machines

Article Inter-Stage Dynamic Performance of an Axial † Compressor of a Twin-Shaft Industrial Gas Turbine

Samuel Cruz-Manzo 1,*, Senthil Krishnababu 2, Vili Panov 2 and Chris Bingham 1

1 School of Engineering, University of Lincoln, Lincoln LN6 7TS, UK; [email protected] 2 Siemens Energy Industrial Turbomachinery Ltd., Lincoln LN5 7DF, UK; [email protected] (S.K.); [email protected] (V.P.) * Correspondence: [email protected] This paper is an extended version of our paper published in “Cruz-Manzo, S.; Krishnababu, S.; Panov, V.; † Zhang, Y. Performance calculation of a multistage axial compressor through a semi-empirical modelling framework. In Proceedings of the Global Power and Propulsion Society 2019, Beijing, China, 16–18 September 2019; Paper No. GPPS-BJ-2019-123”.



Received: 10 November 2020; Accepted: 7 December 2020; Published: 9 December 2020


Abstract: In this study, the inter-stage dynamic performance of a multistage axial compressor is simulated through a semi-empirical model constructed in the Matlab Simulink environment. A semi-empirical 1-D compressor model developed in a previous study has been integrated with a 0-D twin-shaft gas turbine model developed in the Simulink environment. Inter-stage performance data generated through a high-fidelity design tool and based on throughflow analysis are considered for the development of the inter-stage modeling framework. Inter-stage performance data comprise pressure ratio at various speeds with nominal variable stator guide vane (VGV) positions and with hypothetical offsets to them with respect to the gas generator speed (GGS). Compressor discharge pressure, fuel flow demand, GGS and power turbine speed measured during the operation of a twin-shaft industrial gas turbine are considered for the dynamic model validation. The dynamic performance of the axial-compressor, simulated by the developed modeling framework, is represented on the overall compressor map and individual stage characteristic maps. The effect of extracting air through the bleed port in the engine center-casing on transient performance represented on overall compressor map and stage performance maps is also presented. In addition, the dynamic performance of the axial-compressor with an offset in VGV position is represented on the overall compressor map and individual stage characteristic maps. The study couples the fundamental principles of axial compressors and a semi-empirical modeling architecture in a complementary manner. The developed modeling framework can provide a deeper understanding of the factors that affect the dynamic performance of axial compressors.

Keywords: multistage axial compressor; inter-stage dynamic modeling; twin-shaft gas turbine; Simulink model

1. Introduction The performance of multiaxial compressors plays an important role in the operation of industrial gas turbines (IGTs). The pressure of the air entering the compressor is increased through a series of stages. Each stage comprises a set of rotor blades followed by a set of stator blades. The kinetic energy of air is increased by the rotor blades, and the stator blades transform the kinetic energy of air into static pressure. A variation of the angle of the stator blades across the different stages of an axial compressor ensures correct air diffusion at different operating conditions [1], where a simple actuator changes the angle of the variable stator guide vane (VGVs). The closure and opening of VGVs are a

Machines 2020, 8, 83; doi:10.3390/machines8040083 www.mdpi.com/journal/machines Machines 2020, 8, 83 2 of 20 function of the demanded inlet guide vane (IGV) position. The IGV position is typically scheduled with respect to the gas generator speed (GGS) of gas turbines [2]. Various studies in the literature have reported multistage axial compressor models [2–10]. The aforementioned studies developed a model to predict the inter-stage performance of the axial-compressor at steady-state conditions. In addition, the reported models predicted the variation in the IGV and VGV in the compressor performance. Simulink is a powerful programming environment for modeling and simulation of multidomain dynamical systems such as gas turbines. This simulation tool offers the possibility to represent governing thermodynamic equations of the IGT in the Simulink environment. Several studies have reported gas turbine models developed in the Simulink environment [11–18]. In a previous study [19], a thermodynamic model to predict the performance of a twin-shaft IGT at different operating conditions was reported. The modeling architecture considered a lumped axial compressor model and was developed using a commercial thermodynamic toolbox (Thermolib, EUtech), which is compatible with the Simulink environment. Damiani et al. [20] reported a one-dimensional stage-by-stage model to predict the design and off-design performance of a multistage axial flow compressor. The model was developed in the Matlab Simulink environment and was able to reproduce the compressor performance at different operating conditions. It was possible to predict the blade cooling system performance since the total pressure and total temperature values of the cooling air bleed offs were correctly calculated. The aforementioned studies do not consider the inter-stage dynamic performance of the axial-compressor during different engine operating conditions in the field. In a previous study [21], a 1-D semi-empirical compressor model, based on design parameters, compressor performance data set, and theoretical relations (velocity triangles for blade design) from an 11-stage axial compressor, was developed in the Matlab® environment. A high-fidelity design tool (HFDT) was used to simulate multistage axial compressor steady-state operation and assisted the development of the proposed modeling framework. HFDT is a Siemens standard streamline curvature through flow analysis [22] code that computes the aerodynamic parameters across the compressor. The model predicted the inter-stage performance of the compressor at full and part load steady-state conditions. This manuscript is an extension from the authors’ previous study [21] and considers the integration of the 1-D compressor modeling architecture that predicts stagewise performance at steady-state conditions with the 0-D twin-shaft dynamic gas turbine model [23]. The 0-D gas turbine model, reported by Panov [23], enables the prediction of the dynamic gas turbine performance at different operating conditions. The overall modeling architecture is developed in the Simulink environment and allows inter-stage dynamic performance simulation of a multistage axial compressor during engine transient operation. In this study, the architecture of the 1-D model previously reported [21] has been extended by considering HFDT data set at different rotational speeds and with an offset in IGV position outside of its scheduled nominal position with respect to GGS. The developed 1-D compressor modeling architecture is interfaced with a 0-D twin-shaft engine model in the Simulink environment to predict the dynamic inter-stage performance of the axial compressor during a step load change, air bleed off from the engine center-casing, and offset in IGV position outside of its scheduled nominal position with respect to GGS. It is possible to simulate the dynamic performance of a multistage axial compressor during a twin-shaft IGT operation and represent it in the overall compressor map and individual stage characteristic maps. The overall modeling framework is validated with real-world test data of a twin-shaft engine measured during the load acceptance tests. The representation of the dynamic compressor performance on stage-characteristic maps can provide valuable information during the dynamic operation of an axial compressor in the field. For example, during the dynamic transition between two operating conditions, a given stage may perform close to its stability limit, which could cause instability of the whole compressor during the dynamic operation, particularly at low speeds. Such instability in the compressor could compromise the performance and operability of the whole IGT. Machines 2020, 8, 83 3 of 20

This manuscript is organized as follows: A brief description of the 1-D compressor modeling architecture from a previous study [21] is presented in Section2. HFDT data set at di fferent rotational speeds and with the offset in IGV position outside of its scheduled nominal position with respect to GGS structure are presented as well. In Section3, a Simulink model of an IGT interfaced with the 1-D compressor model is presented. The dynamic validation of the modeling architecture with real-world data measured during load increase in an IGT is presented in Section4. The results are presented in Section5, which demonstrates the dynamic performance of the compressor represented in the overall compressor performance map and stage-performance maps. The manuscript is concluded with Section6, which summarizes the main results.

