are developed progressively, from the simple to the morecomplex ones, by satisfying two interrelated goals that ensurethe best possible agreement between measured and calculatedresponses: first, the selection of the most suitable form of theVOP function; and second, the empirical development of theDE algorithm-based methods to determine the VOP functionparameters.

Fig. 1. Kundur’s turbine model appropriate for SRT.

Fig. 2. Model KU_KS—Kundur’s turbine model modified with Kosterev’sapproach (the solid lines represent Kundur’s turbine model while the dashedlines represent an additional expression proposed by Kosterev).

II. GENERAL SRT MODEL

The general, basic model of an SRT is shown in Fig. 1—Kundur’s turbine model [3].

Kundur’s model is described by (1)–(6).1) The rate of change of water velocity in the conduit:

dU/dt = (−1/TW ) · (H − H0). (1)

2) The relation between the ideal gate opening G and realgate opening g:

G = Atg (2)

where At is the turbine gain given by

At = 1/(gF L − gN L ). (3)

3) No-load water velocity:

UN L = AtgN L

√H0 . (4)

4) The turbine mechanical power:

Pm = (U − UN L )H. (5)

5) Flow equation (water velocity in the penstock):

U = G√

H. (6)

III. REVIEW OF DRT MODELS CREATED FROM SRT MODELS

In this section, two different approaches are described thatextend/modify the SRT model to obtain the DRT model.

A. Kosterev’s Approach Applied to Kundur’s Model

A simple approach that adds an empirically deduced expres-sion to the SRT model to develop the DRT turbine model wasproposed by Kosterev [9]. The aim of the additional expressionis to catch the runner’s influence on the output discharge andsubsequently on the output power of the applied SRT model.This is achieved by multiplying the actual input of the SRTmodel (i.e., the wicket gate opening) with the runner angle andthe empirically deduced parameter KR . Fig. 2 shows Kosterev’sapproach applied to the SRT model developed by Kundur (de-scribed in Section II)—the model KU_KS.


Fig. 3. Model BKT—the SRT model modified with discharge and efficiencyfunctions determined in the look-up tables from the TC data.

Kosterev’s additional expression is given as follows:

GVOP = G(1 + KRβ). (7)

B. Approach Based on TC Data

Brezovec et al. [10] have proposed an approach where theDRT model is obtained by modifying the structure (i.e., modelequations) of the SRT model with discharge and efficiency func-tions, determined by TC data.

The model developed according to such an approach (modelBKT) is presented in Fig. 3, which shows the SRT model mod-ified with the discharge Q(G, β) and efficiency ηt(G, β) func-tions determined in the look-up tables.

Equations (8)–(12) describe the model BKT.1) The rate of change of flow in the conduit:

Q − Qo

H − H0= − 1

TW s. (8)

2) The turbine mechanical torque:

M =QHηt

ω. (9)

3) Neglecting the speed deviation, the discharge and effi-ciency of the DRT are functions of the head, wicket gateopening, and runner blades angle:

Q = Q(H,G, β) (10)

ηt = ηt(H,G, β). (11)

Equations (10) and (11) represent surfaces that can be de-termined from the TC data based on discrete measurements ofcrucial turbine parameters (e.g., power, efficiency, discharge,and so on) acquired at few runner blades angles and wicket gateopenings. Fig. 4 shows an example of the 3-D characteristic ofmeasured electrical power approximated by a surface from TCdata [11], [12]:

In this particular turbine model, the surfaces (10) and (11)are determined by preprocessing the TC data with the cubicspline interpolation technique [13] to determine the look-uptables with sufficient point density. Since the turbine dischargeQ(G, β), determined from the TC data, is defined for a specifichead Hchar , then, according to the laws of similitude [14], theturbine discharge at some head H is given as follows:

Q = Q(G, β)

⎛⎝

√H

Hchar

⎞⎠ . (12)

Fig. 4. Example of measured characteristic of electrical power given as afunction of TC data approximated by a surface.

Fig. 5. DRT model developed according to the VOP concept.

