I am trying to do a linearization of the one state wind turbine model: J*(d_omega/dt)=0.5CprhoAV^3/omega-Ntao_g. In this expression, J is the total inertia of all parts of rotation. It should be expressed like this:
J=J_L+J_HN^2;
J_H is the inertia of the high speed side which is corresponding to the variable ‘GenIner’ in FAST Input file and N is the gearbox ratio. J_L is the inertia of low speed side. I think J_L should include the inertia of all blades, hub and low speed rotor. But from the explanation, the variable ‘HubIner’ seems only represent the inertia of Hub.

So my question is if J_L=HubIner is right? If not, how should I calculate the J_L?

In FAST, HubIner represents the inertia of the hub (without blades) about the low-speed shaft (LSS). In your equation, you want J_L to represent the inertia of the entire rotor about the LSS. This inertia would equal HubIner plus the inertia of each blade about the LSS. This rotor inertia is called RotIner in FAST and is calculated and reported in the FAST summary (*.fsm) file.

But when I check the relation among RotIner, inertia of each blade about the LSS and the HubIner, there is still a little confusion to me. Just taking the WindPACT 1.5MW Wind Turbine for example:
HubIner=34600(kg*m^2);RotIner=2962443.900;Inertia of each blade about the LSS=800799.916(it’s called second mass moment in .fsm file, I suppose it to be the blade internia), then:
34600+800799.9163=2437000!=2962443.9

In FAST’s .fsm file, the center of mass, 1st mass moment, and 2nd mass moment are all output relative to the blade root (not the rotor center). It is the transfer of the blade inertia from the blade root to the LSS that you are missing in your calculation.

Sorry to dig up this old thread, but I have similar questions regarding the rotational inertia of the 5MW offshore spar-buoy (OC3) rotor. I’m trying to determine the inertia about the LSS so I’m starting with the “Rotor Inertia” from my .fsm file (38759236 kg-m^2).

I see that this is ‘at the blade root’ and needs to be translated to the LSS, I assume via parallel-axis. However there is no orientation about this given Rotor Inertia at the root. Is it parallel to the LSS? And would my distance (in the parallel-axis theorem) simply be the radius of the hub?

Perhaps more straight to the point is: Is the rotor inertia (hub and blades as per http://forums.nrel.gov/t/rotor-inertia/850/1) reported in the .fsm file an inertia about the LSS? Thanks for any help!

The rotor inertia written to the FAST summary file of FAST v7 (.fsm) or the ElastoDyn summary file of FAST v8 (.ED.sum) is reported about the LSS, not about the blade root. Only the individual blade center of mass, 1st mass moment, and 2nd mass moment are output relative to the blade root.

For the OC3-Hywind, starting with my rotor inertia of 38759236 kg-m^2 and adding in the generator inertia of 5025500 kg-m^2, I get a drivetrain inertia about the LSS of 43,784,736 kg-m^2. Can anyone verify that number? I’ve been attempting to utilize the baseline variable-speed controller (in separate, non-FAST code) and I suspect that my current inertia setting is wrong. Thanks for any help!

How is Rotor inertia calculated through .fsm file? I have calculated my
Rotor inertia = Hub inertia + 3*Blade inertia

through this formula i calculated my Rotor inertia = 5229.34 kgm2. But my rotor inertia calculated through FAST .fsm file is 5861.016. Even though Rotor blade mass is calculated correctly in .fsm file. What about Second Mass moment? is this rotor blade inertia? if it is then with formula i got rotor blade inertia = 1721.78kgm2, but with .fsm my Second Mass moment calculated as 1758.745 kgm2.

Second, i have attached code, My wind turbine is 50kw power but i am getting Power of around 80kw. Can you point me what i am doing wrong? i am stuck here. This will be very helpfull.

