quantifying the maintenance costs associated with variable ... maint costs.pdfin greek mythology,...

Copyright © 2003 by Hydro Resource Solutions LLC. All rights reserved.

Quantifying the Maintenance Costs Associated with Variable Operating Conditions

By Patrick A. March, Hydro Resource Solutions LLC

Introduction

In Greek mythology, the three Fates determine each person’s destiny. Klotho, the Spinner, selects and prepares the thread representing a human life. Lachesis, the Lot-Caster, considers each thread carefully, feels its fibers, judges its strength, and weaves it into the great tapestry of life. Atropos, the Inexorable, halts the weaving, cuts off the thread, and ends the life (see Figure 1).

Figure 1: The Three Fates (Bulfinch, 1898)


The destiny of industrial equipment is controlled by humans, but through forces akin to the three Fates. Atropos provides a long life when Klotho prepares a strong thread in the form of excellence in design and manufacturing and Lachesis weaves a harmonious tapestry of proper operating conditions and maintenance. In an increasingly deregulated power industry, politics, economics, and uncertainty take the place of the Fates in determining how equipment is used and maintained.

For hydro plant owners, the ability to estimate the expected life of power-generating equipment under varying conditions has enormous value because it allows the economic assessment and optimization of operational parameters and the cost-effective performance of condition-based maintenance. This is the purpose for the maintenance-cost methodology described in this paper. As part of its research program to develop knowledge-based tools for the hydro industry, Hydro Resource Solutions has adapted methodologies from fatigue and accelerated life testing to hydropower applications.

Cumulative Damage Theory

Cumulative damage theory, which was originally developed in conjunction with fatigue testing of metals (Palmgren, 1924; Miner, 1945; Brook and Parry, 1969), provides the foundation for this maintenance-cost methodology. The life of a fatigue specimen, over a wide range of stresses, follows a relationship of this form:

Life = Aσ(-B) [1]

where A is a constant of proportionality, σ is the level of stress in the specimen under test, and B is a numerical constant.

Rabinowicz et al. (1970) applied cumulative damage theory to the accelerated life testing of a variety of mechanical and electromechanical systems, including bearings, light bulbs, electric motors, and electric tools. Their results suggest that cumulative damage theory is widely applicable to complex, real-world situations where the precise laws and mechanisms of deterioration are not explicitly known.

Application to Hydroelectric Generating Equipment

The maintenance-cost methodology, as applied to hydroelectric generating equipment, utilizes and expands the cumulative damage approach. The “life” for a component of a hydroelectric generating unit or an entire unit is assumed to follow the relationship:

L = αS(-β) [2]

where L is the life of the unit or component under consideration; α is a constant of proportionality; S is the generalized “stress” (which may include vibration level, cavitation intensity, bearing oil quality, an inefficiency parameter, a measured mechanical stress, a calculated mechanical stress, etc.) encountered in the operating environment of the equipment; and the exponent β is a numerical constant.


Hydroelectric plant owners and operators continually make decisions about when, and at what power level, to operate each generating unit. The effect of these decisions on a unit’s life expectancy should play a role in the operational planning and the real-time operation of a hydro plant. Hydraulic performance for a hydroelectric unit is typically characterized by curves showing hydraulic efficiency as a function of gate opening or output power for a series of heads representing the expected operating range, as shown in Figure 2. This convention is adopted here.

Efficiency vs. Power

40

50

60

70

80

90

100

0 10 20 30 40 50

Power (MW)

Effic

ienc

y (%

)

200 ft

190 ft

180 ft

170 ft

Figure 2: Hydraulic Efficiency vs. Power over a Range of Heads

In the more general case of a hydro unit operating under a variety of “environmental stressors,” Sij, then

Lij = αijSij(-β

ij) [3]

where Lij is the life of the component or unit resulting from stressor Si at operating condition (such as head or power) j, and αij and βij are the proportionality constant and the exponent, respectively, for stressor Si at operating condition j.

