Energy Conversion and Management 74 (2013) 417–425

Contents lists available at SciVerse ScienceDirect

Energy Conversion and Management

journal homepage: www.elsevier .com/ locate /enconman

Shape and operation optimisation of a supercritical steam turbine rotor

0196-8904/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.enconman.2013.06.037

⇑ Corresponding author. Tel.: +48 322372822.E-mail addresses: grzegorz.nowak@polsl.pl (G. Nowak), andrzej.rusin@polsl.pl

(A. Rusin).

G. Nowak ⇑, A. RusinSilesian University of Technology, ul. Konarskiego 18, 44-100 Gliwice, Poland

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 October 2012Accepted 13 June 2013Available online 26 July 2013

Keywords:Ultra-supercritical turbineOptimisationThermal stress

The presented study discusses the problem of shape optimisation of selected areas of the rotor of the highpressure part of an ultra-supercritical steam turbine together with the optimisation of the turbine start-up method, using the maximum stress objective. The analysis relates to the rotor of a conceptual ultra-supercritical turbine which is characterised by high parameters of operation. The consequence is that themachinery components are subjected to significant stress, which further results in a substantial reductionin its life and reliability. These adverse effects can be contained in two ways, i.e. by optimising the shapeof the rotor areas characterised by high stress values and by optimising the method of the turbine start-up. In the case of the rotor under analysis, it is the thermal stress caused by large temperature gradientsoccurring in unsteady states of operation that has a predominant impact on the stress level.

The performed research prove that the manner in which the power unit start-up is initiated and carriedout depends largely on the limitations of the materials used to make the machinery components. This, inturn, has an impact on the assessment of the power unit in terms of energy and economy. The obtainedoptimisation results translate directly into the power unit energy effectiveness.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The need to meet environmental requirements concerning thereduction in greenhouse gas emissions necessitates the develop-ment of highly efficient technologies of power units for supercrit-ical steam parameters. This in turn entails a further developmentin the design of both boilers and turbines. The components of thesemachines operate not only under a greater load but also in highertemperatures. To manufacture them, it is then necessary to usenew materials resistant to high-temperature failure processes,such as creep and fatigue. Apart from the selection of the appropri-ate material, the choice of the design form is also essential [1]. Theshape and the size of a component determine the level of thermaland mechanical stress that will arise at different phases of theoperation, which has a decisive impact on the component life [2–4]. From the point of view of operational safety, the turbine rotoris an element of particular importance. The temperature gradientsappearing in it in unsteady states are the cause of significant stress[5]. One of the ways to reduce the stress and to improve the rotorlife is the optimisation of its shape. In the further part of the paper,a mathematical model for the optimisation of the turbine compo-nents is defined. The basic objective function assumed here is therotor life. The optimised values are selected rotor dimensions,especially in stress concentration zones. The fundamental problem

in the optimisation of the shape of components as complex as therotor is the need to model the stress state in their subsequentlychanging forms. Consequently, the modelling has to be done witha variable numerical grid. In this study, the response surface meth-od is used, which makes the problem easier to solve thanks toappropriate approximations.

2. Optimisation problem formulation

Each structure treated as an object of design is characterised bya number of features which are given specific values in the designprocess. The selection of those features has a decision-making nat-ure and is conditioned, to a large extent, by the system of objec-tives. To facilitate the decision-making processes and to makethem more objective, the technique referred to as the optimum de-sign method is often used. The optimum designing of a structure isaimed at the creation of the optimum design, i.e. one that not onlymakes it possible to meet all the requirements the structure isfaced with, but also ensures that the structure is the best in respectof the selected optimisation objective.

By assuming the optimisation objectives it is possible to buildthe mathematical model of the structure which includes:

– design variables X = (x1, x2, . . ., xn), i.e. the values to beoptimised;

– constraint functions wiðx1; x2; :::; xnÞ 6 0, defining the permissi-ble area of the variability of the optimised values;

418 G. Nowak, A. Rusin / Energy Conversion and Management 74 (2013) 417–425

– objective function V(x1, x2, . . ., xn), which depends on the vari-ables under optimisation and which constitutes the mathemat-ical notation of the optimisation objective.