[code]--------------------------------------------------------------------------------
------- FAST INPUT FILE --------------------------------------------------------
FAST certification Test #08: AOC 15/50 with many DOFs with fixed yaw error and steady wind.
Many parameters are pure fiction. Compatible with FAST v7.02.00.
---------------------- SIMULATION CONTROL --------------------------------------
False Echo - Echo input data to “echo.out” (flag)
1 ADAMSPrep - ADAMS preprocessor mode {1: Run FAST, 2: use FAST as a preprocessor to create an ADAMS model, 3: do both} (switch)
1 AnalMode - Analysis mode {1: Run a time-marching simulation, 2: create a periodic linearized model} (switch)
3 NumBl - Number of blades (-)
35.0 TMax - Total run time (s)
0.005 DT - Integration time step (s)
---------------------- TURBINE CONTROL -----------------------------------------
0 YCMode - Yaw control mode {0: none, 1: user-defined from routine UserYawCont, 2: user-defined from Simulink/Labview} (switch)
9999.9 TYCOn - Time to enable active yaw control (s) [unused when YCMode=0]
0 PCMode - Pitch control mode {0: none, 1: user-defined from routine PitchCntrl, 2: user-defined from Simulink/Labview} (switch)
9999.9 TPCOn - Time to enable active pitch control (s) [unused when PCMode=0]
0 VSContrl - Variable-speed control mode {0: none, 1: simple VS, 2: user-defined from routine UserVSCont, 3: user-defined from Simulink/Labview} (switch)
1481.98 VS_RtGnSp - Rated generator speed for simple variable-speed generator control (HSS side) (rpm) [used only when VSContrl=1]
318.31 VS_RtTq - Rated generator torque/constant generator torque in Region 3 for simple variable-speed generator control (HSS side) (N-m) [used only when VSContrl=1]
0.0001379 VS_Rgn2K - Generator torque constant in Region 2 for simple variable-speed generator control (HSS side) (N-m/rpm^2) [used only when VSContrl=1]
9999.9 VS_SlPc - Rated generator slip percentage in Region 2 1/2 for simple variable-speed generator control (%) [used only when VSContrl=1]
1 GenModel - Generator model {1: simple, 2: Thevenin, 3: user-defined from routine UserGen} (switch) [used only when VSContrl=0]
True GenTiStr - Method to start the generator {T: timed using TimGenOn, F: generator speed using SpdGenOn} (flag)
True GenTiStp - Method to stop the generator {T: timed using TimGenOf, F: when generator power = 0} (flag)
9999.9 SpdGenOn - Generator speed to turn on the generator for a startup (HSS speed) (rpm) [used only when GenTiStr=False]
6.0 TimGenOn - Time to turn on the generator for a startup (s) [used only when GenTiStr=True]
9999.9 TimGenOf - Time to turn off the generator (s) [used only when GenTiStp=True]
1 HSSBrMode - HSS brake model {1: simple, 2: user-defined from routine UserHSSBr, 3: user-defined from Labview} (switch)
9999.9 THSSBrDp - Time to initiate deployment of the HSS brake (s)
9999.9 TiDynBrk - Time to initiate deployment of the dynamic generator brake [CURRENTLY IGNORED] (s)
9999.9 TTpBrDp(1) - Time to initiate deployment of tip brake 1 (s)
9999.9 TTpBrDp(2) - Time to initiate deployment of tip brake 2 (s)
9999.9 TTpBrDp(3) - Time to initiate deployment of tip brake 3 (s) [unused for 2 blades]
9999.9 TBDepISp(1) - Deployment-initiation speed for the tip brake on blade 1 (rpm)
9999.9 TBDepISp(2) - Deployment-initiation speed for the tip brake on blade 2 (rpm)
9999.9 TBDepISp(3) - Deployment-initiation speed for the tip brake on blade 3 (rpm) [unused for 2 blades]
9999.9 TYawManS - Time to start override yaw maneuver and end standard yaw control (s)
9999.9 TYawManE - Time at which override yaw maneuver reaches final yaw angle (s)
0.0 NacYawF - Final yaw angle for yaw maneuvers (degrees)
9999.9 TPitManS(1) - Time to start override pitch maneuver for blade 1 and end standard pitch control (s)
9999.9 TPitManS(2) - Time to start override pitch maneuver for blade 2 and end standard pitch control (s)
9999.9 TPitManS(3) - Time to start override pitch maneuver for blade 3 and end standard pitch control (s) [unused for 2 blades]
