21, rue d’Artois, F-75008 PARIS
http : //www.cigre.org
A1-102
CIGRE 2012
Inertia of Hydro Generators. Influence on the Dimensioning, Cost, Efficiency
and Performance of the Units
H. D. Piriz
IMPSA Hydro
Argentina
A. R. Cannatella
IMPSA Hydro
Argentina
E. Guerra
IMPSA Hydro
Argentina
D. A. Porcari
IMPSA Hydro
Argentina
SUMMARY
The inertia constant is a parameter of rotating electrical machinery often required to suppliers by
customers as guaranteed data. Besides, it is commonly used by power systems analysts who use it
as input data for simulation programs. The parameter is defined as kinetic energy per apparent
power unit. It has the advantage of conferring simplicity to the motion equation of the unit
connected to a power system, with the torques and speeds expressed per unit, and the time
expressed in seconds.
With regards to this parameter and the motion equation, there seems to be a certain level of
confusion, mainly among people without the necessary expertise related to both, power systems
stability and electrical rotating machines. Apart from being a parameter used exclusively in
complex stability simulation programs, it is a useful parameter that can be used in simple
calculations for obtaining a rapid and reliable idea about the dynamic behavior of the rotating
unit, without the need of neither using sophisticated computing programs, nor memorizing values
of torques and angular speeds in the traditional units.
Considering the communication between the different parties involved in the problem, the aims of
this paper are:
-
to refresh the concept of inertia constant for those people who are not familiar with it;
to clarify some eventual misunderstandings,
to spread out its use by means of some simple examples which make the concept more
understandable;
to contribute to having better communication among power systems analysts, generator
manufacturers and turbine designers.
Besides, this paper is intended to show the impact of the inertia on the generator investment cost,
performance, dimensioning and the settings of some associated equipment. Therefore, the
following items are analyzed:
-
The Investment cost of a hydro-generator as the value of the specified GD2 or H varies
from a minimum “natural value” to a specified higher value, having the same level of
efficiency.
hector.piriz@impsa.com.
-
The influence of the inertia on the dimensioning of some associated systems is also
analyzed; for example, the case of the dimensioning of the starting system of the unit
when it works in pump mode at a pumped storage facility.
-
The risk of resonance between the natural oscillation frequency of the generator rotor and
the exciting frequencies coming from the turbine.
The treatment of the inertia constant value after the unit has been refurbished with a new
rated power.
The optimal settings of governor parameters for those cases in which the hydro-generator
operates in islanding mode.
-
Different cause-effect relationships are analyzed around the inertia of the unit. The impact of GD2
specifications are shown in terms of the cost and efficiencies obtained, and some recommended
practices are suggested when special calculations are required.
KEYWORDS
Inertia Constant, Acceleration time, Accelerating Torque, Flywheel Effect
2
1. INTRODUCTION
The quotient between the kinetic energy stored in the rotating parts of a hydro generator unit at rated
speed, and the rated capacity is the Inertia Constant H. The use of this parameter in the movement
equation of the unit in pu allows finding out, in a simple way, the times involved in the process of
acceleration and de-acceleration of rotating parts every time that the equilibrium between the
mechanical torque and electrical torque is broken by some perturbation. It means that calculations can
be achieved quickly to estimate accelerating times without the need of remembering numerical
quantities of torques in Nm, GD2 in tm2 and rotating speeds in rad/s.
2. THE NATURAL FLYWHEEL EFFECT
The inertia constant H depends on the flywheel effect according to the following equation:
2
1    GD2  n 2
H    
2  60 
kVA
(1)
Where:
GD2
is the Flywheel effect of Generator [tm2]
n
is the Rated rotating speed [rpm]
kVA
is the Rated Capacity [kVA]
H
is the Inertia constant [kW.s/kVA]
The generator manufacturers have developed statistical formulas for the possible inertia to be obtained
in different generator units and from these statistical values, the Natural GD2 has been defined, i.e. the
minimum GD2 that could be obtained complying with all the regular electromagnetic requirements.
One example of this Natural GD2 is the one obtained through equation 1 applied to the Hn detailed
below. This Hn is the inertia constant corresponding to the Natural GD2 indicated by the NEMA MG
5.1 standard.


Hn  0.54  ln kVA  10 3  0.30
(2)
3. BLOCK DIAGRAM OF MOVEMENT EQUATION
The movement equation [2] describes the effect of equilibrium loss between the electromagnetic
torque and mechanical torque of a generator. It is useful to represent this equation in the form of a
block diagram in order to get the concept in more understandable way as shown in Fig. 1.