IV. DEFINITION OF VOP CONCEPT

The basic idea of the VOP concept is described by theblock diagram given in Fig. 5. It shows the DRT model de-veloped as an extension of the SRT model obtained through theVOP function (13). The algorithms and methods used to de-termine the parameters of the VOP functions are described inSections V–VIII:

GVOP = fVOP(G, β). (13)

V. INTRODUCTION TO THE DE ALGORITHM

The first DE algorithm was proposed by Price et al. [15].A good explanation of the basic DE algorithm, given in pseu-docode, is presented in [16] and summarized here.

The DE algorithm is a tool capable of solving an optimizationproblem of finding a minimum value of the scalar function givenas follows (i.e., the objective function):

minx

f(X), L ≤ X ≤ H,X ∈ �D (14)

where X is the vector of parameters xi, i = 1, . . . , D, subjectto boundary constraints L ≤ X ≤ H .

Before starting the optimization procedure, the population isinitialized by (15) and the criterion for optimization (i.e., fitness)is evaluated as follows:

Popij = L + (H − L) · rndij [0, 1), i = 1, . . . , D,

j = 1, . . . , NP (15)

Fitj = f(Popj ) (16)


where rndij [0, 1) is the uniformly distributed random number atthe interval [0,1). The optimization is performed in GEN itera-tions, where each consists of the following four steps performedfor the entire population (j = 1, . . . , NP ).

1) Mutation—three mutually different vectors r1 , r2 , r3 arerandomly chosen from the population (17). The selectedvectors are also different from the current vector j:

r1,2,3 ∈ [1, . . . , NP ], r1 �= r2 �= r3 �= j. (17)

2) The creation of the new trial vector X of parameters xi—the vector is created according to the probabilistic rule(18). The initial value of the mutation parameter Rd israndomly determined in the [1, . . . , D] range:

xi ={

xi,r3 +F · (xi,r1 −xi,r2) if (rndij [0, 1) < Cr)∨ (Rd = i)

xij otherwise

i = 1, . . . , D. (18)

3) Verification of boundary constraints by

if (xi /∈ [L,H]) then xi = L + (H − L) · rndi [0, 1).(19)

4) The selection of the best vector—if the fitness functionf(X) of the trial vector is less than or equal to the fitnessfunction of the current vector in the population Fitj , thenaccording to (20), the current vector in the populationis replaced by the trial vector. Furthermore, if the newmember of a population, i.e., the previously validated trialvector, is better than the existing best vector, the best vectorindex is updated as well:

if (f(X) ≤ Fitj ) then Popj ← X, Fitj ← f(X)

and if (f(X) ≤ FitiBest) then iBest ← j. (20)

The optimal solution of the DE algorithm is PopiBest with itsfitness value FitiBest .

VI. DESCRIPTION OF THE DE ALGORITHM-BASED

IDENTIFICATION OF VOP CONCEPT-BASED MODELS

In the following Section (VII), new DRT models developedaccording to the VOP concept are presented. To identify thesemodels the parameters of the VOP function are determined bypurposely developed DE algorithm-based methods on the basisof measured time behaviors acquired during NOC. These meth-ods are determined empirically from an analysis of the obtainedresults with the aim of ensuring the best possible agreement be-tween the measured and calculated responses. The parametersof the applied SRT model in the presented VOP concept-basedmodels are not affected by DE, since they are calculated priorto its application from the turbine technical data. The measuredtime behaviors used in the DE algorithm-based identificationare the wicket gate opening, the runner blades angle and theelectrical power, that may be considered as mechanical powergenerated by the hydraulic turbine.

The DE algorithm searches for the VOP function parameterswith respect to the objective function f , defined by (21) as thesquare difference et(t) (22) between the measured Pelmeas andthe calculated power Pcalc in the time interval of observation

t ∈ [t1 , t2 ]:

f =∫ t2

t1

e2t (τ)dτ (21)

et(t) = Pelmeas(t) − Pcalc(t). (22)

The power Pcalc in (22), representing the output from theVOP concept-based model, is calculated in each iteration of theDE algorithm from the measured time behaviors of the wicketgate opening and runner blades angle, i.e., the actual inputs tothe VOP function.