9999.9 TPitManE(1) - Time at which override pitch maneuver for blade 1 reaches final pitch (s)
9999.9 TPitManE(2) - Time at which override pitch maneuver for blade 2 reaches final pitch (s)
9999.9 TPitManE(3) - Time at which override pitch maneuver for blade 3 reaches final pitch (s) [unused for 2 blades]
1.54 BlPitch(1) - Blade 1 initial pitch (degrees)
1.54 BlPitch(2) - Blade 2 initial pitch (degrees)
1.54 BlPitch(3) - Blade 3 initial pitch (degrees) [unused for 2 blades]
1.54 BlPitchF(1) - Blade 1 final pitch for pitch maneuvers (degrees)
1.54 BlPitchF(2) - Blade 2 final pitch for pitch maneuvers (degrees)
1.54 BlPitchF(3) - Blade 3 final pitch for pitch maneuvers (degrees) [unused for 2 blades]
---------------------- ENVIRONMENTAL CONDITIONS --------------------------------
9.80665 Gravity - Gravitational acceleration (m/s^2)
---------------------- FEATURE FLAGS -------------------------------------------
False FlapDOF1 - First flapwise blade mode DOF (flag)
False FlapDOF2 - Second flapwise blade mode DOF (flag)
False EdgeDOF - First edgewise blade mode DOF (flag)
False TeetDOF - Rotor-teeter DOF (flag) [unused for 3 blades]
True DrTrDOF - Drivetrain rotational-flexibility DOF (flag)
True GenDOF - Generator DOF (flag)
False YawDOF - Yaw DOF (flag)
False TwFADOF1 - First fore-aft tower bending-mode DOF (flag)
False TwFADOF2 - Second fore-aft tower bending-mode DOF (flag)
False TwSSDOF1 - First side-to-side tower bending-mode DOF (flag)
False TwSSDOF2 - Second side-to-side tower bending-mode DOF (flag)
True CompAero - Compute aerodynamic forces (flag)
False CompNoise - Compute aerodynamic noise (flag)
---------------------- INITIAL CONDITIONS --------------------------------------
0.0 OoPDefl - Initial out-of-plane blade-tip displacement (meters)
0.0 IPDefl - Initial in-plane blade-tip deflection (meters)
0.0 TeetDefl - Initial or fixed teeter angle (degrees) [unused for 3 blades]
0.0 Azimuth - Initial azimuth angle for blade 1 (degrees)
0.0 RotSpeed - Initial or fixed rotor speed (rpm)
0.0 NacYaw - Initial or fixed nacelle-yaw angle (degrees)
0.0 TTDspFA - Initial fore-aft tower-top displacement (meters)
0.0 TTDspSS - Initial side-to-side tower-top displacement (meters)
---------------------- TURBINE CONFIGURATION -----------------------------------
7.490 TipRad - The distance from the rotor apex to the blade tip (meters)
0.230 HubRad - The distance from the rotor apex to the blade root (meters)
1 PSpnElN - Number of the innermost blade element which is still part of the pitchable portion of the blade for partial-span pitch control [1 to BldNodes] [CURRENTLY IGNORED] (-)
0.0 UndSling - Undersling length [distance from teeter pin to the rotor apex] (meters) [unused for 3 blades]
0.0 HubCM - Distance from rotor apex to hub mass [positive downwind] (meters)
1.341 OverHang - Distance from yaw axis to rotor apex [3 blades] or teeter pin [2 blades] (meters)
0.0 NacCMxn - Downwind distance from the tower-top to the nacelle CM (meters)
0.0 NacCMyn - Lateral distance from the tower-top to the nacelle CM (meters)
0.6 NacCMzn - Vertical distance from the tower-top to the nacelle CM (meters)
30 TowerHt - Height of tower above ground level [onshore] or MSL [offshore] (meters)
0.61 Twr2Shft - Vertical distance from the tower-top to the rotor shaft (meters)
0.0 TwrRBHt - Tower rigid base height (meters)
0.0 ShftTilt - Rotor shaft tilt angle (degrees)
0.0 Delta3 - Delta-3 angle for teetering rotors (degrees) [unused for 3 blades]
0.0 PreCone(1) - Blade 1 cone angle (degrees)
0.0 PreCone(2) - Blade 2 cone angle (degrees)
0.0 PreCone(3) - Blade 3 cone angle (degrees) [unused for 2 blades]
0.17 AzimB1Up - Azimuth value to use for I/O when blade 1 points up (degrees)
---------------------- MASS AND INERTIA ----------------------------------------
0.0 YawBrMass - Yaw bearing mass (kg)
2440.25 NacMass - Nacelle mass (kg)
800.0 HubMass - Hub mass (kg)
0 TipMass(1) - Tip-brake mass, blade 1 (kg)
0 TipMass(2) - Tip-brake mass, blade 2 (kg)
0 TipMass(3) - Tip-brake mass, blade 3 (kg) [unused for 2 blades]
417.106 NacYIner - Nacelle inertia about yaw axis (kg m^2)