1
Tdt
2H
(3)
Where:
ω
is the angular speed [pu]
t
is the time [s]
T
is the accelerating torque [pu]
Tm
is the mechanical torque [pu]
Te
is the electromagnetic torque [pu]
s
is the laplace Operator
3
Tm +
-
Te
1
2Hs
ω
Fig. 1 Block Diagram
Here is one example of the use of this equation: considering H=4.5 kW.s/kVA for a generator motor of
206 MVA – 300 rpm to be started as motor and accelerated from standstill to rated speed in 180 s from
one static frequency converter, we could find out the accelerating torque necessary for staring this
motor.
T= Tm –Te= Constant
Then, equation 3 results in:

Tt
2H
When ω=1, after 180 sec., the acceleration time shall be complied with. From the above equation, T
results in 0.05 pu. It means that the capacity of Static Frequency Converter Te shall be defined in order
to produce an accelerating torque of T of 0,05 pu compensating, at the same time, the loss that is
present during the process ( friction, windage, joule stator, joule rotor, etc)
4. INFLUENCE OF THE INERTIA ON COST AND EFFICIENCY
Recent investigation [8] shows that GD2 has influence on the loss and efficiency with a general
tendency to decrease the level of efficiency as GD2 is increased.
The minimum necessary value of inertia is generally indicated in the technical specifications of a
project. The specified inertia generally comes from estimations achieved, considering the impact of
the new unit on the power system and on the hydraulic conditions of the project. Achieving maximum
values of over pressure in the penstock and over speed of the unit sets a minimum of inertia, which for
hydro applications, is mostly concentrated on the generator rotor.
The requirement of a high GD2 with regards to the Natural GD2 will lead to a cost increase of the
generator [10]. In this current analysis, three cases were studied considering hydro generators of low,
medium and high rated speed.
In each case the analysis was carried out bearing in mind the following considerations:
-
It is a new project with no restrictions in the definition of air-gap diameter of the unit.
Steps of 10% of the natural GD2 ranging from 100% to 180% are the objectives of
dimensioning. The dimensioning of the unit and the resulting cost for each GD2 step were
calculated using the ARGEN program [4].
Correlations between the different cases studied were investigated in order to identify common
characteristics and differences.
All the analysed scenarios where calculated keeping the efficiency constant as the GD2
changed.
The results show that the cost variation originated by a GD2 variation is approximately linear with the
rated speed, and it can be characterized by equation 4.
4
C 
n
 GD 2
2570
(4)
Where
∆C
is the Cost Variation [pu]
n
is the rated speed [rpm]
∆GD2 is the GD2 Variation [pu]
This result shows convergence with the general rule of thumb of reference [10] only in the cases of
high speed generators, i.e. where the rated speed is equal to or higher than 514.3 rpm
The analysis shows that, when you have an excessive requirement of inertia, it will result in
cost increase, especially when the rated speed of the machine is medium or high. The
generator is an equipment that has a significant impact on the value of the entire system;
therefore, it is an important detail to bear in mind when it comes to specifying.
This cause-effect relation is more evident when the specifications have high efficiency requirements
combined with high GD2.
5. INFLUENCE OF THE INERTIA ON THE FREQUENCY OF ROTOR OSCILATIONS
The simplified block diagram of Fig. 3, shows the movement equation including, for simplicity of the
conceptual analysis, only the influence of electromagnetic torque with regards to the loading angle δ
of the generator. It means that the constant flux linkages in the d axis are considered [9]. Neither the
influence of the load damping, nor the damper windings and excitation systems action are considered
in this analysis.
Tm +
1
2Hs
Te


s

K1
Fig.3. Block Diagram for Rotor Electromechanical Oscillations
The typical equation of this closed loop system is:
1  K1 
2f
0
Ms 2
(4)
From a typical system of second order the natural oscillation frequency can be obtained as follows:
n 
K1  2    f
M
(5)
Where:
M  2H
K1= Synchronizing Coefficient [9], i.e. the slope of the curve power – angle of the generator.