Before the DE algorithm is started, the measured time be-haviors are preselected to contain typical turbine start-up orshut-down sequence that last from 15 min to 1 h. The sequencesare preselected to consider such transients that adequately excitethe model in the whole (normal) operating range determined bythe cam [17], i.e., the optimal relationship between wicket gateopening and runner blades angle. The preselection criteria areaccurately described in the following sections.

VII. DEVELOPMENT OF NEW DRT MODELS

ACCORDING TO THE VOP CONCEPT

In this section, Kundur’s model, modified by the approachproposed by Kosterev (i.e., the model KU_KS described inSection III-A), is used as a starting point for development of theDRT models based on the VOP concept. The aim is to improvethe Kosterev’s approach which is based on a simple empiricallydeduced expression that modifies the SRT model into the DRTmodel. This is achieved by substituting the empirically deducedexpression with more sophisticated functions (the VOP func-tions) whose parameters are then determined as described inSection VI. In the further reading, the phrases “determinationof function parameters by DE,” “DE-based determination ofparameters” or “DE-based identification” are used as a simpli-fied description that denotes determination of the VOP functionparameters according to the DE algorithm-based identificationprocedure described in Section VI.

A. Upgrading of Kosterev’s Approach With Introductionof Polynomial Functions

If the model KU_KS is validated with the NOC measuredtime behaviors it shows only partial agreement between thecalculated and the measured power. This model is accurate inthe limited range due to the parameter KR (7) that determinesthe degree of the runner’s influence on the output power. Outsidethis range the disagreement between measured and calculatedpower can be quite large, meaning that the model accuracy isobtained at certain operating point determined by the parameterKR .

However, the agreement between measured and calculatedpower improves significantly over almost the entire range ofpower, if KR in (7), i.e., Kosterev’s modification, is substitutedwith a polynomial given by (23), whose coefficients are deter-mined with DE on the basis of measured time behaviors acquiredduring NOC:

Prun(β) = b5β5

+ b4β4

+ b3β3

+ b2β2

+ b1β + b0 . (23)


Fig. 6. Offset at lower values of power—simulation results of the modelKU_KS modified by (23) (HPP Mariborski otok).

Considering this change (7), i.e., the VOP function becomes:

GVOP = G · Prun(β). (24)

Nevertheless, a notable offset is still present at lower valuesof calculated power as shown in Fig. 6.

This offset occurs, since the DE-based identification is ap-plied to the polynomial (23), dependent on the runner bladesangle variable, and thus the model [i.e., the model KU_KSmodified with (23)] can be identified in the operating range thatis determined only by the runner blades movement. Since therunner moves from its initial position when wicket gates areopened about 50% (according to cam [17]) the power responsebefore the runner blades movement (see Fig. 6) is unidentifiedwith the proposed DE-based identification of the polynomial(23) and subsequently the described offset appears. The offsetcan be eliminated by modifying the VOP function with an addi-tional polynomial, given by (25), that considers the wicket gatemovement as well:

Pwg (G) = g5G5

+ g4G4

+ g3G3

+ g2G2

+ g1G + g0 . (25)

Considering both polynomials (23) and (25), the VOP func-tion becomes

GVOP = Pwg (G) · Prun(β). (26)

To simplify for further reading, the polynomials (23) and (25)are called the “runner polynomial” and the “gate polynomial,”respectively. Fig. 7 depicts a model that includes both of thepolynomials—the model K_DP.

To understand the role of the wicket gate polynomial inthe aforementioned model, the following question must be an-swered: What happens with the power response if the mea-sured time behaviors of the wicket gate opening, acquired at the

Fig. 7. Model K_DP—upgrading the model KU_KS with the use of the gateand the runner polynomial.

Fig. 8. BLPD of the SRT model.

double-regulated Kaplan turbine unit during NOC, are appliedto the input of the standard SRT model (i.e., Kundur’s turbinemodel)?

Depicted in Fig. 8, the result of such an experiment showssatisfactory agreement between the power calculated by the SRTmodel and the power acquired at the double-regulated Kaplanturbine unit for values below 20% of the rated power (approx.).