Assume that a unit operates for a time tij under the effects of stressor Si at operating condition j. The fraction of its life (lij) at these conditions is the ratio of the operating time tij to the expected life Lij:

lij = tij/Lij. [4]

At the time of failure, the following relationship applies:

Σ lij = 1 (at failure). [5]


The life of the component is “used up” by the summation of effects from the environmental stressors, as illustrated in Figure 3. This theoretical relationship is supported by experimental data from the accelerated testing research of Rabinowicz et al. (1970).

Time

Life

0

1

t2jt1j t3j

l3j

l2j

l1j

⊗Failure

Figure 3: Cumulative Damage Methodology

At some time before failure, the remaining life R, expressed in nondimensional terms, is:

R = (1 - Σ lij). [6]

When expressed as a percentage, Equation [6] becomes:

R (in %) = 100(1 - Σ lij). [7]

Calculation of Maintenance Costs

In applying these relationships to the maintenance-cost methodology, a maintenance cost M is assumed at failure. The fraction of this total cost due to operation at a condition corresponding to environmental stressor Si at operating condition j is:

Fractional cost = lijM. [8]

Similarly, the fractional maintenance cost per unit time at the same operating conditions is:


Fractional cost per unit time = lijM/tij. [9]

When Equations [3] and [4] are substituted, this expression becomes:

Fractional cost per unit time = M/Lij [10]

Fractional cost per unit time = M/αijS(-βij). [11]

In practice, the maintenance cost M and the typical unit or component lifetimes under “best” conditions are generally derived from the experience of maintenance personnel, from statistical analyses of plant or industry data, and from numerical simulations. Typical unit or component lifetimes under “extreme” conditions are generally derived from root-cause analyses of specific failure events, from on-line monitoring of operating data during failure events, and from maintenance experience under prolonged and severe operating conditions. The values for αij and βij can be determined from the “best” and the “extreme” maintenance data, using Equation [3]. The equations given above are for a single unit. A multi-unit application is a straightforward summation of the effects on individual units. Also, for simplicity, this analysis omits the time-varying aspects of maintenance costs, such as escalation, outage duration, technological advances, etc.

Illustrative Example – Turbine Guide Bearings

In practical application, computer-based data acquisition and analysis systems can be used to monitor operational parameters, characterize unit performance and environmental stressors (such as inefficiency, turbine guide bearing vibrations, and cavitation-related acoustic emissions), and report expected maintenance costs associated with current operating conditions, calculated by using Equation 11. Such systems are installed at Chelan County Public Utility District's 11-unit, 1,280 MW Rocky Reach Hydro Plant and 18-unit, 660 MW Rock Island Hydro Plant, located on the Columbia River near Wenatchee, Washington, and, under a Cooperative Research and Development Agreement (CRADA) with the U. S. Bureau of Reclamation, at Unit G-24 of Reclamation’s 6-unit, 4,500 MW Grand Coulee Third Powerhouse, located on the Columbia River near Coulee Dam, Washington.

An example, in which these procedures are applied to turbine guide bearings, is provided below. Figure 4 displays measured turbine guide bearing vibrations as a function of power level for a hydro plant with three identical Francis-type generating units. The “one unit” data is based on measurements, and the “two unit” and “three unit” curves are constructed from the single unit data under the assumption of equal load distribution among units for multi-unit operations. Typical Francis unit vibrations are visible in the figure, including the peak due to draft tube swirl at 13 MW and the vibration minimum at the best efficiency point near 33 MW.

For this plant, the operating head varies insignificantly throughout the year, so the output power effectively represents the operating conditions. Maintenance personnel estimate an expected life of twenty years for continuous operation under the “best” (i.e.,


maximum efficiency, lowest vibration) conditions. The assumed cost for a typical unscheduled repair is $100,000 for a single turbine guide bearing. Also, based on previous experience in an “extreme” situation, a similar unit operated for ten hours at a turbine guide bearing vibration of 50 mils before a bearing failure occurred.