In the case of heat turbines operating in creep and fatigue con-ditions, one of the basic design objectives is the life objective,which can be written as:

V ¼ te ð1Þ

where te is the component life, i.e. working time to failure.Therefore the aim of the optimisation is to maximise the com-

ponent life.

V ! Vmax ð2Þ

Certain dimensions of the component may be the variables tobe optimised. This means that the desired structure form of the ro-tor is the one with the longest operation time. This time is limitedby the damage done to the component due to life degradation pro-cesses. Because of the fact that both creep and fatigue processesdepend on the stress level, the optimisation objective related toworking time may be replaced with the optimisation objective re-lated to stress minimisation. Therefore the following may beassumed:

V ¼ r ð3Þ

V ! Vmin ð4Þ

The dimensions to be optimised will only be those that have themost significant impact on the stress level of the component. Typ-ically, these are the dimensions in stress concentration zones, e.g.curvature radii. The material properties, operation conditions, aswell as the main dimensions of the component may be the optimi-sation parameters.

Solving the optimisation problem presented above with mathe-matical methods, it is necessary to perform multiple calculations ofthe value of the objective function, which in this case is the maxi-mum stress level in a given component. To determine the stress,the calculations of transient temperature distributions in the entireworking cycle have to be made, and then, based on their results,stress distributions are calculated. Each time the calculations haveto be performed for a different rotor form, i.e. for a differentnumerical grid. Therefore, obtaining the optimum solution is verytime-consuming. One of the ways to avoid this particular inconve-nience is the application of certain approximation techniques dis-cussed in the further part of the paper.

Fig. 1. Central composite design.

3. Response surface methodology

The optimisation methodology used in the calculations is basedon the Response Surface Method (RSM), which is a set of methodsof the mathematical analysis and statistics [6,7]. For this purpose,experimental numerical studies are used, i.e. numerical simula-tions of the real process which is subjected to optimisation. Thesimulations, called the design of experiments (DoE) consist in find-ing the response (of the initial values) of a given process to inde-pendent input parameters (design variables) which affect theprocess [7]. In other words, the design of the experiments consistsin a series of numerical simulations for changing input data with aview to identifying their impact on the output values. The proce-dure is to allow the approximation of the real (modelled) processwith the use of an assumed approximation function, which in turnwill be the object of the optimisation process. The aim of such anapproach is to reduce the computational cost related to the optimi-sation processes which need time-consuming numerical analyses.

3.1. The design of the experiments

The process begins with setting the range of variability of indi-vidual design parameters of the simulation model. Based on that,the design of experiments is created that will be used to determinethe response surface of the model [8]. Depending on the analysedproblem and on the character of the correlation between designparameters and output values, various designs of experimentsare assumed. The simplest is what is referred to as the screeningdesign. It makes it possible to examine the impact of the assumeddesign variables on the system response, and to select for furtheranalyses only those variables which affect the response in a signif-icant way. In this design, permutations of the upper and lower con-straints of all design variables are used. Because for each variablecalculations are made for two levels of its value (the upper andthe lower constraint), the design is referred to as two-level andmarked as 2N, where N is the number of design variables. However,it allows a linear description of the input–output effect only. Also,the possible impact of the interaction between design variables onthe result values is lost here. These inconveniences are removed bythe application of three-level designs (3N) and multi-level, typi-cally five-level ones (5N), which allow a non-linear description ofthe impact of parameters. The number of conducted experiments(in this case – numerical simulations) rises because it is equivalentto the number of levels raised to the power equal to the number ofdesign variables. Thus, an increase in the number of design vari-ables for a given design of experiments results in an exponentialrise in the number of their possible combinations. The designs ofexperiments mentioned above are referred to as full factorial de-signs which include all possible combinations of the input param-eter values. However, they become impractical for a bigger numberof design variables and then designs referred to as fractional facto-rial designs are employed. These designs no longer include all pos-sible combinations of the input values, but only a part of them. Thereduction in the number of combinations of the values of variablesmay entail obtaining results of a poorer quality. Therefore, in orderto minimise this effect, it becomes necessary to carefully select thecombinations of the values of design variables, as well as theirappropriate number. One of the fractional factorial designs whichis often used is the Central Composite Design (CCD) [6], which isa two-level design expanded with an axial experiment design. Thismeans that for each variable there are two extreme realisationsavailable (at the edge of the area), plus a central realisation as wellas axial ones in the middle of the intervals of all variables. In thiscase, the number of experiments is reduced compared to the fullfactorial design, e.g. for three design variables the fall is from 27to 15 (Fig. 1).