10.0 GenIner - Generator inertia about HSS (kg m^2)
64.0 HubIner - Hub inertia about rotor axis [3 blades] or teeter axis [2 blades] (kg m^2)
---------------------- DRIVETRAIN ----------------------------------------------
98.0 GBoxEff - Gearbox efficiency (%)
89.4 GenEff - Generator efficiency [ignored by the Thevenin and user-defined generator models] (%)
23.23 GBRatio - Gearbox ratio (-)
False GBRevers - Gearbox reversal {T: if rotor and generator rotate in opposite directions} (flag)
9999.9 HSSBrTqF - Fully deployed HSS-brake torque (N-m)
9999.9 HSSBrDt - Time for HSS-brake to reach full deployment once initiated (sec) [used only when HSSBrMode=1]
“unused” DynBrkFi - File containing a mech-gen-torque vs HSS-speed curve for a dynamic brake [CURRENTLY IGNORED] (quoted string)
1.76E5 DTTorSpr - Drivetrain torsional spring (N-m/rad)
1.0E5 DTTorDmp - Drivetrain torsional damper (N-m/(rad/s))
---------------------- SIMPLE INDUCTION GENERATOR ------------------------------ Crude approximation of torque/speed curve.
2.222 SIG_SlPc - Rated generator slip percentage (%) [used only when VSContrl=0 and GenModel=1]
1500.0 SIG_SySp - Synchronous (zero-torque) generator speed (rpm) [used only when VSContrl=0 and GenModel=1]
318.31 SIG_RtTq - Rated torque (N-m) [used only when VSContrl=0 and GenModel=1]
1.75 SIG_PORt - Pull-out ratio (Tpullout/Trated) (-) [used only when VSContrl=0 and GenModel=1]
---------------------- THEVENIN-EQUIVALENT INDUCTION GENERATOR -----------------
60.0 TEC_Freq - Line frequency [50 or 60] (Hz) [used only when VSContrl=0 and GenModel=2]
4 TEC_NPol - Number of poles [even integer > 0] (-) [used only when VSContrl=0 and GenModel=2]
4.92E-02 TEC_SRes - Stator resistance (ohms) [used only when VSContrl=0 and GenModel=2]
5.34E-04 TEC_RRes - Rotor resistance (ohms) [used only when VSContrl=0 and GenModel=2]
480.0 TEC_VLL - Line-to-line RMS voltage (volts) [used only when VSContrl=0 and GenModel=2]
1.00E-04 TEC_SLR - Stator leakage reactance (ohms) [used only when VSContrl=0 and GenModel=2]
1.00E-04 TEC_RLR - Rotor leakage reactance (ohms) [used only when VSContrl=0 and GenModel=2]
4.49E-03 TEC_MR - Magnetizing reactance (ohms) [used only when VSContrl=0 and GenModel=2]
---------------------- PLATFORM ------------------------------------------------
0 PtfmModel - Platform model {0: none, 1: onshore, 2: fixed bottom offshore, 3: floating offshore} (switch)
“unused” PtfmFile - Name of file containing platform properties (quoted string) [unused when PtfmModel=0]
---------------------- TOWER ---------------------------------------------------
11 TwrNodes - Number of tower nodes used for analysis (-)
“AOC_Tower.dat” TwrFile - Name of file containing tower properties (quoted string)
---------------------- NACELLE-YAW ---------------------------------------------
0.0 YawSpr - Nacelle-yaw spring constant (N-m/rad)
0.0 YawDamp - Nacelle-yaw damping constant (N-m/(rad/s))
0.0 YawNeut - Neutral yaw position–yaw spring force is zero at this yaw (degrees)
---------------------- FURLING -------------------------------------------------
False Furling - Read in additional model properties for furling turbine (flag)
“unused” FurlFile - Name of file containing furling properties (quoted string) [unused when Furling=False]
---------------------- ROTOR-TEETER --------------------------------------------
0 TeetMod - Rotor-teeter spring/damper model {0: none, 1: standard, 2: user-defined from routine UserTeet} (switch) [unused for 3 blades]
0.0 TeetDmpP - Rotor-teeter damper position (degrees) [used only for 2 blades and when TeetMod=1]
0.0 TeetDmp - Rotor-teeter damping constant (N-m/(rad/s)) [used only for 2 blades and when TeetMod=1]
0.0 TeetCDmp - Rotor-teeter rate-independent Coulomb-damping moment (N-m) [used only for 2 blades and when TeetMod=1]
0.0 TeetSStP - Rotor-teeter soft-stop position (degrees) [used only for 2 blades and when TeetMod=1]
0.0 TeetHStP - Rotor-teeter hard-stop position (degrees) [used only for 2 blades and when TeetMod=1]
