Transient Behavior in Variable Geometry Industrial Gas Turbines: A Comprehensive Overview of Pertinent Modeling Techniques
Abstract
:1. Introduction
1.1. Literature Survey of Transient Modeling Domains
1.2. Research Gaps
 To the author’s best knowledge, there is a lack of organized literature review so far that may cover all the possible techniques and methods for the development of transient models of industrial gas turbine regarding fault detection and diagnostics (FDD).
 The pertinent literature for variable geometry features (i.e., VIGVs, VSVs, variable bleed and VAN) that play significant role in engine’s reliability preventing engine from surging during transient behavior, remained shallow.
 There is no such existing document that provides accurate data for shaft’s polar moment of inertia required for accurate transient model development
 To date, there is no authentic document that aids in selection of proper VIGV and bleed schedules for a particular IGT engine based on its inherent configuration, i.e., single shaft, twin shaft, and triple shaft
2. Classifications of Transient Regimes in IGT
2.1. Startup
2.2. Load Change (Acceleration and Deceleration)
2.3. Shutdown
2.4. Secondary Transient Effects
3. Methods and Techniques for Transient Models
3.1. White Box Models
3.2. Black Box Models
3.3. Other Models
4. Transient Model Development Portfolio of IGTs
4.1. Shaft Dynamics
Authors  Polar Moment of Inertia (kgm^{2})  Configuration of Engine  

$\mathbf{GG}\text{}\left({\mathit{J}}_{\mathit{G}\mathit{G}\mathit{S}}\right)$  $\mathbf{PT}\left({\mathit{J}}_{\mathit{P}\mathit{T}\mathit{S}}\right)$  
Gaudet, [145]  0.08  0.05  Twin shaft (Marine) 
Janikovic, [140]  30–50  50  Twin shaft (Turbofan) 
Novikov, [55]  0.060334  1.3694  Twin shaft (Turboshaft) 
Silva, [146]  0.55  0.35  Twin shaft (Turboshaft) 
Barbosa et al. [147]  0.0125    Single shaft (Turbojet) 
Kim et al. [64]  1.14  1.60  Three shaft (Turbofan) 
Kim et al. [72]  0.02    Single shaft (IGT) 
4.2. Volume Dynamics
4.2.1. Constant Mass Flow Method
4.2.2. Inter Component Volume (ICV) Method
4.3. Inlet and Exhaust Duct Modeling
4.4. Compressor Modeling
4.5. Combustor Modeling
4.6. Turbine Modeling Methods
5. Control Strategies for Dynamic Operations
5.1. Simplified PID Control Scheme
5.2. Model Based Control Schemes
5.3. Fuel Flow Actuation
5.4. VIGV Actuation
5.4.1. VIGV Schedule Selection Framework
5.4.2. VIGV Modulation Correction Factors
5.5. Variable Bleed Actuation
6. Software Tools for Transient Modeling
6.1. ZeroDimensional Simulation Programs
6.2. MultiDimensional Simulation Programs
7. Future Recommendations
8. Conclusions
 Although, variety of pertinent transient models exits in the literature, there is a scarcity in transient models for variable geometry IGT.
 Control mechanisms associated with VIGV and VBV are indispensable for different transient regimes, i.e., startup load change and shutdown.
 The VIGV schedule selection framework along with VIGV schedule database play a vital role in academic modeling and real time operation and maintenance of IGT. For instance, framework can be beneficial during maintenance for proper calibration of a drifted VIGV schedule.