5
In case of disturbances caused by perturbations or frequencies coming from the turbine (as for the case
of partial power operation in Francis Turbines) or from the power system, the equilibrium between the
mechanical torque Tm and the electrical torque Te is broken. In this way, the unit rotor will oscillate
with accelerations and de-accelerations around synchronous speed at a rate imposed by this natural
frequency ωn. For Francis turbines applications, this phenomenon is analyzed during the stage of
generator dimensioning. At this stage, the degree of proximity between frequency coming from the
turbine and the natural frequency of the generator is analyzed in order to avoid the risk of resonance.
A security criterion applicable to this situation is to obtain a magnification factor lower than 3.1 [3],
which implies a distance between natural frequencies and frequencies coming from perturbation equal
to or higher than 15%.
If frequencies are close to one another, it is necessary, during the dimensioning stage, to vary
parameters of equation (5) in order to obtain the minimum required distance between frequencies. The
parameters that influence equation 5 are.



H (inertia constant)
Xq (quadrature synchronous reactance)
X´d (direct transient reactance)
A particular case was studied in order to find out what impact the variations of each parameter have on
the variation of the natural frequency, in order to obtain the adequate distance between natural
frequency and the one coming from perturbation. This study was conducted on a generator of 90
MVA. The analysis was carried out using the non-linear mathematical model of the generator, with the
unit operating at rated conditions and producing a perturbation consisting of a step- type variation of
3% on the mechanical torque.
The original design parameters: H=2,46 kW.s/kVA, xq=0,67 pu and x´d=0,36 pu, when the unit
operates at rated capacity, contribute to the rotor natural frequency of 1,60 Hz. During the analysis,
these parameters were varied one at a time in order to obtain a shift of the natural frequency with a
final value away from 1, 60 Hz, at least, 15% lower or higher.
During that process it was discovered that by moving the H value, positive results were obtained with
the frequency shift, increasing or reducing H by 35 % of its original value. With the other parameter
variations, very few variations of natural frequency were achieved, while producing greater changes of
x´d and xq.
A sensitivity analysis was carried out in order to check the influence of each parameter on obtaining a
100 % of variation in the natural frequency. Fig. 4 shows that 90% of the change in the frequency is
contributed by H and 5% of the contribution is achieved by each reactance variation.
Fig.4. Time response of Frequency after a loading step
The results showed that H is definitely the parameter with the greatest degree of effective influence on
this cause-effect relationship. In the case that was studied, with an H variation of 35 %, a distance
between the original natural frequency and the resulting natural frequency of 15% was obtained.
6
The influences of the other parameter variations are of little significance. For instance, a variation of
Xq of 57% from its original value produced a shift in the natural frequency of only 1.87 %.
On the other hand, if a parameter like xq or x´d is forced to produce a shift in the frequency, the
variation obtained will be limited and the final value of xq, corresponding to that action, will threaten
the best practices in electromagnetic design.
The equivalent analyses achieved on other applications have shown similar results.
6. INERTIA CONSTANT IN REFURBISHED GENERATORS
In rehabilitation applications it is very common to find results without producing an impact on the
existing civil work, mainly if the power house is in good conditions. When required to maintain the
dimensions and speed of the units, the GD2 remains practically the same after rehabilitation. When the
rehabilitation implies increasing the output power, without changing rotor dimensions and speed, the
result will be a reduction in the inertia constant H and the same GD2 will be achieved.
Carrying out the performance studies of this new up- graded unit, using the movement equation in p.u.
could lead to the erroneous conclusion that the up- graded unit is less stable with regards to its
previous condition because its new H is lower. In order to clarify the concept of this situation 2
particular cases are analyzed.
6.1 Performance of the unit in Island Mode Operation
When studying the isolated mode operation, the refurbished unit (with the same GD2 and lower H)
would present a greater frequency deviation with regards to the original generator model for the same
load value in pu imposed to both models as perturbation in the time domain analysis.
The following considerations are suggested to be taken into account for that study:
-
The electric loading power step in pu is always expressed in reference to the base pu power of
the unit. Then the same step in pu will represent a higher value in kW for the refurbished unit
with regards to the original
The damping factor of the load D [1] is the power variation in reference to the rated active
power for a frequency variation. Then, in a comparative analysis between the original unit and
the refurbished one, the value of D shall also be varied when the Active Power pu base of the
refurbished unit has been changed with regards to the original. So the relation H/D should be
constant for the original unit and the refurbished one.
Table IV shows an example of refurbishment, in which the frequency variation was simulated in the
time domain for one step of the same value in kW for the original unit and the refurbished one. It can
be observed that taking the above suggestions into consideration, the time response of both units will
be the same.