2. Compressor Modeling Architecture In a previous study [21], the 1-D model of the axial-compressor was developed. A brief summary of the devised model is described in this section. The inter-stage modeling framework considered theoretical velocity triangles, which relate the power input to the axial compressor stage. An analysis of these velocity triangles utilizing trigonometric relations and thermodynamic principles can derive the temperature and pressure rise across the compressor stages. The following equations comprise the overall modeling framework. The difference in total temperature between the outlet and inlet of the compressor stage is defined as:

UλCX ∆T2 1 = (tan α2 tan α1) (1) − CP − where U = rω, U is the blade velocity, r is the tip radius, and ω is the angular velocity; α1,2 represents . m the angle of the air absolute velocity from the axial direction CX entering and leaving the rotor, CX = ρA , . m is the airflow rate, ρ is the air density, and A is the annulus area. λ is the work-done factor and accounts for the reduction in work capacity across the compressor axial stage [1]. The pressure ratio through the stage can be defined as:

 γ 1  CPTin γ− tan α2 tan α1 = PR 1 (2) − ηUλCX − where η is the stage efficiency, and Tin is the stage inlet total temperature. The derivation of Equations (1) and (2) and the relation between the velocity triangles and thermodynamic principles have been reported by Saravanamutto et al. [24]. The 1-D inter-stage compressor model developed in the previous study [21] is mainly comprised . of Equations (1) and (2) and requires the rotational speed ω, flow rate m, efficiency η and ambient temperature Tamb and ambient pressure Pamb as inputs. The outputs from the 1-D compressor model are the pressure ratio PR and temperature Tout across the different stages of the axial compressor. In addition, compressor design parameters and the Equations (1) and (2) were required to develop the semi-empirical 1-D modeling architecture. The annulus area A, tip radius r and blade angles across the different stages of the compressor were provided by a compressor manufacturer. Thermodynamic parameters represented in Equations (1) and (2) such as air density ρ, heat capacity CP, and the ratio between heat capacities γ across the different stages are calculated through data representing stagewise temperature and pressure ratio collected from an HFDT. HFDT is a Siemens standard through flow analysis code that computes the performance of the multistage compressor as an axisymmetric model. HFDT is essentially a two-dimensional computational fluid dynamics code based on the streamline curvature method, as reported by Wu [22]. For a given operating condition defined by the speed and throttle set by the compressor exit pressure, the code calculates the primary flow parameters such as pressure, the temperature at defined axial stations. Typically, these stations are located at the inlet and exit of the blade rows. The use of throughflow analysis for estimation of inter-stage pressure and temperature in a multistage axial compressor was reported in the study of Machines 2020, 8, 83 4 of 20

Damiani et al. [20]. Schnoes et al. [25] considered throughflow analysis for the design optimization of a Machines15-stage 2020, 8 axial, x FOR compressor. PEER REVIEW 4 of 19 A constant efficiency η is considered in the calculation of pressure ratio across the stages study.(Equation Schnoes (2)). et Theal. [25] effi ciencyestimatedη is obtained the efficiency from a 0-Dη across model the of compressorstages of a map15-stage (pressure axial ratio compressor and usingspeed throughflow as input and analysis efficiency and as demonstrated output) related a to variation the 11-stage of axialthe efficiency compressor η considered across the in different this stages.study. In this Schnoes study, et a al. constant [25] estimated efficiency the eηffi isciency consideredη across inthe the stages calculation of a 15-stage of pressure axial compressorratio (Equation (2)). usingThe parameter throughflow λ analysisin Equation and demonstrated (2) accounts a variationfor the variation of the effi ciencyof efficiencyη across η the across different the stages. stages to matchIn the this pressure study, a constant ratio estimated efficiency fromη is considered HFDT and in estimated the calculation by Equation of pressure (2). ratio (Equation (2)). λ η TheIn this parameter study, thein Equation pressure (2) ratio accounts for each for the stag variatione has ofalso effi ciencybeen collectedacross the from stages the to HFDT match theduring pressure ratio estimated from HFDT and estimated by Equation (2). compressor simulation at different rotational speeds and at a different offset in the IGV position. In this study, the pressure ratio for each stage has also been collected from the HFDT during Stagewise pressure ratio collected from the HFDT allows the validation of the compressor model at compressor simulation at different rotational speeds and at a different offset in the IGV position. steady-stateStagewise conditions. pressure ratio This collected has been from demonstrated the HFDT allows in a previous the validation study of [21]. the compressor The IGV demand model at angle was steady-statescheduled with conditions. respect This to hascompressor been demonstrated speed. The in aposition previous (in study degrees) [21]. The of IGVVGVs demand is a function angle of IGV wasposition. scheduled An offset with in respect the IGV tocompressor position is considered speed. The positionas a position (in degrees) of the IGV of VGVs outside is a its function scheduled positionof IGV with position. respect Anto the offset GGS. in theFigure IGV 1 position shows the is considered total pressure as a positionratio calculated of the IGV by outsidethe HFDT its and representedscheduled in position the overall with respect compressor to the GGS. performanc Figure1 showse map the of total an pressure11-stage ratio axial calculated compressor. by the The compressorHFDT and map represented was validated in the overallwith real-world compressor comp performanceressor data. map The of an total 11-stage pressure axial compressor.ratio estimated fromThe the compressor HFDT is map calculated was validated by withmultiplying real-world the compressor individual data. Thepressure total pressure ratio across ratio estimated the stages comprisingfrom the the HFDT 11-stage is calculated axial compressor. by multiplying The the comp individualressor pressure map shown ratio across in Figure the stages 1 considers comprising 9 “beta the 11-stage axial compressor. The compressor map shown in Figure1 considers 9 “beta lines” across lines” across each speed. These beta lines will allow the interpolation of pressure ratio between each speed. These beta lines will allow the interpolation of pressure ratio between different speeds different speeds and flow rates during the dynamic compressor simulation. The running line and flow rates during the dynamic compressor simulation. The running line representing the overall representingoperating the range overall of the operating compressor range is shown of the in comp Figureressor1. It shows is shown the total in Figure pressure 1. ratioIt shows vs. flow the total pressurerate atratio different vs. flow operational rate at speeds.different The operatio pressurenal ratio speeds. and flow The rate pressure were normalized ratio and with flow respect rate were normalizedto the pressure with respect ratio and to flow the ratepressure values ratio at 100% and speed flow and rate at values nominal at conditions. 100% speed The and operational at nominal conditions.speed was The normalized operational with speed respect was to normalized the maximum with speed respectωmax .to Figure the maximum1 also shows speed the pressure ωmax. Figure 1 alsoratio shows calculated the pressure by the HFDT ratio and calculated considering by anthe o ffHFsetDT in theand position considering of the VGVs an offset outside in itsthe nominal position of the VGVsscheduled outside position its nominal with respect scheduled to GGS. position with respect to GGS.

FigureFigure 1. Pressure 1. Pressure ratio ratio estimated estimated by by high-fidelity high-fidelity design tooltool (HFDT) (HFDT) in in overall overall compressor compressor performanceperformance map. map.

The 1-D modeling procedure can be executed in the Matlab® environment. It consists of an optimization process to estimate the parameter λ and a calculation process to predict stagewise temperature and pressure ratio. More information about the optimization process and calculation process from the 1-D compressor modeling architecture can be found in the previous study [21]. White et al. [26] presented a flow diagram to depict the 1-D modeling architecture of a multistage axial compressor. Their modeling architecture considers subroutines and estimates the blade angles across the stages in the compressor to predict overall compressor performance maps. This is in contrast to this study in which the geometry of the 11-stage compressor was provided by the compressor manufacturer. Figure 2 shows the estimation of the parameter λ related to the performance running line (Figure 1) at 100% rotational speed and considering the optimization process from the 1-D compressor

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The 1-D modeling procedure can be executed in the Matlab® environment. It consists of an optimization process to estimate the parameter λ and a calculation process to predict stagewise temperature and pressure ratio. More information about the optimization process and calculation process from the 1-D compressor modeling architecture can be found in the previous study [21]. White et al. [26] presented a flow diagram to depict the 1-D modeling architecture of a multistage axial compressor. Their modeling architecture considers subroutines and estimates the blade angles across the stages in the compressor to predict overall compressor performance maps. This is in contrast to this study in which the geometry of the 11-stage compressor was provided by the compressor manufacturer. MachinesFigure 20202, 8 shows, x FOR thePEER estimation REVIEW of the parameter λ related to the performance running line (Figure5 of 119) Machines 2020, 8, x FOR PEER REVIEW 5 of 19 at 100% rotational speed and considering the optimization process from the 1-D compressor modeling architecturemodelingmodeling architecturearchitecture [21]. Figures [21].[21].3 and FiguresFigures4 show 33 the andand resulting 44 showshow stagewise thethe resultingresulting pressure stagewisestagewise and temperature pressurepressure andand ratio temperaturetemperature calculated fromratioratio thecalculatedcalculated parameter fromfromλ and thethe Equations parameterparameter (1) λλ andandand (2) EquationsEquations and considering (1)(1) andand the(2)(2) calculationandand consideringconsidering process. thethe The calculationcalculation pressure andprocess.process. temperature TheThe pressurepressure ratio and shownand temperaturetemperature in Figures 3ratioratio and shown4shown were normalizedinin FiguresFigures 33 withandand 44 respect werewere normalizednormalized to the maximum withwith respectrespect value oftoto thethethe HFDTmaximummaximum stagewise valuevalue data ofof thethe from HFDTHFDT the totalstagewisestagewise pressure datadata ratio fromfrom of thethe the totaltotal compressor pressurepressure performance ratioratio ofof thethe running compressorcompressor line (designperformanceperformance point) runningrunning at 100% lineline rotational (design(design speed. point)point) atat 100%100% rotationalrotational speed.speed.