 The polar moment of inertia for several configuration of IGTs delineates a paramount importance for accurate modeling of transient behavior in variable geometry IGT
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Nomenclature
Nomenclature  $Max$  Maximum  
$e$  Internal energy per unit mass  $Min$  Minimum 
$F$  Force  $M$  Mean value 
$\mathit{F}$  Force Vector  $to$  total 
$H$  Total enthalpy  $to,in$  total inlet 
$\dot{m}$  mass flow rate  $cc$  combustion chamber 
$\mathit{n}$  unit vector normal to the surface  $DP$  Design Point 
$p$  static pressure  $VIGV$  VIGV corrected value 
$\dot{Q}$  Heat transfer  $FG,map$  Fixed Geometry compressor map 
$S$  Surface  $a,b,c,{c}_{1},{c}_{2},{c}_{3},$ ${K}_{v},{K}_{T},{K}_{m},{K}_{E}$  VIGV correction factors 
$t$  time  $OD$  Off design 
$A$  Area  $Max,\eta $  Maximum efficiency value 
$u$  Axial velocity  $corr$  corrected value 
$\mathit{u}$  velocity vector  $corr,map$  corrected values from maps 
$V$  Volume  $dis$  discharge value 
$\dot{W}$  power  $v,max$  maximum extent of bleed valve opening 
$\omega $  angular velocity  Greek Letters  
$J$  Polar moment of inertia  $\rho $  density 
$N$  shaft rotational speed  ${\tau}_{I}$  Inertial time constant 
$m$  mass  $\mathsf{\Delta}$  Change 
$r$  Engine radius  ${\tau}_{c}$  Compressor Torque 
$P$  Total Pressure  $\eta $  isentropic efficiency 
$T$  Total temperature  $\gamma $  polytropic index 
$R$  Gas Constant  $\varphi $  flow coefficient 
${C}_{p}$  specific heat at constant pressure  $\psi $  pressure coefficient 
$d$  diameters  $\xi $  temperature rise coefficient 
$U$  tangential blade speed  $\alpha $  flow angle relative to a stator 
$P{R}_{s}$  stage pressure ratio  $\beta $  flow angle relative to a rotor 
$G$  Generalized functions  ${\tau}_{cc}$  burner time constant 
$a$  numbers of variables at each control surface  ${\theta}_{VIGV}$  VIGV angle 
$Z=3\left(n+1\right)$  total number of variables  ${\theta}_{VAN}$  VAN angle 
${C}_{D}$  Drag Coefficient  $\overline{\omega}$  Total pressure loss coefficient 
$h$  enthalpy  $\sigma $  Pressure ration between compressor inlet and sea level 
$K$  Stodola Constant  Abbreviations  
$\kappa $  Choking constant  SF  shape factor 
$u\left(t\right)$  input signal  CCPP  combined cycle power plant 
$y\left(t\right)$  Output signal  FDD  fault detection and diagnostics 
$r\left(t\right)$  Demanded signal  GA  Genetic algorithm 
$e\left(t\right)$  Error signal  GG  Gas Generator 
${K}_{p}$  proportional controller gain  HRSG  Heat Recovery steam generator 
${K}_{i}$  integral controller gain  ICV  Intercomponent volume 
${K}_{d}$  derivative controller gain  CMF  Constant Mass Flow 
${C}_{g}$  extent of bleed valve opening  IGT  Industrial gas turbines 
Subscripts  VIGV  Variable inlet guide vane  
$i$  control volume index  VSV  Variable Stator vane 
$in$  inlet  VAN  Variable area nozzle 
$out$  outlet  VBV  variable bleed valve 
$t$  turbine  NGV  Nozzle guide vane 
$c$  compressor  ODEs  Ordinary Differential equations 