This behavior in the low power domain (BLPD) is consideredwhen coefficients of the gate polynomial are determined withDE—the coefficients are searched for in such a manner so asto obtain linearity of the initial part of the polynomial curve(as described in Section VII-B). By doing so, the BLPD ofthe SRT model is unaffected (i.e., transparently transferred) bythe gate polynomial. In addition, the identified gate polynomialalso corrects the rest of the calculated power response for wicketgate openings above the BLPD of the SRT model, resulting inthe elimination of the previously described offset error at lowervalues of power

The model K_DP shows good agreement between measuredand calculated power, as long as it is validated with NOC mea-sured time behaviors acquired at the value of the initial head(i.e., the head value taken at the start of the measuring pro-cess) close to that of the NOC measured time behaviors usedfor DE-based determination of the gate and runner polynomialcoefficients. However, if this does not occur, it may also cause a


Fig. 9. Model K_MDE—“outcome” of the method MDE.

large disagreement between the calculated and measured powerresponse. Because of this inadequacy, the here-presented modelK_DP is not complete; therefore, further improvements are pre-sented in the following Sections VII-B and VII-C.

B. Model K_MDE—Identification of the Gate and RunnerPolynomials Dependence on the Initial Head

The incompleteness of the model K_DP (see previous sec-tion), caused by the unidentified dependence of the power re-sponse on the initial head (H0) is corrected here, with the useof a DE-based method, which determines the coefficients of thegate (23) and runner (25) polynomials with respect to the initialhead. This method, called method MDE, consists of multipleDE-based identifications conducted separately, each on the ba-sis of NOC measured time behaviors acquired at different valueof the initial head. The selection of the head values is determinedwithin the head operating range, given between 0.9 and 1 p.u.The basic idea behind the method MDE is to capture the depen-dence of the cam characteristics [17] on the initial head. Theoutcome of the method MDE requires a new model; therefore,the model K_MDE shown in Fig. 9 is introduced as a slightmodification of the model K_DP.

In the model K_MDE, the runner polynomial is substitutedwith the 2-D look-up table used to define the influence of theinitial head on the power response. The parameters of the ap-plied look-up table are the calculated values of three runnerpolynomials determined by the MDE method on the basis ofNOC measured time behaviors acquired at three different initialheads. It is necessary to emphasize that the MDE method isapplied to the coefficients of the gate and runner polynomialsof the model K_DP, while the model K_MDE only contains theresult of the performed method MDE—the values of the threeestimated runner polynomials are placed in the look-up table ofthe model K_MDE. The method itself can be described in thefollowing steps.

1) The DE-based determination of polynomials coefficientsg5, g4, g3, g2, b5, b4, b3, b2, and b1 in (23) and (25) is per-formed on the basis of NOC measured time behaviorsacquired at H0 = 0.95 p.u.. The coefficient g1 is not de-termined by DE. It is set to 1 to obtain the linearity of theinitial part of the gate polynomial, which consequently sat-isfies the BLPD presented in Section VII-A. Despite this,the linearity of the initial part of the gate polynomial is notideal, since it is also influenced by other polynomial coef-ficients (i.e.,g5, g4, g3, g2). These coefficients produce aslight deviation from the expected linearity (i.e., deviationfrom the BLPD), which in consequence produces a slightoffset at low values of the power response. For conve-nience, this offset is called offset due to the optimization

(ODTO). The ODTO can be eliminated if the DE proce-dure is performed in respect to the objective function f ,given by (27), as the sum of two objective functions (21),determined at different time intervals:

f = f1 · w + f2 (27)

wheref1 = the first objective function applied to the time intervalwhere the ODTO occurs.f2 = the second objective function applied to the timeinterval not affected by the ODTO.w = empirically deduced weight (w = 1000).

The applied weight parameter w in (27) forces the DEto focus on the objective function f1 . By doing so, theoptimization gives higher priority to the ODTO area andthus reduces the errors caused by the determination of co-efficients g5, g4, g3, g2. With the similar purpose of con-trolling of the optimization outcome, the coefficients g0and b0 are set to 0 and 1 to ensure that the gate and run-ner polynomials start at the values 0 and 1, respectively.These coefficients are not determined by DE since DE, ifapplied to them, may influence the “natural” start of thegate and runner polynomials and consequently the powerresponse. The “natural start” means that the curve of thegate polynomial transparently transfers the BLPD and thushas to start at the value 0, while the curve of the runnerpolynomial, applied with the aim of influencing the gatepolynomial, starts at the value 1.