Figure 2. Estimation of parameter λ from HFDT at design point and 100% ωmax. FigureFigure 2. 2.Estimation Estimation ofof parameterparameterλ λfrom from HFDT HFDT at at design design point point and and 100% 100%ω ωmaxmax.

Figure 3. Comparison between stagewise pressure ratio from the 1-D model and HFDT at design point FigureFigure 3.3. ComparisonComparison betweenbetween stagewisestagewise pressurepressure ratiratioo fromfrom thethe 1-D1-D modelmodel andand HFDTHFDT atat designdesign andpoint 100% and ω100%max. ωmax. point and 100% ωmax.

FigureFigure 4.4. ComparisonComparison betweenbetween stagewisestagewise temperaturetemperature ratiratioo fromfrom thethe 1-D1-D modelmodel andand HFDTHFDT atat designdesign point and 100% ωmax. point and 100% ωmax.

Machines 2020, 8, x FOR PEER REVIEW 5 of 19

modeling architecture [21]. Figures 3 and 4 show the resulting stagewise pressure and temperature ratio calculated from the parameter λ and Equations (1) and (2) and considering the calculation process. The pressure and temperature ratio shown in Figures 3 and 4 were normalized with respect to the maximum value of the HFDT stagewise data from the total pressure ratio of the compressor performance running line (design point) at 100% rotational speed.

Figure 2. Estimation of parameter λ from HFDT at design point and 100% ωmax.

Machines 2020Figure, 8, 83 3. Comparison between stagewise pressure ratio from the 1-D model and HFDT at design 6 of 20

point and 100% ωmax.

FigureFigure 4. Comparison 4. Comparison between between stagewise stagewise temperature temperature rati ratioo from from the the 1-D 1-D model model and and HFDT HFDT at design at design pointpoint and 100%and 100%ωmax ωmax. .

3. Engine Modeling Architecture Machines 2020, 8, x FOR PEER REVIEW 6 of 19 A 0-D engine model constructed in a Simulink environment [23] was considered for the simulation of the3. dynamic Engine Modeling compressor Architecture performance of a twin-shaft IGT system at different load conditions. TheA 0-D 0-D twin-shaft engine model engine constructed model comprisesin a Simulink gas environment generator components [23] was considered and power for the turbine components,simulation and of theythe dynamic are connected compressor through performance an interduct of a component. twin-shaft IGT The system gas generator at different components load compriseconditions. the compressor, combustor, gas generator turbine (GGT), and a shaft connecting the compressorThe and 0-D GGT, twin-shaft as shown engine in Figuremodel 5comprises. Power turbinegas generator components components comprise and thepower power-turbine turbine (PT),components, load, and mechanical and they are shaft connected connecting through the an PTinterduct and load. component. As inputs, The gas the generator model requires components the load and thecomprise temperature the compressor, and pressure combustor, of the gas air generator entering theturbine compressor. (GGT), and Conservation a shaft connecting of mechanical the compressor and GGT, as shown in Figure 5. Power turbine components comprise the power-turbine energy for shafts, heat-soaking effects for metal parts, and conservation of thermodynamic energy are (PT), load, and mechanical shaft connecting the PT and load. As inputs, the model requires the load includedand the in thetemperature component and models. pressure Theof the dynamic air enteri modelng the ofcompressor. the gas turbine Conservation engine of can mechanical be expressed with aenergy system for of shafts, nonlinear heat-soaking differential effects equations. for metal pa Therts, following and conservation equation of isthermodynamic considered to energy model the dynamicare included response in of the pressure component in the models. individual The modeldynamic components: model of the gas turbine engine can be expressed with a system of nonlinear differential !equations. The following equation is considered to model the dynamic response of dPpressureout in∂ thePout individualX ∂modelPout ∂ςcomponents:j = + (3) dt ∂t ∂ςj ∂t dP ∂P v j ∂P ∂ς = + (3) dt ∂t ∂ς ∂t where ςj is the internal thermodynamic state. The first term on the right-hand side of Equation (3) representswhere volumeςj is the gasinternal dynamics, thermodynamic and the state. second The term first onterm the on right-hand the right-hand side side describes of Equation the behavior (3) of componentsrepresents volume using nonlinear gas dynamics, algebraic and the equations. second term More on the details right-hand of Equation side describes (3) and the the behavior differential of components using nonlinear algebraic equations. More details of Equation (3) and the differential equations representing the dynamic model of the twin-shaft gas turbine were reported in the study of equations representing the dynamic model of the twin-shaft gas turbine were reported in the study Panovof [Panov23]. [23].

FigureFigure 5. Representation 5. Representation of of 0-D 0-D engine engine modelmodel de developedveloped in in the the Simulink Simulink environment. environment.

The 0-D modeling architecture comprises a lumped axial compressor model and considers compressor performance maps for flow and efficiency as a function of pressure ratio and rotational speed. The 0-D engine model can predict the total flow, efficiency, pressure and temperature of air discharged by the compressor at steady and transient conditions. The normalized compressor flow map defined in the 0-D compressor model is shown in Figure 1. As discussed in the previous section, the 1-D model requires as inputs the temperature at ambient conditions, the airflow rate, efficiency, speed and lambda (λ work-done factor) values. The 0-D compressor developed in the Simulink environment [23] is interfaced with the 1-D model, as shown in Figure 6. The modeling architecture presented in this study combines the 1-D compressor model reported in a previous study [21] and developed in the Matlab® with the 0-D twin-shaft gas turbine model developed in the Simulink environment and reported by Panov [23]. As shown in Figure 6, both modeling architectures can be interfaced in the Simulink environment by feeding parameters from the 0-D twin-shaft engine model into the 1-D compressor model. The 1-D model requires flow, efficiency, and speed from the 0-D compressor (twin-shaft gas turbine model) and calculates the pressure and temperature rising across the different stages of the multistage axial compressor.

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The 0-D modeling architecture comprises a lumped axial compressor model and considers compressor performance maps for flow and efficiency as a function of pressure ratio and rotational speed. The 0-D engine model can predict the total flow, efficiency, pressure and temperature of air discharged by the compressor at steady and transient conditions. The normalized compressor flow map defined in the 0-D compressor model is shown in Figure1. As discussed in the previous section, the 1-D model requires as inputs the temperature at ambient conditions, the airflow rate, efficiency, speed and lambda (λ work-done factor) values. The 0-D compressor developed in the Simulink environment [23] is interfaced with the 1-D model, as shown in Figure6. The modeling architecture presented in this study combines the 1-D compressor model reported in a previous study [21] and developed in the Matlab® with the 0-D twin-shaft gas turbine model developed in the Simulink environment and reported by Panov [23]. As shown in Figure6, both modeling architectures can be interfaced in the Simulink environment by feeding parameters from the 0-D twin-shaft engine model into the 1-D compressor model. The 1-D model requires flow, efficiency, and speed from the 0-D compressor (twin-shaft gas turbine model) and calculates the pressure and temperature rising across

Machinesthe diff erent2020, 8 stages, x FOR PEER of the REVIEW multistage axial compressor. 7 of 19

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Figure 6. InterfaceInterface between between 0-D 0-D and and 1-D 1-D compresso compressorr models models in in the the Simulink environment. Figure 6. Interface between 0-D and 1-D compressor models in the Simulink environment. Values of the parameter λ are required in the 1-D compressor compressor mo model,del, as discussed in the the previous previous Section 22.. AValues 3-D matrixof the parameter defineddefined λ inin are thethe required SimulinkSimulink in the environmentenvironment 1-D compressor waswas mo defineddefineddel, as discussed toto includeinclude in all allthe the theprevious valuesvalues ofof the parameter Section 2. Aλ. 3-DThe The matrix 3-D 3-D matrix matrix defined calculates calculates in the Simulink 11 11 values values environment for for λλ fromfrom was a defined given tospeed, include a given all the valuevalues ofof IGV offsetoffset theand parameter a defined defined λ point . The 3-D on the matrix beta calculates line from 11 ththe valuese compressor for λ from map a given shown speed, in Figure a given1 1. .value TheThe of calculated IGV vectoroffset ofof λλ andvalues values a defined is is provided provided point on to theto the betathe 1-D line1-D model frommodel to th calculatee tocompressor calculate stagewise map stagewise shown pressure in pressure Figure and 1. temperature Theand calculated temperature across vector of λ values is provided to the 1-D model to calculate stagewise pressure and temperature acrossthe 11 stagesthe 11 ofstages the compressor.of the compressor. The overall The overal modelingl modeling architecture architecture comprises comprises the 0-D the engine 0-D engine model across the 11 stages of the compressor. The overall modeling architecture comprises the 0-D engine and the 1-D compressor model, as shown in Figure7. modelmodel and the and 1-D the 1-Dcompressor compressor mode model, asl, as shown shown in in FigureFigure 7.7.