$f$  fictional power loss  PDEs  partial differential equations 
$el$  electric load  OOP  Object oriented programming 
$s$  shaft  TIT  Turbine inlet temperature 
$d$  design  TET  Turbine exhaust temperature 
$target,Eng$  Target Engine  HPC  High pressure compressor 
$ref,Eng$  Reference Engine  HPT  High pressure turbine 
$GGS$  gas generator shaft  LPT  Low pressure turbine 
$PTS$  Power turbine shaft  ANN  Artificial neural network 
$1$  transition stage between to control volume  CFD  Computational fluid dynamics 
$a$  air  CMs  continuity models 
$g$  Gas  TFMs  Transfer function models 
$in,d$  intake design point value  LFMs  Linear function models 
$1,2,3,4$  station numbers 
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Author  Year  Variable Geometry Features  

VIGVs or VSVs  Variable Bleed  NGVs or VGVs for PT  
Mohammadian and Saidi, [56]  2019  ✓  ✓  
MontazeriGh et al. [57]  2018  ✓  
Schobeiri, [58]  2018  ✓  
Mehrpanahi et al. [34]  2018  ✓  
Wang et al. [59]  2018  ✓  
Silva et al. [60]  2018  ✓  
Wang et al. [61]  2018  ✓  
Enalou, [62]  2017  ✓  
MontazeriGh, [63]  2017  ✓  
Wang et al. [43]  2017  ✓  
Kim et al. [45]  2016  ✓  
Kim et al. [64]  2015  ✓  ✓  
Barbosa et al. [65]  2012  ✓  
Chacartegui et al. [66]  2011  ✓  
Barbosa et al. [67]  2011  ✓  
Panov et al. [68]  2009  ✓  
Silva et al. [69]  2007  ✓  
Sekhon et al. [36]  2006  ✓  
Bringhenti et al. [70]  2006  ✓  ✓  ✓ 
Camporeale et al. [71]  2006  ✓  
Kim et al. [72]  2001  ✓  
Kim et al. [73]  2001  ✓  
Kim and Soudarev, [74]  2000  ✓  
Blanco and Henricks, [75]  1998  ✓  
Boumedmed, [76]  1997  ✓  
Perretto, [77]  1997  ✓  
Bettocchi et al. [33]  1996  ✓  
Nava et al. [78]  1995  ✓  
MehrHomji and Bhargava, [39]  1992  ✓ 
Method  Researchers  Respective Equations  Benefits  Overall Drawbacks 

Map Scaling Method  [68,149,151,152,153]  $PR=\frac{{\left(PR\right)}_{design}1}{{\left(PR\right)}_{map,design}1}\left[{\left(PR\right)}_{map}1\right]+1$ $\dot{m}=\frac{{\dot{m}}_{design}}{{\dot{m}}_{map,design}}\times {\dot{m}}_{map}$ $\eta =\frac{{\eta}_{design}}{{\eta}_{map,design}}\times {\eta}_{map}$  1. Quite easy and simplest method to develop compressor map 2. Time saving  1. Not Applicable for variable geometry compressor 2. Limitation in selection of reference map 3. Not accurate for offDesign operation 4. This method overlooks the compressibility factor 
Sequential Stage Stacking method  [137]  Flow coefficient, $\varphi =\frac{{C}_{x}}{u}$ Pressure Coefficient, $\psi ={C}_{P}{T}_{t,in}\left(P{R}_{s}^{\frac{\gamma 1}{\gamma}}1\right)/{U}^{2}$ Temperature Rise,$\xi =\frac{{C}_{p}\Delta {T}_{t}}{{U}^{2}}$ Stage Efficiency, $\eta =\frac{\psi}{\xi}$  1. Accurate performance prediction 2. Applicable for both fixed and variable geometry compressors  1.Problem in off design operations 2. problems during stalling and choking 3. Time consuming 4. It requires Gas path geometric data like stage mean radius and annulus area that are not provided by manufacturer 