2) The DE-based determination of polynomial coefficientsb5, b4, b3, b2, and b1 based on NOC measured time be-haviors acquired at H0 = 0.919 p.u.

3) The DE-based determination of polynomial coefficientsb5, b4, b3, b2, and b1 based on NOC measured time be-haviors acquired at H0 = 0.986 p.u.

4) The values of the determined runner polynomials are cal-culated with the MATLAB function polyval and placed inthe look-up table.

The method MDE determines the coefficients of the runnerpolynomial with respect to three values of the initial head, whichmay lead to the conclusion that the coefficients of the gate poly-nomial should also be determined in the same way. However,this is not necessary since the gate polynomial, by being deter-mined to consider the BLPD (see step 1 of the method MDE),acts naturally to the change of the initial head parameter, as theSRT model alone does.

The model K_MDE is not quite accurate, since the searchedpolynomials are determined against the value of the initialhead parameter, while, in reality, the head value varies dur-ing each measurement interval (±0.1–0.2 p.u.). Nevertheless,the results of the method MDE may be quite satisfactory (seeSection VIII) and the approaches described in Sections VII-Aand VII-B (i.e., BLPD and the elimination of ODTO) are use-fully applied in the development of the model presented in thefollowing section.

C. Model K_VOP—A Surface Data Fitting Solution

In this section, a model founded entirely on the VOP concept(Section IV) is proposed: the model K_VOP (see Fig. 10). The


Fig. 10. Model K_VOP—VOP function given as the surface polynomial.

Fig. 11. Graphical representation of two measured time behaviors of poweracquired at different initial heads and given in 3-D space.

VOP function of the model K_VOP is represented by the surfacepolynomial (28) of the third degree [18]:

PVOP(G, β) = (a1G3

+ a2G2

+ a3G + a4)β3

+ (a5G3

+ a6G2

+ a7G + a8)β2

+ (a9G3

+ a10G2

+ a11G + a12)β

+ (a13G3

+ a14G2

+ a15G + a16). (28)

Similarly, as in the previous section, the coefficients of theVOP function (28) are determined by a DE-based method thatrelies on measured time behaviors acquired during NOC.

The aim of the method is to determine the “searched surface,”which represents the functional relationship between the wicketgate opening and runner blades angle, by using (28). The methoditself requires enough data from the available measured timebehaviors to successfully approximate the searched surface. Toconfirm the availability and applicability of the data needed forthe approximation a graphical analysis of the NOC-measuredtime behaviors is made. Fig. 11 shows two characteristics ofNOC-measured time behaviors acquired at different values ofhead, where power is presented in the 3-D space as a functionof the wicket gate opening and runner blades angle. The twomeasured time behaviors depicted in Fig. 11 are near the lowerand upper margins of the normal operating range determined bycam [17]—the head operating range is given between 0.9 and1 p.u.

It is evident from the graphical representation given in Fig. 11that the characteristics of the NOC measured time behaviors,acquired at different heads provide a good “frame” that may be

approximated by the VOP function, although the amount of datahere is less than the TC data used to determine the surfaces (10)and (11) (see Fig. 4). On the contrary, a single NOC measuredtime behavior is insufficient for the satisfactory approximationof the searched surface, since it does not provide enough datain the whole operating range.

Based on this insight, the applied DE-based method mustconsider more NOC measured time behaviors, where each isacquired at a different value of the initial head. To accomplishsuch a method, a similar approach may be used as the one usedto eliminate the ODTO (see Section VII-B). Here, the parame-ters of the VOP function (28) are determined in respect to theobjective function given as the sum of three weighted objectivefunctions, where each of them focuses on the NOC measuredtime behavior acquired at a different value of the initial head.Such an approach leads to quite satisfactory results, but it alsohas the following disadvantage: the DE only minimizes theresulting objective function and not the individual sums. There-fore, even with the weight parameters included, the influence ofeach individual objective function on the final approximation ofthe searched surface by (28) is unpredictable. Consequently, theresulting (searched) surface may not be equally well approxi-mated in the whole area.