0

5

10

15

20

25

30

0 10 20 30 40 50 60 70 80 90 100 110 120

Power (MW)

TGB

Vib

ratio

ns (m

ils, p

tp)

Figure 4: Turbine Guide Bearing Vibrations vs. Power

By defining the vibration level as the stressor S in Equation [3], values for α and β can be determined from these estimates as shown in Figure 5.

TGB Vibration (mils, ptp) Life (hours) Proportionality Constant Exponent S L α β

2.3 175200 2462761 3.1733 50 10 2462761 3.1733

Figure 5: Computation of Vibration-Related Values from Equation [3]

The turbine guide bearing vibration data from Figure 4, the proportionality constant α and the exponent β shown in Figure 5, hourly operating data for a 566-day period, and Equation [7] were used to determine the amount of the three turbine bearings’ life “used up” during the evaluation period of the analysis (see Figure 6). During the 566-day evaluation period, about 74% of the three bearings’ total life (for three turbines, the total life is 300%) has been “used up,” with much of the damage occurring in the less efficient but economically attractive operating range between 110 MW and 120 MW, where the units are typically operated. The maintenance cost over this 566-day evaluation period was computed in a similar manner, and the results (which are the products of each percentage in Figure 6 and the repair cost M, as shown in Equation [8]), are provided in Figure 7.

One Unit Two Units Three Units


Operating Period = 566 DaysLife "Used Up" = 74 %

Life Remaining = 226 %(300 % total life for 3 units)

0.0

5.0

10.0

15.0

0 10 20 30 40 50 60 70 80 90 100 110 120

Power (MW)

Turb

ine

Gui

de B

earin

g Li

fe "

Use

d U

p"

(%)

Figure 6: Computation of Bearing Life “Used Up” under Varying Operating Conditions

Operating Period = 566 DaysTotal Maintenance Cost = $73,746

$0

$5,000

$10,000

$15,000

0 10 20 30 40 50 60 70 80 90 100 110 120Power (MW)

Mai

nten

ance

Cos

t ($)

Figure 7: Vibration-Related Maintenance Costs under Varying Operating Conditions


Illustrative Example – Cavitation Damage

A similar analysis can be used to compute the cavitation-related costs under varying operating conditions, using the measured output from a cavitation monitor (Jones et al., 1989; Derakhshan et al., 1990; March and Jones, 1991) as a function of power level. Figure 8 presents output from an acoustic emissions-based cavitation monitoring system as a function of power level for old (1930s vintage) runners and for new runners of modern design. The “one unit” data is based on measurements, and the “two unit” and “three unit” curves are constructed from the single unit data under the assumption of equal load distribution among units for multi-unit operations. For comparison purposes, the higher output for the new turbine design has been normalized to provide an equivalent best efficiency near 33 MW. Note that the new runners have cavitation levels that are significantly lower throughout the operating range.

Cavitation levels for the most efficient single unit load and for the maximum single unit load were used in Equation [3] as the environmental stressor. Corresponding cavitation repair intervals were estimated and used to determine values for α and β, as shown for the old and new runners in Figures 9 and 10, respectively.

0.000

0.500

1.000

1.500

0 10 20 30 40 50 60 70 80 90 100 110 120

Power (MW)

Out

put f

rom

Cav

itatio

n M

onito

r (Vo

lts)

Figure 8: Output from Cavitation Monitor vs. Power (Old and New Runners)

One Unit Two Units Three Units


Output from Cavitation Monitor (volts)

Life (hours)

Proportionality Constant

Exponent

S L α β 0.7 26280 10352 2.612 1.39 4380 10352 2.612

Figure 9: Computation of Cavitation-Related Values for Old Runners

Output from Cavitation Monitor (volts)

Life (hours)

Proportionality Constant

Exponent

S L α β 0.19 87600 9814 1.318 1.09 8760 9814 1.318

Figure 10: Computation of Cavitation-Related Values for New Runners

The cavitation monitoring data from Figure 8, the values of the proportionality constant α and the exponent β shown in Figures 9 and 10, the hourly operating data for the 566-day evaluation period, Equation [7], and an assumed single unit cavitation repair cost of $50,000 were used to determine the amount of the cavitation “life” used up during the evaluation period and the corresponding total cavitation damage cost for the evaluation period. These results are presented for the old runners and new runners in Figure 11.