FigureFigure 7. Overall 7. Overall modeling modeling architecture architecture for for dynamicdynamic simulation of of compressor compressor performance. performance.

FigureThe gas 7. Overallturbine modelingdata measured architecture during for the dynamic engine testsimulation of a twin-shaft of compressor IGT are performance. provided to the different components of the 0-D engine model. The deployed gas turbine 0-D model does not contain Thea control gas turbine system dataand measuredmechanical duringsystem thecompon engineents test such of asa twin-shaftthe GG shaft IGT and are the provided PT shaft. to the differentTherefore, components GGS, power of the turbine 0-D engine speed model. (PTS) and Th econt deployedrol system gas variables turbine such0-D modelas fuel flowdoes demand not contain a control(FFDEM), system VGV and demand mechanical (VGVDEM) system and componbleed\blow-offents such valve as demand the GG (BOVDEM) shaft and positions the PT are shaft. Therefore,provided GGS, to powerthe 0-D turbine gas turbine speed as (PTS) a time and history cont rolobtained system from variables engine suchtest. Theas fuel engine flow model demand (FFDEM),calculates VGV the demand pressure (VGVDEM) ratio of the and 0-D bleed\blow-offcompressor model valve from demand the measured (BOVDEM) engine positionsdata and are providedprovides to the the 0-Dflow gasto the turbine 1-D compressor as a time model. history The obtained 1-D model from calculates engine the test. dynamic The engineinter-stage model calculatescompressor the pressure performance ratio at ofdifferent the 0-D engine compressor load conditions. model This from will the be demonstrated measured engine in the results data and section. provides the flow to the 1-D compressor model. The 1-D model calculates the dynamic inter-stage compressor4. Compressor performance Dynamic at different Validation engine load conditions. This will be demonstrated in the results section. Measured engine data from a twin-shaft gas turbine operated during load acceptance test were considered for the validation of the developed model. The engine test data were logged during a step 4. Compressor Dynamic Validation change in engine load from 13% to 40% of the maximum engine load. Two sets of measurements Measured engine data from a twin-shaft gas turbine operated during load acceptance test were considered for the validation of the developed model. The engine test data were logged during a step change in engine load from 13% to 40% of the maximum engine load. Two sets of measurements

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The gas turbine data measured during the engine test of a twin-shaft IGT are provided to the different components of the 0-D engine model. The deployed gas turbine 0-D model does not contain a control system and mechanical system components such as the GG shaft and the PT shaft. Therefore, GGS, power turbine speed (PTS) and control system variables such as fuel flow demand (FFDEM), VGV demand (VGVDEM) and bleed blow-off valve demand (BOVDEM) positions are provided to the \ 0-D gas turbine as a time history obtained from engine test. The engine model calculates the pressure ratio of the 0-D compressor model from the measured engine data and provides the flow to the 1-D compressor model. The 1-D model calculates the dynamic inter-stage compressor performance at different engine load conditions. This will be demonstrated in the results section.

4. Compressor Dynamic Validation Measured engine data from a twin-shaft gas turbine operated during load acceptance test were considered for the validation of the developed model. The engine test data were logged during a step Machines 2020, 8, x FOR PEER REVIEW 8 of 19 change in engine load from 13% to 40% of the maximum engine load. Two sets of measurements whenwhen increasingincreasing engineengine loadload fromfrom 13%13% toto 40%40% werewere considered.considered. The firstfirst setset ofof measurementsmeasurements considersconsiders engineengine operationoperation withwith airair bleedbleed fromfrom center-casingcenter-casing forfor thethe purposepurpose ofof emissionemission control–COcontrol–CO turndown.turndown. The second set of measurementsmeasurements was takentaken during engine operationoperation withoutwithout air bleed,bleed, whichwhich correspondscorresponds toto naturalnatural turndownturndown engineengine operation.operation. InIn addition,addition, measurementsmeasurements duringduring engineengine teststests withwith center-casingcenter-casing airair bleedbleed duringduring loadload acceptanceacceptance fromfrom 26%26% toto 53%53% ofof thethe maximummaximum loadload havehave alsoalso beenbeen considered.considered. TheThe engineengine datadata comprisecomprise FFDEM,FFDEM, GGS,GGS, PTS,PTS, VGVDEMVGVDEM position,position, BOVDEMBOVDEM positionposition andand compressorcompressor dischargedischarge pressurepressure (CDP).(CDP). TheThe FFDEM,FFDEM, BOVDEM,BOVDEM, VGVDEM,VGVDEM, GGSGGS andand PTSPTS datadata areare consideredconsidered asas inputsinputs inin thethe modelingmodeling architecturearchitecture asas shownshown inin FigureFigure7 .7. The The measured measured CDP CDP allowed allowed thethe validationvalidation ofof thethe compressorcompressor model.model. Figure 88 shows shows thethe normalizednormalized GGSGGS withwith respectrespect to to the the measuredmeasured GGSGGS at at steady-state steady-state at at 53% 53% load. load. A decreaseA decrease in measuredin measured GGS GGS is observed is observed when when no air no bled air frombled from center-casing center-casing is considered, is considered, and with and increasingwith increasing load demandload demand from 13%from to 13% 40%. to 40%.

FigureFigure 8.8. Normalized gas generatorgenerator speedspeed (GGS)(GGS) consideredconsidered asas inputinput inin thethe engineengine model.model.

FiguresFigures9 9and and 10 10 show show a comparisona comparison between between the the measured measured CDP CDP and and the predictedthe predicted CDP CDP from from the 0-Dthe and0-D 1-Dand compressor 1-D compressor models. models. The volume The gasvolume dynamic gas expresseddynamic inexpressed Equation (3)in wasEquation considered (3) was in theconsidered 0-D compressor in the 0-D model compressor to simulate model the transientto simulate response the transient of the CDP,response as shown of the in CDP, Figures as 9shown and 10 in. TheFigures interaction 9 and 10. between The interaction the 0-D and between 1-D compressor the 0-D and models 1-D compressor is shown in Figuremodels6. is The shown 1-D compressor in Figure 6. modelThe 1-D requires compressor as inputs model the requires air mass as flow inputs and ethefficiency air mass calculated flow and by efficiency the 0-D compressor calculated modelby the and0-D calculatescompressor the model stagewise and calculates pressure ratiothe stagewise across the pressu 11 stages.re ratio The across total the pressure 11 stages. ratio The estimated total pressure by the 1-Dratio model estimated is calculated by the 1-D by multiplyingmodel is calculated all the individual by multiplying stagewise all the pressure individual ratio acrossstagewise the 11pressure stages. Inratio addition, across pressurethe 11 stages. loss related In addition, to the compressor pressure lo dissff relateduser and to outlet the compressor guide vanes diffuser was considered and outlet to approximateguide vanes was the simulatedconsidered 1-D to compressorapproximate model the simulated response 1-D with compressor the measured model data. response with the measured data.

Figure 9. Compressor discharge pressure with increasing load from 13% to 40% of the maximum load and with no bleed.