This disadvantage can be diminished if the approximationof the searched surface by (28) is solved as a data fitting prob-lem, where NOC measured time behaviors, acquired at differentinitial heads, are used as the input data. This data fitting prob-lem is solved by applying the following DE-based method tothe previously described model K_DP (see Section VII-A andFig. 7).

1) First, three NOC measured time behaviors acquired atdifferent initial heads are preselected within the head op-erating range given between 0.9 and 1 p.u., e.g.: H0 =0.93 p.u.,H0 = 0.95 p.u., and H0 = 0.975 p.u.. The pre-selecting of the measured time behaviors is made in sucha way to obtain as much relevant data as necessary forthe best possible approximation of the searched surfaceby (28) (see Fig. 11).

2) Then, the coefficients of the polynomials (23) and (25)of the model K_DP are determined with DE for each ofthe preselected NOC measured time behaviors, separately.On the basis of the determined coefficients, the outputof the VOP function, given by (13), is known at threedifferent heads. In other words, three VOP characteristics,presented as a function of the wicket gate opening andrunner blades angle, are determined at different valuesof heads. These characteristics are similar to the powercharacteristics depicted in Fig. 11.

3) The three VOP characteristics are joined together toform one single characteristic that represents m input–output data points, given by {(Gi, βi,GVOP i

)}mi=1 , where

GVOP i= f(Gi, βi).

4) The coefficients of the VOP function (28) are searched bythe DE with respect to the objective function f , defined by(29) as the square difference between the input data GVOP i

and the VOP function PVOP(Gi, βi). In other words, byminimizing the objective function f the DE approximatesthe searched surface with (28) on the basis of the input


TABLE IBASIC TECHNICAL TURBINE DATA

data prepared in step 3:

f =m∑

i=1

[GVOP i− PVOP(Gi, βi)]

2 . (29)

From the modeling point of view, the model K_VOP is similarto the model BKT (see Section III-B), yet the principle behindmodel K_VOP is simplified. This can be explained if the headsof both models are mathematically compared. Thus, when con-sidering the relation Q = AU , the head of the model BKT canbe expressed as follows:

H = U2(

Hchar

(U(G, β))2

)(30)

while the head of model K_VOP is given by

H = U2(

1(PVOP(G, β))2

). (31)

The aforementioned comparison clearly shows that the rightparts of equations (30) and (31), if water velocity is excluded,have the same functional meaning. In fact, according to themathematical definition of the model BKT [see (10) and (11)]and of the model K_VOP [see (28)], they both determine surface.However, the following differences can be found between thesetwo models, which also represent the fundamental contributionof this paper:

1) The surface U(G, β) in the model BKT modifies the ap-plied SRT model through an additional similitude lawequation (12). This is simplified in the model K_VOP,where the SRT model remains unchanged, since the VOPfunction [surface polynomial (28)] is applied according tothe VOP concept, without the need for additional mathe-matical expressions.

2) The searched surface U(G, β) in the model BKT is in-terpolated from the TC data. In the model K_VOP, how-ever, the searched surface is approximated by a DE-basedmethod on the basis of few measured time behaviors ac-quired during NOC.

VIII. SIMULATION RESULTS FOR THE MODELS

K_MDE AND K_VOP

This section presents the time-domain simulation results forthe models K_MDE and K_VOP on the basis of the measuredtime behaviors acquired at HPP Mariborski otok under NOC.All identification and simulation procedures were performedwithin the MATLAB/Simulink programming environment. Thebasic turbine technical data and calculated model parametersare given in Tables I and II, respectively.

TABLE IICALCULATED MODEL PARAMETERS

Fig. 12. Turbine start-up, H0 = 0.93 p.u.—simulation results.

Figs. 12–15 show the power responses of the models K_MDEand K_VOP, determined according to the methods described inSections VII-B and VII-C.

This results show excellent agreement between the measuredpower and power calculated by both models, with the resultsof the model K_MDE being slightly worse due to the incom-pleteness of the applied method MDE (see Section VII-B). Thequality of the obtained results of both models depends on theselection of the appropriate measured time behaviors that ade-quately identify the model in the whole operating range.