Modified Stage Stacking Method  [154]  ${\dot{m}}_{i+1}={\dot{m}}_{i}$ ${\dot{m}}_{i+1}{C}_{{x}_{i+1}}+{P}_{i+1}{A}_{i+1}={\dot{m}}_{i}{C}_{{x}_{i}}+{P}_{i}{A}_{i}+{F}_{s}$ ${\dot{m}}_{i+1}{H}_{{t}_{i+1}}={\dot{m}}_{i}{H}_{{t}_{i}}+{\dot{W}}_{s}$  1. Flexibility in boundary conditions 2. Time saving calculations due to simultaneous solutions 3. Variable geometry treatment by variation in setting angle 4. Applicable for transient modeling due to stability in numerical methods  1.Gas path geometric data like stage mean radius and annulus area are not provided by Manufacturer 2. Unavailability of reference data at Max efficiency 
[12,72,73,90,155,156]  ${\psi}^{*}={\psi}_{Max}^{*}\frac{{\psi}_{Max}^{*}1}{{\left({\varphi}_{\psi}^{*}{}_{Max}1\right)}^{2}}\times {\left({\varphi}_{\psi}^{*}{}_{Max}{\varphi}^{*}\right)}^{2}$ ${\eta}_{Min}^{*}=1\frac{1{\eta}_{{\left(\frac{\psi}{\varphi}\right)}_{Min}}^{*}}{{\left[1{\left(\frac{{\psi}^{*}}{{\varphi}^{*}}\right)}_{Min}\right]}^{3.5}}{\left(1\frac{{\psi}^{*}}{{\varphi}^{*}}\right)}^{3.5},where\frac{{\psi}^{*}}{{\varphi}^{*}}\in \left[{\left(\frac{{\psi}^{*}}{{\varphi}^{*}}\right)}_{Min},1\right]$ ${\eta}_{Max}^{*}=1\frac{1{\eta}_{{\left(\frac{\psi}{\varphi}\right)}_{Max}}^{*}}{{\left[{\left(\frac{{\psi}^{*}}{{\varphi}^{*}}\right)}_{Max}1\right]}^{3.5}}{\left(\frac{{\psi}^{*}}{{\varphi}^{*}}1\right)}^{2},where\frac{{\psi}^{*}}{{\varphi}^{*}}\in \left[1,{\left(\frac{{\psi}^{*}}{{\varphi}^{*}}\right)}_{Max}\right]$  Thermodynamic cycle program combined with performance maps generated through stage stacking helps in searching the operating point values (thermodynamic performance parameters) like compressor PR, SFC and TET that is not possible in the rest of the methods  The Minimization Algorithm adopted in this method is not stable for large number of unknowns. So a robust technique i.e., Genetic Algorithm is suggested for future work  
Blade Element Method  [75,157,158]  $\Delta {h}_{t}=({U}_{2}^{2}{U}_{2}{C}_{x2}\mathrm{tan}{\beta}_{2}{U}_{1}{C}_{x1}\mathrm{tan}{\alpha}_{1})/{g}_{c}$ $\overline{\omega}=\frac{{P}_{t1}{P}_{t2}}{{P}_{t1}{P}_{1}}$ ${C}_{D}=\frac{\overline{\omega}}{\sigma}\left(\frac{{\mathrm{cos}}^{3}{\alpha}_{m}}{{\mathrm{cos}}^{2}{\alpha}_{1}}\right),Where\mathrm{tan}{\alpha}_{m}=\left(\frac{\mathrm{tan}{\alpha}_{1}+\mathrm{tan}{\alpha}_{2}}{2}\right)$  1. Little dependency on the cascade data 2. Applicable for variety of stages of compressors according to the desired compressor 3. Holds good for VIGV adjustment due to simulation of each blade element  The loss and deviation correlation curves obtained from the literature are not robust in terms of accuracy 
Turbine Modeling Methods  Respective Mathematical Expression  Significance 

Choking Equation  $\frac{{\dot{m}}_{in}\sqrt{{T}_{in}}}{\kappa A{P}_{in}}=Constant$ Where $\kappa =\sqrt{\frac{\gamma}{R}}{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{\gamma 1}}$ 

Stodola Ellipse Equation  $\frac{{m}_{in}\sqrt{{T}_{Tin}}}{{P}_{in}}=K\text{}\sqrt{1{\left(\frac{{P}_{out}}{{P}_{in}}\right)}^{2}}$  Useful for estimating turbine characteristics during off design condition 