G. Nowak, A. Rusin / Energy Conversion and Management 74 (2013) 417–425 419

This kind of design makes it possible to use a polynomial of thesecond order to build the system response model.

3.2. Response surface

The performance of an appropriate type of the design of exper-iments allows the transition to another optimisation stage, namely– to the approximation of the obtained system responses. The stan-dard response surface is made on the basis of polynomials of thesecond order with the use of the regression analysis [8]. This anal-ysis is based on finding the mathematical dependence which de-scribes the relationship between data. Generally, themathematical model assumes the form:

Y ¼ f ðX;AÞ þ e ð5Þ

where X is the vector of independent variables, A denotes the vectorof regression coefficients, and e is the random error with expectedvalue E(e) = 0. Notation f(X, A) in the equation is the set of basicfunctions of regression.

A slightly different approach is realised in the Kriging algorithm[9], which assumes a relationship between variables in the follow-ing form of the polynomial and deviation combinations:

Y ¼ f ðXÞ þ ZðXÞ ð6Þ

In the relation given above f(X) is the set polynomial function,while Z(X) is the realisation of a random process with a Gaussiandistribution for an average equal to 0, variance r2 and non-zerocovariance. The polynomial proposed in the above formula is a‘‘global’’ model of the relationship between variables, whereasthe other factor creates ‘‘local’’ deviations of approximated points.

3.3. Optimisation method

The search for optimum solutions may be based on variousoptimisation methods, depending on the problem to be solved.One of the most commonly used optimisation algorithms, due toits universal nature, is the evolutionary approach in the form ofthe genetic algorithm [10]. It makes it possible to solve single-and multi-objective optimisation tasks with a continuous and dis-creet objective function. Genetic algorithm is given extensiveattention in many works concerning engineering optimisationtasks [11,12]. This particular method is used in this paper to findthe minimum of the objective function described in [12]. The ap-proach generally requires that very many potential solutions beconsidered, especially at a large number of decision variables,which may involve a huge computational cost. This computationalcosts depends mainly on the computational model under consider-ation. In this case the genetic algorithm operates combined withthe response surface method and therefore it is not the FEM modelof the object under analysis that is directly applied for the optimi-sation purposes, but a meta-model which is a functional approxi-mation of its response to a change in the decision variables ofthe task. Thus, even if it is necessary to analyse thousands of pos-sible solutions, the computational cost related to this task is slight.

4. Shape optimisation of the rotor selected areas

The object of the presented analyses is a conceptual rotor of thehigh pressure part of a supercritical steam turbine. The workingsteam parameters at the turbine inlet are assumed as: Tin = 650 �Cand pin = 30 MPa. The rotor is a drum structure of a reaction typeturbine with 25 stages (Fig. 2). The steam inflow is radial. Theexpansion line of steam in the flow system is determined throughthermal-flow analyses [13]. Based on it, the conditions of the heattransfer on individual surfaces of the rotor are defined. These data

are used further to carry out the presented optimisation calcula-tions which are performed using the Ansys software.

The solving of the optimisation task starts with preliminarythermal-strength calculations of the turbine rotor. As a result,two stress concentration zones are selected: the blade groovesand the shaft undercut in the balance piston area.