0.0 TeetSSSp - Rotor-teeter soft-stop linear-spring constant (N-m/rad) [used only for 2 blades and when TeetMod=1]
0.0 TeetHSSp - Rotor-teeter hard-stop linear-spring constant (N-m/rad) [used only for 2 blades and when TeetMod=1]
---------------------- TIP-BRAKE -----------------------------------------------
0.0 TBDrConN - Tip-brake drag constant during normal operation, CdArea (m^2)
0.0 TBDrConD - Tip-brake drag constant during fully-deployed operation, CdArea (m^2)
0.5 TpBrDT - Time for tip-brake to reach full deployment once released (sec)
---------------------- BLADE ---------------------------------------------------
“AOC_Blade.dat” BldFile(1) - Name of file containing properties for blade 1 (quoted string)
“AOC_Blade.dat” BldFile(2) - Name of file containing properties for blade 2 (quoted string)
“AOC_Blade.dat” BldFile(3) - Name of file containing properties for blade 3 (quoted string) [unused for 2 blades]
---------------------- AERODYN -------------------------------------------------
“aerodyn1.ipt” ADFile - Name of file containing AeroDyn input parameters (quoted string)
---------------------- NOISE ---------------------------------------------------
“unused” NoiseFile - Name of file containing aerodynamic noise input parameters (quoted string) [used only when CompNoise=True]
---------------------- ADAMS ---------------------------------------------------
“AOC_ADAMS.dat” ADAMSFile - Name of file containing ADAMS-specific input parameters (quoted string) [unused when ADAMSPrep=1]
---------------------- LINEARIZATION CONTROL -----------------------------------
“AOC_Linear.dat” LinFile - Name of file containing FAST linearization parameters (quoted string) [unused when AnalMode=1]
---------------------- OUTPUT --------------------------------------------------
True SumPrint - Print summary data to “.fsm” (flag)
1 OutFileFmt - Format for tabular (time-marching) output file(s) (1: text file [.out], 2: binary file [.outb], 3: both) (switch)
True TabDelim - Use tab delimiters in text tabular output file? (flag)
“ES10.3E2” OutFmt - Format used for text tabular output (except time). Resulting field should be 10 characters. (quoted string) [not checked for validity!]
5.0 TStart - Time to begin tabular output (s)
10 DecFact - Decimation factor for tabular output {1: output every time step} (-)
5.0 SttsTime - Amount of time between screen status messages (sec)
0.0 NcIMUxn - Downwind distance from the tower-top to the nacelle IMU (meters)
0.0 NcIMUyn - Lateral distance from the tower-top to the nacelle IMU (meters)
0.0 NcIMUzn - Vertical distance from the tower-top to the nacelle IMU (meters)
0.5 ShftGagL - Distance from rotor apex [3 blades] or teeter pin [2 blades] to shaft strain gages [positive for upwind rotors] (meters)
0 NTwGages - Number of tower nodes that have strain gages for output [0 to 9] (-)
0 TwrGagNd - List of tower nodes that have strain gages [1 to TwrNodes] (-) [unused if NTwGages=0]
5 NBlGages - Number of blade nodes that have strain gages for output [0 to 9] (-)
2,4,6,8,9 BldGagNd - List of blade nodes that have strain gages [1 to BldNodes] (-) [unused if NBlGages=0]
OutList - The next line(s) contains a list of output parameters. See OutList.xlsx for a listing of available output channels, (-)
“TotWindV” - Wind speed
“TipDxb3,TipDyb3” - Blade 3 flapwise and edgewise tip deflections
“TipRDxb3,TipRDyb3” - Blade 3 roll and pitch tip deflections
“Spn5ALxb1,Spn5ALyb1” - Blade 1 Gage #5 (Node 9) local flapwise and edgewise accelerations
“RotSpeed,LSSGagV,HSShftV” - Rotor, LSS (GBox side), and HSS speed
“RootFxb3,RootFyb3” - Blade 3 root flapwise and edgewise shear forces
“RootMEdg3,RootMFlp3” - Blade 3 root edgewise and flapwise bending moments
“Spn4MLxb1,Spn4MLyb1” - Blade 1 Gage #4 (Node 8) local edgewise and flapwise bending moments
“LSSGagFxs,LSSGagFys,LSSGagFzs” - Non-rotating LSS thrust and shear forces
“LSShftTq,HSShftTq,GenTq” - LSS, HSS, and generator torque
“LSShftPwr,HSShftPwr,GenPwr” - LSS, HSS, and generator power
END of FAST input file (the word “END” must appear in the first 3 columns of this last line).