Machines 2020, 8, x FOR PEER REVIEW 8 of 19

when increasing engine load from 13% to 40% were considered. The first set of measurements considers engine operation with air bleed from center-casing for the purpose of emission control–CO turndown. The second set of measurements was taken during engine operation without air bleed, which corresponds to natural turndown engine operation. In addition, measurements during engine tests with center-casing air bleed during load acceptance from 26% to 53% of the maximum load have also been considered. The engine data comprise FFDEM, GGS, PTS, VGVDEM position, BOVDEM position and compressor discharge pressure (CDP). The FFDEM, BOVDEM, VGVDEM, GGS and PTS data are considered as inputs in the modeling architecture as shown in Figure 7. The measured CDP allowed the validation of the compressor model. Figure 8 shows the normalized GGS with respect to the measured GGS at steady-state at 53% load. A decrease in measured GGS is observed when no air bled from center-casing is considered, and with increasing load demand from 13% to 40%.

Figure 8. Normalized gas generator speed (GGS) considered as input in the engine model.

Figures 9 and 10 show a comparison between the measured CDP and the predicted CDP from the 0-D and 1-D compressor models. The volume gas dynamic expressed in Equation (3) was considered in the 0-D compressor model to simulate the transient response of the CDP, as shown in Figures 9 and 10. The interaction between the 0-D and 1-D compressor models is shown in Figure 6. The 1-D compressor model requires as inputs the air mass flow and efficiency calculated by the 0-D compressor model and calculates the stagewise pressure ratio across the 11 stages. The total pressure ratio estimated by the 1-D model is calculated by multiplying all the individual stagewise pressure ratio across the 11 stages. In addition, pressure loss related to the compressor diffuser and outlet Machinesguide2020 vanes, 8, 83 was considered to approximate the simulated 1-D compressor model response with the9 of 20 measured data.

FigureFigure 9. 9.Compressor Compressor discharge discharge pressure pressure withwith increasingincreasing load from from 13% 13% to to 40% 40% of of the the maximum maximum load load Machinesandand with 2020 with, no 8 ,no x bleed. FOR bleed. PEER REVIEW 9 of 19

FigureFigure 10. 10.Compressor Compressor discharge discharge pressure pressure with with increasing increasing load load from from 26% 26% to 53%to 53% of theof the maximum maximum load andload considering and considering bleed. bleed.

TableTable1 shows 1 shows the errorthe error between between the 0-D the compressor 0-D compressor model model and measured and measured data. The data. error The between error thebetween 1-D compressor the 1-D compressor model and model measured and measured data are data also are shownalso shown in Table in Table1. The1. The error error in in the the 1-D1- compressorD compressor model model is higher is higher than than the the error error in in thethe 0-D0-D compressor model. model. From From Figures Figures 9 and9 and 10, it10 , it cancan bebe observedobserved thatthat thethe biggestbiggest discrepancy be betweentween the the measured measured data data and and the the 1-D 1-D compressor compressor modelmodel is duringis during the the dynamic dynamic response. response. ThisThis is considered to to be be due due to to the the steady steady nature nature of oflosses losses predictedpredicted by by HFDT HFDT steady-state steady-state calculations. calculations. The 1-D compressor model model was was developed developed from from the the HFDTHFDT tool, tool, which which represents represents the performancethe performance of the multiaxialof the multiaxial compressor compressor at steady-state at steady-state conditions. Theconditions. increase in The error increase between in error the simulatedbetween the data simulated from the data 1-D from model the and1-D model the measured and the datameasured during data during transient engine performance is mainly attributed to the fact that the 1-D model cannot transient engine performance is mainly attributed to the fact that the 1-D model cannot accurately accurately predict the measured data at transient conditions. The 1-D model was developed using predict the measured data at transient conditions. The 1-D model was developed using simulated data simulated data from the HFDT, as shown in Figure 1. The HFDT data represent the performance of from the HFDT, as shown in Figure1. The HFDT data represent the performance of the compressor at the compressor at steady-state conditions. steady-state conditions. Table 1. Average error in simulated compressor discharge pressure from 0-D and 1-D compressor Tablemodels. 1. Average error in simulated compressor discharge pressure from 0-D and 1-D compressor models.

% Error% Error Bleed Bleed % Error % Error No Bleed No Bleed Load Load 0-D Model0-D Model 1-D 1-D Model Model 0-D 0-D Model Model 1-D Model 1-D Model 13–40%13–40% 1.111.11 3.073.07 2.95 2.95 4.24 4.24 26–53%26–53% 1.131.13 3.05 3.05 - - - -

5. Results The transient compressor responses previously described in the validation section can be represented on the overall compressor map. Figure 11 shows the compressor transient excursion when the engine load is increased from 13% to 40% of the maximum load with and without center casing bleed. Figure 11 shows the effect of air bleeding during compressor operation on the compressor component map. The compressor transient excursion neglecting the air bleed effect is drifted towards the stability line.

Machines 2020, 8, 83 10 of 20

5. Results The transient compressor responses previously described in the validation section can be represented on the overall compressor map. Figure 11 shows the compressor transient excursion when the engine load is increased from 13% to 40% of the maximum load with and without center casing bleed. Figure 11 shows the effect of air bleeding during compressor operation on the compressor component map. The compressor transient excursion neglecting the air bleed effect is drifted towards Machines 2020, 8, x FOR PEER REVIEW 10 of 19 Machinesthe 2020 stability, 8, x FOR line. PEER REVIEW 10 of 19

Figure 11. Representation of transient compressor response on compressor performance map for 13– FigureFigure 11. Representation 11. Representation of transient of transient compressor compressor response response on on compressor compressor performance performance mapmap forfor 13– 40% load with/without bleed air. 40% 13–40%load with/without load with/without bleed bleed air. air. The Thetransient transient response response during during load load acceptance acceptance maneuver from from 13–40%, 13–40%, and and 26% 26% to 53% to 53% with with The transient response during load acceptance maneuver from 13–40%, and 26% to 53% with centercenter case case bleed bleed effect effect is shown is shown in in Figure Figure 12.12. center case bleed effect is shown in Figure 12.

FigureFigure 12. Representation 12. Representation of transient of transient compressor compressor response response on on compressor compressor performance performance map forfor 13– Figure 12. Representation of transient compressor response on compressor performance map for 13– 40% 13–40%load and load 26–53% and 26–53% load with load withbleed bleed air. air. 40% load and 26–53% load with bleed air. 5.1. Offset in IGV Position 5.1. Offset in IGV Position The position of the VGVs is a function of the IGV position. An offset in the IGV position is The position of the VGVs is a function of the IGV position. An offset in the IGV position is considered as a position outside of its scheduled position with respect to the GGS. A positive IGV considered as a position outside of its scheduled position with respect to the GGS. A positive IGV offset position closes the VGVs, and a negative IGV offset position opens the VGVs. The engine offset position closes the VGVs, and a negative IGV offset position opens the VGVs. The engine model, constructed using specialized Simulink library reported by Panov [23], is considered for the model, constructed using specialized Simulink library reported by Panov [23], is considered for the simulation of the engine performance with an offset in the IGV position of the axial compressor. A simulation of the engine performance with an offset in the IGV position of the axial compressor. A controller implemented within the engine model, as shown in Figure 13, regulates the amount of fuel controller implemented within the engine model, as shown in Figure 13, regulates the amount of fuel demand with respect to the temperature and pressure from the different stations of the engine and demand with respect to the temperature and pressure from the different stations of the engine and load demand. load demand.

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5.1. Offset in IGV Position The position of the VGVs is a function of the IGV position. An offset in the IGV position is considered as a position outside of its scheduled position with respect to the GGS. A positive IGV offset position closes the VGVs, and a negative IGV offset position opens the VGVs. The engine model, constructed using specialized Simulink library reported by Panov [23], is considered for the simulation of the engine performance with an offset in the IGV position of the axial compressor. A controller implemented within the engine model, as shown in Figure 13, regulates the amount of fuel demand

Machineswith respect 2020, 8 to, x theFOR temperature PEER REVIEW and pressure from the different stations of the engine and load demand.11 of 19

Figure 13. 13. SimulinkSimulink model model for forsimulation simulation of engine of engine performance performance with withinlet guide inlet guidevane (IGV) vane offset (IGV) position.offset position.