Fig. 2 presents the contours of the maximum stress in stressconcentration zones. Preliminary calculations were performed forthe reference start-up marked as ‘‘initial curve’’ in Fig. 9. The re-sults of these calculations indicate that the values of the maximumunsteady stress in the groove exceed 550 MPa, whereas in theundercut they are close to 500 MPa. Considering the materialswhich are used at present, such big values of stress are unaccept-able in terms of safety and reliability of the turbine operation.

4.1. Shape optimisation of the blade groove

It is preliminarily assumed that the blades are mounted on therotor by means of typical hammer grooves. However, the thermal-strength numerical simulations of the rotor heating process showthat the obtained maximum stress level is very high (Fig. 2), whichmight impair the machine operation reliability and limit its life.The maximum stress values are obtained in the bottom cornersof the groove. The appearing stress results from the radial gradientof temperature in the rotor. Due to it, the deformations close to therotor external surface, where the temperature is higher, are muchbigger than those inside. In this way, the bending of the grooves oc-curs and the maximum bending moment appears on the groovebottom edge. The corner causes an additional stress concentrationin its area. To minimise this effect, measures are taken to reducethis stress by redesigning (optimising the shape of) the groove.The shape optimisation task is solved using the response surfacemethodology. The determination of the response surface is imple-mented by means of the Kriging model based on the CCD type ofthe design of experiments. Depending on the number of designvariables, the plan requires calculations for a different number ofdesign points: 15 in the case of the smallest number of 3 variables,and 151 for the biggest case under analysis with 11 variables. Con-sequently, for each design point, i.e. for the set values of designvariables, thermal-strength simulations of the analysed rotor arecarried out with a view to defining the dependence (or its approx-imation) of the stress resulting during the turbine start-up processon the decision variables. Example curves illustrating the depen-dence of stress in the groove bottom on selected design variablesare shown in Fig. 3.

Each of the curves is defined for its own domain, and its shape isdetermined at fixed values of the remaining decision variables ofthe task. As all decision variables have an impact on the groovebottom shape, a change in any of them results in a change in theshape of the curves presented in Fig. 3.

The optimisation of the obtained response surface is carried outusing the genetic algorithm implemented in Ansys software.

The groove bottom redesigning process calls for assuming adescription of the shape of this area by means of design variables,whose values become the object of the optimisation. The idea ofthe optimisation is to obtain the biggest possible reduction in max-imum stress in the groove bottoms which occurs at the turbinestart-up only by means of changes in the design form of theseareas. To achieve this, numerous different variants of the optimisa-tion calculations are performed, using a different mathematicaldescription of the groove shape. Fig. 4 shows the analysed groovemodelling variants with marked dimensions which constitute deci-sion variables of the optimisation task. The constraints are in thiscase defined as constants in the form of the upper and lowerbounds of the domain of each variable to eliminate shapes of thegroove surface which are physically impossible. This is essential

Fig. 2. The rotor geometrical form with marked areas to be optimised.

Fig. 3. Dependence of maximum stress on selected design variables (markings as inFig. 4f).

420 G. Nowak, A. Rusin / Energy Conversion and Management 74 (2013) 417–425

from the point of view of the automation of optimisation computa-tions, where one incorrect (e.g. loop) configuration causes the com-putational process to stop.

The objective function in the presented optimisation task is tominimise the maximum values of von Mises stress during the un-steady phase of the turbine operation – the start-up.

reqmðx1; x2; . . . ; xn; tÞ !min ð7Þ

Fig. 5 is an overall presentation of the results of this optimisa-tion for the configuration shown in Fig. 4.