The presented results clearly show that the VOP concept maybe successfully applied for simulations of the NOC. However, toadditionally confirm the validity of the concept and the relatedmethods proposed in this paper, the model K_VOP is validatedagainst TC data. The validation method may be described in thefollowing steps.

1) First, the TC data, given in discrete form (see Fig. 4),must be preprocessed to generate continuous time behav-iors suitable for the DE-based identification. Since onlytwo different sets of TC data are provided from field mea-surements, the first set of TC data, obtained at H0 = 1 p.u.,is preprocessed by a cubic spline interpolation [13] to givegenerated time behaviors of the wicket gate opening, therunner blades angle, and the electrical power. The second


Fig. 13. NOCs, H0 = 0.961 p.u.—simulation results.

Fig. 14. Turbine shut down, H0 = 0.977 p.u.—simulation results.

set of TC data, obtained at H0 = 0.935 p.u., is thus leftfor the model validation.

2) The TC generated time behaviors, prepared in step 1, areused as input data to determine the coefficients of (28) byminimizing the objective function f defined by (21) withDE—the model is identified.

Fig. 15. Detail of turbine shut down, H0 = 0.975 p.u.—simulation results.

Fig. 16. Output-power characteristics of the model K_VOP identified withpreprocessed TC data.

3) The validity of the identified model is confirmed withthe second (unprocessed) set of the TC data, obtained atH0 = 0.935p.u., and the result is shown in Fig. 16.

Fig. 16 shows excellent agreement between the output-powercharacteristics calculated by the model K_VOP and the TC data,and therefore, this result additionally confirms the validity of theVOP concept.

The parameters of the model K_MDE determined by DE onthe basis of NOC measured time behaviors are presented inTable III.

The parameters of the model K_VOP determined by DE onthe basis of both NOC and TC generated time behaviors arepresented in Table IV.


TABLE IIIPARAMETERS OF THE “GATE” AND “RUNNER” POLYNOMIALS—K_MDE

TABLE IVPARAMETERS OF THE VOP FUNCTION—K_VOP

IX. CONCLUSION

In this paper, a concept VOP is presented, which determinesthe double-regulated Kaplan turbine model as an extension ofthe SRT model with an approximation function that defines therelationship between the wicket gate opening and the runnerblades angle. The parameters of the approximation functionare determined in the identification procedure using the VOPconcept-based dynamic model, new DE algorithm-based meth-ods and measured time behaviors acquired during NOC. TheDE-based methods are empirically determined from the analysisof the simulation results and applied to different approximationfunctions in the proposed models with the aim of obtaining thebest possible agreement between the measured and calculatedresponses. The measured time behaviors consider such tran-sients that adequately excite the model in the whole operatingrange.

The main contribution of this paper lies in two aspects: first,the proposed concept is quite flexible, since it does not requiremodification of the basic model equations. Thus, it may also besuccessfully applied to other more sophisticated SRT models,which consider the effect of water compressibility and pipeelasticity; and second, the model is identified on the basis of afew preselected measured time behaviors collected during NOC.

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Dalibor Kranjcic was born in Koper, Slovenia,in 1972. He received the B.S.E.E. degree in elec-trical engineering from the University of Maribor,Maribor, Slovenia, in 1999.

Over the last ten years, he has been with the electri-cal generation company DEM—Drava River PowerCompany, Maribor, Slovenia, in the fields of infor-mation technology and process control systems. Hiscurrent research interests include modeling and thesimulation of hydroelectric power units.

Gorazd Stumberger (M’92) was born in Ptuj,Slovenia, in 1964. He received the B.S.E.E., M.Sc.,and Ph.D. degrees in electrical engineering from theUniversity of Maribor, Maribor, Slovenia, in 1989,1992, and 1996, respectively.

In 1989, he joined the Faculty of ElectricalEngineering and Computer Science, University ofMaribor, where he is currently a Professor. His currentresearch interests include design, modeling, analysis,and control of electrical machines.

Dr. Stumberger is a member of the InternationalCompumag Society and the Slovenian Committee CIGRE.

differential evolution-based identification of the nonlinear kaplan turbine model

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