The engine model shown in in Figure 13 was tuned to predict the steady-state response of the compressor (start (start and and endpoints of of transient excurs excursion)ion) shown shown in in Figure Figure 1212 whenwhen the the engine load was increased increased from from 26–53% 26–53% of of the the maximum maximum load. load. A posi A positivetive offset o ffofset 3 degrees of 3 degrees and a andnegative a negative offset ofoff set3 degrees of 3 degrees in the in IGV the position IGV position at 53% at 53%of the of load the load was was considered considered during during engine engine simulation. simulation. A nominalA nominal position position in inthe the IGV IGV (scheduled (scheduled IGV IGV position position for for that that GGS) GGS) was was considered considered at at 26% 26% load. load. Figure 14 shows the simulated GGS when the load is increased from 26–53% load and considering thethe positive and negative offset offset of 3 degrees in thethe IGVIGV positionposition atat 53%53% load.load. The simulated GGS shown in in Figure 14 was normalized with with respect respect to to the the GGS GGS at at 53% 53% load load and and at at nominal nominal conditions. conditions. Figure 14 shows an increase in in the GGS when a a positi positiveve offset offset of 3 degrees is considered in the IGV position. A A decrease decrease in in the the GGS GGS is is also also shown shown in in Fi Figuregure 14,14, when when a a negative negative offset offset of of 3 degrees is considered in in the IGV position. The The difference difference in in GGS for negative and positive IGV offset offset position with respectrespect toto thethe nominal nominal condition condition shown shown in in Figure Figure 14 is14 coined is coined as GGS as GGS offset. offset. The GGSThe oGGSffset offset value valueis implemented is implemented in the in engine the engine modeling modeling structure structur previouslye previously shown shown in Figure in Figure7 to simulate 7 to simulate the e ff theect effectof the IGVof the off setIGV position offset onposition the compressor on the co performancempressor performance map. The engine map. modelingThe engine architecture modeling is architectureconstructed inis theconstructed Simulink in environment. the Simulink It isenviro possiblenment. to add It is or possible subtract valuesto add betweenor subtract the signalsvalues betweenconnecting the the signals different connecting modules the in different the Simulink modu environment.les in the Simulink The GGS environment. offset value The is added GGS offset to the valuesignal is connecting added to the the signal look-up connecting table for thethe measuredlook-up table GGS for at nominalthe measured conditions GGS at and nominal the engine conditions model, andas shown the engine in Figure model, 15. as In shown addition, in Figure the simulated 15. In addition, signals forthe FFDEMsimulated and signals PT speed for FFDEM at positive and and PT speednegative at positive IGV offset and are negative considered IGV inoffset the engineare cons modelidered shown in the inengine Figure model 15. shown in Figure 15.

Figure 14. Simulated GGS response for 26–53% load and considering a positive and negative IGV offset of 3 degrees at 53% load.

Machines 2020, 8, x FOR PEER REVIEW 11 of 19

Figure 13. Simulink model for simulation of engine performance with inlet guide vane (IGV) offset position.

The engine model shown in Figure 13 was tuned to predict the steady-state response of the compressor (start and endpoints of transient excursion) shown in Figure 12 when the engine load was increased from 26–53% of the maximum load. A positive offset of 3 degrees and a negative offset of 3 degrees in the IGV position at 53% of the load was considered during engine simulation. A nominal position in the IGV (scheduled IGV position for that GGS) was considered at 26% load. Figure 14 shows the simulated GGS when the load is increased from 26–53% load and considering the positive and negative offset of 3 degrees in the IGV position at 53% load. The simulated GGS shown in Figure 14 was normalized with respect to the GGS at 53% load and at nominal conditions. Figure 14 shows an increase in the GGS when a positive offset of 3 degrees is considered in the IGV position. A decrease in the GGS is also shown in Figure 14, when a negative offset of 3 degrees is considered in the IGV position. The difference in GGS for negative and positive IGV offset position with respect to the nominal condition shown in Figure 14 is coined as GGS offset. The GGS offset value is implemented in the engine modeling structure previously shown in Figure 7 to simulate the effect of the IGV offset position on the compressor performance map. The engine modeling architecture is constructed in the Simulink environment. It is possible to add or subtract values between the signals connecting the different modules in the Simulink environment. The GGS offset value is added to the signal connecting the look-up table for the measured GGS at nominal conditions Machinesand the2020 engine, 8, 83 model, as shown in Figure 15. In addition, the simulated signals for FFDEM and12 of PT 20 speed at positive and negative IGV offset are considered in the engine model shown in Figure 15.

Figure 14.14. SimulatedSimulated GGS GGS response response for for 26–53% 26–53% load load and and considering considering a positive a positive and negativeand negative IGV o ffIGVset Machinesofoffset 3 2020 degrees of, 83, degreesx FOR at 53% PEER at load. 53% REVIEW load. 12 of 19

FigureFigure 15.15. EngineEngine modelmodel consideringconsidering aa GGSGGS changechange duringduring ooffsetffset inin IGVIGV position.position.

FigureFigure 16 16 showsshows thethe simulatedsimulated compressorcompressor performanceperformance consideringconsidering positivepositive andand negativenegative ooffsetffset ofof 33 degrees degrees in in the the IGV IGV position position at 53% at 53% load. load. An increase An increase in pressure in pressure ratio when ratio considering when considering a negative a onegativeffset in theoffset IGV in positionthe IGV position (opening (opening VGVs) is VGVs) observed is observed in Figure in 16Figure. A decrease16. A decrease in pressure in pressure ratio whenratio when considering considering a positive a positive IGV IGV offset offset position position (closure (closure of VGVs)of VGVs) is is shown shown in in Figure Figure 16 16 asas well. HashmiHashmi etet al. al. [ 27[27]] simulated simulated the the eff effectect of of VGV VGV drift dr inift ain low-pressure a low-pressure compressor compressor (LPC) (LPC) on a three-shafton a three- gasshaft turbine gas turbine (GE LM1600). (GE LM1600). The results The results demonstrated demonstrated that anthat updrift an updrift schedule schedule (positive (positive offset) offset) in the in VGVthe VGV position position decreased decreased the pressure the pressure discharge discharge by the by LPC, the andLPC, a downdriftand a downdrift schedule schedule (negative (negative offset) increaseoffset) increase the LPC the pressure LPC pressure discharge. discharge. Urasek etUrasek al. [28 ]et reported al. [28] thatreported by closing that by the closing IGV and the VGVs IGV and for aVGVs given for speed a given in an speed axial in flow an compressor,axial flow compressor, the total pressure the total ratio pressure is decreased. ratio is decreased.

Figure 16. Simulated compressor dynamic response for 26–53% load and considering positive and negative IGV offset of 3 degrees at 53% load.

5.2. Inter-Stage Dynamic Performance Simulation A representation of a stage performance map is shown in Figure 17. The stage performance map is comprised of two regions: stable and unstable operation. Ideally, a compressor is designed to operate on the stable operation region, but under certain conditions, the compressor operation may

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Figure 15. Engine model considering a GGS change during offset in IGV position.

Figure 16 shows the simulated compressor performance considering positive and negative offset of 3 degrees in the IGV position at 53% load. An increase in pressure ratio when considering a negative offset in the IGV position (opening VGVs) is observed in Figure 16. A decrease in pressure ratio when considering a positive IGV offset position (closure of VGVs) is shown in Figure 16 as well. Hashmi et al. [27] simulated the effect of VGV drift in a low-pressure compressor (LPC) on a three- shaft gas turbine (GE LM1600). The results demonstrated that an updrift schedule (positive offset) in the VGV position decreased the pressure discharge by the LPC, and a downdrift schedule (negative Machinesoffset) increase2020, 8, 83 the LPC pressure discharge. Urasek et al. [28] reported that by closing the IGV13 ofand 20 VGVs for a given speed in an axial flow compressor, the total pressure ratio is decreased.

FigureFigure 16.16. SimulatedSimulated compressorcompressor dynamicdynamic responseresponse forfor 26–53%26–53% loadload andand consideringconsidering positivepositive andand negativenegative IGVIGV ooffsetffset ofof 33 degreesdegrees atat 53%53% load.load.