The analysis starts with the optimisation of the fillet radii of thecorners of the hammer groove and of its height. It is quickly foundthat it is the radius of the bottom corner that has the biggest im-pact on the stress value, and, which seems fairly obvious, the big-ger the radius, the smaller the stress values (b). However,increasing this radius is limited due to the need to fix the bladeroot mounted in the groove. Therefore, in the next step it is decidedto perform deeper undercuts of the groove corners, thus lengthen-ing the curvature radii (c). In this case, the undercut centre is lo-cated at the same place as the centre of the initial groovecurvature and it is the radius which is optimised. The use of anundercut with a constant radius does not bring the desired resultsand even though it is possible to use curvatures with longer radiithan before, the stress level is a bit higher. This is probably the ef-fect of the rise in the groove bending moment caused by an in-crease in the arm of the acting force. The observations made in

the process lead to an attempt to model this area with a curve witha variable curvature, without deepening the groove unnecessarily,i.e. without increasing stress. Therefore, an analysis is performed ofthe possibility of modelling the groove bottom by means of splinecurves whose shape is controlled using a specific number of controlpoints which belong to these curves. A variant with two curves(one curve in each corner (d)) and a variant with one curve describ-ing the entire groove bottom (e and f) are analysed. Both ap-proaches prove to be equally good. Nevertheless, the lowestmaximum stress values are obtained for variant (f) with one curvedefined by 7 control points. This gives 11 design variables (coordi-nates of the points) because 3 out of 14 coordinates are pre-deter-mined (the axial coordinates of the extreme points and of thecentral point of the curve). In variant (e) 6 coordinates are pre-determined, which gives 8 design variables, and the obtained stressvalues are higher than in the case with two curves.

The variant with the lowest stress level (f) is selected as the fi-nal solution to the set optimisation task. In it, relatively gentle cur-vatures in the stress concentration zones are obtained owing to theassumed way of the groove bottom modelling. This shape of thegroove bottom causes that the obtained level of the maximumstress of 336 MPa during the analysed start-up is by almost200 MPa lower than in the initial phase (an almost 36% improve-ment). It should be emphasised that the presented stress valuesdo not take account of the loads caused by the blade inertial forceand they include only the thermal stress resulting from heat ex-change between steam and the shaft. Taking account of the loadscaused by the blade impact causes a reduction in von Mises stressin the first stage of the rotor at nominal rotation speed by approx-imately 45 MPa. This results from the fact that the dominatingthermal and inertial components feature opposite senses. Thegroove shape assumed as the final one is presented in Fig. 6 to-gether with maximum von Mises stress distribution.

The groove shape obtained in the optimisation process is asym-metrical, which results from the temperature gradient asymmetryon both sides of the groove. This shape ensures almost identicalstress values on both sides of the groove. These values are the high-est at the flattest parts of the bottom. Therefore the relatively bigcurvatures at the ends and in the centre of the curve which modelsthe bottom do not cause the effect of stress concentration. It shouldalso be noted here that the thermal stress occurring at the groovebottom is a compressive stress which is partly lessened by tensilecomponents resulting from the blade inertial forces.

4.2. The shaft undercut optimisation

The next step after the design structure of the blade groove isdetermined is to redesign the shaft undercut in the balance pistonarea. In this case, the change in the rotor diameter is more thandouble. Moreover, this is an area that separates the hot inlet zone

Fig. 4. Groove bottom modelling variants with marked decision variables.

350

400

450

500

300

550

Mis

es s

tres

s M

Pa

a

b

c

d

e

f

1 2 3 5 6

Fig. 5. Optimisation results for different variants of the groove shape description.

Fig. 6. The final groove shape with maximum stress contour.

G. Nowak, A. Rusin / Energy Conversion and Management 74 (2013) 417–425 421

of the rotor from the much cooler zone of the end sealing and of thebearing, which results in considerable gradients of temperature.Due to that, stress concentration arises in the analysed corner.The stress values may be close to the material yield point (at smallradii of the fillet curvature). The optimisation process starts withthe selection of the fillet curvature radius of the corner under con-sideration. This, however, does not give satisfying results, i.e. a sen-sible increase in the curvature radius does not result in asignificant reduction in stress. Therefore it is decided to modelthe undercut by means of a spline curve based on 4 control points(Fig. 7). A modification of the location of these points causes achange in the corner undercut shape, which results in – contraryto a simple fillet – a fillet with a variable curvature.