TABLE IV. GENERATOR DATA
Original
Generator
Refurbished
Generator
Rated Capacity [kVA]
75300
78000
GD2 [Tm2]
10809
10809
H [kW.s/kVA]
2,83
2,46
Loading Step [kW]
Loading Step [pu]
D
5556
0,074
0,750
5556
0,064
0,652
H/D
3,777
3,777
7
1.01
1.00
0.99
Máquina Original
FRECUENCIA [pu]
Máquina Repotenciada
0.98
0.97
0.96
0.95
0.94
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
TIEMPO [s]
Fig.4. Time response of Frequency after a loading step
4.2 Braking of the unit
A similar situation arises when the braking time of the unit is studied during a regular downtime of the
unit, using the movement equation in pu [2]. The H reduction in the model of the refurbished
generator could lead to the erroneous conclusion that the refurbished unit would brake faster than the
original unit. Actually, the kinetic energy of the rotor is the same and the refurbished unit will be
stopped at the same time as the original one, if the loss is the same. It is suggested paying special
attention to this matter and taking into account, in the study, that the same loss in kW shall be
represented with different pu values when studying the original and the refurbished units.
5. INFLUENCE OF H IN THE ISLANDED OPERATION MODE
For stable operation under islanded conditions, the optimum setting of the speed regulator parameters
has remarkable influence. The adequate setting depends on the water time constant Tw and the
mechanical time constant Tm [7]. The selection of the Temporary droop RT and the reset time TR of
the governor depends on the values of Tw and Tm and, therefore, it depends on H. Although these
parameters are typical of the old speed mechanical–hydraulic regulators, their same functions are
performed electrically in modern electro-hydraulic regulators. In references [1] and [7] the equations
for the optimal tuning of TR and RT are found, and in reference [6] the equations that associate RT
and TR with proportional, derivative and integrative constant of a PID regulator are found.
The analysis indicates that it is possible to find adequate speed regulator tuning for stable operation
using the recommended equations in [7] even in the cases which are considered unstable in reference
[10], in which the Tm/Tw ratio is lower than 2.
6. CONCLUSIONS
Different studies carried out about generator inertia show that this parameter, which is independently
represented by H, GD2 or J, is a dimensioning parameter with great degree of influence in the design,
cost and performance of the unit. Due to the fact that the inertia value is normally conditioned by
specification requirements, this parameter should result in a good trade-off which will ensure the
compliance with best practices, allowing an adequate dimensioning of the unit within reasonable cost
and, at the same time, the compliance with the requirements of the technical specifications. Every time
that the inertia value is specified, it is advisable to revise it well enough in order to avoid antagonist
8
requirements that could force the generator dimensioning to be excessively different from its optimum
design.
BIBLIOGRAPHY
1 Prabha Kundur, Power System Stability and Control, McGraw –Hill, Inc. 1994
[2] A.R.Cannatella, H.D.Piriz, “Sistemas de Frenado Eléctrico y Frenado Regenerativo en Centrales
Hidroeléctricas”, en XIII ENCUENTRO REGIONAL IBEROAMERICANO DE CIGRÉ, Mayo de
2009
3 M. P. Kostenko, L. M. Piotrovski, Máquinas Eléctricas II, Editorial MIR, Moscú, 1979
[4] E.J.Guerra, A.O.García, F.M.Graffigna, C.A.Verdú, “Optimizing Generators”, International
Water Power and Dam Construction, November 1994
5 Rotating Electrical Machines – Methods for Determining Synchronous Machines Quantities from
Tests. IEC 60034-4: 2008
6 Woodward, The Controlled System, PMCC22
7 P.L Dandeno, P. Kundur, J. P. Bayne, “Hydraulic Unit Dynamic Performance Under Normal and
Islanding Conditions” in IEEE Transactions on Power Apparatus and Systems, Vol. PAS-97, Nro
6, Nov / Dec 1978.
8 WG A1.12 – CIGRE “State of the Art in Efficiency of Hydro Generators Commissioned since
1990”, Technical Brochure - September 2011.
9 Concepts of Synchronous Machine Stability as Affected by Excitation Control. - F. P. DeMello,
Charles Concordia. IEEE TRANSACTION ON POWER APPARATUS AND SYSTEMS, VOL.
PAS-88. NRO.4. APRIL 1969.
10 Generator Inertia for Isolated Power Systems, J.L. Gordon and H. D. Withman - Can. J. Civ.
Eng. Vol. 12. 1985
9