5.2.5.2. Inter-Stage Dynamic Performance SimulationSimulation AA representationrepresentation of of a a stage stage performance performance map map is is shown shown in in Figure Figure 17 17.. The The stage stage performance performance map map is Machines 2020, 8, x FOR PEER REVIEW 13 of 19 comprisedis comprised of twoof two regions: regions: stable stable and unstableand unstable operation. operation. Ideally, Ideally, a compressor a compressor is designed is designed to operate to onoperate the stable on the operation stable operation region, but region, under but certain unde conditions,r certain conditions, the compressor the compressor operation mayoperation be drifted may be driftedtowards towards the stability the limitstability condition. limit condition. The use of VGVs The use and of air VGVs bleed during and air compressor bleed during operation compressor can operationensure can that ensure compressor that can compressor operate with can su ffioperatecient stability with sufficient margin. The stability compressor margin. stage The performance compressor stagemap performance represented map in Figure represented 17 can allowin Figure the assessment17 can allow of the dynamicassessment compressor of the dynamic performance compressor at performancedifferent engine at different load conditions. engine Theload use conditions. of stage-characteristic The use mapsof stage-characteristic to assess compressor performancemaps to assess compressorwas considered performance in different was studiesconsidered [20,29 in–31 different]. studies [20,29–31].

FigureFigure 17. 17. RepresentationRepresentation of stage performanceperformance map. map.

The equations to predict the flow and load coefficients for the stage performance map shown in The equations to predict the flow and load coefficients for the stage performance map shown in Figure 17 are: Figure 17 are: C ϕ = X (4) UC φ= (4) where Equation (4) is defined as the flow coefficient U where Equation (4) is defined as the flow coefficient γ 1  C T Pr γ− 1 P in − ψ = (5) C T Pr2 −1 U (5) ψ= U and Equation (5) is the load coefficient. The nomenclature of Equations (4) and (5) was described in Section 2 on theory. The 1-D compressor model can predict the stage-characteristic maps predicted by the HFDT data using Equations (4) and (5) describing the flow and load coefficients. Figure 18 shows a comparison between the stage-performance map (Stage 8) predicted by the HFDT and by the 1-D compressor model. Figure 18 shows that the flow coefficients and load coefficients calculated by Equations (4) and (5) from 70–100% speed follow the same trend [32].

Figure 18. Comparison between stage-performance map (stage 8) predicted by HFDT and predicted by 1-D compressor model from 70–100% speed.

Machines 2020, 8, x FOR PEER REVIEW 13 of 19 be drifted towards the stability limit condition. The use of VGVs and air bleed during compressor operation can ensure that compressor can operate with sufficient stability margin. The compressor stage performance map represented in Figure 17 can allow the assessment of the dynamic compressor performance at different engine load conditions. The use of stage-characteristic maps to assess compressor performance was considered in different studies [20,29–31].

Figure 17. Representation of stage performance map.

The equations to predict the flow and load coefficients for the stage performance map shown in Figure 17 are:

C φ= (4) U where Equation (4) is defined as the flow coefficient C T Pr −1 (5) Machines 2020, 8, 83 ψ= 14 of 20 U and Equation (5) is the load coefficient. The nomenclature of Equations (4) and (5) was described in and Equation (5) is the load coefficient. The nomenclature of Equations (4) and (5) was described in SectionSection 2 on2 ontheory. theory. The The 1-D 1-D compressor compressor model cancan predict predict the the stage-characteristic stage-characteristic maps maps predicted predicted by theby HFDT the HFDT data data using using Equations Equations (4) (4) and and (5) (5) describing thethe flow flow and and load load coe coefficients.fficients. Figure Figure 18 18 showsshows a comparison a comparison between between the the stage-performance stage-performance map map (Stage (Stage 8) predicted 8) predicted by the by HFDT the HFDT and by theand by the 1-D1-D compressor compressor model.model. Figure Figure 18 18 shows shows that that the the flow flow coe fficoefficientscients and loadand coeloadffi cientscoefficients calculated calculated by by EquationsEquations (4) (4) and and (5) fromfrom 70–100% 70–100% speed speed follow follow the the same same trend trend [32]. [32].

FigureFigure 18. Comparison 18. Comparison between between stage-performance stage-performance map map (stage (stage 8) predicted8) predicted by HFDT by HFDT and predicted and predicted by by 1-D1-D compressor compressor model fromfrom 70–100% 70–100% speed. speed. 5.2.1. Center-Casing Bleed Effect The transient compressor responses shown in Figures 11, 12 and 16 can also be represented in the individual stage performance maps. The transient compressor response on the individual stage-characteristics map does not consider phenomena related to the individual compressor stage, e.g., transient heat transfer [33] and stage tip clearance [34]. Figure 19 shows the transient response of the compressor on the stage characteristic map when the engine load is increased from 13% to 40% of the maximum load with and without center-casing bleed. The 1-D model calculates the stage-characteristic maps predicted by the HFDT, as demonstrated in Figure 18. It can be observed in Figure 19a that the stage performance of stage-11 at all speeds collapse together [32]. The stage-11 performance associated with the running line (RL) considering air bleed and for all speeds is also shown in Figure 19a. This figure demonstrates that the compressor operates on stable conditions when the air bleed effect is considered. As the load is increased from 13 to 40%, the transient compressor response is moving towards the reduced flow coefficient ϕ and towards the stability limit of the stage characteristic map. The transient compressor response that neglects air bleed approaches closer to the stability limit, as shown in Figure 19a. Figure 19b shows the stage characteristic map associated with the second stage. The calculated flow and load coefficients associated with the second stage are sparse and do not collapse into a single trend. This could be associated with different compressor geometry attributed to the VGVs located at the front-end of the compressor. Figure 19b shows that the transient compressor response is moving towards the increased flow coefficient ϕ with increasing load from 13–40%. As expected, the transient compressor response that does not consider air bleed moves closer towards the stability limit and above the response that considers air bleed. Machines 2020, 8, x FOR PEER REVIEW 14 of 19

5.2.1. Center-Casing Bleed Effect The transient compressor responses shown in Figures 11, 12 and 16 can also be represented in the individual stage performance maps. The transient compressor response on the individual stage- characteristics map does not consider phenomena related to the individual compressor stage, e.g., transient heat transfer [33] and stage tip clearance [34]. Figure 19 shows the transient response of the compressor on the stage characteristic map when the engine load is increased from 13% to 40% of the maximum load with and without center-casing bleed. The 1-D model calculates the stage- characteristic maps predicted by the HFDT, as demonstrated in Figure 18. It can be observed in Figure 19a that the stage performance of stage-11 at all speeds collapse together [32]. The stage-11 performance associated with the running line (RL) considering air bleed and for all speeds is also shown in Figure 19a. This figure demonstrates that the compressor operates on stable conditions when the air bleed effect is considered. As the load is increased from 13 to 40%, the transient compressor response is moving towards the reduced flow coefficient φ and towards the stability limit of the stage characteristic map. The transient compressor response that neglects air bleed approaches closer to the stability limit, as shown in Figure 19a. Figure 19b shows the stage characteristic map associated with the second stage. The calculated flow and load coefficients associated with the second stage are sparse and do not collapse into a single trend. This could be associated with different compressor geometry attributed to the VGVs located at the front-end of the compressor. Figure 19b shows that the transient compressor response is moving towards the increased flow coefficient φ with increasing load from 13–40%. As expected, the transient compressor response that does not considerMachines air2020 bleed, 8, 83 moves closer towards the stability limit and above the response that considers15 of 20 air bleed.

FigureFigure 19. 19.CompressorCompressor transient transient response response on on the the stagestage characteristiccharacteristic map map @ 13–40%@ 13–40% load load and and with/withoutwith/without bleed bleed air, air, (a) (a stage) stage 11, 11, ( (bb)) stage stage 2.

5.2.2. Load Effect The transient compressor response when the engine load is increased from 13% to 40% and from 26 to 53% load and considering CO Turndown operation effect is shown in Figure 20. Figure 20a shows that the transient compressor response for the case at 13–40% load is moving towards the reduced flow coefficient ϕ or towards the stability limit of the stage characteristic map compared to the transient compressor response for the case at 26–53% load. Figure 20b shows that the transient compressor response at higher load is shifted towards the increased flow coefficient ϕ– higher load conditions. Machines 2020, 8, x FOR PEER REVIEW 15 of 19

5.2.2. Load Effect The transient compressor response when the engine load is increased from 13% to 40% and from 26 to 53% load and considering CO Turndown operation effect is shown in Figure 20. Figure 20a shows that the transient compressor response for the case at 13–40% load is moving towards the reduced flow coefficient φ or towards the stability limit of the stage characteristic map compared to the transient compressor response for the case at 26–53% load. Figure 20b shows that the transient Machinescompressor2020, 8 ,response 83 at higher load is shifted towards the increased flow coefficient φ– higher16 ofload 20 conditions.