Flugel Formula  $\frac{{m}_{in}}{{m}_{in,ref}}=\frac{\sqrt{{\left({p}_{in}{p}_{out}\right)}^{2}}}{\sqrt{{\left({p}_{inref}{p}_{outref}\right)}^{2}}}\times \frac{\sqrt{{T}_{inref}}}{\sqrt{{T}_{in}}}\text{}$  Gives a correlation of mass flow, pressure and temperature for turbine in off design condition 
Researcher  Correlations  Correction Factors 

Celis et al. [193]  Mass flow ${\left[\frac{\left(\frac{\dot{m}}{{p}_{in}}\times \frac{\sqrt{R{T}_{in}}}{\gamma}\right)}{{\left(\text{}\frac{\dot{m}}{{p}_{in}}\times \frac{\sqrt{R{T}_{in}}}{\gamma}\right)}_{DP}}\right]}_{VIGV}=a{\left[\frac{\left(\frac{\dot{m}}{{p}_{in}}\times \frac{\sqrt{R{T}_{in}}}{\gamma}\right)}{{\left(\text{}\frac{\dot{m}}{{p}_{in}}\times \frac{\sqrt{R{T}_{in}}}{\gamma}\right)}_{DP}}\right]}_{n}$ Pressure ratio ${\left[\frac{\frac{{p}_{out}}{{p}_{in}}1}{{\left(\frac{{p}_{out}}{{p}_{in}}1\right)}_{DP}}\right]}_{VIGV}=b\text{}{\left[\frac{\frac{{p}_{out}}{{p}_{in}}1}{{\left(\frac{{p}_{out}}{{p}_{in}}1\right)}_{DP}}\right]}_{n}$ Efficiency ${\left[\frac{\eta}{{\left(\eta \right)}_{DP}}\right]}_{VIGV}=c\text{}{\left[\frac{\eta}{{\left(\eta \right)}_{DP}}\right]}_{n}$  Correction factor a, b and c represents change in mass flow, pressure ratio and efficiency, respectively. 
Kurzke, [194]  ${\dot{m}}_{VIGV}={\dot{m}}_{FG,map}\times \left(1+\frac{{c}_{1}\Delta {\theta}_{VIGV}}{100}\right)\text{}$ ${\left(PR1\right)}_{VIGV}={\left(PR1\right)}_{FG,map}\times \left(1+\frac{{c}_{2}\Delta {\theta}_{VIGV}}{100}\right)$ ${\eta}_{VIGV}={\eta}_{FG,map}\times \left(1\frac{{c}_{3}\Delta {\theta}_{VIGV}^{2}}{100}\right)$  ${c}_{1}$,$\text{}{c}_{2}$,$\text{}{c}_{3}$ 
Knopf, [195]  Mass flow correction ${\dot{m}}_{OD}={\dot{m}}_{DP}\times \frac{{P}_{1,OD}}{{P}_{1,DP}}\times \frac{{T}_{1,DP}}{{T}_{1,OD}}\left({K}_{V}\Delta {\theta}_{VIGV}\right)\left[1+{K}_{T}\left(\frac{{T}_{1,OD}{T}_{1,DP}}{{T}_{1,DP}}\right)\text{}\right]$ Efficiency correction ${\eta}_{OD}={\eta}_{Max}\left(1\left\frac{{\dot{m}}_{DP}{\dot{m}}_{OD}}{{\dot{m}}_{DP}}\right\left({K}_{m}\right)\right)\left[1+{K}_{E}\left\frac{{N}_{OD}{N}_{Max,\eta}}{{N}_{Max,\eta}}\right\right]$  ${K}_{V}$,$\text{}{K}_{T}$,$\text{}{K}_{m}$,$\text{}{K}_{E}$ 
Plis and Rusinowski, [196]  Mass flow correction ${\dot{m}}_{VIGV}={\dot{m}}_{VIGV,Max}\left[1{K}_{V}\left({\theta}_{VIGV,Max}{\theta}_{VIGV}\right)\right]$ Efficiency empirical correlation developed from Wirkowski’s [197] experimental data ${\eta}_{VIGV}={\alpha}_{0}+{\alpha}_{1}\xb7{\dot{m}}_{corr}+{\alpha}_{2}\xb7{\dot{m}}_{corr}^{2}+{\alpha}_{3}\xb7{N}_{corr}+{\alpha}_{4}\xb7{N}_{corr}^{2}+{\alpha}_{5}\xb7{\dot{m}}_{corr}\xb7{N}_{corr}+{\alpha}_{6}\xb7{\theta}_{VIGV}+{\alpha}_{7}\xb7{\theta}_{VIGV}^{2}+{\alpha}_{8}\xb7{\dot{m}}_{corr}\xb7{\theta}_{VIGV}+{\alpha}_{9}\xb7{N}_{corr}\xb7{\theta}_{VIGV}$  ${K}_{V}$ 
Software  Developer/Owner  Type  Variable Geometry  Range of Flexibility  Pros  Cons  Reason for Utilization in Various Studies 