The presented method of the corner modelling leads to the for-mulation of the shape optimisation task using 6 design variableswhich are the coordinates of the control points belonging to the

curve; the extreme points can move in one plane only. The locationof the inside control points is determined in relation to the locationof the extreme points so that correct curves (without loops) may beobtained at any configuration of the coordinates. Fig. 7 presents allthe design variables on the optimised geometry of the rotor under-cut. Owing to this shape, the maximum von Mises stress may bereduced by about 30%.

4.3. Optimisation of the rotor inner chambers

Despite the conducted optimisation processes of the grooveshape and of the rotor undercut, which result in a significantreduction in stress during the unsteady phases of operation, themaximum stress level in the rotor during the turbine start-up is

Fig. 7. The final shape of the rotor undercut with marked design variables.

Fig. 8. The rotor with hollowed out chambers.

422 G. Nowak, A. Rusin / Energy Conversion and Management 74 (2013) 417–425

still high, which may reduce its life. In order to prevent this, stepshave to be taken with a view to a further reduction in the level ofunsteady stress. Because the large gradients of temperatures dur-ing the initial heating phase are directly responsible for the ther-mal stress values, methods have to be found to level thetemperature fields. One of the solutions may be a reduction inthe amount of material in the areas of the highest gradients oftemperature.

Therefore it is decided that the rotor should be hollowed out atthe steam inlet area. Two cases are analysed, i.e. with one biggerchamber and with two smaller ones. The introduction of thesechambers causes that in their areas the temperatures are levelledrelatively fast in the radial direction, which substantially reducesthe thermal stress observed in the rotor blade grooves. Becausethe location and the size of the inner chambers have an impacton the thermal and strength state of the rotor, an optimisation taskis formulated in this case as well. The object of the task is to findthe optimum location and size of the chambers. The optimisationobjective, like previously, is to minimise the stress during the tur-bine start-up.

The introduction of the chambers results in a new stress con-centration zone – the corners of these chambers. Therefore the ra-dii of the curvatures of the rotor inner chambers are included in theset of the design variables of the optimisation task which is beingsolved. Fig. 8 presents a model of the rotor fragment with two in-ner chambers with additionally marked design variables of theoptimisation task. The presented configuration of the chambers is

already an optimisation result. An analysis of the cases with oneand two chambers gives very similar results in terms of the maxi-mum stress values. However, in order to ensure the rotor appropri-ate rigidity, the length of a single chamber is limited and, for thisreason, the use of two chambers seems to be more favourable inthis respect. The reach of the chambers (in the axial direction)determines the areas where stress occurs in the blade grooves be-cause the maximum thermal stress appears in the first groove rightafter the chambers, i.e. in the presented case – in the groove of thefifth stage. Moving this maximum further into the flow systembrings measurable benefits because lower and lower gradients ofthe temperature in the rotor are observed in subsequent stages(a smaller heat flux is received by the rotor due to a smaller tem-perature difference between the flowing steam and the rotor). Be-sides, the maximum stress occurs at a lower temperature of therotor material. Because the material features better mechanicalproperties in a lower temperature, the safety margin in this caseis bigger.

5. Optimisation of the turbine start-up curve

After the optimisation of the critical areas of the turbine rotor,the next task is to optimise the turbine start-up curve. The analysisin this case is focused on the curve of the change in temperature atthe turbine inlet. The aim of the task is to shorten the time of thestart-up process, maintaining safe operation of the turbine.

Fig. 9. The curves of the changes in the live steam temperature during start-up. Fig. 10. Dependence of the safety factor on the value of multipliers.

G. Nowak, A. Rusin / Energy Conversion and Management 74 (2013) 417–425 423

Shortening the start-up process is of significant importance be-cause it limits the non-productive phase of the power unit opera-tion. On the other hand, it intensifies the unsteady processes ofthe heat transfer between the working medium and the materialof the machinery components, which entails higher thermal loadsof these components. Therefore, from the point of view of the oper-ational safety and reliability, the start-up process should be runmore slowly. This means that two contradictory objectives haveto be dealt with here. The solution to this problem has to be foundby means of optimisation in such a way that the start-up is carriedout fast enough without exposing the power unit components toexcessive (hazardous) loads.