Figure 11: Cavitation-Related Maintenance Costs under Varying Operating Conditions

0

5000

10000

15000

20000

25000

0 10 20 30 40 50 60 70 80 90 100 110 120

Power (MW)

Cav

itatio

n-R

elat

ed M

aint

enan

ce C

osts

($)

Old Runners

New Runners

Operating Period = 566 DaysCavitation Damage Cost (Old Units) = $100,000Cavitation Damage Cost (New Units) = $ 20,000


Using data for the 566-day evaluation period, the cavitation damage cost for the old runners is about $100,000, while the corresponding cavitation damage cost for the new runners would be about $20,000. Most of the cavitation damage costs correspond to operation in the less efficient but economically attractive range between 110 MW and 120 MW, where the units are typically operated.

Illustrative Example – Inefficiency

For these analyses, three years of data were examined for a 21-unit, 600 MW, main-river hydroplant that is typically operated for load control. The years of data include: (1) 1998, a representative wet year; (2) 1999, a normal year of rainfall; and (3) 2000, a dry year. A WaterView® optimization module (March, 2001) was used with a DataWolff™ analysis automation engine to compute the optimized plant dispatch (i.e., the most efficient allocation of available units to provide the specified load) at 15-minute intervals for the same three years. The revenue gains from the optimized plant dispatch, relative to the actual revenues, were calculated by assuming that the additional energy resulting from the more efficiently optimized operation was sold at the market price available at the time it could have been produced (Wolff and March, 2002).

The maintenance-cost benefits from the optimized unit operation were also computed, using the WaterView maintenance-cost module (see Figure 12). The life of a unit was assumed to depend on a single “environmental stressor,” in this case the unit inefficiency. These additional assumptions were used in the analyses:

• a unit life of 40 years while operating at the best efficiency point (i.e., a “best” reference inefficiency);

• a unit life of 10 years while operating at the maximum power output (i.e., an “extreme” reference inefficiency); and

• a unit replacement cost of $5,000,000.

Figure 13 presents the computed energy gains and the corresponding revenue gains that the optimized plant dispatch would have provided, compared to the actual plant dispatch. Figure 13 also presents the separately computed maintenance-cost reductions, based on the assumptions listed above. An additional $2,600,000 in reduced maintenance costs over the units’ “lives” could have been achieved by optimized plant operation over the three-year period.

The simplified analyses summarized in Figure 13 present a methodology to evaluate operational decisions within a quantitative economic framework. For example, although Kaplan hydroturbines are well suited for automatic generation control because of their relatively flat performance curves, in many cases they are not used for this purpose because of concerns about the extra wear that may occur on the blade and gate linkages. The maintenance-cost methodology can be used to evaluate whether or not this is an economically sound decision. Similarly, the additional maintenance costs due to unit start-ups and shut-downs can be evaluated.


Figure 12: Summary Screen and Unit Curves for Maintenance-Cost Module

600 MW, 21-Unit Hydroplant (1998-2000)

$0

$500,000

$1,000,000

$1,500,000

$2,000,000

$2,500,000

$3,000,000

$3,500,000

01/98 07/98 01/99 07/99 01/00 07/00 12/00Date

Cum

ulat

ive

Rev

enue

Incr

ease

($)

0

20,000

40,000

60,000

80,000

100,000

120,000

140,000

Cum

ulat

ive

Ener

gy In

crea

se (M

WH

)

Revenue Increase - 1998Revenue Increase - 1999Revenue Increase - 2000Maint.-Cost Savings - 1998Maint.-Cost Savings - 1999Maint.-Cost Savings - 2000Energy Increase - 1998Energy Increase - 1999Energy Increase - 2000