FigureFigure 20. 20.Transient Transient compressor compressor response response on on the the stage stage characteristic characteristic map map @ @ 13–40% 13–40%load loadand and 26–53%26–53% loadload with with bleed bleed air, air, ( a()a) stage stage 11, 11, ( b(b)) stage stage 2. 2.

5.2.3.5.2.3. IGVIGV OOffsetffset E Effectffect FigureFigure 21 21 shows shows the the transient transient response response of of the the compressor compressor on on the the stage stage characteristic characteristic map map when when thethe engine engine load load is is increased increased from from 26% 26% to to 53% 53% of of the the maximum maximum load load at at nominal nominal IGV IGV position position and and with with anan ooffsetffset in in the the IGV IGV position position at at 53% 53% load. load. AsAs thethe loadload isis increasedincreased fromfrom 26%26% toto 53%,53%, thethe transient transient compressorcompressor response response is is moving moving towards towards the the reduced reduced flow flow coe coefficientfficient ϕ φand and towards towards the the stability stability limit limit ofof the the stage stage characteristic characteristic map map as as shown shown in in Figure Figure 21 21a.a. TheThe steady-statesteady-state flowflowcoe coefficientfficient at at 53% 53% load load increasesincreases when when considering considering a a positive positive IGV IGV o offsetffset (endpoint (endpoint of of the the green green line) line) and and moves moves away away from from thethe stability stability limit limit in in the the stage-performance stage-performance map.map. ThisThis eeffectffect cancan alsoalsobe becorroborated corroboratedin inthe the overall overall compressorcompressor map map shown shown in in Figure Figure 16 16,, considering considering the the typical typical compressor compressor stability stability margin margin for for the the case case atat positivepositive IGV offset. offset. Figure Figure 21b 21 bshows shows the the stage stage characteristic characteristic map map associated associated with withthe second the second stage. stage.Figure Figure 21b shows 21b shows that thatthe thetransient transient compressor compressor response response is ismoving moving towards towards the the increased flowflow coefficient ϕ with increasing load from 26–53%. As expected, the transient compressor response that considers positive IGV offset moves away from the stability limit and below the nominal transient compressor response. Machines 2020, 8, x FOR PEER REVIEW 16 of 19

coefficient φ with increasing load from 26–53%. As expected, the transient compressor response that Machinesconsiders2020 positive, 8, 83 IGV offset moves away from the stability limit and below the nominal transient17 of 20 compressor response.

FigureFigure 21.21. SimulatedSimulated compressorcompressor dynamicdynamic responseresponse onon stagestage performanceperformance mapmap forfor 26–53%26–53% loadload andand consideringconsidering positivepositive andand negativenegative IGVIGV ooffsetffset ofof 33 degreesdegrees at at 53% 53% load, load, ( a(a)) stage stage 11, 11, ( b(b)) stage stage 2. 2.

ThisThis study has has integrated integrated the the 1-D 1-D compressor compressor model model developed developed in a previous in a previous study study[21] within [21] withina whole a gas whole turbine gas turbinemodel [23] model containing [23] containing 0-D comp 0-Donent component models to models predict to the predict inter-stage the inter-stage dynamic dynamiccompressor compressor performance performance at load atacceptance load acceptance maneuvers maneuvers and with and an with offset an o ffinset IGV in IGV position. position. In Inaddition, addition, the the effect effect of of the the center-casing center-casing air air blee bleedd was was analyzed analyzed on on the the overall compressor mapmap andand stage-characteristicstage-characteristic maps.maps. At low speeds and consideringconsidering no center-casingcenter-casing airair bleed,bleed, aa givengiven stagestage maymay bebe performedperformed closeclose toto itsits stabilitystability limit,limit, whichwhich couldcould causecause instabilityinstability ofof thethe wholewhole compressorcompressor duringduring thethe dynamicdynamic engineengine operation.operation. TheThe transienttransient compressorcompressor responseresponse exhibitsexhibits operationoperation withwith aa higherhigher stability margin when the the ce center-casingnter-casing air air bleed bleed effect effect is is active. active. A Ahigher higher stability stability margin margin on onthe the overall overall compressor compressor map map and and stage-characteristic stage-characteristic maps maps is is also also predicted predicted when closingclosing VGVsVGVs (positive(positive ooffsetffset in in IGV IGV position) position) during during compressor compressor operation. operation. In In the the previous previous study study [21 [21],], the the eff ecteffect of IGVof IGV mal-function mal-function and and the etheffect effect of fouling of fouling on stagewise on stagewise performance performance at steady-state at steady-state conditions conditions were demonstrated.were demonstrated. The deposition The deposition of dirt of and dirt dust and particles dust particles on the compressoron the compressor bladesincreases blades increases surface roughnesssurface roughness on the blades on the and blades changes and the changes velocity the triangles velocity representing triangles representing the compressor the compressor blades [35]. Theblades dynamic [35]. The change dynamic in flow change capacity in flow and ecapacityfficiency and in the efficiency compressor in the during compressor fouling conditionsduring fouling and duringconditions VGV and malfunction during VGV was malfunction simulated throughwas simulated a Simulink through 0-D a gas Simulink turbine 0-D model gas [turbine36]. In futuremodel work,[36]. In the future developed work, the modeling developed architecture modeling from architecture this study from will this simulate study andwill validatesimulate theand dynamic validate stagewisethe dynamic performance stagewise duringperformance compressor during fouling compressor conditions fouling and conditions VGV malfunction. and VGV malfunction.

Machines 2020, 8, 83 18 of 20

6. Conclusions A semi-empirical 1-D modeling framework comprising theoretical relations, design parameters, and stagewise data of a multistage axial compressor was integrated with a 0-D engine model developed in the Simulink environment. The overall modeling architecture was validated with engine test data of a twin-shaft gas turbine measured during the load acceptance maneuvers. The engine test data comprise data measured during the load step change and considering natural and CO turndown engine operation. The modeling architecture was able to predict the measured transient response of the compressor discharge pressure with step-change of engine load and considering and neglecting center-casing air bleed effect. The effect of the center-casing air bleed was analyzed on the overall compressor map and stage-characteristic maps. Further analysis by considering an offset in the IGV position with respect to its scheduled position with GGS was carried out. The effect of offset in the IGV position on compressor performance was demonstrated in the overall compressor map and stage-characteristic maps. The transient compressor response exhibits operation with a higher stability margin when the center-casing air bleed effect is active and with closing the VGVs (positive offset in IGV position). This study has demonstrated that it is possible to analyze the dynamic performance of the compressor during the transition between two operating conditions by assessing the stability margin at a given compressor stage. The developed theoretical framework can provide a better prediction of dynamic compressor performance, especially during transient engine operation, at which the compressor running line may get closer to its stability limit.

Author Contributions: Conceptualization, V.P.; formal analysis, S.C.-M.; funding acquisition, C.B.; methodology, V.P., S.K.; supervision, S.K., V.P.; validation, S.C.-M.; writing—original draft, S.C.-M.; writing—review and editing, S.K., V.P. and C.B. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Siemens Energy Industrial Turbomachinery Ltd., Lincoln. The APC was funded by Siemens Energy Industrial Turbomachinery Ltd., Lincoln. Acknowledgments: The authors would like to thank Siemens Energy Industrial Turbomachinery Ltd., Lincoln, U.K., for providing research support and access to real-world data to support the research outcomes. Conflicts of Interest: The authors declare no conflict of interest.

Nomenclature

List of Symbols A annulus area (m2) CP heat capacity at constant pressure (J/kgK) CX air axial velocity (m/s) . m air flow rate (kg/s) PR pressure ratio (dimensionless) r radius of the annulus (m) U blade velocity (m/s) T temperature (K) Greek α air angle from the axial direction λ work-done factor (dimensionless) η polytropic efficiency (dimensionless) ρ air density kg/m3 γ heat capacity ratio (dimensionless)

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