GasTurb  Dr. Joachim Kurzke  0 D, OOP  VIGV + Bleed schedule  1. Turbomachinery fouling and erosion, 2. Inlet flow distortion, 3. Optimization, 4. MonteCarlo  1. Need limited information from user, 2. User friendly for the GT operators due to predefined engine configuration  The model cannot be saved and transferred. For some cases it is very hard to import excel file, VIGV schedule only with respect to speed  Validation [43,50,145,226,227,228] 
GSP  National Aerospace Laboratory NLR, Netherlands  0D, OOP  VIGV +Bleed schedule  Turbomachinery fouling and erosion, shaft dynamics,  1. Easy saving and transporting of the model to other PCs, 2. Effects of ambient and flight conditions, installation losses, and malfunctions of control can be simulated  It is not user friendly for the gas turbines operators because its need every detail from the user even configuration need to be structured by user  Validation [44,49,222] 
PROOSIS  Alexiou from National Technical University of Athens  1D, OOP  No VIGV and Bleed  Parametric study, Optimization, Diagnostics  1. Model adaptation to specific engine using measured data 2. Frequency response analysis can be performed 3. Availability of extra auxiliary components such as gear box, generator and propeller  creation of libraries in the EL language require good expertise in the mathematical formulation of the components that limit this software to only academic community not to operators  validation [9,23,45,229] 
GTAnalysis  Gas Turbine Group, ITA, Brazil  Modular with interactive block structuring  VIGV and Bleed schedule, VAN for turbines  Deterioration  Due to its modular characteristic, any required modification can be easily incorporated to the program, making it very friendly  For variable geometry effect study, another in house program AFCC need to utilize  Modeling [14,60,69,70,230] 
NPSS  NASA Glenn Research Center  multiD  No  1. Integration of components for large systems and subsystems 2. effects of aerothermal and structural loadings on geometry and efficiency can be simulated  1. Additional codes can be appended 2. Zooming can give more details of the component performance inside the engine 3. High fidelity variable complexity analysis can be performed during design problems  Only available to partner research institutes of NASA  Modeling [184,231,232,233] 
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Hashmi, M.B.; Lemma, T.A.; Ahsan, S.; Rahman, S. Transient Behavior in Variable Geometry Industrial Gas Turbines: A Comprehensive Overview of Pertinent Modeling Techniques. Entropy 2021, 23, 250. https://doi.org/10.3390/e23020250
Hashmi MB, Lemma TA, Ahsan S, Rahman S. Transient Behavior in Variable Geometry Industrial Gas Turbines: A Comprehensive Overview of Pertinent Modeling Techniques. Entropy. 2021; 23(2):250. https://doi.org/10.3390/e23020250
Chicago/Turabian StyleHashmi, Muhammad Baqir, Tamiru Alemu Lemma, Shazaib Ahsan, and Saidur Rahman. 2021. "Transient Behavior in Variable Geometry Industrial Gas Turbines: A Comprehensive Overview of Pertinent Modeling Techniques" Entropy 23, no. 2: 250. https://doi.org/10.3390/e23020250