I’m not sure why your hand calculation of the rotor blade inertia does not match the FAST v7 summary file, but your equation for the rotor inertia (RotIner) is not correct because you miss (1) the influence of the hub radius (HubRad) and (2) the influence of the rotor precone (PreCone(k)). The former is important because the second mass moment of inertia of the blade (SecondMom(k)) reported in the FAST v7 summary file is calculated about the blade root, not about the centerline of the low-speed shaft. With these adjustments, the correct equation is:

(That said, I see that you’ve set PreCone(k) = 0 in your FAST v7 primary input file, so, COS( PreCone(k) )^2 = 1.)

I see that you are using the Simple Induction Generator (SIG) model (VSContrl=0, GenModel=1). My guess is you are getting more than 50 kW because your generator speed is spinning faster than 1500*(1.02222) rpm (and thus your generator torque is higher than 318.31 Nm).

Thank you so much for your help. i solved my rotor inertia. I still have problem with my Generator speed. I am using (VSContrl = 0 and GenModel = 1).

---------------------- DRIVETRAIN ----------------------------------------------
98.0 GBoxEff - Gearbox efficiency (%)
92.8 GenEff - Generator efficiency [ignored by the Thevenin and user-defined generator models] (%)
23.23 GBRatio - Gearbox ratio (-)
False GBRevers - Gearbox reversal {T: if rotor and generator rotate in opposite directions} (flag)
9999.9 HSSBrTqF - Fully deployed HSS-brake torque (N-m)
9999.9 HSSBrDt - Time for HSS-brake to reach full deployment once initiated (sec) [used only when HSSBrMode=1]
"unused" DynBrkFi - File containing a mech-gen-torque vs HSS-speed curve for a dynamic brake [CURRENTLY IGNORED] (quoted string)
1.76E5 DTTorSpr - Drivetrain torsional spring (N-m/rad)
1.0E5 DTTorDmp - Drivetrain torsional damper (N-m/(rad/s))
---------------------- SIMPLE INDUCTION GENERATOR ------------------------------ Crude approximation of torque/speed curve.
1e-2 SIG_SlPc - Rated generator slip percentage (%) [used only when VSContrl=0 and GenModel=1]
1500.0 SIG_SySp - Synchronous (zero-torque) generator speed (rpm) [used only when VSContrl=0 and GenModel=1]
318.31 SIG_RtTq - Rated torque (N-m) [used only when VSContrl=0 and GenModel=1]
1.0 SIG_PORt - Pull-out ratio (Tpullout/Trated) (-) [used only when VSContrl=0 and GenModel=1]

My rotor speed is 1500/23.23 = 64.5 rpm. i put inital condition " 64.5 RotSpeed - Initial or fixed rotor speed (rpm)". Now Generator speed should be (1500*1.00001 ) But when simulate my model with FAST. i got rotor speed (LSSGaV) around 100-120 rpm and GenSpeed(HSShftV) around 2500 rpm!!!

Why is there so much difference? How are LSSGaV & HSShftV are calculated with FAST?

You’ve set your rated torque (SIG_RtTq) to 318.31 Nm and the pull-out ratio (SIG_PORt) to 1.0, so, the generator torque can’t exceed 318.31 Nm. if the aerodynamic torque (relative to the high-speed shaft) ever exceeds 318.13 Nm, then the generator / rotor speeds will accelerate well above the rated speed.

Dear Dr Jason
I need some clarification about the red part in the equation below:
RotIner = HubIner + SUM(( SecondMom(K) + BldMass (K)( HubRad + BldCg(k) )^2 - BldCg(k)^2)( CosPreC(K)**2 )), k = 1,2,3 )
According to the Parallel Axis Theorem, the red part of equation should represent the square distance between the blade root and the centerline of the low-speed shaft. I think this distance may be equal to HubRad^2.
Why you add and subtract BldCg(k)? OR
Where is the location of blade root? (Is it at point along pitch axis with a distance of HubRad from Apex of cone rotation?)

Yes, the blade root is located a distance of HubRad from the apex along the pitch axis, so ( HubRad + BldCg(K) ) is the distance from the apex to the blade center of mass. The -BldMass(K)*BldCg(K)^2 term is included so as not to double count the inertia effect already accounted for in SecondMom(K).