The task is solved through a modification of the previously as-sumed curve of the steam temperature change at the turbine inlet.A characteristic with 76 time steps (3 min each) is divided intothree ranges: 0–15, 15–30, 30–76, in which the rate of the increasein the steam temperature is optimised (Fig. 9). In each of the de-fined ranges, the curve local slope coefficient is multiplied by anappropriate multiplier (ARG1, ARG2 and ARG3) which is differentfor each range, and the resulting stress is observed. The value ofeach multiplier may vary in the range of 0.8–1.5.

The task is thus defined by three design variables – the start-upcurve multipliers in individual ranges, and the optimisation objec-tive is the safety factor defined as:

s ¼ ry

reqmð8Þ

where ry is the material yield point (Fig. 10) for the nominal work-ing temperature (650 �C), and reqv denotes the operational vonMises stress.

The minimum safety factor assumed during the calculations is1.2. Besides the safety factor, all three design variables are the taskadditional objectives. In this way, the optimisation is carried outfor objectives related to the safety factor and the curve multipliersmaximisation.

Like before, the set task is solved using the RSM with the Krigingmeta-model which precisely maps the relationships between theinput data and the safety factor (correlation coefficient R = 1).The relations between the start-up curve multipliers and the stresscoefficient are presented in Fig. 10. The course of the curve for eachmultiplier is shown at a fixed value of the other two, for the lowerbound of the domain (LB = 0.8), mean value from the interval(MEAN = 1.15) and for the upper bound of the set of solutions

(UB = 1.5). The impact of decision variables on each other and –in consequence – on the safety factor can be seen in the figure. Itis therefore impossible to present the system response in the formof a single function. Instead, a set of approximation functions hasto be employed to do it. The search for the optimum configurationis conducted using a screening method which results in sets ofmultipliers that satisfy the task objective of the safety factor level.

The obtained results compared to the initial start-up curve arepresented in Fig. 9. The chart also shows the area where the opti-mised curve could appear. The boundaries of this area present thestart-up curves for extreme values of the multipliers (0.8 and 1.5).The optimised curve features a higher rate of the increase in thesteam temperature (ARG1 = 1.4) in the initial phase of the start-up and a lower rate – in the subsequent phases (ARG2 = 0.85,ARG3 = 0.89). With the start-up process run in this way, the safetyfactor reaches the assumed limit value of s = 1.2. As it can be seenin the presented chart, the shortening of the entire start-up processis slight. The live steam nominal temperature at the turbine inlet isreached only a few minutes faster. However, the stress maximumis shifted in time and occurs at a lower temperature of the rotormaterial. In respect of the deterioration in the strength parametersresulting from temperature, the time shift mentioned above is ben-eficial because the safety margin between the operational and per-missible stress values is bigger.

Globally, the highest von Mises stress level is observed in thecorner of Chamber 1, but the other selected areas also feature highstress values (Fig. 12).

The optimisation does not result in stress reduction. In somecases the stress level is even higher, but in each case there is atleast an hour’s shift of the stress maximum towards the start-upbeginning. Analysing the heating process, it can be seen that thisshift results in the occurrence of the biggest loads at the materialtemperature which is by at least 150 �C lower than in the case be-fore optimisation (Fig. 11).

Such a temperature difference raises for example the materialyield point by approximately 20% (Fig. 13).

6. Summary

The paper presents an optimisation of the rotor shape and of themethod of running the start-up process of the steam turbine for ul-tra-supercritical parameters. The analysed turbine is a conceptual

Fig. 11. The temperature and von Mises curves in the rotor selected areas for the initial and optimised start-ups.

Fig. 12. Maximum von Mises stress in the rotor after shape and start-up optimisation.

Fig. 13. The rotor material yield point depending on temperature.