Figure 13: Production, Revenue, and Maintenance-Cost Improvements due to

Optimized Operations


Summary

This paper describes the technical background and provides illustrative examples of a simple technique for quantifying the maintenance costs associated with variable operating conditions. The maintenance-cost methodology assists owners and operators in making sound economic decisions about the operation and maintenance of hydroelectric generating facilities. Specifically, a user of the methodology can:

• Quantify the effects of operating conditions on maintenance costs;

• Determine real-time maintenance costs associated with current operating conditions for use in planning and in operational decision-making;

• Facilitate and support condition-based maintenance by estimating and tracking the “remaining life” in equipment and components;

• Gather quantitative information for use in defining avoidance zones and cavitation limits; and

• Develop a quantitative basis for operations-based maintenance planning, maintenance budgets, and maintenance practices.

This maintenance-cost methodology represents a departure from traditional approaches to predictive maintenance monitoring because the primary focus is on quantifying the effects of varying operating conditions on the maintenance costs. This will be an increasingly important concern as market forces, particularly in the ancillary services area, push hydro owner/operators into non-traditional operational modes and present new challenges for effective asset management while providing new opportunities to profit from hydropower’s unique characteristics.

References

Brook, R. H. W., and J. S. C. Parry, “Cumulative Damage in Fatigue: A Step Toward Its Understanding,” Journal of Mechanical Engineering Science, Vol. 11, 1969, pp. 243-255.

Bulfinch, T., Age of Fable, or Beauties of Mythology, edited by Rev. J. L. Scott, Philadelphia, PA: David McKay, Publisher, 1898.

Derakhshan, O., J. R. Houghton, R. K. Jones, and P. A. March, “Cavitation Monitoring of Hydroturbines with True-RMS Acoustic Emission Measurement,” ASTM Special Publication 1077, Philadelphia: American Society for Testing and Materials, October 1990.

Jones, R. K., P. A. March, and J. M. Epps, “Monitoring Hydroturbines for Efficiency and Cavitation,” Hydro Review, June 1989, pp. 72-79.


March, P. A., “Knowledge Management System for Water Resources Promotes Beneficial Paradigm Shifts to Optimize Energy and Environment,” Proceedings of the Fourth Inter-American Dialogue on Water Management, Foz do Iguaçu, Paraná, Brazil, September 2001.

March, P. A., and R. K. Jones, “Laboratory and Field Experience with Cavitation Monitoring of Hydroturbines,” Proceedings of Waterpower 91, July 1991.

Miner, M. A., “Cumulative Damage in Fatigue,” Journal of Applied Mechanics, Transactions of the ASME, Vol. 12, June 1945, pp. 159-164.

Palmgren, A., “Die Lebensdauer von Kugellagern,” Z. Verein. Deutschland Ingeniur, Vol. 50, 1924, pp. 339-341.

Rabinowicz, E., R. H. McEntire, and B. Shiralkar, “A Technique for Accelerated Life Testing,” Journal of Engineering for Industry, Transactions of the ASME, August 1970, pp. 706-710.

Wolff, Paul J., and P. A. March, “New Methodology for Evaluating the Benefits of Hydro Automation,” Norris, Tennessee: Tennessee Valley Authority, TVA ER&TA Project Report, Revision 1, September 2002.

Author

Patrick March is Senior Product Development Manager for the Tennessee Valley Authority's Resource Management business and General Manager of Hydro Resource Solutions LLC, which is jointly owned by TVA and Voith Siemens Hydro. Mr. March is a member of ASME Power Test Code Committee 18 (Hydraulic Turbines and Pump-Turbines) and a board member for the Hydro Research Foundation. Mr. March has over thirty years of hydropower and water resources experience, primarily in the areas of performance improvements, water quality improvements, condition assessment and monitoring, flow measurement, hydraulic modeling, cavitation erosion, and vibration analysis.

quantifying the maintenance costs associated with variable ... maint costs.pdfin greek mythology,...

Documents