424 G. Nowak, A. Rusin / Energy Conversion and Management 74 (2013) 417–425

machine and the presented calculations relate to the rotor of itshigh pressure part. The very high working parameters of the ma-chine result in the occurrence of hazardous thermal stress in thetransient phases of the operation. Within the study, the stress con-centration zones (the blade grooves and the shaft undercut in thebalance piston area) are selected, and an optimisation of the rotor

shape in these areas is performed. These measures allow a substan-tial reduction in the maximum stress that arises during the turbinestart-up. Additionally, chambers are hollowed in the front part ofthe rotor, which results in a further decrease in stress.

In the next stage, the method of running the turbine start-upprocess is optimised so that, keeping a safe stress level in the rotor,this non-productive phase of operation can be shortened as muchas possible. Although the conducted optimisation does not result ina significant shortening of the time of the start-up process, anotherbeneficial effect of the start-up characteristic modification can beobserved. The start-up process may be run faster in its initial phaseand then it may slow down. This results in a shift of the stress max-imum towards the beginning phase of the start-up with a lowertemperature of the rotor material. At a lower temperature the rotorfeatures higher strength parameters, which makes its operationsafer.

The issues discussed here are really essential because, as expe-rience shows, the first supercritical units made recently are besetby continual failures and their operation continually creates signif-icant problems. Therefore, further research and testing in this areaare by all means desired and each new experience gained in thefield is very important.

Acknowledgements

The results presented in this paper were obtained from researchwork co-financed by the Polish National Centre of Research andDevelopment in the framework of Contract SP/E/1/67484/10 –Strategic Research Programme – Advanced technologies for

G. Nowak, A. Rusin / Energy Conversion and Management 74 (2013) 417–425 425

obtaining energy: Development of a technology for highly efficientzero-emission coal-fired power units integrated with CO2 capture.

References

[1] Fukuda Y. Development of advanced ultra supercritical fossil power plants inJapan: materials and high temperature corrosion properties. Mater Sci For2011;696(September):236–41.

[2] Li Yong, Sun Haoran, Nie Yuhuo. Thermal stress analysis of 600 MW steamturbine rotor in different governing modes. 978-1-422-4813-5 28-31-March2010 IEEE.

[3] Zhang C, Hu N, Wang J, Chen Q, He F, Wang X. Thermal stress analysis for rotorof 600 MW supercritical steam turbine. In: 2011 International Conference onConsumer Electronics, Communications and Networks, CECNet 2011 –Proceedings; 2011. p. 2824–7.

[4] Hai-jun Xuan, Jian Song. Failure analysis and optimization design of acentrifuge rotor. Eng Fail Anal 2007;14(1):101–9.

[5] Kosman W. Thermal analysis of cooled supercritical steam turbinecomponents. Energy 2010;35(2):1181–7.

[6] Ansys 13 Software Documentation, Ansys Inc.[7] Bradley N. The response surface methodology, Master Thesis, Department of

Mathematical Sciences, Indiana University of South Bend; 2007.[8] Bethea R, Duran B, Boullion T. Statistical methods for engineers and

scientists. New York and Basel: Marcel Dekker Inc.; 1984.[9] Li M, Li G, Azarm S. A Kriging metamodel assisted multi-objective genetic

algorithm for design optimization. J Mech Des 2008;130(3).[10] Deb K. Multi-objective optimization using evolutionary algorithms. Wiley &

Sons; 2001.[11] Mueller S. Bio-inspired optimization algorithms for engineering application,

PhD thesis, Swiss Federal Institute of Technology, Zurich; 2002.[12] Nowak G, Wróblewski W. Cooling system optimisation of turbine guide vane.

Appl Therm Eng 2009;19:567–72.[13] Rusin A. Lipka M. Łukowicz H. Thermal and stress states in unsteady

conditions of operation of the rotors of ultra-supercritical parametersturbines. In: VII Int. Conf. Power Engineering, Wrocław; 2012, 7–9November 2012.